# Theory and Practice of Determining the Dynamic Performance of Traction Rolling Stock
## 6 Drawing up Equations of Vertical Oscillations of a Diesel Locomotive
In Equation (5), the summation of \(i\) and \(j\) is performed algebraically, considering the sign of the force and the moment. To determine the sign of the force and moment, it is convenient to use the following rule: if the direction of the force or moment coincides with the direction of the forces of inertia or the moment of the forces of inertia; hence, when summed, they take the \"plus\" sign; else, they take a \"minus\" sign. The forces of inertia, according to [PERSON]'s second law, are equal to the mass times the acceleration, and are directed in the direction opposite to the acceleration [1]. In our example, due to the arbitrary choice of the positive direction of coordinates, the positive acceleration is directed to the right, and the inertial force \(m\ddot{x}\) is the directed to the left. The same can be said about the direction of inertial forces (\(m\ddot{y},m\ddot{z}\)) and moments of inertia forces (\(J_{x}\ddot{\theta}\), \(J_{y}\ddot{\varphi}\), \(J_{z}\ddot{\psi}\)). Reactions \(R_{j}\) on the design scheme are directed as if they acted with a positive deformation of elastic or elastic/viscous elements. For positive deformation, you can take either tension or compression at your discretion [5]. In the diagram, the reactions are directed as if the named elements are experiencing compressive deformations. Equations within Equation (5) are differential equations for the equilibrium of a body as it moves in a space defined by six coordinates. External forces \(P=P(t)\) are, as a rule, functions of time \(t\); reactions \(R\) are functions of time, coordinates and their derivatives, i.e., \(R=R\left(t,x,y,z,\theta,\varphi,\dot{v},\dot{x},\dot{y},\dot{z},\dot{\theta}, \dot{\varphi}\right)\). These equations are written for any small period of time during which it can be assumed that the inertia forces, external forces \(P_{i}\) and reactions \(R_{j}\) are constants. To determine the trajectory of a body in a given space under the action of external forces, it is necessary to find the functions of \(x=x(t)\), \(y=y(t)\), \(z=z(t),\theta=\theta(t)\), \(\varphi=\varphi(t)\), \(\psi=\psi(t)\), which, under given initial conditions, would turn the system of differential equations into an identity [3;4]. Along with the d'Alembert-Lagrange equations written in the Cartesian coordinate system; another form of notation is often used in mechanics--in the so-called generalized coordinate system. In this case, the variational equations of analytical mechanics are called the Lagrange equations of the second kind [11]. The body and bogie of a diesel locomotive are solid bodies with two degrees of freedom and are connected by elastic and dissipative bonds. Wheel pairs move without separation from the rails. Perturbations from the right and left rails will be taken the same, which allows us to consider plane oscillations [6]. This formulation of the problem is quite sufficient to consider the main dynamic processes in the system [7; 8; 9]. Model six degrees of freedom. To study oscillations in the vertical longitudinal plane, we will compose a system of six second-order differential equations describing oscillations in the vertical longitudinal plane. After the transformation, we obtain Equations (6)-(8). Body oscillation equations:  
\[ \begin{gathered} M_{k}\cdot\ddot{z}_{k}+b_{k}\big{(}2\dot{z}_{k}- \dot{z}_{1}-\dot{z}_{2}\big{)}+c_{k}(2z_{k}-z_{1}-z_{2})=0\\ J_{k}\cdot\ddot{\varphi}_{k}+z_{k}b_{k}\big{(}2a_{k}\dot{\varphi}_{k}- \dot{z}_{1}+\dot{z}_{2}\big{)}+a_{k}c_{k}(2a_{k}\varphi_{k}-z_{1}+z_{2})=0\end{gathered}  \tag{6}\]  
The oscillation equations of the first bogie:  
\[ \begin{gathered} M_{T}\ddot{z}_{1}-b_{k}\big{(}\dot{z}_{k}-\dot{ z}_{1}+a_{k}\varphi_{k}\big{)}-c_{k}(z_{k}-z_{1}+a_{k}\varphi_{k})+2b_{T}\dot{z}_{1}+2 c_{T}=b_{T}\big{(}\dot{\eta}_{1}+\dot{\eta}_{2}\big{)}+c_{T}(\eta_{1}+\eta_{2})\\ J_{T}\cdot\ddot{\varphi}_{1}+2b_{T}a_{T}^{2}\dot{\varphi}_{1}+2c_{T} a_{T}^{2}\varphi_{1}=a_{T}\big{[}b_{T}\big{(}\dot{\eta}_{1}-\dot{\eta}_{2} \big{)}+c_{T}(\eta_{1}-\eta_{2})\big{]}\end{gathered}