Patent Publication Number: US-2021173974-A1

Title: Method for determining parameters of high-frequency vibrating mill with three grinding drums

Description:
TECHNICAL FIELD 
     The present invention belongs to the technical field of vibrating mill device and relates to a method for determining parameters of a high-frequency vibrating mill with three grinding drums. 
     BACKGROUND 
     A vibrating mill is an ultra-fine crushing device with balls or rods as the medium, which can grind 2 mm materials to several microns. The vibrating mill has the advantages of high efficiency, energy-saving, space-saving, uniform particle size, etc., which plays an important role in the field of ultra-fine crushing, and has been widely used in engineering. The working principle of the vibrating mill is to make a drum body vibrate at a high frequency by an exciting force generated by the eccentric block and crush material by a grinding medium. 
     The present invention is a mill with three grinding drums, which can work at high frequency to achieve finer grinding. The number of exciters used in an ordinary mill is generally less than three. Compared with the ordinary mill, the present invention has the following advantages: 
     1. To motivate the total vibrating system, the requirements for exciters are generally strict in the system with a small number of exciters. This is because that to ensure the rated power, the technical requirements of the exciters are very high. Whereas the present invention uses three exciters, which lowers the technical requirements of exciters, reduces the loss of the exciters, and increases the service life of the mill while ensuring that the rated power and standard working requirements are met. 
     2. The present invention uses three motors for driving, which can work at a high frequency. When the three motors resonate, a high amplitude can be reached, so that the mill has a better crushing effect, small noise, no pollution, and high reliability. 
     SUMMARY 
     Taking a dynamic model with four mass bodies driven by three exciters as the research object, the mathematic modeling of the system is established by the average parameter method and transfer function method. Besides, the characteristic curves of synchronization and stability capability coefficient, the dimensionless coupling torque maximum, are obtained by characteristic analysis of synchronization and stability. Finally, the curves of the rotational velocity of the motors, phase difference between the exciters, and displacements of the mass bodies are obtained by simulation. By the comparison between the characteristic analysis and simulation, the correctness of the method is verified. 
     The present invention has the following specific technical solution: 
     A method for determining parameters of a high-frequency vibrating mill with three grinding drums, wherein a dynamic model of the vibrating mill comprises three exciters, four mass bodies, and springs; mass body  1 , mass body  2 , and mass body  3  are respectively three grinding drums connected with mass body  4  through springs, and mass body  4  is connected with a base through spring; the three exciters are respectively located on mass body  1 , mass body  2  and mass body  3 ; the three exciters rotate in the same direction, and each exciter rotates about its own rotation axis; a method for determining parameters of the exciters, comprising the following steps: 
     Step 1: Establishing a Dynamic Model and the Motion Differential Equations of the System 
     A dynamic model of the vibrating mill is shown in  FIG. 1 . Establishing two rectangular coordinate systems, the three exciters rotate in the same direction, and the motion of the mill body is divided into vibration in x and y directions and oscillation around the mass center, represented by x, y and ψ respectively. 
     Selecting x, y, ψ, φ 1 , φ 2  and φ 3  as generalized coordinates, solving the kinetic energy, potential energy, and energy dissipation functions of a system, and substituting the functions into Lagrange equations to obtain motion differential equations of the system. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                               4 
                             
                           
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                       - 
                       
                         
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                           2 
                         
                          
                         
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                                 . 
                               
                               2 
                             
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                                 . 
                               
                               4 
                             
                           
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                       - 
                       
                         
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                           2 
                         
                          
                         
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                           3 
                         
                          
                         
                           ( 
                           
                             
                               
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                                 . 
                               
                               3 
                             
                             - 
                             
                               
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                                 . 
                               
                               4 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           k 
                           3 
                         
                          
                         
                           ( 
                           
                             
                               y 
                               3 
                             
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                               y 
                               4 
                             
                           
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                       + 
                       
                         
                           f 
                           4 
                         
                          
                         
                           
                             y 
                             . 
                           
                           4 
                         
                       
                       + 
                       
                         
                           k 
                           4 
                         
                          
                         
                           y 
                           4 
                         
                       
                     
                     = 
                     0 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         J 
                          
                         
                           ψ 
                           ¨ 
                         
                       
                       + 
                       
                         
                           f 
                           ψ 
                         
                          
                         
                           ψ 
                           . 
                         
                       
                       + 
                       
                         
                           k 
                           ψ 
                         
                          
                         ψ 
                       
                     
                     = 
                     
                       
                         l 
                         0 
                       
                        
                       r 
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           
                             m 
                             i 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     ϕ 
                                     ¨ 
                                   
                                   i 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         ϕ 
                                         i 
                                       
                                       - 
                                       
                                         β 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     ϕ 
                                     . 
                                   
                                   2 
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       
                                         ϕ 
                                         i 
                                       
                                       - 
                                       
                                         β 
                                         i 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           J 
                           oi 
                         
                          
                         
                           
                             ϕ 
                             ¨ 
                           
                           i 
                         
                       
                       + 
                       
                         
                           f 
                           di 
                         
                          
                         
                           
                             ϕ 
                             . 
                           
                           i 
                         
                       
                     
                     = 
                     
                       
                         m 
                         oi 
                       
                        
                       
                         r 
                          
                         
                           [ 
                           
                             
                               
                                 
                                   y 
                                   ¨ 
                                 
                                 i 
                               
                                
                               
                                   
                               
                                
                               cos 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 i 
                               
                             
                             - 
                             
                               
                                 
                                   ϰ 
                                   ¨ 
                                 
                                 i 
                               
                                
                               
                                   
                               
                                
                               sin 
                                
                               
                                   
                               
                                
                               
                                 ϕ 
                                 i 
                               
                             
                             - 
                             
                               
                                 l 
                                 0 
                               
                                
                               
                                 ψ 
                                 ¨ 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       ϕ 
                                       i 
                                     
                                     - 
                                     
                                       β 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 l 
                                 0 
                               
                                
                               
                                 
                                   ψ 
                                   . 
                                 
                                 2 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       ϕ 
                                       i 
                                     
                                     - 
                                     
                                       β 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     wherein 
     m 0i —mass of each exciter (i=1˜3); m i —mass of each mass body (i=1˜4); 
     M 1 =m 1 +m 01 , M 2 =m 2 +m 02 , M 3 =m 3 +m 03 , and M 4 =m 4 +m 01 +m 02 +m 03    
     J ψ =J m4 +(m 01 +m 02 +m 03 )(r 2 +l 2 )—moment of inertia of a vibrating rigid body; 
     J oi =m oi r i   2 —moment of inertia of each exciter (i=1˜3); 
     
       
         
           
             
               
                 k 
                 ψ 
               
               = 
               
                 
                   1 
                   2 
                 
                  
                 
                   
                     k 
                     4 
                   
                    
                   
                     ( 
                     
                       
                         l 
                         
                           x 
                            
                           3 
                         
                         2 
                       
                       + 
                       
                         l 
                         
                           y 
                            
                           3 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
             
             , 
           
         
       
     
     k i (i=1˜4)—spring stiffness coefficient; 
     
       
         
           
             
               
                 f 
                 ψ 
               
               = 
               
                 
                   
                     1 
                     2 
                   
                    
                   
                     ( 
                     
                       
                         f 
                         1 
                       
                       + 
                       
                         f 
                         2 
                       
                       + 
                       
                         f 
                         3 
                       
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         l 
                         
                           x 
                            
                           1 
                         
                         2 
                       
                       + 
                       
                         l 
                         
                           x 
                            
                           2 
                         
                         2 
                       
                       + 
                       
                         l 
                         
                           y 
                            
                           1 
                         
                         2 
                       
                       + 
                       
                         l 
                         
                           y 
                            
                           2 
                         
                         2 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     1 
                     2 
                   
                    
                   
                     
                       f 
                       4 
                     
                      
                     
                       ( 
                       
                         
                           l 
                           
                             x 
                              
                             3 
                           
                           2 
                         
                         + 
                         
                           l 
                           
                             y 
                              
                             3 
                           
                           2 
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
           
         
       
     
     f i (i=1˜4)—damping coefficient; 
     
       
         
           
             
               
                 ϕ 
                 
                   0 
                    
                   
                       
                   
                    
                   i 
                 
               
                
               – 
             
             - 
             
               
                 1 
                 2 
               
                
               
                 ( 
                 
                   
                     f 
                     1 
                   
                   + 
                   
                     f 
                     2 
                   
                   + 
                   
                     f 
                     3 
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     
                       l 
                       
                         x 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       l 
                       
                         x 
                          
                         
                             
                         
                          
                         2 
                       
                     
                   
                   + 
                   
                     
                       l 
                       
                         y 
                          
                         
                             
                         
                          
                         1 
                       
                     
                      
                     
                       l 
                       
                         y 
                          
                         
                             
                         
                          
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     phase angle of exciter i (i=1˜3); {dot over (φ)} i —angular velocity of exciter i (i=1˜3); {circumflex over (φ)} i —angular acceleration of exciter i (i=1˜3); 
     Step 2: Synchronization Analysis 
     Letting the phase difference between exciter  1  and exciter  2  be 2α 1 , the phase difference between exciter  2  and exciter  3  be 2α 2 , and the average phase difference among the three exciters be φ. 
     
       
         
           
             
               
                 
                   ϕ 
                   1 
                 
                 - 
                 
                   ϕ 
                   2 
                 
               
               = 
               
                 2 
                  
                 
                     
                 
                  
                 
                   α 
                   1 
                 
               
             
             , 
             
               
                 
                   ϕ 
                   2 
                 
                 - 
                 
                   ϕ 
                   3 
                 
               
               = 
               
                 2 
                  
                 
                     
                 
                  
                 
                   α 
                   2 
                 
               
             
             , 
             
               ϕ 
               = 
               
                 
                   1 
                   3 
                 
                  
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     3 
                   
                    
                   
                     ϕ 
                     i 
                   
                 
               
             
           
         
       
     
     Rearranging the above equations to obtain 
     
       
         
           
             
               
                 
                   
                     
                       ϕ 
                       1 
                     
                     = 
                     
                       
                         ϕ 
                         + 
                         
                           
                             4 
                             3 
                           
                            
                           
                             α 
                             1 
                           
                         
                         + 
                         
                           
                             2 
                             3 
                           
                            
                           
                             α 
                             2 
                           
                         
                       
                       = 
                       
                         ϕ 
                         + 
                         
                           υ 
                           1 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ϕ 
                       2 
                     
                     = 
                     
                       
                         ϕ 
                         - 
                         
                           
                             2 
                             3 
                           
                            
                           
                             α 
                             1 
                           
                         
                         + 
                         
                           
                             2 
                             3 
                           
                            
                           
                             α 
                             2 
                           
                         
                       
                       = 
                       
                         ϕ 
                         + 
                         
                           υ 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ϕ 
                       3 
                     
                     = 
                     
                       
                         ϕ 
                         - 
                         
                           
                             2 
                             3 
                           
                            
                           
                             α 
                             1 
                           
                         
                         - 
                         
                           
                             4 
                             3 
                           
                            
                           
                             α 
                             2 
                           
                         
                       
                       = 
                       
                         ϕ 
                         + 
                         
                           υ 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Using the transfer function method to obtain the response of the system: 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       1 
                     
                     = 
                     
                       
                         
                           F 
                           1 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           5 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 5 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           9 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 9 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       x 
                       2 
                     
                     = 
                     
                       
                         
                           F 
                           2 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           6 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 6 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           10 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 10 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       x 
                       3 
                     
                     = 
                     
                       
                         
                           F 
                           3 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           7 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 7 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           11 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 11 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       x 
                       4 
                     
                     = 
                     
                       
                         
                           F 
                           4 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 4 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           8 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 8 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           12 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 12 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       1 
                     
                     = 
                     
                       
                         
                           F 
                           1 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           5 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 5 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           9 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 9 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       2 
                     
                     = 
                     
                       
                         
                           F 
                           2 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           6 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 6 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           10 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 10 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       3 
                     
                     = 
                     
                       
                         
                           F 
                           3 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 3 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           7 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 7 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           11 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 11 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       y 
                       4 
                     
                     = 
                     
                       
                         
                           F 
                           4 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 1 
                               
                               - 
                               
                                 γ 
                                 4 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           8 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 2 
                               
                               - 
                               
                                 γ 
                                 8 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           F 
                           12 
                         
                          
                         r 
                          
                         
                             
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 3 
                               
                               - 
                               
                                 γ 
                                 12 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     ψ 
                     = 
                     
                       
                         
                           
                             r 
                             m 
                           
                            
                           
                             r 
                             l 
                           
                            
                           r 
                         
                         
                           
                             μ 
                             ψ 
                           
                            
                           
                             l 
                             e 
                           
                         
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 ϕ 
                                 i 
                               
                               - 
                               
                                 β 
                                 i 
                               
                               + 
                               
                                 γ 
                                 13 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Determining intermediate parameters: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             a 
                             = 
                             
                               
                                 
                                   
                                     M 
                                     4 
                                   
                                    
                                   
                                     M 
                                     0 
                                     3 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     8 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                       
                                       + 
                                       
                                         9 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           0 
                                           2 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                          
                                         
                                           M 
                                           0 
                                           3 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           M 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                       + 
                                       
                                         
                                           k 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     6 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                       
                                       + 
                                       
                                         6 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           0 
                                           2 
                                         
                                       
                                       + 
                                       
                                         15 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           M 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                          
                                         
                                           k 
                                           0 
                                           2 
                                         
                                       
                                       + 
                                       
                                         f 
                                         0 
                                         4 
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           4 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     4 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           0 
                                           2 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           4 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                          
                                         
                                           k 
                                           0 
                                           2 
                                         
                                       
                                       + 
                                       
                                         
                                           M 
                                           4 
                                         
                                          
                                         
                                           k 
                                           0 
                                           3 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           0 
                                           3 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                       
                                       + 
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           4 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     4 
                                   
                                    
                                   
                                     k 
                                     0 
                                     3 
                                   
                                    
                                   
                                     
 
                                   
                                    
                                   b 
                                 
                               
                               = 
                               
                                 
                                   
                                     
                                       - 
                                       
                                         ( 
                                         
                                           
                                             3 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               M 
                                               4 
                                             
                                              
                                             
                                               M 
                                               0 
                                               2 
                                             
                                              
                                             
                                               f 
                                               0 
                                             
                                           
                                           + 
                                           
                                             4 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               f 
                                               0 
                                             
                                              
                                             
                                               M 
                                               0 
                                               3 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                       7 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           6 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                         
                                         + 
                                         
                                           15 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                            
                                           
                                             M 
                                             0 
                                             2 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                             2 
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                         
                                         + 
                                         
                                           6 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                             3 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             f 
                                             0 
                                             3 
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                       5 
                                     
                                   
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           6 
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                         
                                         + 
                                         
                                           12 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                             2 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                             3 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                            
                                           
                                             k 
                                             0 
                                             2 
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             f 
                                             0 
                                             3 
                                           
                                            
                                           
                                             k 
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                       3 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             f 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                             3 
                                           
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             4 
                                           
                                            
                                           
                                             k 
                                             0 
                                             2 
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                     
                                      
                                     
                                       
 
                                     
                                      
                                     
                                       c 
                                       1 
                                     
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       - 
                                       
                                         M 
                                         4 
                                       
                                     
                                      
                                     
                                       M 
                                       0 
                                       2 
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                       6 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                         + 
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                             M 
                                             0 
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                             f 
                                             0 
                                             2 
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                         
                                         + 
                                         
                                           6 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                             2 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             k 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                       4 
                                     
                                   
                                   - 
                                   
                                       
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             M 
                                             4 
                                           
                                            
                                           
                                             k 
                                             0 
                                             2 
                                           
                                         
                                         + 
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             0 
                                             2 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                         
                                         + 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             f 
                                             0 
                                             2 
                                           
                                            
                                           
                                             k 
                                             4 
                                           
                                         
                                         + 
                                         
                                           5 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                             2 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       ω 
                                       
                                         m 
                                          
                                         
                                             
                                         
                                          
                                         0 
                                       
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       k 
                                       4 
                                     
                                      
                                     
                                       k 
                                       0 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       k 
                                       0 
                                       3 
                                     
                                      
                                     
                                       
 
                                     
                                      
                                     
                                       
                                         d 
                                         1 
                                       
                                       = 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                         
                                         + 
                                         
                                           4 
                                            
                                           
                                               
                                           
                                            
                                           
                                             f 
                                             0 
                                           
                                            
                                           
                                             M 
                                             0 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                          
                         
                           ω 
                           
                             m 
                              
                             
                                 
                             
                              
                             0 
                           
                           5 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 f 
                                 0 
                               
                                
                               
                                 k 
                                 4 
                               
                                
                               
                                 M 
                                 0 
                               
                             
                             + 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 f 
                                 0 
                               
                                
                               
                                 M 
                                 4 
                               
                                
                               
                                 k 
                                 0 
                               
                             
                             + 
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 f 
                                 0 
                                 3 
                               
                             
                             + 
                             
                               10 
                                
                               
                                   
                               
                                
                               
                                 f 
                                 0 
                               
                                
                               
                                 M 
                                 0 
                               
                                
                               
                                 k 
                                 0 
                               
                             
                           
                           ) 
                         
                          
                         
                           ω 
                           
                             m 
                              
                             
                                 
                             
                              
                             0 
                           
                           3 
                         
                       
                       + 
                       
                         2 
                          
                         
                             
                         
                          
                         
                           f 
                           0 
                         
                          
                         
                           k 
                           4 
                         
                          
                         
                           k 
                           0 
                         
                       
                       + 
                       
                         4 
                          
                         
                             
                         
                          
                         
                           f 
                           0 
                         
                          
                         
                           k 
                           0 
                           2 
                         
                          
                         
                           
 
                         
                          
                         
                           e 
                           1 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             f 
                             0 
                             2 
                           
                            
                           
                             M 
                             0 
                           
                            
                           
                             ω 
                             
                               m 
                                
                               
                                   
                               
                                
                               0 
                             
                             4 
                           
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 
                                   k 
                                   0 
                                   2 
                                 
                                  
                                 
                                   M 
                                   0 
                                 
                               
                               + 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   f 
                                   0 
                                   2 
                                 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                             
                             ) 
                           
                            
                           
                             ω 
                             
                               m 
                                
                               
                                   
                               
                                
                               0 
                             
                             2 
                           
                         
                         + 
                         
                           
                             k 
                             0 
                             3 
                           
                            
                           
                             
 
                           
                            
                           
                             f 
                             1 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       f 
                                       0 
                                     
                                      
                                     
                                       M 
                                       0 
                                     
                                      
                                     
                                       k 
                                       0 
                                     
                                   
                                   + 
                                   
                                     f 
                                     0 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ω 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 0 
                               
                               3 
                             
                           
                           + 
                           
                             3 
                              
                             
                                 
                             
                              
                             
                               f 
                               0 
                             
                              
                             
                               k 
                               0 
                               2 
                             
                              
                             
                               ω 
                               
                                 m 
                                  
                                 
                                     
                                 
                                  
                                 0 
                               
                             
                              
                             
                               
 
                             
                              
                             
                               h 
                               1 
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 f 
                                 0 
                                 2 
                               
                                
                               
                                 M 
                                 0 
                               
                                
                               
                                 ω 
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   0 
                                 
                                 4 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       k 
                                       0 
                                       2 
                                     
                                      
                                     
                                       M 
                                       0 
                                     
                                   
                                   + 
                                   
                                     3 
                                      
                                     
                                         
                                     
                                      
                                     
                                       f 
                                       0 
                                       2 
                                     
                                      
                                     
                                       k 
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ω 
                                 
                                   m 
                                    
                                   
                                       
                                   
                                    
                                   0 
                                 
                                 2 
                               
                             
                             + 
                             
                               
                                 k 
                                 0 
                                 3 
                               
                                
                               
                                 
 
                               
                                
                               
                                 p 
                                 1 
                               
                             
                           
                           = 
                           
                             
                               
                                 
                                   - 
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                       + 
                                       
                                         f 
                                         0 
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   ω 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     0 
                                   
                                   3 
                                 
                               
                               + 
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   f 
                                   0 
                                 
                                  
                                 
                                   k 
                                   0 
                                   2 
                                 
                                  
                                 
                                   ω 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     0 
                                   
                                 
                                  
                                 
                                   
 
                                 
                                  
                                 
                                   u 
                                   1 
                                 
                               
                             
                             = 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           k 
                                           0 
                                         
                                          
                                         
                                           M 
                                           0 
                                           2 
                                         
                                       
                                       + 
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     4 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         3 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                           2 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                       + 
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           0 
                                           2 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     0 
                                     3 
                                   
                                    
                                   
                                     
 
                                   
                                    
                                   
                                     z 
                                     1 
                                   
                                 
                               
                               = 
                               
                                 
                                   
                                     f 
                                     0 
                                   
                                    
                                   
                                     M 
                                     0 
                                     2 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     5 
                                   
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         f 
                                         0 
                                         3 
                                       
                                       + 
                                       
                                         4 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           0 
                                         
                                          
                                         
                                           M 
                                           0 
                                         
                                          
                                         
                                           k 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                     3 
                                   
                                 
                                 + 
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     f 
                                     0 
                                   
                                    
                                   
                                     k 
                                     0 
                                     2 
                                   
                                    
                                   
                                     ω 
                                     
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       0 
                                     
                                   
                                    
                                   
                                     
 
                                   
                                    
                                   
                                     
                                       c 
                                       2 
                                     
                                     = 
                                     
                                       e 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
                       d 
                       2 
                     
                     = 
                     
                       f 
                       1 
                     
                   
                   , 
                   
                     
                       e 
                       2 
                     
                     = 
                     
                       c 
                       1 
                     
                   
                   , 
                   
                     
                       f 
                       2 
                     
                     = 
                     
                       d 
                       1 
                     
                   
                   , 
                   
                     
                       h 
                       2 
                     
                     = 
                     
                       e 
                       1 
                     
                   
                   , 
                   
                     
                       p 
                       2 
                     
                     = 
                     
                       f 
                       1 
                     
                   
                   , 
                   
                     
                       u 
                       2 
                     
                     = 
                     
                       u 
                       1 
                     
                   
                   , 
                   
                     
                       z 
                       2 
                     
                     = 
                     
                       z 
                       1 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       c 
                       3 
                     
                     = 
                     
                       e 
                       1 
                     
                   
                   , 
                   
                     
                       d 
                       3 
                     
                     = 
                     
                       f 
                       1 
                     
                   
                   , 
                   
                     
                       e 
                       3 
                     
                     = 
                     
                       e 
                       1 
                     
                   
                   , 
                   
                     
                       f 
                       2 
                     
                     = 
                     
                       f 
                       1 
                     
                   
                   , 
                   
                     
                       h 
                       3 
                     
                     = 
                     
                       c 
                       1 
                     
                   
                   , 
                   
                     
                       p 
                       3 
                     
                     = 
                     
                       d 
                       1 
                     
                   
                   , 
                   
                     
                       u 
                       3 
                     
                     = 
                     
                       u 
                       1 
                     
                   
                   , 
                   
                     
                       z 
                       3 
                     
                     = 
                     
                       
                         
                           z 
                           1 
                         
                          
                         
                           
 
                         
                          
                         
                           γ 
                           1 
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     arctan 
                                      
                                     
                                       
                                         
                                           bc 
                                           1 
                                         
                                         - 
                                         
                                           ad 
                                           1 
                                         
                                       
                                       
                                         
                                           ac 
                                           1 
                                         
                                         + 
                                         
                                           bd 
                                           1 
                                         
                                       
                                     
                                   
                                   , 
                                 
                               
                               
                                 
                                   
                                     
                                       ac 
                                       1 
                                     
                                     + 
                                     
                                       bd 
                                       1 
                                     
                                   
                                   &gt; 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   
                                     π 
                                     + 
                                     
                                       arctan 
                                        
                                       
                                         
                                           
                                             bc 
                                             1 
                                           
                                           - 
                                           
                                             ad 
                                             1 
                                           
                                         
                                         
                                           
                                             ac 
                                             1 
                                           
                                           + 
                                           
                                             bd 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                   , 
                                 
                               
                               
                                 
                                   
                                     
                                       ac 
                                       1 
                                     
                                     + 
                                     
                                       bd 
                                       1 
                                     
                                   
                                   &lt; 
                                   0 
                                 
                               
                             
                           
                           , 
                           
                             
 
                           
                            
                           
                             
                               γ 
                               2 
                             
                             = 
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             arctan 
                                              
                                             
                                               
                                                 
                                                   be 
                                                   1 
                                                 
                                                 - 
                                                 
                                                   af 
                                                   1 
                                                 
                                               
                                               
                                                 
                                                   ae 
                                                   1 
                                                 
                                                 + 
                                                 
                                                   bf 
                                                   1 
                                                 
                                               
                                             
                                           
                                           , 
                                         
                                       
                                       
                                         
                                           
                                             
                                               ae 
                                               1 
                                             
                                             + 
                                             
                                               bf 
                                               1 
                                             
                                           
                                           &gt; 
                                           0 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             π 
                                             + 
                                             
                                               arctan 
                                                
                                               
                                                 
                                                   
                                                     be 
                                                     1 
                                                   
                                                   - 
                                                   
                                                     af 
                                                     1 
                                                   
                                                 
                                                 
                                                   
                                                     ae 
                                                     1 
                                                   
                                                   + 
                                                   
                                                     bf 
                                                     1 
                                                   
                                                 
                                               
                                             
                                           
                                           , 
                                         
                                       
                                       
                                         
                                           
                                             
                                               ae 
                                               1 
                                             
                                             + 
                                             
                                               bf 
                                               1 
                                             
                                           
                                           &lt; 
                                           0 
                                         
                                       
                                     
                                   
                                    
                                   
                                     
 
                                   
                                    
                                   
                                     γ 
                                     3 
                                   
                                 
                                 = 
                                 
                                   { 
                                   
                                     
                                       
                                         
                                           
                                             
                                               arctan 
                                                
                                               
                                                 
                                                   
                                                     bh 
                                                     1 
                                                   
                                                   - 
                                                   
                                                     ap 
                                                     1 
                                                   
                                                 
                                                 
                                                   
                                                     ah 
                                                     1 
                                                   
                                                   + 
                                                   
                                                     bp 
                                                     1 
                                                   
                                                 
                                               
                                             
                                             , 
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ah 
                                                 1 
                                               
                                               + 
                                               
                                                 bp 
                                                 1 
                                               
                                             
                                             &gt; 
                                             0 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               π 
                                               + 
                                               
                                                 arctan 
                                                  
                                                 
                                                   
                                                     
                                                       bh 
                                                       1 
                                                     
                                                     - 
                                                     
                                                       ap 
                                                       1 
                                                     
                                                   
                                                   
                                                     
                                                       ah 
                                                       1 
                                                     
                                                     + 
                                                     
                                                       bp 
                                                       1 
                                                     
                                                   
                                                 
                                               
                                             
                                             , 
                                           
                                         
                                         
                                           
                                             
                                               
                                                 ah 
                                                 1 
                                               
                                               + 
                                               
                                                 bp 
                                                 1 
                                               
                                             
                                             &lt; 
                                             0 
                                           
                                         
                                       
                                     
                                     , 
                                     
                                       
 
                                     
                                      
                                     
                                       
                                         γ 
                                         4 
                                       
                                       = 
                                       
                                         { 
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         arctan 
                                                          
                                                         
                                                           
                                                             
                                                             bu 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             az 
                                                             1 
                                                             
                                                           
                                                           
                                                             
                                                             az 
                                                             1 
                                                             
                                                             + 
                                                             
                                                             bu 
                                                             1 
                                                             
                                                           
                                                         
                                                       
                                                       , 
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           az 
                                                           1 
                                                         
                                                         + 
                                                         
                                                           bu 
                                                           1 
                                                         
                                                       
                                                       &gt; 
                                                       0 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         π 
                                                         + 
                                                         
                                                           arctan 
                                                            
                                                           
                                                             
                                                             
                                                             bu 
                                                             1 
                                                             
                                                             - 
                                                             
                                                             az 
                                                             1 
                                                             
                                                             
                                                             
                                                             
                                                             az 
                                                             1 
                                                             
                                                             + 
                                                             
                                                             bu 
                                                             1 
                                                             
                                                             
                                                           
                                                         
                                                       
                                                       , 
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           az 
                                                           1 
                                                         
                                                         + 
                                                         
                                                           bu 
                                                           1 
                                                         
                                                       
                                                       &lt; 
                                                       0 
                                                     
                                                   
                                                 
                                               
                                                
                                               
                                                 
 
                                               
                                                
                                               
                                                 γ 
                                                 5 
                                               
                                             
                                             = 
                                             
                                               γ 
                                               2 
                                             
                                           
                                           , 
                                           
                                             
                                               γ 
                                               6 
                                             
                                             = 
                                             
                                               γ 
                                               1 
                                             
                                           
                                           , 
                                           
                                             
                                               γ 
                                               7 
                                             
                                             = 
                                             
                                               γ 
                                               2 
                                             
                                           
                                           , 
                                           
                                             
 
                                           
                                            
                                           
                                             
                                               γ 
                                               8 
                                             
                                             = 
                                             
                                               γ 
                                               4 
                                             
                                           
                                           , 
                                           
                                             
                                               γ 
                                               9 
                                             
                                             = 
                                             
                                               γ 
                                               2 
                                             
                                           
                                           , 
                                           
                                             
                                               γ 
                                               10 
                                             
                                             = 
                                             
                                               γ 
                                               2 
                                             
                                           
                                           , 
                                           
                                             
                                               γ 
                                               11 
                                             
                                             = 
                                             
                                               γ 
                                               1 
                                             
                                           
                                           , 
                                           
                                             
                                               γ 
                                               12 
                                             
                                             = 
                                             
                                               γ 
                                               4 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     wherein γ i (i=1˜4)—lag angle; 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 M 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                 M 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 M 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 M 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       K 
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 k 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                 k 
                                 0 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 k 
                                 0 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   0 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   k 
                                   0 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   0 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     0 
                                   
                                 
                                  
                                 
                                     
                                 
                               
                             
                             
                               
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     k 
                                     0 
                                   
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       Δ 
                        
                       
                         ( 
                         
                           ω 
                           2 
                         
                         ) 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 k 
                                 0 
                               
                               - 
                               
                                 
                                   M 
                                   0 
                                 
                                  
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                               
                                 k 
                                 0 
                               
                               - 
                               
                                 
                                   M 
                                   0 
                                 
                                  
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                               
                                 k 
                                 0 
                               
                               - 
                               
                                 
                                   M 
                                   0 
                                 
                                  
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 0 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 0 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 0 
                               
                             
                           
                           
                             
                               
                                 3 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   0 
                                 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               - 
                               
                                 
                                   M 
                                   0 
                                 
                                  
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     wherein, M—mass coupling matrix, K—stiffness coupling matrix, and Δ(ω 2 ) is an eigenvalue equation.
 
Letting the eigenvalue equation be equal to 0, i.e., Δ(ω 2 )=0:
 
         k   0   3   k   4   −k   0   3 ω m0   2   M   4 −3ω m0   2   M   0   k   0   3 −3 k   0   2 ω m0   2   M   0   k   4 +3 k   0   2 ω m0   4   M   0   M   4 +6ω m0   4   M   0   2   k   0   2 +3 k   0 ω m0   4   M   0   2   k   4 −3 k   0 ω m0   6   M   0   2   M   4 −3ω m0   6   M   0   3   k   0 −ω m0   6   M   0   3   k   4 +ω m0   8   M   0   3   M   4 =0
 
     Obtaining inherent frequency: 
     
       
         
           
             
               
                 
                   
                     
                       ω 
                       1 
                     
                     = 
                     
                       
                         ω 
                         2 
                       
                       = 
                       
                         
                           ω 
                           0 
                         
                         = 
                         
                           
                             
                               k 
                               0 
                             
                             
                               M 
                               0 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ω 
                       3 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     0 
                                   
                                    
                                   
                                     k 
                                     0 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                     M 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                     M 
                                     4 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                   
                                     
                                       M 
                                       0 
                                     
                                      
                                     
                                       
                                         k 
                                         4 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             6 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               M 
                                               0 
                                             
                                              
                                             
                                               k 
                                               0 
                                             
                                           
                                           + 
                                           
                                             
                                               M 
                                               0 
                                             
                                              
                                             
                                               k 
                                               4 
                                             
                                           
                                           - 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                               M 
                                               4 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             M 
                             0 
                           
                            
                           
                             M 
                             4 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ω 
                       4 
                     
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   3 
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     0 
                                   
                                    
                                   
                                     k 
                                     0 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                     M 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     0 
                                   
                                    
                                   
                                     M 
                                     4 
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           3 
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             0 
                                           
                                            
                                           
                                             k 
                                             0 
                                           
                                         
                                         + 
                                         
                                           
                                             k 
                                             0 
                                           
                                            
                                           
                                             M 
                                             4 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                   
                                     
                                       M 
                                       0 
                                     
                                      
                                     
                                       
                                         k 
                                         4 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             6 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               M 
                                               0 
                                             
                                              
                                             
                                               k 
                                               0 
                                             
                                           
                                           + 
                                           
                                             
                                               M 
                                               0 
                                             
                                              
                                             
                                               k 
                                               4 
                                             
                                           
                                           - 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               0 
                                             
                                              
                                             
                                               M 
                                               4 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             M 
                             0 
                           
                            
                           
                             M 
                             4 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     When the three exciters can operate synchronously, {dot over (φ)}=ω m0 . Taking the second derivative of x 1 , x 2 , x 3 , y 1 , y 2 , y 3  and ψ to time t, which are substituted into the equations with respect to φ i  in Eq. (1), and then integrating the equations on φ=0˜2π, after taking the average value and arranging the equations, the equilibrium equations of the three exciters are obtained as follows: 
         T   e01   −f   01 ω m0   = T     L1  
 
         T   e02   −f   02 ω m0   = T     L2  
 
         T   e03   −f   03 ω m0   = T     L3   (7)
 
       wherein 
           T     L1   =T   u {2 F   2 [sin(2α 1 +2α 2 )+sin(2α 1 )]cos γ 2 +2 F   2 [cos(2α 1 +2α 2 )+cos(2α 1 )]sin γ 2   −l   0   F   13 [sin(2α 1 +2α 2 −β 1 +β 3 )+sin(2α 1 −β 1 +β 2 )]cos γ 13 +2 F   1  sin γ 1   −l   0   F   13 [cos(2α 1 +2α 2 −β 1 +β 3 )+cos(2α 1 −β 1 +β 2 )]sin γ 13   −l   0   F   13  sin γ 13 }
 
           T     L2   =T   u {2 F   2 [sin(2α 2 )−sin(2α 1 )]cos γ 2 +2 F   2 [cos(2α 2 )+cos(2α 1 )]sin γ 2   +l   0   F   13 [sin(2α 1 −β 1 +β 2 )−sin(2α 2 −β 2 +β 3 )]cos γ 13 +2 F   1  sin γ 1   −l   0   F   13 [cos(2α 1 −β 1 +β 3 )+cos(2α 2 −β 2 +β 3 )]sin γ 13   −l   0   F   13  sin γ 13 }
 
           T     L3   =T   u {−2 F   2 [sin(2α 1 +2α 2 )+sin(2α 2 )]cos γ 2 +2 F   2 [cos(2α 1 +2α 2 )+cos(2α 2 )]sin γ 2   +l   0   F   13 [sin(2α 1 +2α 2 −β 1 +β 3 )+sin(2α 2 −β 2 +β 3 )]cos γ 13 +2 F   1  sin γ 1   −l   0   F   13 [cos(2α 1 +2α 2 −β 1 +β 3 )+cos(2α 2 −β 2 +β 3 )]sin γ 13   −l   0   F   13  sin γ 13 }  (8)
 
     In the above integration process, compared with the variation degree of φ ({dot over (φ)}=ω m0 ) with time t, the changes of 2α 1  and 2α 2  are very small, and the two parameters can be regarded as slow-changing parameters thereof. According to the direct motion separation method, in the integration process, 2α 1  and 2α 2  are represented by the mean values 2 α   1  and 2 α   2  for integrals thereof. 
     The differences of output torques among the exciters are: 
       Δ T   12   = T     L1   − T     L2   =T   u {4 F   2  sin(2α 1 )cos γ 2 −2 l   0   F   13  sin(2α 1 −β 1 +β 2 )cos γ 13   +l   0   F   13 [sin(2α 2 −β 2 +β 3 )−sin(2α 1 +2α 2 −β 1 +β 3 )]cos γ 13   +l   0   F   13 [cos(2α 2 −β 2 +β 3 )−cos(2α 1 +2α 2 −β 1 +β 3 )]sin γ 13 +2 F   2 [sin(2α 1 +2α 2 )−sin(2α 2 )]cos γ 2 +2 F   2 [cos(2α 1 +2α 2 )−cos(2α 2 )]sin γ 2 }  (9)
 
       Δ T   23   = T     L2   − T     L3   =T   u {4 F   2  sin(2α 2 )cos γ 2 −2 l   0   F   13  sin(2α 2 −β 2 +β 3 )cos γ 13   +l   0   F   13 [sin(2α 1 −β 1 +β 2 )−sin(2α 1 +2α 2 −β 1 +β 3 )]cos γ 13   −l   0   F   13 [cos(2α 1 −β 1 +β 2 )−cos(2α 1 +2α 2 −β 1 +β 3 )]sin γ 13 +2 F   2 [sin(2α 1 +2α 2 )−sin(2α 1 )]cos γ 2 −2 F   2 [cos(2α 1 +2α 2 )−cos(2α 1 )]sin γ 2 }  (10)
 
     Arranging the above two equations to obtain: 
                         Δ                   T   012         T   u       =       τ     c                 12            (         α   _     1     ,       α   _     2       )         ,         Δ                   T   023         T   u       =       τ     c                 23            (         α   _     1     ,       α   _     2       )                 (   11   )               wherein, 
       τ c12 =4 F   2  sin(2α 1 )cos γ 2 −2 l   0   F   13  sin(2α 1 −β 1 +β 2 )cos γ 13   +l   0   F   13 [sin(2α 2 −β 2 +β 3 )−sin(2α 1 +2α 2 −⊕ 1 +β 3 )]cos γ 13   +l   0   F   13 [cos(2α 2 −β 2 +β 3 )−cos(2α 1 +2α 2 −β 1 +β 3 )]sin γ 13 +2 F   2 [sin(2α 1 +2α 2 )−sin(2α 2 )]cos γ 2 +2 F   2 [cos(2α 1 +2α 2 )−cos(2α 2 )]sin γ 2   (12)
 
       τ c23 =4 F   2  sin(2α 2 )cos γ 2 −2 l   0   F   13  sin(2α 2 −β 2 +β 3 )cos γ 13   +l   0   F   13 [sin(2α 1 −β 1 +β 2 )−sin(2α 1 +2α 2 −β 1 +β 3 )]cos γ 13   −l   0   F   13 [cos(2α 1 −β 1 +β 2 )−cos(2α 1 +2α 2 −β 1 +β 3 )]sin γ 13 +2 F   2 [sin(2α 1 +2α 2 )−sin(2α 1 )]cos γ 2 −2 F   2 [cos(2α 1 +2α 2 )−cos(2α 1 )]sin γ 2   (13)
 
     In the above derivation, τ c12  (α 1 , α 2 ) and τ c23  (α 1 , α 2 ) are the dimensionless coupling torques between motors  1  and  2  and between motors  2  and  3  respectively, and the constraint functions thereof are deduced as follows: 
       |τ c12 ( α   1 , α   2 )|≤τ c12 max  
 
       |τ c23 ( α   1 , α   2 )|≤τ c23 max   (14)
 
     To sum up, and combined with the above formula, the synchronization criterion of the three exciters can be obtained: 
     
       
         
           
             
               
                 
                   
                     
                        
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             T 
                             012 
                           
                         
                         
                           T 
                           u 
                         
                       
                        
                     
                     ≤ 
                     
                       τ 
                       
                         c 
                          
                         
                             
                         
                          
                         12 
                          
                         max 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                        
                       
                         
                           Δ 
                            
                           
                               
                           
                            
                           
                             T 
                             023 
                           
                         
                         
                           T 
                           u 
                         
                       
                        
                     
                     ≤ 
                     
                       τ 
                       
                         c 
                          
                         
                             
                         
                          
                         23 
                          
                         max 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     It can be seen from the above formulas that to implement synchronization, the absolute value of the difference between the dimensionless residual torques of any two exciters should be less than or equal to the maximum value of the dimensionless coupling torque. 
       T   L1 ,  T   L2  and  T   L3  are summed and divided by 3T u  to obtain the average dimensionless load torque of the three exciters, which is expressed as τ a  (α 1 , α 2 ). 
     
       
         
           
             
               
                 
                   
                     
                       
                         τ 
                         a 
                       
                       = 
                       
                         
                           
                             1 
                             
                               3 
                                
                               
                                   
                               
                                
                               
                                 T 
                                 u 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               3 
                             
                              
                             
                               
                                 T 
                                 _ 
                               
                               Li 
                             
                           
                         
                         = 
                         
                           
                             
                               1 
                               3 
                             
                              
                             
                               { 
                               
                                 4 
                                  
                                 
                                     
                                 
                                  
                                 
                                   F 
                                   2 
                                 
                                  
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     γ 
                                     2 
                                   
                                   [ 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               α 
                                               1 
                                             
                                           
                                           + 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               α 
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             α 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       cos 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           - 
                         
                       
                     
                       
                   
                    
                   
                       
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           l 
                           0 
                         
                          
                         
                             
                         
                          
                         
                           F 
                           13 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           
                             γ 
                             13 
                           
                            
                           
                             [ 
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         1 
                                       
                                     
                                     + 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         2 
                                       
                                     
                                     - 
                                     
                                       β 
                                       1 
                                     
                                     + 
                                     
                                       β 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         1 
                                       
                                     
                                     - 
                                     
                                       β 
                                       1 
                                     
                                     + 
                                     
                                       β 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         α 
                                         2 
                                       
                                     
                                     - 
                                     
                                       β 
                                       2 
                                     
                                     + 
                                     
                                       β 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                       + 
                       
                         6 
                          
                         
                             
                         
                          
                         
                           F 
                           1 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           γ 
                           1 
                         
                       
                       - 
                       
                         3 
                          
                         
                             
                         
                          
                         
                           l 
                           0 
                         
                          
                         
                           F 
                           13 
                         
                          
                         sin 
                          
                         
                             
                         
                          
                         
                           γ 
                           13 
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     The constraint function of the average dimensionless load torque of the three exciters is obtained as follows: 
       |τ a (α 1 ,α 2 )|≤τ a max   (17)
 
     The synchronization capability coefficient is defined as ζ ij  (i, j=1, 2, 3, 4), defined as: 
     
       
         
           
             
               
                 
                   
                     
                       ζ 
                       12 
                     
                     = 
                     
                       
                         τ 
                         
                           c 
                            
                           
                               
                           
                            
                           12 
                            
                           max 
                         
                       
                       
                         τ 
                         
                           a 
                            
                           
                               
                           
                            
                           max 
                         
                       
                     
                   
                   , 
                   
                     
                       ζ 
                       23 
                     
                     = 
                     
                       
                         τ 
                         
                           c 
                            
                           
                               
                           
                            
                           23 
                            
                           max 
                         
                       
                       
                         τ 
                         
                           a 
                            
                           
                               
                           
                            
                           max 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The greater the synchronization capability coefficient is, the stronger the system to achieve synchronous operation. 
     Step 3: Stability Criterion of Synchronization Status 
     The kinetic energy T and potential energy V of the whole system are deduced as: 
     
       
         
           
             
               
                 
                   T 
                   = 
                   
                     
                       1 
                       2 
                     
                     [ 
                     
                       
                         
                           M 
                           1 
                         
                          
                         
                           ( 
                           
                             
                               
                                 x 
                                 . 
                               
                               1 
                               2 
                             
                             + 
                             
                               
                                 y 
                                 . 
                               
                               1 
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                   ( 
                   20 
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     Average kinetic energy E T  and average potential energy E V  can be obtained in a single cycle. 
     Hamilton&#39;s average action amplitude over one periodic, denoted by I, is presented by: 
     
       
         
           
             
               
                 
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                   = 
                   
                     
                       
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                            
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                         T 
                       
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                         v 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Expressing the Hesse matrix of I as H to obtain: 
                             H   =       [           d   11           d   12               d   21           d   22           ]                   wherein            
            d   11     =           ∂   2        I       ∂       α   _     1   2         =       -   2                     r   2          {         [       4                   k   0          F   1          F   4          cos        (       γ   1     -     γ   4       )         +     8                   k   0          F   2          F   4          cos        (       γ   2     -     γ   4       )                    +              4                             ω     m                 0     2                     M   0          F   1          F   2          cos        (       γ   1     -     γ   2       )         -     4                   k   0          F   1          F   2          cos        (       γ   1     -     γ   2       )         +     2                   ω     m                 0     2          M   0          F   2   2       +     2                   ω     m                 0     2          M   4          F   4   2       -     2                   k   0          F   2   2       -     6                   k   0          F   4   2         ]     ×                            [       cos        (     2                   α   1       )       +     cos        (       2                   α   1       +     2                   α   2         )         ]       +                          ω     m                 0     2          JF   13   2          cos        (       2                   α   1       -     β   1     +     β   2       )         +     cos        (       2                   α   1       +     2                   α   2       -     β   1     +     β   3       )         ]             }          
            d   12     =       d   21     =           ∂   2        I         ∂       α   _     1            ∂       α   _     2           =         -   2                     r   2          {         [       4                   ω     m                 0     2          M   0          F   1          F   2          cos        (       γ   1     -     γ   2       )         +             4                   k   0          F   1          F   4          cos        (       γ   1     -     γ   4       )         +     8                   k   0          F   2          F   4          cos        (       γ   2     -     γ   4       )         -                      4                   k   0          F   1          F   2          cos        (       γ   1     -     γ   2       )         +     2                   ω     m                 0     2          M   4          F   4   2       +     2                   ω     m                 0     2          M   0          F   2   2       -     2                   k   0          F   2   2       -     6                   k   0          F   4   2         ]          cos        (       2                   α   1       +     2                   α   2         )         +       ω     m                 0     2          JF   13   2          cos        (       2                   α   1       +     2                   α   2       -     β   1     +     β   3       )                   }          
            d   22     =           ∂   2        I       ∂       α   _     2   2         =       -   2                     r   2          {     [       8                   k   0          F   2          F   4          cos        (       γ   2     -     γ   4       )         +               4                   ω     m                 0     2          M   0          F   1          F   2          cos        (       γ   1     -     γ   2       )         +     4                   k   0          F   1          F   4          cos        (       γ   1     -     γ   4       )         -     4                   k   0          F   1          F   2          cos        (       γ   1     -     γ   2       )         +     2                   ω     m                 0     2          M   0          F   2   2       +     2                   ω     m                 0     2          M   4          F   4   2       -     2                   k   0          F   2   2       -     6                   k   0          F   4   2         ]     ×                      [       cos        (     2                   α   2       )       +     cos        (       2                   α   1       +     2                   α   2         )         ]         +       ω     m                 0     2            JF   13   2     [                  cos        (       2                   α   2       -     β   2     +     β   3       )       +     cos        (       2                   α   1       +     2                   α   2       -     β   1     +     β   3       )         ]         }                     (   22   )               Letting 
     
       
      
       H 
       1 
       =d 
       11  
      
     
     
       
      
       H 
       2 
       =d 
       11 
       d 
       22 
       −d 
       12 
       d 
       21  
      
     
     In order to make the Hesse matrix of I positive definite, i.e., the H matrix positive definite, the following shall be satisfied: 
         H   1 &gt;0,  H   2 &gt;0  (23)
 
     Defining H 1  and H 2  as the stability capability coefficients of the system in the synchrous condition, the above formulas can be used to describe the stability of the system, and when the above formulas in (23) are satisfied, the system is stable. 
     The present invention has the following beneficial effects: the parameters of the high-frequency vibrating mill obtained by the method can lower the technical requirements of exciters, reduce the loss of the exciters, increase the service life of the mill, and ensure that the mill has a better crushing effect, small noise, no pollution and high reliability at the same time, which have a significant directive function for the design of structural parameters of vibrating feeder devices and the selection of working areas. 
    
    
     
       DESCRIPTION OF DRAWINGS 
       Meanings of parameters in dynamic model of system in  FIG. 1 : 
       m 1 —mass of mass body  1 ; 
       m 2 —mass of mass body  2 ; 
       m 3 —mass of mass body  3 ; 
       m 4 —mass of mass body  4 ; 
       m 01 —mass of exciter rotor  1 ; 
       m 02 —mass of exciter rotor  2 ; 
       m 03 —mass of exciter rotor  3 ; 
       φ 1 —rotation angle of exciter rotor  1 ; 
       φ 2 —rotation angle of exciter rotor  2 ; 
       φ 3 —rotation angle of exciter rotor  3 ; 
       k i , i=0˜4—spring stiffness coefficient; 
       β i (i=1˜3)—included angle between the line from the rotary center of exciter i to the mass center of a vibrating rigid body and the horizontal direction; 
       l 0 —distance from the mass center of each exciter to the mass center of the vibrating rigid body; 
       l x3 —horizontal distance from the spring of mass body  4  to the mass center of a vibrating trough; 
       l y3 —vertical distance from the spring of mass body  4  to the mass center of the vibrating trough; 
       l x1 —horizontal distance from the near endpoint of the spring to the mass center of each of the three mass bodies; 
       l x2 —horizontal distance from the far endpoint of the spring to the mass center of each of the three mass bodies; 
       l y1 —vertical distance from the near endpoint of the spring to the mass center of each of the three mass bodies; 
       l y2 —vertical distance from the far endpoint of the spring to the mass center of each of the three mass bodies; 
         FIG. 2 : phase difference among three exciters; 
         FIG. 3 : phase lag angle 
         FIG. 4 : stability capacity coefficient H 1 ; 
         FIG. 5 : stability capacity coefficient H 2 ; 
         FIG. 6 : synchronization capacity coefficient curve; 
         FIG. 7 : maximum coupling torque curve; 
         FIG. 8 : rotational speeds of three motors in area  1 ; 
         FIG. 9 : phase difference among three exciters in area  1 ; 
         FIG. 10 : displacement of area  1  in x direction; 
         FIG. 11 : partial enlarged view of the front part of displacement of area  1  in x direction; 
         FIG. 12 : partial enlarged view of the rear part of area  1  in x direction; 
         FIG. 13 : displacement of area  1  in y direction; 
         FIG. 14 : partial enlarged view of the front part of displacement of area  1  in y direction; 
         FIG. 15 : partial enlarged view of the rear part of displacement of area  1  in y direction; 
         FIG. 16 : oscillating displacement of area  1 ; 
         FIG. 17 : rotational speeds of three motors in area  2 ( a ); 
         FIG. 18 : phase difference among three exciters in area  2 ( a ); 
         FIG. 19 : displacement of area  2 ( a ) in x direction; 
         FIG. 20 : partial enlarged view of the front part of displacement of area  2 ( a ) in x direction; 
         FIG. 21 : partial enlarged view of the rear part of displacement of area  2 ( a ) in x direction; 
         FIG. 22 : displacement of area  2 ( a ) in y direction; 
         FIG. 23 : partial enlarged view of the front part of displacement of area  2 ( a ) in y direction; 
         FIG. 24 : partial enlarged view of the rear part of displacement of area  2 ( a ) in y direction; 
         FIG. 25 : oscillating displacement of area  2 ( a ); 
         FIG. 26 : rotational speeds of three motors in area  2 ( b ); 
         FIG. 27 : phase difference among three exciters in area  2 ( b ); 
         FIG. 28 : displacement of area  2 ( b ) in x direction; 
         FIG. 29 : partial enlarged view of the front part of displacement of area  2 ( b ) in x direction; 
         FIG. 30 : partial enlarged view of the rear part of displacement of area  2 ( b ) in x direction; 
         FIG. 31 : displacement of area  2 ( b ) in y direction; 
         FIG. 32 : partial enlarged view of the front part of displacement of area  2 ( b ) in y direction; 
         FIG. 33 : partial enlarged view of the rear part of displacement of area  2 ( b ) my direction; 
         FIG. 34 : oscillating displacement of area  2 ( b ); 
         FIG. 35 : rotational speeds of three motors in area  3 ; 
         FIG. 36 : phase difference among three exciters in area  3 ; 
         FIG. 37 : displacement of area  3  in x direction; 
         FIG. 38 : partial enlarged view of the front part of displacement of area  3  in x direction; 
         FIG. 39 : partial enlarged view of the rear part of displacement of area  3  in x direction; 
         FIG. 40 : displacement of area  3  in y direction; 
         FIG. 41 : partial enlarged view of the front part of displacement of area  3  in y direction; 
         FIG. 42 : partial enlarged view of the rear part of displacement of area  3  in y direction; 
         FIG. 43 : oscillating displacement of area  3 . 
     
    
    
     DETAILED DESCRIPTION 
     A high-frequency vibrating mill with three grinding drums, wherein a dynamic model thereof is shown in  FIG. 1 , comprising: exciters m 0i (i=1˜3); mass bodies m i (i=1˜4), and springs k i (i=1˜4). The model is composed of three exciters and four mass bodies. The three exciters rotate in the same direction, and each exciter rotates about its rotation axis, represented by φ oi . 
     Embodiment 1: Numerical Verification 
     Giving the parameters first, the masses of mass bodies  1 ,  2 , and  3  are equal, m 1 =m 2 =m 3 =1400 kg, and the mass of mass body  4  is 2000 kg, the masses of exciters  1 ,  2 , and  3  are equal and set as m 0 =10 kg. Assuming l 0 =1.1 m and the spring stiffness are: k 1 =k 2 =k 3 =46000 kN/m, k 4 =10 kN/m, k ψ =4000 kN/rad. Based on the given parameters setting, it can be obtained that ω ψ =52 rad/s, ω 0 =179 rad/s and ω 3 =319 rad/s. 
     Therefore, according to ω 0 =179 rad/s and ω 3 =319 rad/s, three areas can be divided: area  1  is ω m0 &lt;ω 0 , area  2  is ω 0 &lt;ω m0 &lt;ω 3 , and area  3  is ω 3 &lt;ω m0 . 
       FIG. 2  shows the stable phase difference between two exciters. In the sub-resonant area relative to ω 0 , the phase difference has two sets of solutions, one of which is 0° and 120°, and the other is 120°; in the sub-resonant area relative to ω 3 , 0° and 120°, and 120°; and in the super-resonant area relative to ω 3 , 0° and 120°, and 120°. 
       FIG. 3  shows the lag angle of the three mass bodies, and γ is the lag angle. In the first area, γ 1  is 0, γ 2  and γ 4  are 180; in the second area, γ 2  is 360, γ 1  and γ 4  are 180; and in the third area, γ 4  is 360, γ 1  and γ 2  are 180. 
       FIG. 4  and  FIG. 5  show the stability capacity coefficient curve, and it can be seen from an enlarged view of the curve that the stability capacity coefficient curve is greater than or equal to 0 in the whole interval. Moreover, in area  1 , the stability coefficient is increased significantly. 
       FIG. 6  shows the synchronization capability coefficient. It can be seen that the extremum is around at ω ψ , and the synchronization capability coefficient is nearly 0 at points of ω 0  and ω 3 . 
       FIG. 7  shows the maximum coupling torque curve. It can be seen that the extremum of the maximum coupling torque is observed in the vicinity of point ω ψ , and the maximum coupling torque increases significantly when approaches ω 3 . 
     The results of numerical analysis show that when the system is operated in the sub-resonant state relative to ω 0  and super-resonant state of ω 3 , i.e., area  1  (ω m0 &lt;ω 0 ) and area  3  (ω 3 &lt;ω m0 ), the phase difference between two exciters has two sets of stable solutions, and nonlinear system diversity appears; and when the system is in sub-resonant corresponding to ω 3 , or super-resonant state relative to ω 0 , i.e., area  2  (ω 0 &lt;ω m0 &lt;ω 3 ), the working region is divided into a first half and a second half with ω m0 =250 rad/s as the midpoint. In the first half, the system only has one set of stable solutions about phase difference, while in the second half, the system has two sets of stable solutions, which then become one set of stable solutions. 
     Embodiment 2: Simulation of the Vibrating System 
     During the simulation of a vibrating system, a fourth order fourth-order Rouge-Kutta procedure is mainly used, where four sets of simulation is finished corresponding to the three areas previously divided. In actual engineering applications, the same exciters are generally used, and the parameters of four motors are selected to be same, i.e., η=1.0. The overall system parameters are set as follows: rotor resistance R r =3.40Ω, stator resistance R s =3.35Ω, rotor inductance L r =170 mH, stator inductance L s =170 mH, mutual inductance L m =164 mH, and f 1y =f 2y =0.05. Other parameters of the vibrating system: r=0.15 m, m 1 =m 2 =m 3 =1400 kg, m 0 =10 kg, and k 1 =k 2 =k 3 =k 0 . The parameters are adjusted to make the system operate in the sub-resonant and super-resonant states, respectively. 
     For the simulation of area  1 , assuming k 0 =30000 KN/m, k 4 =30 KN/m, and k ψ =4000 KN/m: 
     The rotational speed of the motors corresponding to the simulation result is 894 r/min. Through calculation, the rotational speed is 114 rad/s in the phase difference diagram of numerical analysis in this condition. 
       FIG. 8  shows the stable state of the speed of two exciters. The speed of the two exciters becomes stable quickly in a short time, and the synchronization speed is basically stabilized at about 894 r/min. After interference is added on exciter  2  at 30 s, the rotational speed is not changed significantly. 
       FIG. 9  shows the stable state of the phase difference. In the stable state, the phase difference between exciters  1  and  2 ,  2  and  3 ,  1  and  3  is both in the vicinity of 120°; after interference is added at 30 s, the curve has a small fluctuation and then returns to the original phase difference quickly. 
       FIGS. 10, 11, and 12  show the displacements of mass bodies  1 ,  2 , and  3  in x direction. It can be seen from the partial enlarged view of the displacement relationship that the displacements among mass bodies  1 ,  2 , and  3  are equal, whereas there exists no vibration for mass body  4  in x direction; after interference is added at 30 s, the displacement curve is not changed significantly. 
     Similarly,  FIGS. 13, 14, and 15  show the displacements of mass bodies  1 ,  2 , and  3  in y direction. From  FIGS. 13-15 , the displacements among mass bodies  1 ,  2 , and  3  are equal, whereas the mass body  4  is nearly at rest in y direction. The displacement curve of each mass body is consistent before and after the addition of interference. 
       FIG. 16  shows the simulation curve of oscillating angle of the system. It can be seen that the oscillating angle is near 0° and does not change obviously when implement the interference. 
     For simulation of area  2 ( a ), the stiffness parameters are set as k 0 =9000 KN/m, k 4 =30 KN/m, and k ψ =3000 KN/m. 
       FIG. 17  shows the stable state of the speed of two exciters. The rotational speed of the motors corresponding to the simulation result is 870 r/min. After interference is added on exciter  2  at 30 s, the rotational speed is not changed significantly. Through calculation, the rotational speed corresponds to 205 rad/s in the phase difference diagram of numerical analysis at this time. 
       FIG. 18  shows the stable state of the phase difference. The phase difference between exciters  1  and  2 , exciters  2  and  3 , exciters  1  and  3  is both around 0°; after interference is added at 30 s, the phase difference remains unchanged. 
       FIGS. 19-21  illustrates the displacements of mass bodies  1 ,  2 , and  3  in x direction. According to the partial enlarged view, the displacements among mass bodies  1 ,  2 , and  3  are equal, and the displacement direction of mass body  4  is opposite to the moving direction of mass bodies  1 ,  2 , and  3 ; after interference is added at 30 s, the displacement curve remains stable. 
       FIGS. 22-24  show the displacements of mass bodies  1 ,  2 , and  3  in y direction, where the motion characteristics are similar to that of x direction. Besides, the oscillating angle is plotted as  FIG. 25 , which is almost 0° and nearly unchanged after adding interference to the system. 
     For simulation of area  2 ( b ), the parameters are set as k 0 =9000 KN/m, k 4 =30 KN/m, and k ψ =3000 KN/m. The rotational speed of the motors corresponding to the simulation result is 982 r/min. Through calculation, the rotational speed corresponds to 264 rad/s in the phase difference diagram of numerical analysis in this condition. 
       FIG. 26  shows the stable state of the speed of two exciters. The speed of the two exciters becomes stable quickly in a short time, and the synchronization speed is basically stabilized at 982 r/min; after interference is added on exciter  2  at 30 s, the rotational speed is not changed significantly. 
       FIG. 27  shows the stable state of the phase difference. The phase difference between exciters  1  and  2 , exciters  2  and  3 , exciters  1  and  3  is 0°; after interference is added at 30 s, the phase difference is suddenly changed to 120°. 
       FIGS. 28, 29 and 30  show the displacements of mass bodies  1 ,  2 , and  3  in x direction. It can be seen from the partial enlarged view of the displacement relationship that the displacements among mass bodies  1 ,  2 , and  3  are equal, and the displacement direction of mass body  4  is opposite to the moving direction of mass bodies  1 ,  2 , and  3 ; after interference is added at 30 s, the displacements are changed, mass body  4  is in stationary state, and the displacements of mass bodies  1 ,  2 , and  3  are equal. 
       FIGS. 31, 32, and 33  show the displacements of mass bodies  1 ,  2 , and  3  in y direction. It can be seen from the partial enlarged view of the displacement relationship that the displacements among mass bodies  1 ,  2 , and  3  are equal, and the displacement direction of mass body  4  is opposite to the moving direction of mass bodies  1 ,  2 , and  3 ; after interference is added at 30 s, the displacements are changed, mass body  4  is in stationary state, and the displacements of mass bodies  1 ,  2 , and  3  are equal.  FIG. 34  shows the oscillating angle of the system. It can be seen that the oscillating angle is near 0° and is not changed significantly after the addition of interference. 
     Simulation is performed to area  3 , assuming k 0 =3200 KN/m, k 4 =30 KN/m, and k ψ =2000 KN/m. The rotational speed of the motors corresponding to the simulation result is 982 r/min. Through calculation, the rotational speed corresponds to 376 rad/s in the phase difference diagram of numerical analysis at this time. 
       FIG. 35  shows the stable state of the speed of two exciters. The speed of the two exciters becomes stable quickly in a short time, and the synchronization speed is basically stabilized at 982 r/min; after interference is added on exciter  2  at 30 s, the rotational speed is not changed significantly. 
       FIG. 36  shows the stable state of the phase difference. The phase difference between exciters  1  and  2 , exciters  2  and  3 , exciters  1  and  3  is 0°; after interference is added at 30 s, the phase difference is suddenly changed to 120°. 
       FIGS. 37-39  show the displacements of mass bodies  1 ,  2 , and  3  in x direction. It can be seen from the partial enlarged view of the displacement relationship that the displacements among mass bodies  1 ,  2 , and  3  are equal, and the displacement direction of mass body  4  is opposite to the moving direction of mass bodies  1 ,  2 , and  3 ; after interference is added at 30 s, mass body  4  is in stationary state, and the displacements of mass bodies  1 ,  2 , and  3  are equal. 
       FIGS. 40, 41, and 42  show the displacements of mass bodies  1 ,  2 , and  3  in y direction. It can be seen from the partial enlarged view of the displacement relationship that the displacements among mass bodies  1 ,  2 , and  3  are equal, and the displacement direction of mass body  4  is opposite to the moving direction of mass bodies  1 ,  2  and  3 ; after interference is added at 30 s, the displacements are changed, mass body  4  is in stationary state, and the displacements of mass bodies  1 ,  2 , and  3  are equal. 
       FIG. 43  shows the oscillating angle of the system. It can be seen that the oscillating angle is near 0° and is not changed significantly after the addition of interference. 
     System simulation results show that the phase differences of area  1 , area  3  and the second half of area  2  does not return to the original state after interference, while in the first half of area  2 , the system is still stable after applying interference; at the same time, we can observe that in the working region of the first half of area  2 , the displacements of mass bodies  1 ,  2 , and  3  in x and y directions are equal and equidirectional, while mass body  4  has large displacements in x and y directions and reverse vibration. 
     Conclusion 
     (1) According to the comparison between the results of step 4 and step 5, the numerical and simulation results are the same. Therefore, the method for determining parameters of the present invention is correct. 
     (2) The present invention provides a new model of vibrating mill, which uses three motors to drive four mass bodies. According to the establishment of motion differential equations, numerical analysis, and simulation, it is concluded that the phase differences in the area  1 , area  3 , and the second half of working area  2  in the present invention cannot return to the original state after interference, so the stability is insufficient. Therefore, the working area should be selected in the first half of area  2 , and the motion state of the vibrating mill during operation can be analyzed. 
     (3) It is known from the simulation results that the moving direction of mass bodies  1 ,  2 ,  3  is opposite to that of the mass body  4  (the mill body), and the amplitude of the mill body is large so that medium and materials are fully ground, and the grinding accuracy is improved, thus proving the rationality of the model of the present invention. 
     (4) The research content of the present invention has a significant directive function for the design of structural parameters of vibrating mill devices and the selection of working areas. 
     Embodiment 3: Example Data and Parameters of a Vibrating Feeder. The Present Invention is not Limited to the Design Parameters Herein 
     Spring stiffnesses: k 0 =9000 KN/m, k 4 =30 KN/m, k ψ =3000 KN/m, and k 1 =k 2 =k 3 =k 0 ; 
     Damping coefficient: 
     Masses of mass bodies: m 1 =m 2 =m 3 =1400 kg, and m 4 =2000 kg; 
     r=0.15 m; l 0 =1.1 m; Synchronization speed: ω m0 =790 r/min to 811 r/min; 
     Masses of unbalanced rotors of exciters: m 01 =m 02 =m 03 =10 kg; and 
     Motor parameters: rotor resistance R r =3.40Ω, stator resistance R s =3.35Ω, rotor inductance L r =170 mH, stator inductance L s =170 mH, and mutual inductance L m =164 mH. 
     The two motors are of the same type and are three-phase squirrel-cage motors (VB-1082-W type, 380 V, 50 Hz, 6-pole, Δ-connection, 0.75 kw, 980 r/min, and 39 kg).