This section describes how preloads, offsets and scales enter into bushing force
        computations. You use Preloads, Offsets and Scales to alter the operating point of a
        bushing. You can offset the bushing displacement in any direction, and scale the input
        displacement and velocity. You can also offset the bushing force in any direction by adding
        a preload or scale-output force or moment in any direction.
        Displacement Offsets ans Scales 
            
            Displacement offsets change the origins of the force-deflection curves. Displacement
                scales expand or contract the force-deflection curves along the deflection axis.
            Let:
            
                
                    
                            
                                
                                    
                                        
                                            d 
                                            k 
                                         
                                     
                                    MathType@MTEF@5@5@+=
                                        feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                        hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                        4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                        vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                        fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa
                                        aaleaacaWGRbaabeaaaaa@37FB@  
                                 
                             
                         be the displacement in the kth  displacement direction (x, y,
                            z, ax, ay, az ) between the bushing I- and J-Markers reported in the
                        coordinate system of the J marker. 
                
                
                    
                            
                                
                                    
                                        
                                            H 
                                            k 
                                         
                                     
                                    MathType@MTEF@5@5@+=
                                        feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                        hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                        4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                        vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                        fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa
                                        aaleaacaWGRbaabeaaaaa@37DF@  
                                 
                             
                         is the displacement (horizontal) scaling factor in the kth 
                        direction. 
                
                
                    
                            
                                
                                    
                                        
                                            Q 
                                            k 
                                         
                                     
                                    MathType@MTEF@5@5@+=
                                        feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                        hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                        4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                        vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                        fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa
                                        aaleaacaWGRbaabeaaaaa@37E8@  
                                 
                             
                         is the displacement offset in the kth  direction. 
                
             
            Then the scaled and offset deflection, 
                
                    
                        
                            q 
                            k 
                         
                     
                    MathType@MTEF@5@5@+=
                        feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                        hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                        4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                        vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                        fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa
                        aaleaacaWGRbaabeaaaaa@3808@
                     
                 
             
            
            
                
                    
                        
                            
                                q 
                                k 
                             
                            = 
                                H 
                                k 
                             
                            · 
                                d 
                                k 
                             
                            − 
                                Q 
                                k 
                             
                         
                        MathType@MTEF@5@5@+=
                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaBa
                            aaleaacaWGRbaabeaakiabg2da9iaadIeadaWgaaWcbaGaam4Aaaqa
                            baGccqWIpM+zcaWGKbWaaSbaaSqaaiaadUgaaeqaaOGaeyOeI0Iaam
                            yuamaaBaaaleaacaWGRbaabeaaaaa@4269@
                         
                     
                 
                 
            The velocity in the kth  direction, is given by:
            
                
                    
                        
                            
                                
                                    q 
                                    ˙ 
                                 
                                
                                k 
                             
                            = 
                                H 
                                k 
                             
                            · 
                                
                                    d 
                                    ˙ 
                                 
                                
                                k 
                             
                         
                        MathType@MTEF@5@5@+=
                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyCayaaca
                            WaaSbaaSqaaiaadUgaaeqaaOGaeyypa0JaamisamaaBaaaleaacaWG
                            Rbaabeaakiabl+y6NjqadsgagaGaamaaBaaaleaacaWGRbaabeaaaa
                            a@3F92@
                         
                     
                 
                 
        Preload and Force Scales 
            
            Preloads offset the forces or torques at zero displacement, while the force scale
                applies an amplification factor to the force. If 
                        
                            
                                
                                    
                                        G 
                                        k 
                                     
                                 
                                MathType@MTEF@5@5@+=
                                    feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                    hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                    4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                    vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                    fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa
                                    aaleaacaWGRbaabeaaaaa@37DE@  
                             
                         
                     th  direction, then the force in that direction is given as
                follows:
            Let:
            
                
                    
                        
                            
                                
                                    P 
                                    k 
                                 
                             
                            MathType@MTEF@5@5@+=
                                feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa
                                aaleaacaWGRbaabeaaaaa@37E7@
                             
                         
                     
                    is the preload in the kth  direction. 
                
                
                    
                        
                            
                                
                                    V 
                                    k 
                                 
                             
                            MathType@MTEF@5@5@+=
                                feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa
                                aaleaacaWGRbaabeaaaaa@37ED@
                             
                         
                     
                    is the force (vertical) scaling in the kth   direction. 
                
                
                    
                        
                            
                                
                                    G 
                                    k 
                                 
                             
                            MathType@MTEF@5@5@+=
                                feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa
                                aaleaacaWGRbaabeaaaaa@37DE@
                             
                         
                     
                    is the sum of all force methods in the kth  direction. 
                
                
                    
                        
                            
                                
                                    F 
                                    k 
                                 
                             
                            MathType@MTEF@5@5@+=
                                feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                                hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                                4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                                vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                                fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa
                                aaleaacaWGRbaabeaaaaa@37DD@
                             
                         
                     
                    is the preload in the kth  direction. 
                
             
            Then:
            
                
                    
                        
                            
                                F 
                                k 
                             
                            = 
                                P 
                                k 
                             
                            − 
                                V 
                                k 
                             
                            · 
                                G 
                                k 
                             
                            ( 
                                
                                    
                                        q 
                                        k 
                                     
                                    , 
                                        
                                            q 
                                            ˙ 
                                         
                                        
                                        k 
                                     
                                    , 
                                        x 
                                        k 
                                     
                                    , t  
                                )  
                        MathType@MTEF@5@5@+=
                            feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
                            hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
                            4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
                            vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
                            fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa
                            aaleaacaWGRbaabeaakiabg2da9iaadcfadaWgaaWcbaGaam4Aaaqa
                            baGccqGHsislcaWGwbWaaSbaaSqaaiaadUgaaeqaaOGaeS4JPFMaam
                            4ramaaBaaaleaacaWGRbaabeaakmaabmaabaGaamyCamaaBaaaleaa
                            caWGRbaabeaakiaacYcaceWGXbGbaiaadaWgaaWcbaGaam4Aaaqaba
                            GccaGGSaGaamiEamaaBaaaleaacaWGRbaabeaakiaacYcacaWG0baa
                            caGLOaGaayzkaaaaaa@4D2E@