Track modulus detection by vehicle scanning method

Abstract

An effective technique is proposed for identifying the track modulus, i.e., the foundation stiffness of the rails, using the contact-point response of the moving test vehicle. First, for a simple beam resting on an elastic foundation under a moving sprung mass, closed-form solutions are derived for the responses of the vehicle and its contact point with the rail. It is shown that the track modulus can be well identified via the first rail frequency (RF) extracted from the contact-point response, in that (1) the first RF is most outstanding compared with the other RFs in the vehicle-generated responses; and (2) the contact-point response outperforms the vehicle response as it is free of the disturbing effects of the vehicle and driving frequencies. The proposed technique is then verified numerically along with the key parameters assessed. The main conclusions are: (1) the first RF can be easily recognized as the “cliff” of the RF “plateau” in the contact-point spectrum; (2) the track irregularity has a little effect on the identification of track modulus; (3) a moderate speed is recommended for the test vehicle for a balance between efficiency and accuracy; and (4) the efficacy of the proposed technique remains good even with the presence of track damping and irregularity. Moreover, the application of the proposed technique to detecting the damage modeled by variation in foundation stiffness is demonstrated.

Appendices

Appendix A

The derivation of Eq. (10) is as follows.

Firstly, by substituting Eq. (8) into Eq. (9), one can obtain the following equation:

$$\begin{aligned} \frac{A\left( q_{bn} \right) }{A\left( q_{b1} \right) }=\frac{\frac{k_{f}}{m}+\bar{\omega }_{b1}^{2}-\varOmega _{1}^{2}}{\frac{k_{f}}{m}+\bar{\omega }_{bn}^{2}-\varOmega _{n}^{2}}=\frac{\frac{k_{f}}{m}+\frac{\pi ^{4}EI}{L^{4}m}-\frac{\pi ^{2}v^{2}}{L^{2}}}{\frac{k_{f}}{m}+\frac{n^{4}\pi ^{4}EI}{L^{4}m}-\frac{n^{2}\pi ^{2}v^{2}}{L^{2}}}\le {\alpha }_{\mathrm {lim}}. \end{aligned}$$

(A.1)

Since the term $\bar{\omega }_{b1}^{{2}}-\varOmega _{{1}}^{{2}}$ is sufficiently small compared with $k_{f} /m$, the preceding equation can be approximatively expressed by

$$\begin{aligned} \frac{A\left( q_{bn} \right) }{A\left( q_{b1} \right) }\approx \frac{\frac{k_{f}}{m}}{\frac{k_{f}}{m}+\frac{n^{4}\pi ^{4}EI}{L^{4}m}-\frac{n^{2}\pi ^{2}v^{2}}{L^{2}}}\le {\alpha }_{\mathrm {lim}}. \end{aligned}$$

(A.2)

Then, with some mathematical operations on Eq. (A.2), one can obtain

$$\begin{aligned} \frac{\pi ^{4}EI}{L^{4}m}n^{4}-\frac{\pi ^{2}v^{2}}{L^{2}}n^{2}-\frac{1}{{\alpha }_{\mathrm {lim}}}\frac{k_{f}}{m}+\frac{k_{f}}{m}\ge 0. \end{aligned}$$

(A.3)

As the variable n is a positive integer, its solution can be selected from Eq. (A.3) as

$$\begin{aligned} n\ge \sqrt{\frac{L^{2}v^{2}}{\pi ^{2}}\frac{m}{EI}\left[ \frac{1}{2}+\sqrt{\frac{1}{4}+\frac{k_{f}EI}{m^{2}v^{4}}\left( \frac{1}{\alpha _{\mathrm {lim}}}-1 \right) } \right] }. \end{aligned}$$

(A.4)

Owing to the fact that $\alpha _{\mathrm {lim}}$ is generally as small as less than 0.01, the optimized number of eigenmodes N required for solution convergence can be thus determined from the preceding equation, as

$$\begin{aligned} N=\left\lceil \frac{Lv}{\pi }\sqrt{\frac{m}{EI}} \sqrt{\frac{1}{2}+\sqrt{\frac{k_{f}}{\alpha _{\mathrm {lim}}}\frac{EI}{m^{2}v^{4}}+\frac{1}{4}} } \right\rceil . \end{aligned}$$

(A.5)

Appendix B

The derivations of Eq. (17) are as follows:

The amplitude of the first rail frequency $\omega _{b1}$ of the contact-point acceleration can be extracted from Eq. (11), as:

$$\begin{aligned} A_{c}\left( \omega _{b1} \right) =\left| \frac{\varDelta _{st1}{\cdot }S_{1}}{2\left( 1-S_{1}^{2} \right) }\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2} \right| . \end{aligned}$$

(B.1)

As for the vehicle response, the amplitude of the first rail frequency $\omega _{b1}$ can be extracted from Eq. (13), as:

$$\begin{aligned} A_{v}\left( \omega _{b1} \right) =\left| \frac{\varDelta _{st1}{\cdot }S_{1}}{2\left( 1-S_{1}^{2} \right) }\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2}\frac{\omega _{v}^{2}}{\omega _{v}^{2}-\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2}} \right| . \end{aligned}$$

(B.2)

Then, dividing Eq. (B.2) by Eq. (B.1) yields the relationship equation of $A_{v}\left( \omega _{b1} \right)$ and $A_{c}\left( \omega _{b1} \right)$, as

$$\begin{aligned} \frac{A_{v}\left( \omega _{b1} \right) }{A_{c}\left( \omega _{b1} \right) }=\left| \frac{\omega _{v}^{2}}{\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2}-\omega _{v}^{2}} \right| =\left| \frac{1}{\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2} / \omega _{v}^{2}-1} \right| . \end{aligned}$$

(B.3)

Since the term $\omega _{b1}$ is much larger than ${\pi v}/L$, the preceding equation reduces to

$$\begin{aligned} A_{v}\left( \omega _{b1} \right) =\frac{1}{\left( \omega _{b1}/\omega _{v} \right) ^{2}\mathrm {-1}}{\times }A_{c}\left( \omega _{b1} \right) . \end{aligned}$$

(B.4)

Appendix C

The derivations of Eqs. (18) are as follows:

The amplitude of the driving frequency $\omega _{dn} (= 2n\pi v/L)$ of the contact-point acceleration can be extracted from Eq. (11), as:

$$\begin{aligned} A_{c}\left( \omega _{dn} \right) =\frac{\varDelta _{stn}}{2\left( 1-S_{n}^{2} \right) }\left( \frac{2n\pi v}{L} \right) ^{2}. \end{aligned}$$

(C.1)

Dividing Eq. (C.1) by Eq. (B.1) yields the relationship equation of $A_{c}\left( \omega _{dn} \right)$ and $A_{c}\left( \omega _{b1} \right)$, as

$$\begin{aligned} \left| \frac{A_{c}\left( \omega _{dn} \right) }{A_{c}\left( \omega _{b1} \right) } \right| =\left| \frac{\frac{\varDelta _{stn}}{2\left( 1-S_{n}^{2} \right) }\left( \frac{2n\pi v}{L} \right) ^{2}}{\frac{\varDelta _{st1}{\cdot }S_{1}}{2\left( 1-S_{1}^{2} \right) }\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2}} \right| . \end{aligned}$$

(C.2)

Substituting Eqs. (4a) and (6) into the preceding equation, one can obtain

$$\begin{aligned} \left| \frac{A_{c}\left( \omega _{dn} \right) }{A_{c}\left( \omega _{b1} \right) } \right| =\left| \frac{\frac{1}{\omega _{bn}^{2}}\frac{1}{\left( 1-S_{n}^{2} \right) }\left( \frac{2n\pi v}{L} \right) ^{2}}{\frac{1}{\omega _{b1}^{3}}\frac{\pi v}{L}\frac{1}{\left( 1-S_{1}^{2} \right) }\left( \omega _{b1}-\frac{\pi v}{L} \right) ^{2}} \right| . \end{aligned}$$

(C.3)

Since the term $\omega _{b1}$ is much larger than ${\pi v}/L$, the preceding equation reduces to

$$\begin{aligned} \left| \frac{A_{c}\left( \omega _{dn} \right) }{A_{c}\left( \omega _{b1} \right) } \right| \approx 4\left| \frac{\frac{1}{\left( 1-S_{n}^{2} \right) }\left( \varOmega _{n} \right) ^{2}/\omega _{bn}^{2}}{\frac{1}{\left( 1/S_{1}-S_{1} \right) }} \right| . \end{aligned}$$

(C.4)

Since the term $S_{n}$ is much less than 1, the preceding equation reduces to

$$\begin{aligned} \left| \frac{A_{c}\left( \omega _{dn} \right) }{A_{c}\left( \omega _{b1} \right) } \right| \approx 4\left| \frac{S_{n}^{2}}{S_{1}} \right| \approx 4S_{n}\ll 1. \end{aligned}$$

(C.5)

Appendix D

The three matrices $\mathbf{M}^{e}_{b}$, $\mathbf{K}^{e}_{b}$, and $\mathbf{K}^{e}_{f}$ appearing in Eq. (20) are listed as follows:

$$\begin{aligned} \mathbf{M}^{e}_{b}= & {} \frac{m\,l_e}{420} \begin{bmatrix} 156 &{} 22l_e&{} 54 &{} -13l_e\\ 22l_e &{} 4l_e^2 &{} 13l_e &{} -3l_e^2\\ 54 &{} 13l_e &{} 156&{} -22l_e\\ -13l_e&{} -3l_e^2&{} -22l_e &{} 4l_e^2, \end{bmatrix}, \end{aligned}$$

(D.1)

$$\begin{aligned} \mathbf{K}^{e}_{b}= & {} \frac{EI}{l_e^3} \begin{bmatrix} 12 &{} 6l_e&{} -12 &{} 6l_e\\ 6l_e &{} 4l_e^2 &{} -6_e &{} 2l_e^2\\ -12 &{} -6l_e &{} 12&{} -6l_e\\ 6l_e&{} 2l_e^2&{} -6l_e &{} 4l_e^2, \end{bmatrix}, \end{aligned}$$

(D.2)

$$\begin{aligned} \mathbf{K}_f^e= & {} k_f l_e \begin{bmatrix} \frac{13}{35} &{} \frac{11}{210}l_e&{} \frac{9}{70}&{} -\frac{13}{420}l_e\\ \frac{11}{210}l_e &{} \frac{1}{105}l_e^2&{} \frac{13}{420}l_e&{} -\frac{1}{140}{l_e}^2\\ \frac{9}{70} &{}\frac{13}{420}l_e &{} \frac{13}{35}&{} -\frac{11}{210}l_e\\ -\frac{13}{420}l_e &{}-\frac{1}{140}l_e^2 &{}-\frac{11}{210}l_e &{}\frac{1}{105}{l_e}^2 \\ \end{bmatrix}. \end{aligned}$$

(D.3)