# Theory and Practice of Determining the Dynamic Performance of Traction Rolling Stock
## 6 Drawing up Equations of Vertical Oscillations of a Diesel Locomotive  
When compiling the mathematical model of oscillations of the undercarriage, a flat design scheme of the locomotive was adopted (see Figure 24). Equations of steady-state spatial vertical oscillations of the vehicle-track system under study, with respect to the complex amplitudes of generalized coordinates, compiled the accepted assumptions and described forced oscillations in the longitudinal vertical plane [29]. It is convenient to compose differential equations of oscillations of rail vehicles using the general equation of system dynamics (1) as follows:  
\[Q_{l}^{\alpha}+Q_{l}^{J}=0;\;l=1,2,\ldots,K \tag{1}\]  
where \(Q_{l}^{\alpha}\) is the generalized force from active forces; \(Q_{l}^{J}\) is the generalized force of inertia; \(K\) is the number of degrees of freedom of the system. It is convenient to calculate the generalized inertial forces using the Lagrange operator, which in the problem of small oscillations is written as Equation (2) that follows. \[Q_{l}^{I}=-\frac{d}{dt}\Bigg{(}\frac{\partial T}{\partial\dot{q}_{l}}\Bigg{)} \tag{2}\]  
where \(T=T\big{(}\dot{q}_{l}\big{)}\) is kinetic energy of the system, depending only on generalized velocities. The generalized forces from active forces can be found through the possible power of these forces according to Formula (3). \[Q_{l}^{\alpha}=-\frac{N_{la}^{*}}{\dot{q}_{l}};\;(l=1,2,\ldots,K) \tag{3}\]  
Figure 23: Safety factor against wheel derailment in a curve with a radius of 400 m.  
where the possible power of all given forces \((i=1,2,\ldots,n)\) correspond to the generalized speed \(\hat{q}_{l}\succ 0\) through which the possible speeds \(v_{i}^{*}\) are expressed at each \(i\)-th point of application of the force is equal to  
\[N_{la}^{*}=\sum_{i}^{n}\vec{F}_{i}^{a}\cdot v_{i}^{*}\cdot\left(\hat{q}_{i}^{*}\right) \tag{4}\]  
Figure 24: Kinematic diagram of a diesel locomotive model. After substituting the values of the generalized inertial forces and the generalized active forces in Equation (4) and assuming that \(q_{l}^{2}\leq q_{l}\), a system of linear differential equations of small oscillations is obtained relative to the position of a stable rectilinear uniform motion of the vehicle, in which the gravity forces of individual bodies and external active forces are balanced with static values bond reactions [30]. The values of the generalized coordinates in this position are equal to zero. In translational motion, inertial forces act on the body; in rotational motion, the moments of inertial forces act on the body [2]. Consequently, the equations of the motion of the body in Cartesian coordinates will take the following form:  
\[ \begin{array}{c}m\ddot{x}+\sum_{i=1}^{n}P_{xi}+\sum_{j=1}^{k}Rxj=0\\ m\ddot{y}+\sum_{i=1}^{n}P_{yi}+\sum_{j=1}^{k}Ryj=0\\ m\ddot{z}+\sum_{i=1}^{n}P_{zi}+\sum_{j=1}^{k}Rzj=0\\ J_{x}\ddot{\theta}+\sum_{i=1}^{n}M_{xPi}+\sum_{j=1}^{k}M_{xBi}=0\\ J_{y}\ddot{\theta}+\sum_{i=1}^{n}M_{yPi}+\sum_{j=1}^{k}M_{yBi}=0\\ J_{z}\ddot{\theta}+\sum_{i=1}^{n}M_{xPi}+\sum_{j=1}^{k}M_{zBi}=0\\ \end{array}  \tag{5}\]  
where \(P_{xi}\), \(P_{yi}\), \(P_{zi}\) are the force projections with number \(i\) on the \(x\), \(y\), \(z\) axes, respectively; \(R_{xi}\), \(R_{yi}\), \(R_{zi}\) are the projections of reactions of elastic (or elastoviscous) elements with number \(j\) on the \(x\), \(y\), \(z\) axes, respectively; \(M_{xPi}\), \(M_{yPi}\), \(M_{zPi}\) are the moments of forces \(Pi\) about the \(x\), \(y\), \(z\) axes; \(M_{xRi}\), \(M_{yRi}\), \(M_{zRi}\) are the moments of force \(R_{j}\) about the \(x\), \(y\), \(z\) axes; \(j_{x}\), \(j_{y}\), \(j_{z}\) are the moments of inertia of a rigid body about the \(x\), \(y\), \(z\) axes; \(\ddot{x}\), \(\ddot{y}\), \(\ddot{z}\) are the acceleration of the body along the \(x\), \(y\), \(z\) axes; \(\ddot{\theta}\), \(\ddot{\varphi}\), \(\ddot{\psi}\) are the angular accelerations of the body during its rotation around the \(x\), \(y\), \(z\) axes.