Stiffness Formulation

The stiffness of the spring can be defined multiple ways with each degree of freedom defined differently.

Force and Moment

Linear Stiffness and Damping Behavior

The simplest formulation is a linear elastic spring stiffness, where the internal force is proportional to the relative displacement. In this case, only the constant stiffness parameter K i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadMgaaeqaaaaa@3848@ and optional damping parameter C i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiaadMgaaeqaaaaa@3848@ are entered.

For linear stiffness, the force and moment are:


  • F ( δ ) = K i δ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiTdq2aaWbaaSqabeaacaWGPbaaaaaa@4012@
  • M ( θ ) = K i θ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGnbWaae WaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiUde3aaWbaaSqabeaacaWGPbaaaaaa@403B@
For linear dashpot, the force and moment are:


  • F ( δ ) = C i δ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaWGdbWaaSbaaSqa aiaadMgaaeqaaOGafqiTdqMbaiaadaahaaWcbeqaaiaadMgaaaaaaa@4013@
  • M ( θ ) = C i θ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGnbWaae WaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpcaWGdbWaaSbaaSqa aiaadMgaaeqaaOGafqiUdeNbaiaadaahaaWcbeqaaiaadMgaaaaaaa@403C@
For linear stiffness and dashpot, the force and moment are:


  • F ( δ ) = K i δ i + C i δ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGgbWaae WaaeaacqaH0oazaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiTdq2aaWbaaSqabeaacaWGPbaaaOGaey4kaS Iaam4qamaaBaaaleaacaWGPbaabeaakiqbes7aKzaacaWaaWbaaSqa beaacaWGPbaaaaaa@45B3@
  • M ( θ ) = K i θ i + C i θ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGnbWaae WaaeaacqaH4oqCaiaawIcacaGLPaaacqGH9aqpcaWGlbWaaSbaaSqa aiaadMgaaeqaaOGaeqiUde3aaWbaaSqabeaacaWGPbaaaOGaey4kaS Iaam4qamaaBaaaleaacaWGPbaabeaakiqbeI7aXzaacaWaaWbaaSqa beaacaWGPbaaaaaa@45ED@

Nonlinear Behavior

The force and moment in a spring is computed as:(1)
F i ( δ i ) = f ( δ i A s c a l e i ) [ A i + B i ln ( max ( 1 , | δ ˙ i D i | ) ) + E i g ( δ ˙ i F i ) ] + C i δ ˙ i + H s c a l e i h ( δ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiGacA eadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabes7aKnaaCaaaleqa baGaamyAaaaaaOGaayjkaiaawMcaaiabg2da9iGacAgadaqadaqaam aalaaabaGaeqiTdq2aaWbaaSqabeaacaWGPbaaaaGcbaGaamyqaiaa dohacaWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaadMgaaeqaaa aaaOGaayjkaiaawMcaamaadmaabaGaamyqamaaBaaaleaacaWGPbaa beaakiabgUcaRiaadkeadaWgaaWcbaGaamyAaaqabaGcciGGSbGaai OBamaabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiil amaaemaabaWaaSaaaeaacuaH0oazgaGaamaaCaaaleqabaGaamyAaa aaaOqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaaGccaGLhWUaayjc SdaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaey4kaSIaamyramaaBa aaleaacaWGPbaabeaakiGacEgadaqadaqaamaalaaabaGafqiTdqMb aiaadaahaaWcbeqaaiaadMgaaaaakeaacaWGgbWaaSbaaSqaaiaadM gaaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaiaaywW7 cqGHRaWkcaaMe8Uaam4qamaaBaaaleaacaWGPbaabeaakiqbes7aKz aacaWaaWbaaSqabeaacaWGPbaaaOGaey4kaSIaamisaiaadohacaWG JbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaadMgaaeqaaOGaciiAam aabmaabaWaaSaaaeaacuaH0oazgaGaamaaCaaaleqabaGaamyAaaaa aOqaaiaadAeadaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaayzkaa aaaaa@82E9@

Where, i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@374C@ is the translational degrees of freedom: 1,2,3

(2)
M i ( θ i ) = f ( θ i A s c a l e i ) [ A i + B i ln ( max ( 1 , | θ ˙ i D i | ) ) + E i g ( θ ˙ i F i ) ] + C i θ ˙ i + H s c a l e i h ( θ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiGac2 eadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiabeI7aXnaaCaaaleqa baGaamyAaaaaaOGaayjkaiaawMcaaiabg2da9iGacAgadaqadaqaam aalaaabaGaeqiUde3aaWbaaSqabeaacaWGPbaaaaGcbaGaamyqaiaa dohacaWGJbGaamyyaiaadYgacaWGLbWaaSbaaSqaaiaadMgaaeqaaa aaaOGaayjkaiaawMcaamaadmaabaGaamyqamaaBaaaleaacaWGPbaa beaakiabgUcaRiaadkeadaWgaaWcbaGaamyAaaqabaGcciGGSbGaai OBamaabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiil amaaemaabaWaaSaaaeaacuaH4oqCgaGaamaaCaaaleqabaGaamyAaa aaaOqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaaGccaGLhWUaayjc SdaacaGLOaGaayzkaaaacaGLOaGaayzkaaGaey4kaSIaamyramaaBa aaleaacaWGPbaabeaakiGacEgadaqadaqaamaalaaabaGafqiUdeNb aiaadaahaaWcbeqaaiaadMgaaaaakeaacaWGgbWaaSbaaSqaaiaadM gaaeqaaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaqaaiaaywW7 cqGHRaWkcaaMb8UaaGjbVlaadoeadaWgaaWcbaGaamyAaaqabaGccu aH4oqCgaGaamaaCaaaleqabaGaamyAaaaakiabgUcaRiaadIeacaWG ZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaWGPbaabeaaki GacIgadaqadaqaamaalaaabaGafqiUdeNbaiaadaahaaWcbeqaaiaa dMgaaaaakeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaaaaaOGaayjkai aawMcaaaaaaa@84E0@

Where, i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@374C@ is the rotational degrees of freedom: 4,5,6

The variables in the force and moment equation represent:

f ( δ i A s c a l e i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaae WaaeaadaWcaaqaaiabes7aKnaaCaaaleqabaGaamyAaaaaaOqaaiaa dgeacaWGZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaWGPb aabeaaaaaakiaawIcacaGLPaaaaaa@4238@ Spring force versus displacement function input as fct_ID1i

f ( θ i A s c a l e i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaae WaaeaadaWcaaqaaiabeI7aXnaaCaaaleqabaGaamyAaaaaaOqaaiaa dgeacaWGZbGaam4yaiaadggacaWGSbGaamyzamaaBaaaleaacaWGPb aabeaaaaaakiaawIcacaGLPaaaaaa@4249@ Spring force versus rotation function input as fct_ID1i.

A i B i D i E i  and  F i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaiaaciWacmaadaGabiaaeaGaauaaaO qaaiaadgeadaWgaaWcbaGaamyAaaqabaGccaqGSaGaaeiiaiaadkea daWgaaWcbaGaamyAaaqabaGccaqGSaGaaeiiaiaadseadaWgaaWcba GaamyAaaqabaGccaqGSaGaaeiiaiaadweadaWgaaWcbaGaamyAaaqa baGccaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaadAeadaWgaaWcba GaamyAaaqabaaaaa@4ACC@ Scaling coefficients

ln ( max ( 1 , | δ ˙ i D i | ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGSbGaai OBamaabmaabaGaciyBaiaacggacaGG4bWaaeWaaeaacaaIXaGaaiil amaaemaabaWaaSaaaeaacuaH0oazgaGaamaaCaaaleqabaGaamyAaa aaaOqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaaGccaGLhWUaayjc SdaacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@4785@ Logarithmic function that scales the spring stiffness as the velocity increases


Figure 1. With Parameter Input

g ( δ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbWaae WaaeaadaWcaaqaaiqbes7aKzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DA6@ Scale the stiffness as a function of linear input as fct_ID2i

g ( θ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbWaae WaaeaadaWcaaqaaiqbeI7aXzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DB7@ Scale the moment as a function of rotational velocity input as fct_ID2i

This input can be used to model nonlinear strain rate effects of the spring stiffness.


Figure 2. With g Function Input

C i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGdbWaaS baaSqaaiaadMgaaeqaaaaa@3840@ Linear damping coefficient used to increase the spring stiffness as a function of velocity

h ( δ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGObWaae WaaeaadaWcaaqaaiqbes7aKzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DA7@ or h ( θ ˙ i F i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGObWaae WaaeaadaWcaaqaaiqbeI7aXzaacaWaaWbaaSqabeaacaWGPbaaaaGc baGaamOramaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaaaa a@3DB8@ Nonlinear damping function input as fct_ID4i

Linear or nonlinear damping as a function of velocity can also be added to the spring force using either a linear damping coefficient or a user-defined function.


Figure 3. With h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ Function Input

The functions g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ and h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ both describe the damping behavior of the spring. However, the g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ function scales the spring stiffness function f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ , but the h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ function adds to the spring stiffness function f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGNbaaaa@374B@ .

Time Step

Time step for spring:(3)
Δ t = M K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WG0bGaeyypa0ZaaOaaaeaadaWcaaqaaiaad2eaaeaacaWGlbaaaaWc beaaaaa@3C9A@
Time step for dashpot:(4)
Δ t = M 2 C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WG0bGaeyypa0ZaaSaaaeaacaWGnbaabaGaaGOmaiaadoeaaaaaaa@3C29@
Time step for spring and dashpot:(5)
Δ t = C 2 + K M C K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHuoarca WG0bGaeyypa0ZaaSaaaeaadaGcaaqaaiaadoeadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaWGlbGaeyyXICTaamytaaWcbeaakiabgkHiTi aadoeaaeaacaWGlbaaaaaa@4306@
Where,
M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
Mass of the spring
K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
Linear stiffness or max [ d F d δ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGTbGaai yyaiaacIhadaWadaqaamaalaaabaGaamizaiaadAeaaeaacaWGKbGa eqiTdqgaaaGaay5waiaaw2faaaaa@3F76@ for a nonlinear spring
C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3730@
Linear damping or max [ g ( d δ d t ) ( d δ d t ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGTbGaai yyaiaacIhadaWadaqaamaalaaabaGaeyOaIyRaci4zamaabmaabaWa aSaaaeaacaWGKbGaeqiTdqgabaGaamizaiaadshaaaaacaGLOaGaay zkaaaabaGaeyOaIy7aaeWaaeaadaWcaaqaaiaadsgacqaH0oazaeaa caWGKbGaamiDaaaaaiaawIcacaGLPaaaaaaacaGLBbGaayzxaaaaaa@4AFF@ for nonlinear damping