simulation of torsion moment at the wheel set of the railway vehicle ...

After the insertion of the previous relations into the equationof motion in the time t Δt and after modification the equationobtains the form:6t M 3d +t B + K n $ q2t+Dt=D D6 6= F M q(12)t t q qt+ Dt+ d t+ ot+ 2ptn+2 2D D3Bt q q Dt + d t+ 2 ot+q ptnD2The method of the linear acceleration does not request anystarting procedure because displacements, velocities and accelerationsin the time t Δt are expressed in the dependence on thesame quantities in the time t.3.3 The Wilson θ-methodSimilar to the Newmark method, the Wilson method is implicitas well. Its original version, the method of linear acceleration, is infact the Newmark method, where δ 1/2 and α 1/6. The methodof linear acceleration is a special case of the Wilson θ-method,where θ 2. The linear change of acceleration in the interval is the basic presupposition, as is shown in Fig. 2. In thearbitrary point of the interval the following holds:2 2 2 2i $ Dt i $ Dtqt+ iDt = qt+ i $ Dt$qot+ q pt+ _ qpt+iDt- qpti=2 62 2i $ Dt= qt+ i $ Dt$qot+ _ qpt+iDt+2 $ qpti6These relations lead to the expressions for qp t+iDta qot+iDt6qpt+ iDt = _ qt t-i$ Dt$ qt-qt -2$ q2 2 + iDo i pti $ Dt3qo qt q t q qt+ iDt = ot+ _ t+iDt-i$ D $ ot- ti-i $ D(16)i $ Dt3- i $ Dt$qp q2t q 2 qt+ $ pt = t+iDt- $ ot-i $ D3t q i $ Dt- t-$ qpti $ D 2We insert the previous relations into the equation of motionin the time τ θ ⋅ Δt and after the modification the equation getsthe form:6t M 3d +t B + K n $ q F2 2t+ iDt = t+iDt+i $ D i $ D6 6+ Md$ qt+ $ qot+ 2 $ qptn+(17)2 2i $ Dti $ Dt3i $ Dt+ Be$ qt+ 2 $ qot+$ qptoi $ Dt2The stability and the accuracy of the method depends on thecoefficient θ choice. For the method stability it is necessary to beθ 1.37.3.4 The Newmark methodThis method is an implicit method. It comes out from themethod of the average constant acceleration. The principal retationsof the Newmark method give the relationship between thedisplacement, the velocity and the acceleration in the t and t Δt.They are:Fig. 2 An acceleration dependence on the timexqt p^+ xh= qp (13)t q qt+ _ pt+iDt-ptii $ DVia a double integration we obtain the relations for qt o ^ + xhand qt ^ + xh:2xqt o^+ xh= qo qt q qt+ xpt+ _ pt+iDt-pti2 $ i $ D(14)2 3x xqt ^ + xh= q q q2 6 t q qt+ xot+ pt+ _ pt+iDt-pti$ i $ DAt the end of an integration step for τ θ ⋅ Δt we have:i $ Dt qot+ iDt = qt+ i $ Dt$qpt+ _ q pt+iDt- q pti=2i $ Dt = qot+ _ q pt+iDt-q pti(15)2pq t+Dqo = qo + Dt$ 7d$qp + ^1-dhqpAt+ Dt t t+Dt tdqt q dq do1t qt+ Dt = t+Dt-e- o ot- t-a $ D b a $ Dd-Dtd-1nqpt2 $ a(18)12qt+ Dt = qt+ Dt$ qot+ = a$ qpt+Dt+ d -anqptG$ Dt(19)2From the previous relations we obtain the expressions fort and qot+Dt11qpt+ Dt = _ qt t-Dt$qt-qt - -1qt(20)2 + Do i d npa $ Dt2 $ a(21)Inserting the previous relations into the equation of motion inthe time t Δt and after the modification, the equation has theform:28 ● COMMUNICATIONS 3/2008