Tension Force Detection of Short Suspender by Using Laser Doppler Vibrometer

Abstract

In this study, the vibration frequency of steel rod-type short suspenders simulated by three types of prestressed steel rebars with aspect ratios of 14, 20.8, and 37.8 was detected accurately and remotely by using Laser Doppler Vibrometer (LDV); then the tension force state of short suspenders was evaluated through the quadratic regression relationships between vibration frequency and tension force. The results showed that the vibration frequency of steel rod-type short suspenders increases with the increase in tension force, and there is a good correlation between vibration frequency and prestress. Furthermore, when the prestress applied is the same, the vibration frequency of short suspenders decreases with the increase of their aspect ratios, indicating aspect ratio plays a decisive role on the change of vibration frequency. The errors between the tension force obtained by experiment and tension string theoretical calculation for short suspenders are large, due to the nonrigid boundary condition. The establishment of quadratic regression relationships between vibration frequency and tension force effectively avoids the influence of various factors on the existing calculation model such as mass, stiffness, and constraint type, and makes the detection accuracy of tension force achieve 98%.

1. Introduction

As an important force component, cables with different aspect ratios (those with small aspect ratio are usually called short suspenders) and profiles (strand or steel rod) have been widely used in large-span bridge structures [1,2,3,4]. Maintaining the cable force in design level is key in ensuring structural stability and safety. However, degradation of cable force occurs frequently, due to lack of pretension forces, relaxation of anchor cable, anchor slipping, and environment corrosion, contributing to structural failure and instability [5,6,7,8]. Timely detection of cable tension force can discover problems in time to give early warnings and then ensure the operational safety of structures. There are several methods to measure cable tension force at present, including oil pressure gauge, pressure sensor, magnetic flux, and vibration frequency. The oil pressure gauge method can accurately detect the cable tension force in the construction stage, but it does not function well during the operation stage of cables due to heavy equipment, labor intensity, low efficiency, and high cost [9,10]. Although the pressure sensor method has a good performance in cable force detection, it can only be used in embedded manner, leading to high cost, complexity construction, and inconvenience of replacement [11]. Magnetic flux method [12] is a new test technology developed in recent years, by which the relationships between the cable tension force/temperature and magnetic flux can be established to detect cable force in real time. It is worthwhile to note that the calibration process of magnetic flux method is complicated, seriously affecting its large-scale promotion and application.

The vibration frequency method is an indirect test method based on the development of tension string vibration theory, and it has been widely studied and applied in cable tension force detection due to its high accuracy and simplicity [11,13,14,15,16]. It should be noted that two aspects must be guaranteed to implement this method effectively. On one hand, the cable vibration frequency should be picked up accurately; on other hand, a suitable theoretical calculation model between cable force and vibration frequency should be adopted. Because the relationships between natural vibration frequency of slender cables with high aspect ratio and tension force follows string vibration theory, the tension force of slender cables can be calculated according to measured frequency and theoretical calculation model. However, the string vibration theory for short suspenders is easily affected by boundary conditions, stiffness and inclination [17,18]. When calculating the tension force for short suspender from frequency, it is necessary to solve the simultaneous equations of transcendental equations, and the calculation process is very complicated [19,20,21]. A new theory model needs to be established for tension force detection of short suspenders. Meanwhile, the vibration frequency measured by traditional vibration sensors for short suspender easily suffers from service environmental and stress state, reducing the accuracy of tension force detection. Meanwhile, the short suspenders in bridge structure with high stiffness and high vibration frequency are more prone to fracture under the dual influence of dynamic stress and corrosion compared to slender cables, resulting in partial collapse or direct damage to the bridge [22,23]. There is an urgent need for a fast and accurate measurement for short suspenders. Laser Doppler Vibrometer (LDV) has been proved to be a fast and accurate equipment for testing the vibration frequency of cables [24,25,26]. Huang et al. [25] reported that the vibration frequencies of stayed cable obtained by LDV are in good agreement with those obtained by vibration sensors. The maximum relative errors between these two methods are only 5.5% and 1.0%, respectively, as the first order frequency and the high order frequency is used to calculate fundamental frequency.

Therefore, based on the fact that the durability of steel rod type short suspenders is higher than that of strand type, three types of prestressed steel rebars with aspect ratios of 14, 20.8, and 37.8 were used to simulate steel rod-type short suspenders, and their tension force state was detected in real time by using Laser Doppler Vibrometer (LDV) and vibration frequency analysis. Meanwhile, the theoretical calculation model for tension force of steel rod-type short suspenders according to vibration frequency was established, without considering mass, stiffness, and boundary condition.

2. Experimental Design

2.1. Laser Doppler Vibrometer (LDV)

Laser Doppler Vibrometer (LDV, type of PDV-100), fabricated by German Polytec company, can achieve remote and contactless detection of the vibration frequency of short suspenders, based on the working principle of the laser doppler effect [27]. When vibration occurs, if the object is far away from the laser source, the wavelength of reflected light becomes larger, otherwise, when the object is close to the light source, the wavelength of reflected light becomes shorter. LDV can emit a helium-neon laser beam onto the target surface, and collects the reflected light at the same time. Vibration curve in time domain can be obtained through wavelength variation; then the relationships between vibration mode and inherent frequency can be established after Fast Fourier Transform (FFT) on the basis of peak picking method, as demonstrated in Figure 1. Compared to traditional vibration sensors, LDV has huge advantages as it is remote, non-contact, has high accuracy, and is immune to environmental influence [28,29].

2.2. Raw Materials

Hot-rolled steel rebars (HRB400E) with diameters of 12 mm and different aspect ratios, produced in Chengde iron and steel group, China, were adopted to simulate the steel rod type short suspenders with a range of bending stiffness. In order to verify the influence of aspect ratio and the stability of the experiment, three samples with different lengths were selected for the test, and the specific parameters of the sample were shown in

Table 1. Geometric parameters of the specimens.

NO.	l/mm	l/d	ρ/(kg·m−1)	EI/(N·mm2)	Yield Strength/MPa	Tensile Strength/MPa
1#	168	14	0.89	2.03 × 108	410	645
2#	250	20.8	0.89	2.03 × 108	410	645
3#	453	37.8	0.89	2.03 × 108	410	645

.

2.3. Testing Process

A universal testing machine of the type WEP-600, produced by Chang-chun Test Instrument Co., Ltd., Changchun, China, was used to preload and sustain the load on steel rebars until the design force and the tensioning construction simulation were completed. The maximum range of the machine is 600 kN and the accuracy is 0.01 kN. The vibration frequency of prestressed steel rebars was measured after the load shutdown for 1–2 min in order to reduce the influence of instrument vibration. Meanwhile, the maximum prestress was divided into ten levels for forcing, and the corresponding vibration frequency was collected in sequence. The test procedure and experiment operation site for testing vibration frequency of prestressed steel rebars by using LDV are shown in Figure 2 and Figure 3. That is to say, the tension force for steel rod-type short suspenders during experiments was controlled by a universal testing machine, and the detected results for tension force were obtained on the basis of theoretical calculation and vibration frequency.

The results in previous studies showed that the excitation method of steel rod beating has a higher signal-noise ratio compared to other excitation methods. After applying excitation of steel rod beating, the amplitudes at different positions of short suspender are different, but the vibration frequencies are the same. Because the maximum amplitude is in the middle of theshort suspenders, the vibration frequency of that position was studied in this paper. During the experiments, the temperature and humidity of external environment were 5–10 °C and 30–40%, respectively. The influence of temperature and humidity on the tension force of steel rod-type short suspenders was not been considered in this study, and will be explored in the following research. It is important that the prestressed steel rebars were used to simulate steel rod type short suspender in this study, and the terms “steel rod-type short suspender” or “short suspender” were used for the following analysis.

3. Results

3.1. Vibration Frequency under Different Tension Force

Figure 4 shows the relationships between tension force and vibration frequency for steel rod-type short suspenders of samples 1–3 with different lengths. It can be seen that the vibration frequency of specimens increases with the increase of tension force and decreases with the increase of suspender length.

The typical tension string theory used to describe the relationships between vibration frequency and tension force is demonstrated in Equation (1) [15]. Meanwhile, the modified model considering the bending stiffness of cables is displayed in Equation (2) [18]. These two models have been used to calculate the theoretical value of the vibration frequency of short suspenders of specimens 1–3 on the basis of each level tension force, and the differences between the experimental and calculated values of tension force are listed in

Table 2. First order vibration frequency of sample 1 under different tension force.

Tension/kN	${\mathit{f}}_{\mathit{M}}$/Hz	Typical Model in Equation (1)		Modified Model in Equation (2)
		${\mathit{f}}_{\mathit{C}}/\mathbf{Hz}$	${\mathit{\delta }}_{\mathit{f}}/\%$	${\mathit{f}}_{\mathit{C}}/\mathbf{Hz}$	${\mathit{\delta }}_{\mathit{f}}/\%$
2.3	959.38	151.70	84.19	154.00	83.95
4.4	997.81	208.91	79.06	210.59	78.89
6.8	1105.63	259.90	76.49	261.25	76.37
9.1	1155.94	300.11	74.04	301.28	73.94
11.2	1186.56	333.30	71.91	334.36	71.82
14.1	1214.06	373.97	69.20	374.91	69.12
17.3	1238.44	413.64	66.60	414.49	66.53
23.1	1278.75	478.98	62.54	479.72	62.49
26.2	1293.44	509.29	60.62	509.98	60.57
29.3	1307.50	538.91	58.78	539.56	58.73
32.4	1325.0	566.46	57.25	567.08	57.20

.

$$T=4m{l}^2{f}_{}{}^2$$

(1)

where $T$ is tension force, $m$ is linear density of suspenders, $l$ is length of suspenders, and $f_{}$ is the first order vibration frequency of short suspenders.

$$T=4m{l}^2{f}^2-\frac{{\pi }^{2}EI}{l^2}$$

(2)

where $EI$ is the bending stiffness of short suspenders.

$${\delta }_f=\left|\frac{f_M-f_C}{f_M}\right|\times 100\%$$

(3)

where ${\delta }_f$ is the deviation of the frequency, $f_M$ is the measured frequency, and $f_C$ is the calculated frequency. It can be seen from Equations (1) and (2) that natural frequencies of different orders all can be used to estimate tension force for cables based on vibration theory, but the expression of the formula is different. If the first-order natural frequency can be clearly obtained by LDV, it is not necessary to do a multi-order modal test. Furthermore, the calculated results obtained by using first order and high order vibration frequency have the same accuracy [25]. Hence, the first order vibration frequency has been used to calculate tension force in this study.

It can be seen from Table 2 that there is a large error between the experimental and theoretical values of vibration frequency, and this is consistent with the conclusion obtained by Mehrabi et al. [30]. The big error mainly comes from the following two aspects [1,12,14]. On one hand, the boundary conditions in models are simplified as either hinged or fixed constraints, reducing the calculation accuracy. On other hand, the bending stiffness and sag of short suspenders are also ignored in the derivation process of these two models. Therefore, it is difficult to obtain the accurate calculation results of vibration frequency for short suspenders by using the existing theoretical models.

3.2. Quadratic Regression Relationships for Vibration Frequency and Tension Force for Short Suspenders

It can be found from the experimental results in Figure 5 that there is a high degree of quadratic regression fitting between vibration frequency and tension force for short suspenders with different aspect ratios. Based on this result, the quadratic regression considering the influence of aspect ratios can be used to establish the calculation model between vibration frequency and tension force for short suspenders, as shown in Figure 5. Equation (4) is used to calculate the relative errors between the experimental and calculated values of tension force. It can be observed that when the applied tension force is more than 10 kN, the relative error between the experimental and calculated values of vibration frequency is less than 5%, as demonstrated in Figure 6. Meanwhile, with the increase of tension force, the error between the experimental and calculated values obtained by quadratic regression fitting for vibration frequency presents a declining trend, indicating the accuracy improvement of calculation.

$${\delta }_T=\left|\frac{T_A-T_C}{T_A}\right|\times 100\%$$

(4)

where ${\delta }_T$ is the relative error between the experimental and calculated values of tension force, $T_A$ is the experimental tension force, and $T_C$ is the calculated tension force based on quadratic regression fitting.

It is worthwhile to note that the established quadratic regression relationships between the frequency and tension force was only used to illustrate that the tension force state of steel rod-type short suspenders can be judged according to vibration frequency without considering the influence of mass, stiffness, and constrained type. Meanwhile, different quadratic regression formulas can be used for different types of short suspenders. For example, the quadratic regression formulas for short suspenders with specific features can be established in the design stage, and then this fitting model can be used to detect the tension force of short suspenders in the service stage.

4. Discussion

The above results show that the tension force can be accurately calculated according to the quadratic regression relationships between tension force and vibration frequency obtained by using LDV. This represents the fact that the tension force of short suspenders can be detected in real time to provide guidance for timely maintenance and life prediction. As an accurate and rapid detection method, the advantages of LDV can be listed as follows. First, using LDV to accomplish tension force detection of short suspender belongs to remote and non-contact measurement method, the scope of which is 100 m. Within the effective range, the accuracy of signal can be achieved only by adjusting the angle of the host to aim at the object, and many adverse effects can be avoided. By contrast, conventional sensors need be installed in the object, being more susceptible to the changes in their own properties and surroundings. Second, the sampling frequency and resolution of LDV are all higher than conventional sensors due to its remote and non-contact characteristics, effectively avoiding the errors caused by the additional quality of sensors and wire length in the test. Third, using LDV to obtain the vibration characteristics of short suspender is a nondestructive monitoring method, having no influence on the construction and service process of cables. Fourth, the short suspender surface does not require special processing to obtain enough reflection signal. In particular, the reflector can be pasted in the test position to achieve a strong enough signal [24].

Identifying the vibration frequency of short suspender accurately and then establishing the accurate relationships between tension force and vibration frequency are the key problems for real-time detection of the service condition of cables by using the frequency method. The complex boundary conditions and flexural rigidity of short suspenders produces a large error for the experimental and calculated values of tension force based on tension string vibration frequency. The quadratic relationships between vibration frequency and tension force provide an approach for establishing an accurate method to detect tension force for steel rod-type short suspenders by using vibration frequency.

5. Conclusions

Laser Doppler Vibrometer (LDV) was used to collect the vibration frequency of steel rod-type short suspenders with different aspect ratios, and then the calculation method for tension force based on vibration frequency was established. The following main conclusions can be drawn from this study.

(1)

There is a good correlation between tension force and vibration frequency for short suspenders. With the increase of tension force and aspect ratio, vibration frequency displays an increasing trend, indicating the possibility of using vibration frequency for real-time detection of the tension force state of short suspenders.

(2)

The relationships between vibration frequency and tension force of steel rod-type short suspenders is influenced mainly by boundary constraints and flexibility of suspenders, leading to the inadaptability of typical tension string theory. It has been found that for steel rod-type short suspenders, there is a quadratic regression relationship between vibration frequency and tension force, controlling the error between experimental and calculated tension force within 5%.

Due to its nondestructive, fast, and noncontact characteristics, the use of LDV can effectively improve monitoring accuracy on the service state of short suspenders, and can also achieve lifetime detection with its low-cost and simple operation.