3.1. Single Sensor Measurement Compensation
As depicted in
Figure 4, the algorithm assumes that a measurement system with platform length (or chord length) of
L performs a pitching motion with a longitudinal tilt on the rail with a sinusoidal wave profile of
with an arbitrary wavelength [
11]. In the model, we also assume that the tilted slope from two center points of the wheels is consistent with the slope from the contact points of the front and rear wheel, and the
x coordinates of the center point and contact point of the wheels are the same. Define the contact point of the rear wheel of the platform and the rail to be
and the contact point of the front wheel to be
. Considering a circle equation with the rear wheel as the center of the circle, we can obtain the front wheel contact point
of the inclined measurement platform and calculate the relative measurement value of the displacement sensor fixed to an arbitrary position as follows. Here, the errors generated by the incline of the sensor fixed to the platform and by the vibration are assumed to be very small—and therefore ignored—compared to the error generated by the pitching motion of the platform body.
This assumes that the system measures while moving in the positive
x-axis direction on the coordinate
and that the driving wheel moves while in contact with the rail surface with a sinusoidal wave profile, and thus, the coordinates of the contact of two wheels with the rail can be expressed in Equations (
1) and (
2)
where
Y refers to the amplitude of the sinusoidal wave rail, while
k is
. Equation (
3) is a circle equation with the rear wheel as the center and the radius of
L while the system is inclined.
We can solve the three simultaneous equations through a simulation to obtain the cross-point of the sinusoidal wave rail and the circle, i.e., the coordinates of the front wheel.
Therefore, the sensor position on the rail (
P in
Figure 4) is:
From the
x coordinate of the sensor,
we can calculate the sensor position on the platform (
Q in
Figure 4).
Since the measured value
of the sensor is the distance from the sensor position on the platform to the rail waveform on a vertical line, it can be summarized as follows.
where
Thus, we can calculate the amplitude
and phase
of the value measured by the sensor from an arbitrary sensor position on the platform. Furthermore, the transfer function of the measured value of the rail surface can be defined as follows.
Figure 5 depicts the amplitude and phase angle of the transfer function calculated with Equation (
7).
Here, the wheelbase (L) of the platform is considered as a constant value. As a result of our analysis, the graph of the amplitude and phase angle of the transfer function changes according to the position (d) of the measurement sensor and the wavelength domain of the rail. Assuming the sensor is positioned at the center of the platform (d = L/2), the amplitude of the transfer function is amplified twice when the wavelength of the rail surface becomes -times the wheelbase. In addition, when the wavelength of the rail surface is -times the wheelbase, the amplitude of the transfer function is close to zero, and the wavelength of the rail surface cannot be measured.
Although there is no difference in the phase of the transfer function and the measured value if the sensor is positioned at the center of the platform (d = L/2), the phase angle of the observed transfer function changes when the position of the sensor changes in other cases.
As a result, when using a single sensor to measure rail surface roughness, the measured value differs according to the wavelength of the rail surface, the length (L) of the system, and the sensor position (d), even if we analyze the measured value by reflecting the platform incline, making it difficult to compensate for error in values measured by the existing trolley-type mobile systems. Therefore, it is necessary to reduce these errors by offsetting the amplification or attenuation of measured values through the design of the platform length (L) in consideration of the physical size of the platform and using the position of multiple sensors by considering the wavelength range that generates rolling noise.
3.2. Analysis of Acoustic Roughness by a Convergence of Multiple Sensors
This section describes chord offset synchronization, whereby results can be obtained by compensating the measured values of multiple sensors fixed at different positions to reduce errors in measured values more than those produced by single-sensor measurement systems. Multi-sensor-based chord offset synchronization analyzes the response of the measured rail surface values to obtain the relative positions of the sensors on the platform, so the amplitude and phase angle of the transfer function are not amplified or attenuated and synchronize the measured values to find optimal measurement results. This method results in values differently compensated according to the relative distance between sensors and can produce more accurate measured values on a wide rail surface wavelength spectrum than conventional analytical methods. It can also easily discriminate between cracks and rail corrugations in any analysis. The measured values need to be synchronized in the same position to converge the sensors used in different positions, and at least three sensor values are needed to increase the accuracy of the synchronized results. Moreover, the convergence of the measured values from multiple sensors can reduce the impact of disturbances that affect the measured values by incorporating the difference in sensor measurement times while the mobile platform is moving. In our research, we calculated the optimal sensor arrangement for frequency analysis of the rolling noise section through sensor interval setting and calculating the position of the center sensor to determine the optimal positions of at least three sensors to be fastened to the platform for compensation of rail acoustic roughness.
First, we modeled the multi-sensor-based chord offset synchronization using three sensors to calculate the measured value and phase difference of each sensor, as depicted in
Figure 6. The three sensors were designated as S1, S2, and S3 in order starting with the front sensor, while the sensor position ratios to the inclined platform
were defined as
, respectively. We then calculated the contact point with the driving wheel in consideration of the incline of the railhead in the sinusoidal wave profile and the system and calculated the sensor measurement according to the position ratio.
Since the measured value is the distance from the contact point between sensor and platform, the measured value
of each sensor can be expressed as follows.
Therefore, we can obtain the amplitude and phase angle output by each sensor position and on the platform in the acoustic roughness wavelength range measured by the three sensors. The platform length or chord length (L) is assumed to be constant for the calculation.
To study the correlation between railhead and platform length, we set the important wavelength range of surface roughness to 0∼500 mm and calculated the amplitude (
Z) and phase angle
of the values measured by the sensors. The results shown in
Figure 7 indicate that the transfer function of the wavelength of the value measured on an arbitrary sensor position
and
is amplified or attenuated to 0∼2-times, while the phase angle has an error of
∼
. Therefore, we can assume the value measured in each sensor is a sinusoidal wave of different amplitude and phase angle as shown in Equation (
10) below.
where the measurement position of each sensor can be converged through a mutual compensation, as shown in Equation (
11).
Figure 7 shows the measured values from the three sensors and the converged result value. The results indicate that the converged value has a lower magnitude of amplification or attenuation than the individual sensor values.