Modal utilization method for measuring the track axial force

Abstract

Measurement of thermal forces is an important process that aids in reducing the maintenance cost of continuous welded rails (CWRs). But it is still a very challenging problem. Axial force can be measured through vibration modal changes, and vibration measurement methods have several merits compared to conventional measurement methods using piles, strain sensors, magnetoelastic and acoustoelastic effect sensors, and X-rays. Vibration measurement methods have the ability to measure the absolute thermal force in a local interval and are robust against abrasion, surface residual stress, and material grains. However, they are affected by both axial force and track parameters, thereby restricting their engineering applications. To address these limitations, this study proposes a novel method called ‘modal utilization method of periodic structure’ (MUMPS). Its novelty lies in proving that variations in frequency constitute a random sequence satisfying the constraints of Chebyshev’s law of large numbers. It can determine the influence of track parameters on the vibration modes by averaging across different eigenfrequencies without investigating what they correspond to respectively. First, MUMPS is introduced through a hypothesis that the mean value of the natural frequency variations caused by the track parameter variations approaches a stable value. Second, MUMPS is validated using finite element simulation results, which show that the probability density function of the measurement error of the neutral temperature obey N(0.5,2.6). Finally, the accuracy of MUMPS is investigated through experiments, the results are compared with the strain-gauge method results, and they show good agreement. The experimental results show that as the number of the eigenfrequencies reaches 12, the mean value becomes stable, and the measurement error of neutral temperature is less than 4 $^\circ {\textrm{C}}$. Engineering applicability of MUMPS is demonstrated by conducting real-time measurements on railway tracks. The results show that the proposed method is simple, as sophisticated instruments are not employed to determine the axial forces on railway tracks.

Appendices

Appendix A: Abbreviations

Symbols	Full name
MUMPS	Modal utilization method of periodic structure
CWRs	Continuous welded rails
FEM	Finite element model
RPs	Response points
MPs	Measurement points

Appendix B: Evaluation of stochasticity of natural frequencies

The deviation $\bigtriangleup f_{i,str}$ can be approximated by a generalized form as follows.

$$\begin{aligned} \bigtriangleup f_{i,str}(\xi )= {\textstyle \sum _{m=1}^{L} {\textstyle \sum _{n=1}^{L} A_{mni}(\xi _m,\xi _n)\xi _m \xi _n}} \end{aligned}$$

(B.1)

The expected value of $E[\bigtriangleup f_{i,str}]$ of $\bigtriangleup f_{i,str}$ is expressed by the following equation.

$$\begin{aligned} E[\bigtriangleup f_{i,str}]= & {} \iint _{-\infty }^{+\infty }{\textstyle \sum _{m=1}^{L} {\textstyle \sum _{n=1}^{L} A_{mni}(\xi _m,\xi _n)\xi _m \xi _n}}\nonumber \\{} & {} f_{\xi }(\xi _m)f_{\xi }(\xi _n)d\xi _m d\xi _n \end{aligned}$$

(B.2)

Assume that $\xi$ obeys a Gaussian distribution with mean value zero. $A_{mni}(\xi _m,\xi _n)$ can be approximately replaced by a constant coefficient irrespective of the value of $\xi$ for a small $\sigma _{\xi }^{2}$. Equation B.2 is expressed as follows.

$$\begin{aligned} E[\bigtriangleup f_{i,str}]= & {} {\textstyle \sum _{m=1}^{L}A_{mmi}\int _{-\infty }^{+\infty } \xi _m^2 f_{\xi }(\xi _m) d\xi _m} \nonumber \\= & {} \sigma _{\xi }^2 {\textstyle \sum _{m=1}^{L} A_{mmi}} \end{aligned}$$

(B.3)

where $\sigma _{\xi }^2$ is the variance of track parameter. It means that the mean of $\bigtriangleup f_{i,str}$ also exist.

The variance value of $D[\bigtriangleup f_{i,str}]$ of $\bigtriangleup f_{i,str}$ is expressed by the following equation.

$$\begin{aligned} D[\bigtriangleup f_{i,str}]= & {} E[[\bigtriangleup f_{i,str}]^2]-[E[\bigtriangleup f_{i,str}]]^2\nonumber \\= & {} E[({\textstyle \sum _{m=1}^{L}} {\textstyle \sum _{n=1}^{L}} A_{mni} \xi _m\xi _n)^2]\nonumber \\{} & {} -[\sigma _{\xi }^2 {\textstyle \sum _{m=1}^{L}}A_{mmi}]^2 \end{aligned}$$

(B.4)

It means that the variance of $\bigtriangleup f_{i,str}$ also exist.

Assuming deviation $\xi _m (m=1,2,3,\ldots ,L)$ is not greater than $\xi _{max}$. According to Eqs. (B.1) and (B.3), Eq. (B.4) meets the following inequality.

$$\begin{aligned} D[\bigtriangleup f_{i,str}]\le & {} \xi _{max}^4 E[({\textstyle \sum _{m=1}^{L}} {\textstyle \sum _{n=1}^{L}} A_{mni})^2]\nonumber \\{} & {} -[\sigma _{\xi }^2 {\textstyle \sum _{m=1}^{L}}A_{mmi}]^2 \end{aligned}$$

(B.5)

Because $\xi _{max}$,$A_{mni}$ and $\sigma _{\xi }^2$ exist, Eq. (B.5) indicates the variance $D[\bigtriangleup f_{i,str}]$ is bounded.

Appendix C: Neutral temperature

$$\begin{aligned} F=F_T+F_0=-E\alpha A(T-T_L)+F_0 \end{aligned}$$

(C.1)

Neutral temperature $T_{NT}$ was calculated using Eq. (C.1) by equating $F=0$.

$$\begin{aligned} T_{NT}=\frac{F_0}{E\alpha A}+ T_L \end{aligned}$$

(C.2)

If $F\ne 0$, neutral temperature $T_{NT}$ was calculated using Eqs. (C.1) and (C.2).

$$\begin{aligned} T_{NT}=T+\frac{F}{E \alpha A}=T+\bigtriangleup T_F \end{aligned}$$

(C.3)

where $\bigtriangleup T_F$ is the equivalent temperature variation of the axial force F.

$$\begin{aligned} \bigtriangleup T_F=-\frac{\bigtriangleup f_{i,temp}}{K_i} \end{aligned}$$

(C.4)

where $\bigtriangleup f_{i,temp}$ is the temperature-induced variation of the i-th natural frequency, and $K_i$ is the corresponding temperature sensitivity coefficient, whose unit is Hz/$^\circ {\textrm{C}}$.

$$\begin{aligned} \bigtriangleup f_i=\bigtriangleup f_{i,temp}+\bigtriangleup f_{i,str} \end{aligned}$$

(C.5)

where $\bigtriangleup f_i$ and $\bigtriangleup f_{i,str}$ represent the natural frequency variation and structure-induced frequency variation, respectively.

If N eigenfrequencies are counted, the equivalent temperature variation $\bigtriangleup T_F$ can be expressed using Eqs. (C.4) and (C.5).

$$\begin{aligned} \bigtriangleup T_F= & {} -\frac{1}{N} {\textstyle \sum _{i=1}^{N}} \frac{\bigtriangleup f_{i,temp}}{K_i}\nonumber \\= & {} -\frac{1}{N} {\textstyle \sum _{i=1}^{N}} \frac{\bigtriangleup f_i-\bigtriangleup f_{i,str}}{K_i} \end{aligned}$$

(C.6)

Because all temperature sensitivity coefficients $K_i$ ($1\le i \le N$) are approximately equal, Eq. (C.6) can also be expressed as follows:

$$\begin{aligned} T_{NT}\approx & {} \lim _{N \rightarrow \infty } \left( T-\frac{1}{N {\bar{K}}} {\textstyle \sum _{i=1}^{N}} \bigtriangleup f_i\right. \nonumber \\{} & {} \left. +\frac{1}{{\bar{K}} }\bigtriangleup f_{str} \right) \end{aligned}$$

(C.7)

$$\begin{aligned} {\bar{K}}= & {} \frac{1}{N} {\textstyle \sum _{i=1}^{N}} K_i \end{aligned}$$

(C.8)

where ${\bar{K}}$ is the equivalent temperature sensitivity coefficient.

$$\begin{aligned} \bigtriangleup f_{str}=\lim _{N \rightarrow \infty } \frac{1}{N} {\textstyle \sum _{i=1}^{N}} \bigtriangleup f_{i,str} \end{aligned}$$

(C.9)

where $\bigtriangleup f_{i,str}$ can be obtain through experiments, and N is the number of eigenfrequencies.

Equation (C.7) can also be expressed as follows:

$$\begin{aligned} \bigtriangleup T_F\approx & {} \lim _{N \rightarrow \infty } \left( -\frac{1}{N{\bar{K}}} {\textstyle \sum _{i=1}^{N}} \bigtriangleup f_{i}\right. \left. +\frac{1}{{\bar{K}}} \bigtriangleup f_{str}\right) \nonumber \\ \end{aligned}$$

(C.10)

The neutral temperature $T_{NT}$ can be expressed as follows.

$$\begin{aligned} T_{NT}\approx & {} \lim _{N \rightarrow \infty } \left( T-\frac{1}{N{\bar{K}}} {\textstyle \sum _{i=1}^{N}} \bigtriangleup f_{i}\right. \left. +\frac{1}{{\bar{K}}} \bigtriangleup f_{str}\right) \nonumber \\ \end{aligned}$$

(C.11)

Appendix D: Robustness

Let us define the measurement error $\varepsilon$, which is a random vector, as the difference between the measurement value $T_{N,i,m}$ and the real value $T_{N,i,r}$

$$\begin{aligned} \varepsilon _i=T_{N,i,m}-T_{N,i,r} \end{aligned}$$

(D.1)

The different statistics of $\varepsilon$ can be obtained using Monte Carlo simulations. The mean of the measurement error $\varepsilon$ denoted by ${\bar{\varepsilon }}$ defined as

$$\begin{aligned} {\bar{\varepsilon }} \approx \frac{1}{N} {\textstyle \sum _{i=1}^{N}} \varepsilon _i \end{aligned}$$

(D.2)

The standard deviation of error $\varepsilon$ denoted by ${\hat{\varepsilon }}$

$$\begin{aligned} {\hat{\varepsilon }}=\sqrt{\frac{1}{N}}{\textstyle \sum _{i=1}^{N}}(\varepsilon _i-{\bar{\varepsilon }})^2 \end{aligned}$$

(D.3)