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

The implicit methods, vice versa, use the equation of motionin the time t Δt. They are suitable, first of all, for the solutionof the linear equation of motion with a consistent mass matrix.The stability is a characteristic feature of numerical methods fora direct integration of an equation of motion. The stability meansthat the solutions must not go through limits for the arbitrarystarting conditions. If this condition of an arbitrary ratio of Δt/T nis fulfiled, we describe the method as unconditionally stable andif it is fulfiled for an individual critical ratio Δt/T n only, we talkabout an conditionally stable method. In the following text we willapply some methods that may be used for the mechanical systemvibration analysis in the time domain. They are: the Differencemethod, the Linear acceleration method, the Wilson θ – method,the Newmark method [3], and the HHT-method [3, 6].Its advantage is the possibility to use it for a nonlinear taskssolution as well.3.2 The method of linear accelerationThis method is based on the presupposition of a linear accelerationcourse during each of integration steps, as shown in Fig.1.3.1 The differential methodFor the numerical integration of the equation of motion wesubstitute the time invariant variable for differenties. The simpliestdifferential substitutions of the accelerations vector qp and thevelocities vector qo in the time t are:1qo = $ _ qt + D t-qt -Dti, (2)2 $ Dt1qp = , (3)t$ _ q 2 $ q q2 t+ Dt- t+t-DtiDthat we apply to the equation of motion in the time tM$ qp+ B$ qo+ K$q = F . (4)t t t tAfter insertion and modification we obtain:1 1 2d $ M + $ Bn$ q F K M q2 t+Dt = t-d- $ n$t-2Dt2 $ DtDt(5)1 1-d$ M - $ Bn$ qtt.2+ DDt2 $ DtThe differential method is the type of an explicit method. Themethod has all the advantages of explicit methods, if the dampingmatrix is B 0 or B α ⋅ M. The most effective usage is in thecase of a diagonal mass matrix, id est for solutions of systems withconcentrated massess. The method is only conditionally stable, inthis case the lenght of integration step Δt has to meet the followingdemand:T nDt# , (6)rwhere T n is the smallest period of the vibration. The relatively veryshort length of integration step is needed in order to ensure thatthe difference method gives the correct solution. Another disadvantageis the fact, that we need to use the special procedure inthe first step.2Dtq- Dt = q0- Dt$ qo0+$ qp02(7)Fig. 1 An acceleration courseWe label the variable that changes in the interval withthe mark τ, we express the relation for acceleration with:xqt p^+ xh= qp t q qt+ _ pt +D t-ptiD(8)We obtain the relations for qt o ^ + xh and qt ^ + xhvia thedouble integration:2xqt o^+ xh= x$qp + _t q p q q2+D - pi+ o$ Dt t t. (9)2 3x xqt ^ + xh= $ qp 2 6 t q q q qt+ _ pt+Dt- pti+ x $ ot+t$ DIn the end the integration step for x = Dt we will get:Dt Dt qot+ Dt = Dt$qpt+ _ q pt+ Dt- q pti+ q ot = q ot+ _ q pt+Dt+q pti2 22 2Dt Dtqt+ Dt = q pt+ _ qpt+Dt- qpti+ Dt$qot+ qt=2 62Dt= qt+ Dt$ qot+ _ qpt+Dt+2 $ qpti6(10)From these relations we obtain the relations for qp t+Dta qot+Dt6qpt+ Dt = _ q t q q 2 q2 t+Dt-D$ ot- ti- $ ptDt3q qt q t q q t q Dt ot+ Dt = ot+ _ t+Dt-D$ ot- ti- D $ pt+ q pt=D233 Dt = $ qt+Dt-2$ qot- $ qt-q p(11)tDtDt2COMMUNICATIONS 3/2008 ●27