Patent ID: 11859686
Assignee: HARBIN ENGINEERING UNIVERSITY
Field: Mechanical elements (Mechanical engineering)
Classification: CPC F | IPC F

Claim 4:
5. The method according to claim 4, wherein the step 5 comprises:
step 5.1: establishing a vibration isolation system dynamic formula, and substituting an ideal electromagnetic force form into a dynamic mathematical model of a vibration isolation system by the following formulas:

m{umlaut over (x)}t+c({dot over (x)}t−{dot over (x)}e)+km(xt−xe)+k(xt−xe)=0,

xe=Xe cos(wet),

x=xt−xe,, wherein m represents a vibration-isolated mass; {umlaut over (x)}t represents a vibration response acceleration of a vibration-isolated object; {dot over (x)}t represents a vibration response speed of the vibration-isolated object; xe represents a system excitation; {dot over (x)}e represents a system excitation speed; c represents a system damping; xt represents a vibration response of the vibration-isolated object; k represents a magnitude of a value of the positive stiffness; Xe represents an excitation amplitude; we represents an excitation frequency; and t represents time;
step 5.2: under ideal conditions, when a magnitude of an absolute value of negative stiffness is the same as that of positive stiffness and an overall dynamic stiffness of the system is zero, that is, km+k=0, representing a dynamic mathematic model of the vibration isolation system by the following formula:, x
      ¨
     
     t
    
    +
    
     
      c
      m
     
     ⁢
     
      
       x
       ˙
      
      t
     
    
   
   =
   
    
     
      
       -
       c
      
      ⁢
      
       X
       e
      
      ⁢
      
       w
       e
      
     
     m
    
    ⁢
    sin
    ⁢
    
     (
     
      
       w
       e
      
      ⁢
      t
     
     )
    
   
  
  ,
 

determining a displacement response, and representing a vibration response by the following formulas:, x
       
        t
        ⁢
        11
       
      
      =
      
       
        C
        1
       
       +
       
        
         C
         2
        
        ⁢
        
         e
         
          
           
            -
            c
           
           m
          
          ⁢
          t
         
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       C
       1
      
      =
      
       -
       
        X
        e
       
      
     
     ,
     
      
       C
       2
      
      =
      
       
        
         X
         e
        
        ⁢
        
         m
         2
        
        ⁢
        
         w
         e
         2
        
       
       
        
         
          m
          2
         
         ⁢
         
          w
          e
          2
         
        
        +
        
         c
         2
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       x
       
        t
        ⁢
        12
       
      
      =
      
       
        
         
          C
          3
         
         ⁢
         cos
         ⁢
         
          (
          
           
            w
            e
           
           ⁢
           t
          
          )
         
        
        +
        
         
          C
          4
         
         ⁢
         sin
         ⁢
         
          (
          
           
            w
            e
           
           ⁢
           t
          
          )
         
        
       
       =
       
        
         
          
           C
           3
           2
          
          +
          
           C
           4
           2
          
         
        
        ⁢
        
         sin
          
         [
         
          
           
            w
            e
           
           ⁢
           t
          
          +
          
           arc
           ⁢
           tan
           ⁢
           
            (
            
             
              C
              3
             
             
              C
              4
             
            
            )
           
          
         
         ]
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       C
       3
      
      =
      
       
        
         X
         e
        
        ⁢
        
         c
         2
        
       
       
        
         
          m
          2
         
         ⁢
         
          w
          e
          2
         
        
        +
        
         c
         2
        
       
      
     
     ,
     
      
       C
       4
      
      =
      
       
        c
        ⁢
        
         X
         e
        
        ⁢
        
         mw
         e
        
       
       
        
         
          m
          2
         
         ⁢
         
          w
          e
          2
         
        
        +
        
         c
         2
        
       
      
     
     ,
    
   
  
 

wherein xt11 represents a solution corresponding to a free vibration response; xt12 represents a particular solution corresponding to a forced vibration response; C1 represents a coefficient of a particular solution corresponding to the free vibration response; C2 represents a coefficient of the particular solution corresponding to the free vibration response; C3 represents a coefficient of a particular solution corresponding to the forced vibration response; and C4 represents a coefficient of the particular solution corresponding to the forced vibration response; and
step 5.3: when stiffness of the negative stiffness mechanism is positive stiffness, that is, km+k>0, representing a dynamic mathematic model of the vibration isolation system by the following formula:, x
      ¨
     
     t
    
    +
    
     
      c
      m
     
     ⁢
     
      
       x
       ˙
      
      t
     
    
    +
    
     
      
       
        k
        m
       
       +
       k
      
      m
     
     ⁢
     
      x
      t
     
    
   
   =
   
    
     
      
       
        k
        m
       
       +
       k
      
      m
     
     ⁢
     
      X
      e
     
     ⁢
     cos
     ⁢
     
      (
      
       
        w
        e
       
       ⁢
       t
      
      )
     
    
    -
    
     
      c
      m
     
     ⁢
     
      X
      e
     
     ⁢
     
      w
      e
     
     ⁢
     sin
     ⁢
     
      (
      
       
        w
        e
       
       ⁢
       t
      
      )
     
    
   
  
  ,
 

determining a displacement response, and representing the displacement response by the following formulas:, x
       t
      
      =
      
       
        x
        
         t
         ⁢
         21
        
       
       +
       
        x
        
         t
         ⁢
         22
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       x
       
        t
        ⁢
        21
       
      
      =
      
       
        
         C
         7
        
        ⁢
        
         e
         
          
           
            -
            c
           
           
            2
            ⁢
            m
           
          
          ⁢
          t
         
        
        ⁢
        cos
        ⁢
        
         (
         
          
           
            
             
              k
              m
             
             +
             k
            
            m
           
          
          ⁢
          t
         
         )
        
       
       +
       
        
         C
         8
        
        ⁢
        
         e
         
          
           
            -
            c
           
           
            2
            ⁢
            m
           
          
          ⁢
          t
         
        
        ⁢
        sin
        ⁢
        
         (
         
          
           
            
             
              
               k
               m
              
              +
              k
             
             m
            
            -
            
             
              c
              2
             
             
              4
              ⁢
              
               m
               2
              
             
            
           
          
          ⁢
          t
         
         )
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       C
       8
      
      =
      
       
        
         -
         c
        
        ⁢
        
         X
         e
        
        ⁢
        
         {
         
          
           2
           ⁢
           
            m
            2
           
           ⁢
           
            w
            e
            4
           
          
          +
          
           [
           
            
             
              (
              
               
                k
                m
               
               +
               k
              
              )
             
             ⁢
             
              (
              
               
                k
                m
               
               +
               k
               -
               
                mw
                e
                2
               
              
              )
             
            
            +
            
             
              c
              2
             
             ⁢
             
              w
              e
              2
             
            
           
           ]
          
         
         }
        
       
       
        
         
          
           4
           ⁢
           
            m
            ⁡
            (
            
             
              k
              m
             
             +
             k
            
            )
           
          
          +
          
           c
           2
          
         
        
        [
        
         
          
           (
           
            
             k
             m
            
            +
            k
            -
            
             mw
             e
             2
            
           
           )
          
          2
         
         +
         
          
           c
           2
          
          ⁢
          
           w
           e
           2
          
         
        
        ]
       
      
     
     ,
    
   
  
  
   
    
     
      
       x
       
        t
        ⁢
        22
       
      
      =
      
       
        
         
          C
          9
         
         ⁢
         cos
         ⁢
         
          (
          
           
            w
            e
           
           ⁢
           t
          
          )
         
        
        +
        
         
          C
          
           1
           ⁢
           0
          
         
         ⁢
         sin
         ⁢
         
          (
          
           
            w
            e
           
           ⁢
           t
          
          )
         
        
       
       =
       
        
         
          
           C
           9
           2
          
          +
          
           C
           
            1
            ⁢
            0
           
           2
          
         
        
        ⁢
        
         sin
         [
         
          
           
            w
            e
           
           ⁢
           t
          
          +
          
           arc
           ⁢
           tan
           ⁢
           
            (
            
             
              C
              9
             
             
              C
              
               1
               ⁢
               0
              
             
            
            )
           
          
         
         ]
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       C
       9
      
      =
      
       
        
         X
         e
        
        [
        
         
          
           (
           
            
             k
             m
            
            +
            k
           
           )
          
          ⁢
          
           (
           
            
             k
             m
            
            +
            k
            -
            
             mw
             e
             2
            
           
           )
          
         
         +
         
          
           c
           2
          
          ⁢
          
           w
           e
           2
          
         
        
        ]
       
       
        
         
          (
          
           
            k
            m
           
           +
           k
           -
           
            mw
            e
            2
           
          
          )
         
         2
        
        +
        
         
          c
          2
         
         ⁢
         
          w
          e
          2
         
        
       
      
     
     ,
    
   
  
  
   
    
     
      
       C
       
        1
        ⁢
        0
       
      
      =
      
       
        
         cmX
         e
        
        ⁢
        
         w
         e
         3
        
       
       
        
         
          (
          
           
            k
            m
           
           +
           k
           -
           
            mw
            e
            2
           
          
          )
         
         2
        
        +
        
         
          c
          2
         
         ⁢
         
          w
          e
          2
         
        
       
      
     
     ,
    
   
  
 

wherein xt21 represents a solution corresponding to a free vibration response; xt22 represents a particular solution corresponding to a forced vibration response; C7 represents a coefficient of a particular solution corresponding to the free vibration response; C8 represents a coefficient of the particular solution corresponding to the free vibration response; C9 represents a coefficient of a particular solution corresponding to the forced vibration response; and C10 represents a coefficient of the particular solution corresponding to the forced vibration response.