Abstract:
A method of determining springback in metal forming with a fluid cell press through establishing a computational formulation to determine bend angle and compensated die radius based on factors of geometry of the part being formed, material properties of the sheet material and the forming process, and computing additional iterations of springback until a specific tolerance between the formed part angle and the desired part angle are reached.

Description:
BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   This invention relates to bending methods for forming sheet metal parts and more specifically for calculating springback when forming parts by hydroforming with a fixed die and hydraulic pressure exerted by a flexible diaphragm. 
   2. Description of Prior Art 
   With the advent of metal airplanes, it became necessary to find a method of forming complex sheet metal parts economically. Traditional stamping methods, also referred to as draw forming, involve high tool costs that could not be justified for the relatively small quantities of parts produced. Traditional stamping required two mating precisely aligned dies. Hydroforming, also referred to as fluid forming, involves only a single die wherein the hydraulic pressure was applied against a flexible diaphragm, which forms the sheet material against a die. 
   Various other metal forming methods are utilized in the various industries such as a stretch press, wherein the work piece is stretched over a single die. The brake press is another commonly used method, which deals with more simplified two-dimensional bends. 
   When a metal workpiece is deformed during a metal forming operation, the deformation thus given has two components—elastic and plastic. Upon removal of the forming load, while the plastic component remains unchanged, the elastic component is recovered. The magnitude of this recovery is called springback. All metal forming methods thus have to deal with the springback problem. During sheet bending, this recovery or springback is manifested in the workpiece in the form of an increased part angle and radius of bend than that desired. The shape of the workpiece springs back to a shape, which is almost never the shape optimally desired nor the shape of the bending die. In the prior art this required re-cutting the forming die numerous times, which is very costly and time consuming. 
   One method to solve this problem has been to cause the workpiece to be excessively bent in the bending direction such that upon springback the workpiece can assume the proper shape. This method requires the die designer to guess at the shape of the bending die, which can be very costly if incorrect. 
   Solving the springback problem has been addressed in various other metal forming methods such as the patent to Ewert, U.S. Pat. No. 4,989,439 in a stretch press process; the patent to Jones, U.S. Pat. No. 4,802,357 and Ooenoki et al, U.S. Pat. No. 6,161,408, both of which deal with methods for compensating for springback in the brake press field of metal forming. The patent to Yamano et al, Pub. No. US 2003/0061852 deals with calculated springback in the conventional draw forming method of metal forming. 
   Research to predict springback in hydroforming, has been very limited and is exemplified in a publication entitled “Sheet Metal Forming in the Quintus® Fluid Cell Press” published in April of 1980 and authored by Eric Enroth, published by the Quintus Press Department. This publication provides some rules of thumb for springback allowances for different materials irrespective of geometric or process conditions. Commonly used springback prediction tools in the industry are based on experiments done for a specific bend angle (90 degrees) for specific materials, and springback charts obtained by doing tests on specific geometric conditions. The springback is predicted for different geometric conditions on the basis of limited data and arbitrary extrapolations; therefore, their prediction is not precise. Process parameters are not considered in these methods and any process variation during manufacturing alters springback. There is no single tool available that can accurately predict the total springback for all parts irrespective of material, process and geometric condition. 
   SUMMARY OF THE INVENTION 
   The only way to control springback in any forming process is to be able to accurately predict springback. This requires understanding all the different factors and their respective interactions that influence springback. 
   The significant factors influencing springback can be divided into the following three groups:
         1. Geometric factors: Which include; a) part geometry, b) bend radius, c) part angle, and d) sheet thickness.   2. Process factors: Which include; a) forming pressure, b) type of hydropress, c) type of lubricant used and friction coefficient, and d) hardness of rubber.   3. Material factors: Which include; a) yield strength, b) elastic modulus, c) lot variability, d) material hardening (strain and work hardening), and e) anisotropy.       

   The computational formulation for calculating springback takes into account process, geometric and material factors during the bending operation using hydroforming to predict the amount of springback. 
   The problems in the prior methods of accurately predicting springback are numerous. They don&#39;t include process factors; they don&#39;t consider interactions between the geometric and material factors; they include incorrect assumption of angle of bend resulting in incorrect prediction; they don&#39;t include repetitive physical iterations to the die angle and springback to achieve the part angle after springback. 
   It is, therefore, the principal object of the present invention to provide a method for accurately calculating springback by establishing a computational formulation which includes factors involving geometry, material properties and forming process factors. 
   Another object of the present invention is the method of calculating total compensated springback and compensated die radius based on factors of geometry, material properties and forming process factors and then computing additional iterations of die angle and springback prediction until the formed part angle and design part angle are the same. 

   
     DESCRIPTION OF THE DRAWINGS 
     The accompanying drawings which are incorporated in and constitute a part of the specification illustrate an embodiment of the invention and the steps of the method for practicing the invention and together with the description, serve to explain the principals of the invention. 
       FIG. 1  is a schematic view of a fluid cell hydropress in various stages; 
       FIG. 2  is a block diagram illustrating the steps of the method for determining the amount of over-bend required to compensate for springback in workpiece; 
       FIG. 3  is a schematic illustration of a series of iterations for calculating total compensated springback; 
       FIG. 4  is a schematic for springback measurement; 
       FIG. 5  is schematic illustration of the forces involved in hydroforming; 
       FIG. 6  is a schematic drawing showing the dependency of the die radius on accurate prediction springback; and, 
       FIG. 7  is an example of a web-based application utilizing the disclosed method of present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   Hydroforming, sometimes referred to as fluid forming or rubber diaphragm forming, was developed in response to a need for a low cost method of producing relatively small quantities of a wide variety of sheet metal parts. The principal of forming in a typical hydropress is illustrated in  FIG. 1 . It symbolically illustrates a typical hydropress before the forming starts in Illustration “A”; pressure being applied on the diaphragm or bladder to form the part in Illustration “B”; and removal of the part after the pressure of the bladder is relieved in Illustration “C”. The tool  12  is placed in the hydropress  10  on a fixed table  13 . A sheet metal blank  14  placed on top of the tool  12 . A rubber diaphragm  16  in the “A” illustration is shown retracted with unpressurized fluid  18  positioned above the diaphragm. In Illustration “B” the fluid  18  is pressurized thus causing diaphragm  16  to extend downward forming the blank  14  around the tool  12 . The high pressure fluid above that diaphragm  16  is relieved as shown in Illustration  3 , whereupon table  13  is rolled out of the press  10  and the formed part is removed from the tooling. 
   Once the formed part  20  is unloaded in the press, it tries to regain its original shape. This difference is called springback. In  FIG. 3 , a blank  22  is hydroformed on tool  24  with a design part angle X. Once the hydropress diaphragm is retracted blank  22  springs back to an initial springback angle of S 1 , which is the initial springback prediction. To account for springback, the magnitude of the die angle must be reduced to X−S 1 , as shown in the first iteration with a tool angle of X−S 1 . A theoretical blank is hydroformed as shown in iteration No. 1, which when released from the press has springback angle of S 2 . Therefore, die angle for the next iteration needs to be X−S 2 . With each additional iteration, the predicted springback S increases. The iterations are performed till the sum of the die angle and the predicted springback reaches the precise part angle X. The predicted springback at this point is the total compensated springback, S N  and the formed part is at the precise angle of the designed part. 
   Referring next to  FIG. 2 , there is shown a flow chart and a system diagram of the computational application utilizing the disclosed method. In unit  26 , the user is required to input bend conditions for the bending process that include material, process and geometric parameters. This information is fed into the numerical operation unit  28 . Within unit  28  are two sub-units, computational formulation unit  30  and virtual iteration loop unit  32 . The sub-unit  30  uses a mathematics formulation involving all material, process and geometric parameters and their interactions to predict springback. Unit  32  then proceeds through the iterations to predict the total compensated springback. The format of the formulation is as follows:
 
Springback= A+B (Thickness)+ C (Pressure)+ D (Bend Radius)+ E (Die Angle) +F (Hydropress)+ G (Thickness)(Pressure)+ H (Thickness)(Bend Radius)+ I (Thickness)(Die Angle)+ J (Thickness)(Hydropress)+ K (Pressure)(Bend Radius)+ L (Pressure)(Die Angle)+ M (Pressure)(Hydropress)+ N (Bend Radius)(Die Angle)+ O (Bend Radius)(Hydropress)+ P (Die Angle)(Hydropress) +Q (Thickness)(Pressure)(Bend Radius)+ R (Thickness)(Pressure)(Die Angle)+ S (Thickness)(Pressure)(Hydropress)+ T (Thickness)(Bend Radius)(Die Angle)+ U (Thickness)(Bend Radius)(Hydropress)+ V (Thickness)(Die Angle)(Hydropress) +W (Pressure)(Bend Radius)(Die Angle)
 
   The values of the constants A–W are as follows for the three aluminum sheet stocks listed: 
   
     
       
             
             
             
             
           
             
             
             
             
           
         
             
                 
                 
             
             
                 
                 
               AA 2024-T3 
               AA 2024-T4 
             
             
                 
               Constant 
               (ISO Al Cu 4 Mgl) 
               Tform/2524-T3 
             
             
                 
                 
             
           
           
             
                 
             
           
        
         
             
                 
               A 
               6.5766 
               14.9827 
             
             
                 
               B 
               12.444 
               −247.431 
             
             
                 
               C 
               0.00138 
               0.0000759 
             
             
                 
               D 
               140.113 
               81.4027 
             
             
                 
               E 
               1.1835 
               −0.01518 
             
             
                 
               F 
               0 
               −0.782 
             
             
                 
               G 
               0 
               −0.00412 
             
             
                 
               H 
               1925.3 
               −489.422 
             
             
                 
               I 
               1.7394 
               2.0696 
             
             
                 
               J 
               0 
               0 
             
             
                 
               K 
               0.00149 
               0 
             
             
                 
               L 
               0.000031 
               0 
             
             
                 
               M 
               0 
               0.00006203 
             
             
                 
               N 
               −0.3195 
               −0.35584 
             
             
                 
               O 
               0 
               0 
             
             
                 
               P 
               0 
               0 
             
             
                 
               Q 
               0 
               0 
             
             
                 
               R 
               0 
               0 
             
             
                 
               S 
               0 
               0 
             
             
                 
               T 
               16.352 
               0 
             
             
                 
               U 
               0 
               0 
             
             
                 
               V 
               0 
               0 
             
             
                 
               W 
               −0.000016 
               0 
             
             
                 
                 
             
           
        
       
     
   
   The following variables in the equation in parenthesis are; (thickness) of the blank; (pressure) in the hydropress; die (bend radius); (die angle) and (hydropress) type. 
   The variables in formulation are in parenthesis; for example, (pressure) represent mathematical terms involving pressure utilized in the hydropress. The terms (thickness) (pressure) represent interaction between the respective parameters. The material parameters have been included by formulating individual springback prediction equations for each individual material by performing experiments over a wide range of material production lots for the respective material. 
   The consideration of material, process and geometric parameters and their interactions helps estimate the springback with very high accuracy. In  FIG. 2 , sub-unit  32  tries to minimize the differences between design and formed part angles after unloading of the fluid pressure. It uses design part angle X in  FIG. 3  to obtain initial springback prediction S 1  but then iteratively predicts springback based on the new forming angle until the part angle is equal to the design part angle (∠X*=∠X). These iterations are done on the computer within a few seconds as compared with hours in resources spent on physical shop trials. The iterative springback thus predicted is called the total compensated springback, as shown in the following chart. The numerical operation unit  28  uses the total compensated springback to calculate the compensated die radius. The compensated die radius would be different from the desired part radius owing to compensation of the die by the total compensated springback. The formulation used to calculate the compensated die radius is as follows: 
   
     
       
         
           
             Compensated 
             ⁢ 
             
                 
             
             ⁢ 
             die 
             ⁢ 
             
                 
             
             ⁢ 
             radius 
           
           = 
           
             
               Designed 
               ⁢ 
               
                   
               
               ⁢ 
               part 
               ⁢ 
               
                   
               
               ⁢ 
               radius 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 
                   180 
                   - 
                   
                     Designed 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     part 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     angle 
                   
                 
                 ) 
               
             
             
               180 
               - 
               
                 Designed 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 part 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 angle 
               
               + 
               
                 Total 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 compensated 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 springback 
               
             
           
         
       
     
   
   The calculated total compensated springback and estimated die radius are displayed in user-output unit  34  in  FIG. 2 . 
   The operation of the application is illustrated below: 
   
     
       
             
           
             
             
             
           
         
             
                 
             
             
               User - Input Unit: 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
                 
               Material: 2024-T3 
               Bend radius: 0.250″ 
             
             
                 
               Sheet thickness: 0.032″ 
               Part angle (∠A): 60° 
             
             
                 
               Forming pressure: 8000 psi 
               Hydropress: ASEA 
             
             
                 
                 
             
             
                 
               Assume specified tolerance for formed part (Part angle after springback − Desired part angle) = 0° 
             
           
        
       
     
   
   
     
       
             
           
             
             
             
             
             
           
         
             
                 
             
             
               Computation formulation and iteration (30 and 32): 
             
           
        
         
             
                 
                 
                 
               Formed part 
               Difference between 
             
             
                 
               Die angle 
               Predicted 
               angle after 
               formed and design 
             
             
               Iteration 
               (X-S n ) 
               springback (S n ) 
               springback 
               part angles 
             
             
                 
             
             
                 
               60.00° (X) 
               18.76° (S 1 ) 
                 
                 
             
             
               1 
               41.24° 
               21.166° (S 2 ) 
               62.406° (X 1 ) 
               2.406° 
             
             
               2 
               38.834° 
               21.475° (S 3 ) 
               60.309° (X 2 ) 
               0.309° 
             
             
               3 
               38.525° 
               21.515° (S 4 ) 
               60.040° (X 3 ) 
               0.040° 
             
             
               4 
               38.485° 
               21.519° (SX) 
               60.004° (X 4 ) 
               0.004° 
             
             
               5 
               38.480° 
               21.520° (Total 
               60.000° (X*) 
               0.000° 
             
             
                 
                 
               compensated 
             
             
                 
                 
               springback) 
             
             
                 
             
           
        
       
     
   
   We now refer to  FIG. 2  and the above table to understand the operation of the virtual iteration loop  32 . To begin the virtual iteration loop operation, the sub-unit  30  uses the design part angle (∠X=60°) to obtain the initial springback prediction (∠S 1 =18.76°). 
   Iteration 1: The sub-unit  30  uses the predicted springback (∠S 1 =18.76°) to compensate the die to obtain the forming angle for Iteration 1 (Forming angle=Die angle=Desired part angle, ∠X−Predicted springback, (∠S 1 =41.24°). It then uses the forming angle (41.24°) to predict the new springback (∠S 2 =21.166°). The sub-unit  32  now adds the thus calculated springback (21.166°) to the forming angle (41.24°) to obtain the formed part angle after springback (X 1 =21.166°+41.24°=62.406°). As indicated in  FIG. 2 , the sub-unit  32  then compares the difference between the formed part angle after springback and the part angle to the specified tolerance (0°). Since the difference (2.406°) is greater than the specified tolerance (0°), the process moves back to sub-unit  32  for Iteration 2. 
   Iteration 2: The sub-unit  32  uses the predicted springback (∠S 2 =21.166°) to compensate the die to obtain the forming angle for Iteration 2 (Forming angle=Die angle=Design part angle, ∠X−Predicted springback, (∠S 2 =38.834°). It then uses the forming angle (38.834°) to predict the new springback (∠S 3 =21.475°). The sub-unit  32  now adds the thus calculated springback (21.475°) to the forming angle (38.834°) to obtain the formed part angle after springback (X 2 =21.475°+38.834°=60.309°). The sub-unit  32  then compares the difference between the formed part angle after springback and the desired part angle to the specified tolerance (0°). Since the difference (0.309°) is greater than the specified tolerance (0°), the process moves back to sub-unit  32  for Iteration 3. 
   Iterations 3 through 5: The process is repeated till the formed part angle after springback (X*) is equal to the desired part angle. The predicted springback at end of the final iteration (Iteration 5) is the total compensated springback (21.520°). 
   The unit  28  finally calculates the compensated die radius as follows: 
   
     
       
         
           
             Compensated 
             ⁢ 
             
                 
             
             ⁢ 
             die 
             ⁢ 
             
                 
             
             ⁢ 
             radius 
           
           = 
           
             
               
                 
                     
                 
                 ⁢ 
                 
                   Designed 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   part 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   radius 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       0.25 
                       ′ 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       180 
                       - 
                       
                         Designed 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         part 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         angle 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             60 
                             ⁢ 
                             ° 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       180 
                       - 
                       
                         Designed 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         part 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         angle 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             60 
                             ⁢ 
                             ° 
                           
                           ) 
                         
                       
                       + 
                     
                   
                 
                 
                   
                     
                       Total 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       compensated 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       springback 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           21.52 
                           ⁢ 
                           ° 
                         
                         ) 
                       
                     
                   
                 
               
             
             = 
             
               0.2119 
               ″ 
             
           
         
       
     
   
   
     
       
             
           
             
             
             
           
         
             
                 
             
             
               User-output (34) 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
                 
               Total compensated springback: 
               21.520° 
             
             
                 
               Compensated die radius: 
                0.2119″ 
             
             
                 
                 
             
           
        
       
     
   
     FIG. 4  illustrates springback in terms of the part angle before springback θb and the part angle after springback θp wherein the springback is equal to θb minus θp. 
     FIG. 5  illustrates how hydraulic forming pressure is applied to the blank as it wraps around the die. 
     FIG. 6  is a graphic representation of how the bend radius of the part changes with a change in the springback compensated die surface. 
     FIG. 7  illustrates the user input section  26  in the computer where the various variables such as material thickness, bend angle, bend radius, model of hydropress and pressure are inputted into the computer for calculations of springback and compensated die radius. 
   The hydropress choices include an Asea press and a Quintus press. 
   Additional advantages and modifications will readily occur to those skilled in the art. In the invention in its broader aspects is, therefore, not limited to the specific details, representative apparatus and illustrated examples shown and described. Accordingly, departures may be made from such details without departing from the spirit or scope of applicants&#39; general inventive concept.