# Influence of Structural Stiffness and Loss Factor on Railroad Vehicle Comfort
## 2 Formulation
\begin{aligned} & W_{1}(0)\\ &\vdots\\ & W_{n}(0)\end{aligned} \right\}\quad\mathbf{W}_{l}=\left\{ \begin{aligned} & W_{1}(l)\\ &\vdots\\ & W_{n}(l)\end{aligned} \right\} \tag{7}\]  
matrices \(\mathbf{M}\) and \(\mathbf{B}\) as  
\[\mathbf{M}=\left[ \begin{aligned} & m\\ && I_{g}\\ & m\\ &&\ddots\\ &&& m\end{aligned} \right]\quad\quad\mathbf{B}=\left[ \begin{aligned} & 0\\ & 0\\ & \beta_{3}^{4}\\ &&\ddots\\ &&&\beta_{n}^{4}\end{aligned} \right] \tag{8}\]  
and matrix \(\mathbf{A}\) as \(\mathbf{W}_{0}\mathbf{W}_{0}^{T}+\mathbf{W}_{l}\mathbf{W}_{l}^{T}\), Equations (6) can be grouped in matrix form as  
\[\mathbf{M}\tilde{\mathbf{q}}+c\mathbf{A}\tilde{\mathbf{q}}+\left(k\mathbf{A}+ \frac{EI}{\mu}\,\mathbf{B}\mathbf{M}\right)\mathbf{q}=(kz_{1}+c\dot{z}_{1}) \mathbf{W}_{0}+(kz_{2}+c\dot{z}_{2})\mathbf{W}_{l} \tag{9}\]  
It is worth noting that this system of equations reduces to the dynamic equations of a rigid body on two elastic supports when all modes are discarded except for the first two. For harmonic input, \(z_{1}=Ze^{i\omega t}\), harmonic solutions of Equation (9) may be sought in the form \(\mathbf{q}=\mathbf{Q}e^{i\omega t}\), in which case Equation (9) becomes the following set of algebraic equations:  
\[\left[k\mathbf{A}+i\omega c\mathbf{A}+\frac{EI}{\mu}\,\mathbf{B}\mathbf{M}- \omega^{2}\mathbf{M}\right]\mathbf{Q}=Z(k+i\omega c)\Big{(}\mathbf{W}_{0}+e^{ -i\omega t/\
u}\,\mathbf{W}_{l}\Big{)} \tag{10}\]  
where input \(z_{2}\) lags behind \(z_{1}\) the time it takes the vehicle to travel the distance \(l\), that is \(z_{2}=Ze^{i\omega(t-l/\
u)}\), with \(\
u\), the vehicle speed. Given the Fourier transform \(Z\) of input \(z_{1}\), vector \(\mathbf{Q}\) can be obtained by solving Equation (10) for each frequency \(\omega\). Each component of vector \(\mathbf{Q}\) represents the amplitude with which the corresponding mode participates in beam motion at frequency \(\omega\). Nevertheless, it will soon be clear that, for the purpose of comfort assessment, it is more convenient to solve Equation (10) for the quotient \(\mathbf{Q}/Z\), in which case Fourier transform \(Z\) need not be explicitly specified. This fraction, which will be referred to as \(\mathbf{H}_{1}\) in what follows, can be interpreted as a vector of transfer functions between input \(z_{1}\) and \"modal\" time responses \(\mathbf{q}\). Using the vectors defined in Equation (7), the summation in Equation (3) can be written as  
\[w=\mathbf{W}^{T}\,\mathbf{q} \tag{11}\]  
For the harmonic case (\(\mathbf{q}=\mathbf{Q}e^{i\omega t}\)), defining \(W\) (not to be mistaken with vector \(\mathbf{W}\) nor its components \(W_{i}\)) such that \(w=We^{i\omega t}\), the following relations hold:  
\[w=We^{i\omega t}=\mathbf{W}^{T}\mathbf{q}=\mathbf{W}^{T}\mathbf{Q}\,e^{i \omega t}=\mathbf{W}^{T}\mathbf{H}_{1}Z\,e^{i\omega t}=\mathbf{W}^{T} \mathbf{H}_{1}z_{1} \tag{12}\]  
Therefore, the transfer function defined as \(H_{2}=W/Z\) can be computed as  
\[H_{2}=\mathbf{W}^{T}\mathbf{H}_{1} \tag{13}\]  
This transfer function can now be used to determine the response spectral density in terms of the input spectral density. However, comfort indexes are determined from a filtered or weighted response rather than from the raw signal. The goal is to take into account the effect of the different frequencies on comfort. For a detailed explanation of the calculation procedure, as well as of the meaning, of the comfort index used in this work, see [11]. Filtering in the time domain is tantamount to multiplying by the filter transfer function in the frequency domain. Therefore, multiplying \(H_{2}\) by the comfort filter (let it be referred to as \(H_{c}\)) yields the transfer function between weighted accelerations (\(\bar{w}_{w}\)) and input acceleration (\(\bar{z}_{1}\)) as \(H=H_{2}H_{c}\).