This model can be used for both dynamic and quasi-static tests.
            Figure 1 .   The figure above-left shows a schematic of the bushing model where:
                    X  is the input displacement provided to the bushing.y  and w  are the internal states of the
                        bushing.
                                
                                    
                                        
                                            
                                                k 
                                                0 
                                             
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa
                                            aaleaacaaIWaaabeaaaaa@37CC@  
                                     
                                 
                             
                                
                                    
                                        
                                            
                                                k 
                                                1 
                                             
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa
                                            aaleaacaaIXaaabeaaaaa@37CD@  
                                     
                                 
                             
                                
                                    
                                        
                                            
                                                k 
                                                2 
                                             
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa
                                            aaleaacaaIYaaabeaaaaa@37CE@  
                                     
                                 
                             
                                
                                    
                                        
                                            
                                                c 
                                                0 
                                             
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa
                                            aaleaacaaIWaaabeaaaaa@37C4@  
                                     
                                 
                             
                                
                                    
                                        
                                            
                                                c 
                                                1 
                                             
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa
                                            aaleaacaaIXaaabeaaaaa@37C5@  
                                     
                                 
                             
                                
                                    
                                        
                                            
                                                c 
                                                2 
                                             
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa
                                            aaleaacaaIYaaabeaaaaa@37C6@  
                                     
                                 
                              The governing equations for this bushing are shown above-right where:
                    R  is the cutoff frequency associated with a first order
                        filter that acts on the input X .x  is the dynamic content of the bushing input
                            X . This is the filter output.
                                
                                    
                                        
                                            y 
                                            ˙ 
                                         
                                        MathType@MTEF@5@5@+=
                                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyEayaaca
                                            aaaa@36FD@  
                                     
                                 
                             
                        
                            
                                w 
                                ˙ 
                             
                            
                            MathType@MTEF@5@5@+=
                                feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaca
                                aaaa@36FB@
                             
                         
                     
                    y  and
                            w .K  is the effective stiffness of the bushing.C  is the effective damping of the bushing.Spline (X)  is the static force response of the
                        bushing. The effective stiffness K  and effective damping
                    C  are dependent on nonlinear effects such as friction in the
                bushing material and other nonlinear behavior that cannot be easily represented
                physically.
            
                The effective stiffness K  is 
                            
                                
                                    
                                        
                                            k 
                                            0 
                                         
                                     
                                    MathType@MTEF@5@5@+=
                                        feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                        hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                        4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                        vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                        fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa
                                        aaleaacaaIWaaabeaaaaa@37CC@  
                                 
                             
                 
                    
                        
                            
                                S 
                                y 
                             
                            = ( 
                                
                                    
                                        p 
                                        0 
                                     
                                    + 
                                        p 
                                        1 
                                     
                                    
                                        
                                            |  y  |  
                                        
                                            
                                                
                                                    
                                                        p 
                                                        2 
                                                     
                                                 
                                             
                                         
                                     
                                 
                                )  
                        MathType@MTEF@5@5@+=
                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa
                            aaleaacaWG5baabeaakiabg2da9maabmaabaGaamiCamaaBaaaleaa
                            caaIWaaabeaakiabgUcaRiaadchadaWgaaWcbaGaaGymaaqabaGcda
                            abdaqaaiaadMhaaiaawEa7caGLiWoadaahaaWcbeqaamaaCaaameqa
                            baGaamiCamaaBaaabaGaaGOmaaqabaaaaaaaaOGaayjkaiaawMcaaa
                            aa@4595@
                         
                     
                 
                
 
                Similarly, the effective damping C  is 
                            
                                
                                    
                                        
                                            c 
                                            0 
                                         
                                     
                                    MathType@MTEF@5@5@+=
                                        feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                        hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                        4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                        vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                        fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa
                                        aaleaacaaIWaaabeaaaaa@37C4@  
                                 
                             
                 
                    
                        
                            
                                S 
                                w 
                             
                            = ( 
                                
                                    
                                        q 
                                        0 
                                     
                                    + 
                                        q 
                                        1 
                                     
                                    
                                        
                                            |  
                                                w 
                                                ˙ 
                                             
                                                |  
                                        
                                            
                                                
                                                    
                                                        q 
                                                        2 
                                                     
                                                 
                                             
                                         
                                     
                                 
                                )  
                        MathType@MTEF@5@5@+=
                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa
                            aaleaacaWG3baabeaakiabg2da9maabmaabaGaamyCamaaBaaaleaa
                            caaIWaaabeaakiabgUcaRiaadghadaWgaaWcbaGaaGymaaqabaGcda
                            abdaqaaiqadEhagaGaaaGaay5bSlaawIa7amaaCaaaleqabaWaaWba
                            aWqabeaacaWGXbWaaSbaaeaacaaIYaaabeaaaaaaaaGccaGLOaGaay
                            zkaaaaaa@459D@
                         
                     
                 
                
 
             
            The total force generated by the bushing is the sum of 2 forces:
                    Static force at the operating point: Spline (X)  
                    Force due to the dynamic behavior of the bushing:
                        
                            
                                
                                    K y + C 
                                        w 
                                        ˙ 
                                     
                                 
                                MathType@MTEF@5@5@+=
                                    feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaadM
                                    hacqGHRaWkcaWGdbGabm4Dayaacaaaaa@3A73@