A novel smartphone-based evaluation system of pedestrian-induced footbridge vibration comfort

Abstract

Pedestrian-induced vibration comfort is an important factor affecting the serviceability of footbridges. This article proposes a smartphone-based evaluation system for pedestrian-induced footbridge vibration comfort, and the evaluation system consists of a data acquisition subsystem, management center subsystem, and smartphone client. Four technical challenges in the application of the evaluation system are solved: coordinates transformation, acceleration signal drift correction, signal filtering, and computation of the total weighted root mean square acceleration. To verify the validity of the proposed evaluation system, field experiments are carried out on the Forth Corridor Footbridge in Guangzhou. A comparison of the proposed system and the traditional methodology shows that the total weighted root mean square acceleration errors between smartphones and accelerometers are less than ±5%. In addition, the subjective feelings in the field experiments are in excellent agreement with the corresponding stipulation in ISO 2631-1:1997 (Amendment 1. International Standardization Organization, Geneva, 2010).

Keywords

coordinate transformation drift correction evaluation system pedestrian-induced footbridge vibration comfort total weighted root mean square acceleration

Introduction

Contemporary footbridges often suffer from pedestrian-induced vibrations, which seriously impair the walking comfort of pedestrians. The infamous Millennium Bridge in London is the prime pedestrian-induced vibration example. Studies of video footage revealed up to 50 mm of lateral movement of the south span and 70 mm of the central span (Dallard, 2005; Dallard et al., 2001), and pedestrians were frightened. The T-bridge was the only way that the motorboat stadium and the bus terminal were connected (Nakamura, 2004). The girder response reached 8–10 mm during the congested period (Strogatz et al., 2005), and the walking comfort of pedestrians was also affected seriously. Similarly, there are vibration comfort problems on many footbridges, for example, the Solferino Footbridge in Paris (Gheitasiet al., 2016), the Alexandra Bridge in Ontario (Bruno and Venuti, 2009), the NEC Bridge in Birmingham (Zivanovic et al., 2005), the Queens Park Footbridge in Britain (Huang et al., 2005), Wuhan Yangtze River Bridge (Li, 1975), and Shanghai Railway Station Footbridge (Xiao, 2009). The above-mentioned cases demonstrate that walking comfort is severely affected by pedestrian-induced vibration. Hence, footbridge vibration comfort has become a critical aspect of the vibration serviceability evaluation.

Civil engineers have begun to focus on vibration comfort in the 1930s. Reiher and Meister (Smith, 1988) conducted a landmark experiment and obtained “equal feeling curve.” Subsequently, Helberg and Sperling (1941), Dieckmann (1955), Chang (1967, 1973), and Griffin (1994) conducted detailed research on vibration comfort. In 1994, the International Union of Railways summed up the research results on the ride comfort of trains and then formulated the evaluation criteria for the vibration comfort of passengers in railway vehicles (Li, 2005). The above research results basically define vibration comfort from the viewpoint of the structural vibration acceleration response. However, there is no real-time evaluation system for footbridge vibration comfort at present, and the vibration comfort of footbridges cannot be evaluated timely.

Meanwhile, smartphones have become indispensable tools in people’s daily lives. Since the birth of the first smartphone, IBM Simon, in 1993 (Grush, 2012), smartphones have become increasingly powerful. Mainstream smartphones are basically integrated with acceleration sensors, orientation sensors, distance sensors, Bluetooth, Wi-Fi module, GPS module, and other functional components (Chun and Maniatis, 2009; Laurila et al., 2013). The high-precision acceleration sensor and orientation sensor built in the smartphone make it feasible to collect the pedestrian-induced vibration signals of footbridges in real time and lay a foundation for real-time evaluation of vibration comfort. In addition, smartphones are compact and portable, and the Wi-Fi function guarantees command transmission between smartphones. Consequently, it does not require wiring like traditional accelerometers, which significantly reduces the tedious process of wiring.

Therefore, this article proposes a real-time evaluation system for pedestrian-induced vibration comfort based on smartphones. The rest of this article is organized as follows: first, an evaluation system of pedestrian-induced footbridge vibration comfort is put forward. Second, technical challenges in the evaluation system are solved. Third, the evaluation system of footbridge vibration comfort is verified experimentally.

Design scheme of pedestrian-induced footbridge vibration comfort evaluation system

The design scheme consists of two aspects: (1) the real-time measurement of a single smartphone and (2) the pedestrian-induced footbridge vibration comfort evaluation system.

Real-time measurement of a single smartphone

A call program used for calling the built-in acceleration sensor in smartphone is compiled, and the application code is presented in Table 1.

Table 1.

Application code for calling the built-in acceleration sensor.

//Call acceleration sensor
Intent intent=new intent(Acceleration Store. ACTION_SIGNAL_CAPTURE);
// Set the save position of acceleration signal
File f=new file(Acc, “crack.swf”
Uri u=Uri.fromFile(f)
// Starting acceleration sensor
intent.putExtra(AccelerationStore.Media.ORIENTATION,0);
intent.putExtra(AccelerationStore.EXTRA_OUTPUT,u)
startActivitityForResult(intent,1);

The built-in acceleration sensor can be called to collect the pedestrian-induced vibration acceleration time history signal based on the application code in Table 1. Then, a data acquisition program is designed (see Appendix 1 for the detailed application code), which can be easily installed on a smartphone for signal acquisition, and the data acquisition interface is shown in Figure 1. As shown in Figure 1, when the acceleration along x-axis, y-axis, z-axis, and search peaks is automatically selected, the smartphone will automatically collect the pedestrian-induced vibration acceleration time history signals. Similarly, when the real-time display is selected, the vibration acceleration time history signals will be displayed in real time.

Figure 1.

Data acquisition interface.

Pedestrian-induced footbridge vibration comfort evaluation system

The pedestrian-induced footbridge vibration comfort evaluation system consists of a data acquisition subsystem, management center subsystem, and smartphone client, as presented in Figure 2.

Figure 2.

Pedestrian-induced footbridge vibration comfort evaluation system.

The data acquisition subsystem consists of some smartphones, which are used for acceleration signal acquisition on the basis of the installed data acquisition program. The management center subsystem is composed of a computer, which plays a critical role in the evaluation system. The smartphone client is developed for efficient use of the evaluation system.

The smartphone client sends the data acquisition command to the management center subsystem via Wi-Fi (Chia-Hsin and Ho, 2015; Frezzetti et al., 2015; Xu et al., 2015), and the management center subsystem receives the command and transmits it to the data acquisition subsystem through Wi-Fi. Then, the data acquisition subsystem executes the command and sends back the collected acceleration signal to the management center subsystem. The management center subsystem saves the acceleration time history signal and sends it to the smartphone client. Hence, the pedestrian-induced vibration acceleration time history signals can be displayed on the smartphone in real time, and pedestrians who carry a smartphone can easily check the vibration response of the footbridge.

Technical challenges

Prior to solving the technical challenges in the evaluation system, the built-in acceleration and orientation sensor in smartphone are introduced. The built-in acceleration sensor is made of a highly integrated silicon wafer, which can accurately measure the acceleration along three directions, that is, x-axis, y-axis, and z-axis. The highest acquisition frequency can reach 100 Hz, and resolution can reach 1 × 10⁻⁵ m/s², with the range of ±2 g to ±16 g, which can sufficiently meet the requirements of pedestrian-induced footbridge vibration time history signal acquisition.

In contrast with acceleration sensor, the orientation sensor can accurately measure angle data called azimuth, pitch, and roll. The related parameters of acceleration and orientation sensor built in the smartphone are listed in Table 2.

Table 2.

Parameters of acceleration and orientation sensor built in the smartphone.

Sensor type	Data	Parameter description
Acceleration sensor	A-[0]	Acceleration along the x-axis(including gravity)
	A-[1]	Acceleration along the y-axis(including gravity)
	A-[2]	Acceleration along the z-axis(including gravity)
Orientation sensor	$φ$	Azimuth, the angle between z-axis and the North Pole (0° to 360°)
	$θ$	Pitch, the angle between x-axis and horizontal plane (−180° to 180°)
	$γ$	Roll, the angle between y-axis and horizontal plane (−90° to 90°)

There are four technical challenges in the application of the evaluation system: (1) transformation from smartphone coordinate system to inertial coordinate system, (2) vibration acceleration signal drift correction, (3) signal filtering, and (4) computation of the total weighted root mean square acceleration (TWA). These technical challenges are discussed in the following paragraphs.

Coordinates transformation

The pedestrian-induced footbridge vibration signals are collected by the acceleration sensors built in the smartphone. Hence, the collected raw signals only denote the acceleration quantity based on the smartphone’s own coordinate system. Nevertheless, smartphones are carried by pedestrians, so the acceleration values will be changing simultaneously with different carrying postures. In other words, the collected raw signals cannot express the actual vibration response of footbridges. To solve this problem, the space coordinate transformation is studied to ensure the facticity of acceleration signal under different smartphone postures.

First, there are three different coordinate systems, namely, smartphone coordinate system, the world coordinate system, and the inertial coordinate system. The smartphone coordinate system is a relative coordinate system depending on the screen of the smartphone. The definition is as follows: when the smartphone is placed horizontally and the screen is upward, the center of the screen is the origin of this relative coordinate system, the direction parallel to the short side of the screen is the x-axis, the direction parallel to the long side of the screen is the y-axis, and the z-axis is vertically upward (Zhao et al., 2016). The world coordinate system is an absolute coordinate system, which can be described by longitude, latitude, and height. The inertial coordinate system is an intermediate status between the smartphone coordinate system and the world coordinate system, and the origin of the inertial coordinate system is identical to that of smartphone coordinate system, while the coordinate axes are parallel to the axes of the world coordinate system.

The pedestrian-induced footbridge vibration acceleration signals should be unified in the inertial coordinate system. According to the characteristics of the inertial coordinate system, the smartphone coordinate system can be converted into the inertial coordinate system through rotating coordinate axes operation, while the world coordinate system can be converted into the inertial coordinate system through shift operation (Zhao and Guo, 2016).

Then, the acceleration signal collected by the smartphone is a certain vector, where the acceleration vector is $\vec{A}$ , $x \vec{i}$ , $y \vec{j}$ , and $z \vec{k}$ denote the acceleration in the smartphone coordinate system, and $x' \vec{i}'$ , $y' \vec{j}'$ , and $z' \vec{k}'$ denote the acceleration in the inertial coordinate system, so

\vec{A} = x \vec{i} + y \vec{j} + z \vec{k} = x' \vec{i}' + y' \vec{j}' + z' \vec{k}'

(1)

Formula (1) can be expressed as follows

[\begin{matrix} x' \\ y' \\ z' \end{matrix}] = [\begin{matrix} \vec{i} \cdot \vec{i}' \\ \vec{i} \cdot \vec{j}' \\ \vec{i} \cdot \vec{k}' \end{matrix} \begin{matrix} \vec{j} \cdot \vec{i}' \\ \vec{j} \cdot \vec{j}' \\ \vec{j} \cdot \vec{k}' \end{matrix} \begin{matrix} \vec{k} \cdot \vec{i}' \\ \vec{k} \cdot \vec{j}' \\ \vec{k} \cdot \vec{k}' \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} \cos α_{1} \\ \cos β_{1} \\ \cos γ_{1} \end{matrix} \begin{matrix} \cos α_{2} \\ \cos β_{2} \\ \cos γ_{2} \end{matrix} \begin{matrix} \cos α_{3} \\ \cos β_{3} \\ \cos γ_{3} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}]

(2)

If the cosine matrix in formula (2) is C, it can be seen that the key step to realize the conversion from the smartphone coordinate system to the inertial coordinate system is to determine the cosine matrix C.

The cosine matrix C can be obtained through z-x-y transformation, and it is as follows

C = C (γ) \cdot C (θ) \cdot C (φ) = [\begin{matrix} \cos γ \\ 0 \\ \sin γ \end{matrix} \begin{matrix} 0 \\ 1 \\ 0 \end{matrix} \begin{matrix} - \sin γ \\ 0 \\ \cos γ \end{matrix}] [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \begin{matrix} 0 \\ \cos θ \\ - \sin θ \end{matrix} \begin{matrix} 0 \\ \sin θ \\ \cos θ \end{matrix}] [\begin{matrix} \cos φ \\ - \sin φ \\ 0 \end{matrix} \begin{matrix} \sin φ \\ \cos φ \\ 0 \end{matrix} \begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

(3)

Hence, coordinate transformation from the smartphone coordinate system to the inertial coordinate system can be described as follows

[\begin{matrix} x' \\ y' \\ z' \end{matrix}] = [\begin{matrix} \cos φ \cos γ - \sin φ \sin θ \sin γ \sin φ \cos γ + \cos φ \sin θ \sin γ - \cos θ \sin γ \\ - \sin φ \cos θ \cos φ \cos θ \sin θ \\ \cos φ \sin γ + \sin φ \sin θ \cos γ \sin φ \sin γ - \cos φ \sin θ \cos γ \cos θ \cos γ \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}]

(4)

The acceleration signals can be converted from the smartphone coordinate system to the inertial coordinate system through formula (4).

Third, the orientation angles azimuth $(φ)$ , pitch $(θ)$ , and roll $(γ)$ in formula (4) can be measured by the orientation sensor built in the smartphone. In this study, a call program used for calling the built-in orientation sensor is compiled, and a data acquisition program is designed (see Appendix 1 for the detailed application code). The orientation angles’ acquisition interface is presented in Figure 3.

Figure 3.

Orientation angles’ measurement interface.

Signal drift correction

Vibration signal drift exists in many research fields, which is the limiting factor of improving the measurement accuracy (Botto, 1984; Jacob and Hermance, 2005). The built-in acceleration sensors in smartphones also suffer from drift issue; this problem will be discussed from two aspects: trend item elimination and drift correction.

Trend item elimination

Trend item is an important factor affecting the raw acceleration signal. A trend item elimination program, compiled based on MATLAB language, is presented in Table 3. The trend item elimination can be selected in the client interface, as presented in Figure 3.

Table 3.

Trend item elimination program.

F_s=N=100;
n=0: N-1;
t=n/ F_s;h=1/ F_s;
a_o=random (size(t))
j=1: N
j₁=sum(j); j₂=sum(j²); j₃=sum(j³); j₄=sum(j⁴);
ss=sum(a_o); ss₂=sum(a_o×j’); ss₃ = (a_o×(j²)′);
A=[N, h* j₁, h²* j₂; j₂, h* j₃, h²*j₄];
E=[ss, ss₂, ss₃]; B=inv(A)*E;
BD0=B(1); BD1=B(2); BD2=B(3);
Trend2=BD0+BD1*t+random(BD2)
a=a_o- Trend2;

The trend item in the raw acceleration signal can be eliminated by running the program installed in the smartphone. Figure 4 presents the comparison between the raw acceleration signal and the signal after nonlinear trend elimination.

Figure 4.

Comparison between the acceleration signal before and after nonlinear trend elimination: (a) raw acceleration time history signal and (b) signal after nonlinear trend item elimination.

Drift correction based on mathematical morphology

Mathematical morphology is an image analysis subject based on lattice theory and topology: the most common basic operations are corrosion, expansion, opening operation, and closing operation (Sun et al., 2009; Yang and Qi, 2011). Assuming that the raw signal $f (n)$ is a discrete function defined on $F = (0, 1, 2, \dots, N - 1)$ , structural elements $s (n)$ are discrete functions defined on $S = (0, 1, 2, \dots, M - 1)$ , and $N \geq M$ , and the corrosion and expansion of $f (n)$ on $s (n)$ are as follows

(f ⊖ s) (n) = min [f (n + m) - s (m)]; m = 0, 1, \dots, M - 1

(5)

(f \oplus s) (n) = max [f (n - m) + s (m)]; m = 0, 1, \dots, M - 1

(6)

The opening operation and closing operation are as follows

(f ◯ s) (n) = (f ⊖ s \oplus s) (n)

(7)

(f \cdot s) (n) = (f \oplus s ⊖ s) (n)

(8)

Open-close filter, close-open filter, and hybrid filter can be constructed on the basis of the opening operation and closing operation. In this study, a drift correction filter based on mathematical morphology is proposed. The drift correction consists of the following steps:

First, close-open filtering was conducted on the raw signal $f (n)$ . The output signal after filtering is as follows

g (n) = (f \cdot s ° s) (n)

(9)

2.Second, open-close filtering was conducted on the output signal $g (n)$ , and $h (n)$ can be obtained

h (n) = (g ° s \cdot s) (n)

(10)

3.Third, the correct acceleration time history signal $a (n)$ is obtained after removing the drift component from the raw signal

a (n) = f (n) - h (n)

(11)

Signal filtering based on wavelet analysis

It is inevitable that there are different kinds of noises in the raw acceleration signal, including gravity and other random noises, and gravity has higher interference on the acceleration signal. Meanwhile, gravity in the acceleration signal is relatively stable; thus, it can be separated as deterministic noises through signal filtering method based on wavelet analysis (Jing et al., 2000). Wavelet analysis is an effective signal filtering method different from the traditional signals processing technology, and it decomposes different kinds of frequency elements to non-overlapped frequency bands. The telescopic translation system ${φ_{a, b}}$ of the basic wavelet $φ (t)$ is generally called wavelet function and can be denoted as follows

φ_{a, b} (t) = \frac{1}{\sqrt{a}} φ (\frac{t - b}{a})

(12)

where $a$ is the scale parameter and $b$ is the position parameter (Daubechies, 2015). According to wavelet theory, the wavelet transform of function $f (t)$ is equal to the inner product between function $f (t)$ and wavelet function

W_{f} (a, b) = f (t), φ_{a, b} (t)

(13)

Wavelet transform is a time frequency analysis, generally using the Mallat algorithm to calculate the discrete dyadic wavelet transform (Mallat, 1989). Assuming the discrete sequence of function $f (t)$ is $f (n), n = 1, 2, 3, \dots, N$ , the discrete dyadic wavelet transform of the function is as follows

{\begin{matrix} C_{j + 1} (n) = \sum_{k \in Z} h (k - 2 n) C_{j} (k) \\ D_{j + 1} (n) = \sum_{k \in Z} g (k - 2 n) C_{j} (k) \end{matrix}

(14)

where $h (k)$ and $g (k)$ are the orthogonal conjugate filter coefficient of the wavelet function $φ_{2^{j}, b} (t)$ , $g (k) = (- 1)^{1 - k} h (1 - k)$ , $C_{j}$ is the approximate part of the signals on the scale $j$ , and $D_{j}$ is the detail part of the signals on the scale $j$ .

In this study, the raw signals collected by a smartphone can be regarded as approximation of scale $j = 0$ . $C_{0} (n) = f (n)$ , $D_{1}, D_{2}, D_{3}, \dots, D_{J}$ can be obtained after discrete signals decomposition with scale $j = 1, 2, 3, \dots, J$ . Assuming the frequency of the raw signal is $f$ , the corresponding bands are (2⁻¹f∼f),(2⁻²f∼2⁻¹f), (2⁻³f∼2⁻²f),…, (2^-Jf∼ 2^−J ⁺¹f), (0∼2^−Jf), and the analysis results demonstrate that the raw signals are decomposed into different frequency bands from high frequency to low frequency through discrete dyadic wavelet transform. Meanwhile, each frequency band does not overlap, some frequency bands that are not concerned can be deleted from the different frequency bands according to the analysis target, which is called signals filtering. After signals filtering, the effective signals used for the pedestrian-induced footbridge vibration comfort analysis can be obtained.

Figure 5 presents the comparison between the raw acceleration signal and the signal after filtering. If the original signal is decomposed into six layers, 64 equal width bands can be obtained. The frequency range of each frequency band is 0–16 Hz, 16–32 Hz, 32–48 Hz, …, 1008–1024 Hz. The raw acceleration time history signal is shown in Figure 5(a). Signals without gravity can be obtained after removing the gravity band components and reconstructing the remaining bands, as presented in Figure 5(b).

Figure 5.

Comparison between the acceleration signal before and after filtering: (a) raw acceleration time history signal before wavelet transform and (b) acceleration time history signal after wavelet transform.

TWA

Since the human body has different vibration comfort levels for different vibration frequencies, Chinese national criterion GB 4970-1996 proposes an approximate frequency-weighted function in each direction. The weight values of vertical direction of z-axis are presented in formula (15), and the weight value of x-axis and y-axis are shown in formula (16)

ω_{v} (f) = {\begin{matrix} 0.5 & 0.5 < f \leq 2 \\ f / 4 & 2 < f \leq 4 \\ 1 & 4 < f \leq 12.5 \\ 12.5 / f & 12.5 < f \leq 80 \end{matrix}

(15)

ω_{l} (f) = {\begin{matrix} 1 & 0.5 < f \leq 2 \\ 1 / f & 2 < f \leq 80 \end{matrix}

(16)

where $f$ is the vibration frequency.

Then, the uniaxial weighted acceleration $a_{ω}$ can be calculated as follows

a_{ω} = {[\int_{0.5}^{80} ω^{2} (f) G_{a} (f)]}^{1 / 2}

(17)

where $ω (f)$ is the weight value obtained from equations (15) and (16). $G_{a} (f)$ is the power spectral density function, which can be obtained by power spectrum analysis of acceleration signals.

The TWA is presented as follows

a_{ω T} = \sqrt{{(1.4 a_{ω x})}^{2} + {(1.4 a_{ω y})}^{2} + {a_{ω z}}^{2}}

(18)

where $a_{ω x}$ , $a_{ω y}$ , and $a_{ω z}$ are the uniaxial weighted acceleration obtained from equation (17).

International evaluation criterion of pedestrian-induced footbridge vibration comfort

Pedestrian-induced footbridge vibration is a complex structural vibration mode rather than a simple forced resonance vibration. Michael Griffin et al. (1994) found that different pedestrians have different reactions to the same vibration, and even the same pedestrian will have different feelings at different times for the same vibration. Therefore, the international standard ISO 2631-1:1997 recommended the TWA value to evaluate vibration comfort, and the relationship between the TWA $a_{ω T}$ and human body comfort is presented in Table 4.

Table 4.

Relationship between human body comfort and the TWA.

TWA $a_{ω T}$ (m/s²)	Subjective feelings	Average of TWA (m/s²)	Comfort score
<0.315	Not uncomfortable	<0.315	10
0.315–0.63	Little uncomfortable	0.38	8
0.5–1.0	uncomfortable	0.75	6
0.8–1.6	More uncomfortable	1.2	4
1.25–2.5	Very uncomfortable	1.86	2
>2.0	Extremely uncomfortable	>2.0	0

TWA: total weighted root mean square acceleration.

Experimental verification

Experiment

To verify the validity of the proposed evaluation system, experiments were carried out on the Forth Corridor Footbridge in Guangzhou. The Forth Corridor Footbridge connects Creative Building and Lion Rock Park, with a two-span continuous half-through structural characteristic. The span combination is 64 + 63.2 m, and many pedestrians pass through the footbridge at rush hour, as presented in Figure 6.

Figure 6.

The Forth Corridor Footbridge in Guangzhou.

In this experiment, there are six experimental conditions, from condition 1 to condition 6. Creative Building (near the left span) is the starting point, and Lion Rock Park (near the right span) is the ending point. A total of 40 testers are walking on the footbridge under three different walking modes: slow walking, brisk walking, and marching. There are two different walking speeds in each walking mode, as listed in Table 5. Every tester takes with a smartphone, and the different conditions are shown in Figure 7. When the acceleration response exceeds the limit value of comfort degree, the smartphone will send an alarm sound, which prompts pedestrians to adjust walking mode and walking speed.

Table 5.

Experimental conditions.

Walking mode	Slow walking	Brisk walking	Marching
Experimental conditions	Condition 1 (v = 0.8 m/s)	Condition 3 (v = 1.6 m/s)	Condition 5 (v = 1.2 m/s)
Experimental conditions	Condition 2 (v = 1.0 m/s)	Condition 4 (v = 2.0 m/s)	Condition 6 (v = 1.4 m/s)

Figure 7.

Field experiments on the Forth Corridor Footbridge (image by authors): (a) condition 1 (slow walking, v = 0.8 m/s) and (b) condition 5 (marching, v = 1.2 m/s).

Due to space limitation, the field experiments under conditions 1 and condition 5 are presented in Figure 7.

Meanwhile, the vertical and lateral traditional accelerometers were installed on the left and right spans at the positions of L/4, L/2, and 3L/4, and an INV3018 portable data acquisition instrument was used to collect the vibration acceleration signals, as presented in Figures 8 and 9.

Figure 8.

INV3018 data acquisition instrument and computer (image by authors).

Figure 9.

Vertical and lateral traditional accelerometers (image by authors).

Experimental results

As far as the raw vibration acceleration time history signals collected by smartphones are concerned, the following technical treatments are required. First, coordinate transformation is performed based on formula (4). Second, the acceleration signal is corrected by a drift correction filter. Third, signal filtering based on wavelet is analyzed. Fourth, the TWA computation is based on formula (18). After the above processing, the comparisons of two different measurement results are shown in Figure 10. In view of the vibration acceleration along the z-axis being the maximum value among x-axis, y-axis, and z-axis direction as well as space limitations, the vertical vibration acceleration time history curves are compared in this study.

Figure 10.

Comparisons of two measurement results of condition 1 and condition 5: comparison for (a) condition 1 and (b) condition 5.

Due to space limitation, comparisons of two measurement results of condition 1 and condition 5 are presented in Figure 10.

The TWA of smartphones and accelerometers are calculated based on formula (18). Meanwhile, the subjective feelings in the experiment are recorded, which are compared with the corresponding stipulation in ISO 2631-1:1997, as presented in Table 6.

Table 6.

Comparison of experimental results between smartphones and traditional accelerometers.

	TWA based on smartphones	TWA based on accelerometers	TWA errors (%)	Subjective feelings in the experiment	Stipulation in ISO 2631-1:1997
Condition 1	0.2771	0.2911	−4.8	Not uncomfortable	Not uncomfortable
Condition 2	0.2959	0.3102	−4.6	Not uncomfortable	Not uncomfortable
Condition 3	0.4686	0.4876	−3.9	Little uncomfortable	Little uncomfortable
Condition 4	0.5982	0.6233	−4.1	Little uncomfortable	Little uncomfortable
Condition 5	1.3362	1.4016	−4.7	Very uncomfortable	Very uncomfortable
Condition 6	1.4290	1.4901	−4.1	Very uncomfortable	Very uncomfortable

TWA: total weighted root mean square acceleration.

Table 6 indicates that the TWA errors between smartphones and accelerometers are less than $\pm 5 %$ , as expected. Meanwhile, the subjective feelings under a certain vibration response in the experiment are in excellent agreement with the stipulation in ISO 2631-1:1997. TWA is related to the walking mode and walking speed, which increases with the increase in walking speed under the same walking mode. The main reason for the TWA errors is that the accelerometers are fixed on the footbridge while the smartphones are carried by pedestrians, and there is a certain attenuation in the process of vibration transmission from footbridge to pedestrians. Therefore, the TWA value of the smartphones is somewhat lower than that of the traditional accelerometers.

Conclusion

Pedestrian-induced footbridge vibration leads to an uncomfortable and unsafe feeling. To evaluate the pedestrian-induced footbridge vibration comfort, a real-time evaluation system based on smartphones is proposed.

In this work, four technical challenges in the evaluation system are solved, and a field experiment is conducted to compare the smartphone-based system with the traditional accelerometers. Overall, experimental results show that the TWA errors between smartphones and accelerometers are less than $\pm 5 %$ , which satisfied the engineering application. In addition, the subjective feelings in the experiment are identical with the corresponding stipulation in ISO 2631-1:1997.

In the process of the evaluation system application, the vibration acceleration time history signals can be displayed on the smartphones in a timely manner, and pedestrians can quantitatively understand the footbridge vibration response in real time. In addition, the data acquisition subsystem consists of some smartphones rather than traditional accelerometers; smartphones are compact and portable, while traditional accelerometers require cumbersome wiring. Hence, the proposed evaluation system is more efficient and convenient than the traditional accelerometers in vibration acceleration response measurement.

The aim of this study is to propose a real-time evaluation system for footbridge vibration comfort. A more detailed analysis that accounts for both the pedestrian-induced vibration mechanism and the vibration behavior of the pedestrian–footbridge coupled system is required.

Footnotes

Appendix

Appendix 1

Data acquisition program

Public void onSensorChanged(SensorEvent e) {
if (N==0) {
start=System.currentTimeMillis();
avr=e.values[2];
}
long now=System.currentTimeMillis();
// Displaying acquisition information at the title
String title=““;
if (N>100) {
mFs=(int) (N * 1000 / (now—start));
title=mFs+“hz”+“”:+““+N;
} else {
title=“Point: ”+N;
}
// title=“Sampling number: ”+N;
setTitle(title);
double[] value=new double[3];
value[0]=e.values[0];
value[1]=e.values[1];
value[2]=e.values[2]—avr;
mZAccList.add(value(2)
AccDataaccData=new AccData(value(2)
try {
fos.write(accData.getZData().getBytes());
} catch (IOExceptionexp) {
}
// Display acceleration time history curve
for (inti=0; i<3; i++) {
if (!bDraw(i)) {
continue;
}
myXYSeries(i).add(N, value(i));
SharedPreferencessp=PreferenceManager
.getDefaultSharedPreferences(this);
booleanbRt = sp.getBoolean(“realtime_mode,” true);
if (bRt)// Real-time rendering
{
if (myXYSeries(i).getItemCount()>POINTS_ON_FRAME) {
myXYSeries[i].remove(0);
}
mAcclChartArr(i).update(myXYSeries(i));
} else// Frame by frame
{
if (N % POINTS_ON_FRAME==0) {
mAcclChartArr(i).update(myXYSeries(i));
myXYSeries[i]= new XYValueSeries(““);
}
}
}
// Displayorientation anglesazimuth, pitch and roll
if ((N+1) % POINTS_ON_FRAME==0 &&bDraw[3]) {
Double[]x=new Double(mZAccList.size());
mZAccList.toArray(x);
orientationChart.update(x, mFs, mLowFreq, mHighFreq, m_t1, m_t2,
mIsAutoSearPeak);
}
// Draw an acceleration diagram
if (bDraw(4) {
// mySeries[4].add(now, value(2)
// updateTimeChart(mAccChart, mySeries(4)
}
N++;
}

Acknowledgements

The authors thank the volunteers who participated in the experiment and thank Dr Jingqi Shang and Dr Shuangrui Chen for their valuable advice on this study.

Declaration of Conflicting Interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: The authors declare that this evaluation system is only used for acceleration signal acquisition and analysis for footbridge vibration comfort research. This proposed evaluation system will not be used for any commercial purpose, nor will it collect any user personal information. It should be emphasized that there is not any privacy and security hazards in the process of data acquisition program installation and application. The authors declare that there are no conflicts of interest regarding the publication of this paper.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Project of National Natural Science Foundation of China under Grant Nos 51478193 and 51208208 and the Key Project of Fundamental Research Funds for Central-Level Universities and Institutes under no. 2015ZM114.

ORCID iDs

Deyi Chen

Jie Wu

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