## 1 Introduction  
The arch bridge has become a widely used bridge-type because of its large span, short construction period, and beautiful appearance. As the most critical force transfer component of a cable structure, the designer will stiffen the hangers during the design process [1]. Approximately 25% of the construction cost is spent on the cables [2; 3; 4]. In addition, the cable is subjected to the cyclic action of service and environmental loads and is influenced by the vortex-induced vibration and wind-rain-induced vibration. Hence, it is easy to damage and even cause the bridge to collapse in severe cases [5; 6; 7]. Therefore, the identification of cable damage is of great significance to ensure the safe operation of this kind of bridge [7]. The damage of cable is often detected by local detection methods. These tests are mainly aimed at the cable's free length and anchorage zone, usually using visual inspection, ultrasonic inspection, magnetic flux leakage method, and so on [3]. Visual inspection is the most commonly used method in the annual inspection of bridges. This method is time-consuming and laborious, often exposing the inspectors to risks, and the results are highly dependent on the inspectors' experience [8]. The ultrasonic wave needs to arrange the acoustic emission probe in advance, and the equipment required by the magnetic flux leakage method is complex. The detection signal for the hanger with protective measures is weak [7]. In the field of structural damage identification and health monitoring, vibration-based monitoring provides a relatively low-cost method of inspection. Many researchers identify the damage of cable based on the vibration method, e.g., [PERSON] et al. [9] proposed using the discrete wavelet transform to calculate the acceleration dynamic response signal's total wavelet energy of the measuring point to identify the damage of the hanger. Curvaturemode is one of the sensitive dynamic identification indexes. Many researchers have studied the damage identification of the hanger based on the curvature mode [10; 11; 12]. However, the modal parameters of the structure are affected mainly by operational and environmental conditions, such as traffic, wind, and temperature [5]. Sometimes the changes caused by the damage may be completely masked by the regular changes caused by the environment, which may lead to misjudgment [10; 13]. Therefore, when using the vibration-based method for damage identification, the effects of environment and operating conditions must be separated from the damage. Many researchers are devoted to exploring the relationship between dynamic characteristics and environmental loads, which is undoubtedly difficult [14]. In addition, the change of cable force often indicates the damage of cable. The cable force is one of the most critical parameters of cable-stayed bridges for safety evaluation [15]. The cable damage includes surface damage and structural damage [16; 17; 18]. The surface damage consists of the aging and cracking of the cable protective layer, which can be detected by visual inspection; the structural damage includes broken wire, corrosion, anchor head shedding failure, etc. No matter what kind of structural damage it is, it will reduce the stiffness of the cable [19]. When the stiffness of the hanger decreases, according to the principle of structural mechanics, its tension force will be reduced. When the external load is constant, the reduced force will be shared by the tie-beam and other hangers, increasing the cable force of the adjacent hanger, thus resulting in the redistribution of the cable force. Therefore, the cable damage can be evaluated by measuring the change of cable force or cable stress [5; 20; 21; 22]. A structure's frequency is usually used to measure the cable force in engineering [23], but this method is not accurate in measuring the force of a short cable. The frequency change resulting from slight damage in cables is difficult to identify for long-span bridges [24].