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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 547416, 8 pages
Research on Longitudinal Vibration Characteristic of the Six-Cable-Driven Parallel Manipulator in FAST
Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China
Received 22 August 2013; Accepted 29 October 2013
Academic Editor: Yuanying Qiu
Copyright © 2013 Zhihua Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The first adjustable feed support system in FAST is a six-cable-driven parallel manipulator. Due to flexibility of the cables, the cable-driven parallel manipulator bears a concern of possible vibration caused by wind disturbance or internal force from the fine drive system. The purpose of this paper is to analyze vibration characteristic of the six-cable-driven parallel manipulator in FAST. The tension equilibrium equation of the six-cable-driven parallel manipulator is set up regarding the cables as catenaries. Then, vibration equation is established considering the longitudinal vibration of the cables. On this basis, the natural frequencies are depicted in figures since both analytical and numerical solutions are ineffective. Influence of the sags of the cables on the natural frequencies is discussed. It is shown that the sags of the cables will decrease the natural frequencies of the six-cable-driven parallel manipulator. Simplification to acquire the natural frequencies is proposed in this paper. The results justify effectiveness of the simplification to calculate the first-order natural frequencies. Distribution of the first-order natural frequencies in the required workspace is provided based on the simplification method. Finally, parameters optimization is implemented in terms of natural frequencies for building the six-cable-driven parallel manipulator in FAST.
In 1993, the large telescope was first proposed at the General Assembly of the International Union Radio Science. In 1994, Chinese astronomers carried out the conceptual design of the five-hundred-meter aperture spherical radio telescope (FAST) . Now, the project of FAST is in the process of construction and planned to be accomplished in 2016. There is no solid connection between the reflector and the feed cabin because of the large dimension of the telescope. Instead, the cable-driven parallel manipulator is designed to serve as the first adjustable feed support system.
With the advantages of simple configuration, high-load ability, large workspace, and high speed, the cable-driven parallel manipulator is widely used in applications of handling and assembly operations . With six suspension cables plus a latent cable of gravity, the six-cable-driven parallel manipulator is a kind of completely restrained positioning mechanisms (CRPMs) . Due to flexibility of the long span cable, the cable-driven parallel manipulator bears a concern of possible vibration under wind disturbance and internal force . This motivates the research on analyzing the vibration characteristic of the six-cable-driven parallel manipulator in this paper.
In recent years, problems of the cable-driven parallel manipulator such as motion planning, force distribution, kinematics and calibration, control, and dynamics have been studied by a few researchers [5–10]. The vibration problem of the cable-driven parallel manipulator has received considerably less attention. In cable-driven parallel manipulator systems, vibration of the cables can be longitudinal or transverse . Our previous paper  addressed the natural frequencies of the cable-driven parallel manipulator considering the longitudinal vibration of cables. Liu et al.  reported that the transverse vibration of cables can be observed in experiments when the manipulator is in a high-speed or high-acceleration motion. Diao and Ma [14, 15] analyzed both longitudinal and transverse vibration of cables and pointed out that the transverse vibration influences little the cable-driven parallel manipulator. However, previous studies mainly focus on small span cable-driven parallel manipulator. Further study is needed to analyze the vibration characteristic of large span cable-driven parallel manipulator.
The cable in the small span cable-driven parallel manipulator is often regarded as line. While the effects of sag must be taken into consideration in the large span cable-driven parallel manipulator. Researchers [16, 17] used catenary equations to describe the static shape of the cable under the influence of gravity. There are multiple cables in the cable-driven parallel manipulator, which makes it difficult to acquire the analytical solution of the catenaries. Kozak et al.  presented a numerical approach to compute the static displacement of a homogeneous cable. Yao et al. [19, 20] simplified the catenary to a parabola aiming at parameters optimization of the four-cable-driven parallel manipulator. Dallej et al.  studied the effects of sagging on cable lengths, end-effector positioning, and cable tensions. Effects of cable sagging on the workspace and stiffness of a cable-suspended parallel mechanism are investigated by Arsenault .
To investigate vibration of the cable-driven manipulator systems, researchers modeled cables as linear springs , nonlinear springs , and wave equation [25, 26]. The spring is a simplification of the wave equation in the condition of neglecting the mass of the cable. In order to acquire more precise vibration of the large span cable, the wave equation will be used to model cables of the six-cable-driven parallel manipulator.
This paper is organized as follows. In Section 2, the six-cable-driven parallel manipulator is described in detail. In Section 3, the tension equilibrium equation is established by describing the cables as catenaries. Then, the vibration equation is deduced by modeling the cables as wave equation in Section 4. Section 5 analyzes the vibration characteristic aiming at the natural frequencies of the six-cable-driven parallel manipulator. Finally, conclusions of this paper are given in Section 6.
As shown in Figure 1, the FAST is composed of a reflector and a feed support system (FSS). The FSS contains three mechanisms, which are arranged in series: a six-cable-driven parallel manipulator, an A-B rotator, and a Stewart platform. The six-cable-driven parallel manipulator acts as the preliminary adjustor to achieve the coarse positioning. The position error of the receiver in the cabin is further corrected by the fine drive system (the A-B rotator and the Stewart platform). Both the wind disturbance and the internal force from the fine drive system can result in the vibration of the preliminary adjustor. Therefore, research on the vibration characteristic of the six-cable-driven parallel manipulator is crucial and important.
As shown in Figure 2, the six-cable-driven parallel manipulator is composed of six towers, six cables, and the feed cabin. are the connected points of the cables and the towers and distributed symmetrically in a circle. are the connected points of the cables and the feed cabin and distributed symmetrically in a circle.
Two coordinates are set up in the six-cable-driven parallel manipulator. Let be the inertial frame, located at the center of the reflector’s bottom, with the -axis crossing the midpoint of bottom and the -axis along the opposite of the gravity. Let be the feed cabin frame, attached to the geometric center of the feed cabin, with the -axis crossing the midpoint of and with the -axis perpendicular to the feed cabin plane. Symbols used in this paper are listed in the nomenclature and have been described at the place of first occurrence.
The required workspace of FAST is a sphere crown with the radius of 159.9 m shown in Figure 2. The center of the sphere is concentric with the reflector and the maximum pose angle of the sphere crown is 40 degrees. The normal vector of the receiver needs to pass through the sphere center in the required workspace. According to the result of the cable force optimization , the natural pose angle of the six-cable-driven parallel manipulator should satisfy three-eighth of the measured pose angle.
3. Tension Equilibrium Equation
Since the cable span of the six-cable-driven parallel manipulator in FAST is more than 300 m, the sag of the cable must be taken into consideration. The catenary equation will be used to describe the static displacement of the cables under the influence of gravity. A cable of the six-cable-driven parallel manipulator in its vertical section is shown in Figure 3. and are the horizontal and vertical component of the cable’s tension, and are the horizontal and vertical length of the cable’s span, and is the length of the cable.
The catenary equation of the cable is as follows: By substituting the boundary conditions at and at , we can obtain where Then, the vertical component of the tension can be calculated as Then, the length of the cable can be expressed as The unit vector from to can be obtained: Define the unit vectors , , and as follows: As shown in Figure 4, we can get the unit vector of the slope of the cable as The static equilibrium equations of the six-cable-driven parallel manipulator can be expressed as Substituting (4) into (9), we can obtain the nonlinear equations containing 6 variables of as follows: We can work out the horizontal components of the tensions by solving (10) using numerical iteration method.
4. Vibration Equation
The longitudinal vibration of the cables contributes more to the vibration of the feed cabin, since the tensions of the cables are really great and the feed cabin moves slowly. Therefore, we focus on the longitudinal vibrations of the cables in this paper.
As shown in Figure 5, the governing equations of motion can be expressed as where is the deflection of the cable along its length and .
Using segregation variable method, the governing equations of motion can be expressed as
where , . Taking the boundary conditions on the motion of the cable into consideration, the governing equations of motion can be simplified as
Then, the tension and acceleration at the end of the cable with respect to the static state can be expressed, respectively, as follows: According to the Newton-Euler method, the dynamic equations of the six-cable-driven parallel manipulator can be obtained as where denotes the acceleration of the platform, represents the inertia matrix, and . The nonlinear coriolis and centripetal terms are neglected in the dynamic equation. The gravity term is eliminated by the static tensions of the cables. It should be pointed out that, due to the sags of the cables, the tensions of the cables are along but not along . Then, becomes As the deflection at the end of a cable is along , we can obtain the relation of the velocity of the platform and the velocity of the end point of the cables : Substituting (16) into (14) ignoring the nonlinear terms, we can obtain We are interested in the natural frequencies of the six-cable-driven parallel manipulator. Vibration frequencies of the platform and all the cables are the same in the cable-driven parallel manipulator system. For the sake of the natural frequencies, by substituting (13) into (17) concerning all the cables that vibrate at the same frequency, we can obtain In this paper, we assume that all cables are made of identical material and have the same area of cross section. Hence, density , Young’s modulus , area of cross section , and . Then, (18) can be simplified as follows:
5. Vibration Characteristic
5.1. Natural Frequency
According to (19), the natural frequency of the six-cable-driven parallel manipulator can be obtained by solving , where Figure 6 depicts the relationship between and of the six-cable-driven parallel manipulator at the center of the required workspace. The parameters of the six-cable-driven parallel manipulator are listed in Table 1. It can be seen that the natural frequencies are periodically distributed. In each natural frequency domain, there are six natural frequencies owing to 6 degrees of freedom of the six-cable-driven parallel manipulator.
According to Figure 4, the direction of the tensions and longitudinal vibration is closely related to the sags of the cables. Figure 7 depicts the difference between the catenary cables and line cables in the first-order natural frequencies. It is clear that the sags of the cables decrease the natural frequencies of the six-cable-driven parallel manipulator. The difference in the first-order natural frequencies is listed in Table 2. It can be seen that the difference between the linear cable and the catenary cable is larger as the natural frequency increases. The difference is quite great as the error ratio has exceeded 10%. The results show that cable sag has a great effect on the natural frequencies and ignoring the influence of the cable sag will yield relatively great errors.
The equations of are found to be multivariate nonlinear equations. No analytical method is available to solve these equations. The sine function and cosine function will result in periodic solutions. The numerical iteration method is also feeble as it is difficult to specify the initial values. Further simplification is needed to achieve the natural frequencies of the six-cable-driven parallel manipulator.
Generally, we are more concerned about the first-order natural frequencies. In this case, the value of becomes really small. Retaining only the linear term in the Taylor’s series of the sine function and the cosine function at the point of 0, (20) can be simplified as follows: The analytical solutions of can be expressed as follows: where is the eigenvalue function.
Figure 8 shows the simplified solutions of the first-order natural frequencies of the six-cable-driven parallel manipulator in the center of the required workspace. It is clear that the error greatly increases while becomes larger. Therefore, the simplification is only effective to obtain the first-order natural frequencies.
On the basis of the simplification, we are available to provide the distribution of the first-order natural frequencies in the required workspace shown in Figure 9. The distribution of the natural frequencies is symmetrical about the center since the mechanical structure is symmetrical. The minimal frequency in the first-order frequency domain ranges from 0.21 Hz to 0.32 Hz. The low natural frequencies are easily to be excited by wind disturbance.
5.3. Parameters Optimization
The aim of the optimization is to maximize the minimal natural frequency in case of being excited easily by wind disturbance. In this section, three parameters of the six-cable-driven parallel manipulator will be optimized: diameter of the cables, height of the towers, and the radius of . The influence of the parameters on the minimal frequency is shown in Figure 10. It illustrates distribution of the minimal natural frequency at XOZ section of the required workspace.
Figure 10(a) shows the influence of distributed radius of the towers on the minimal natural frequency. Larger distributed radius has larger natural frequency. When the distributed radius is larger than 300 m, the natural frequency decreases rapidly at the border of the required workspace. Therefore, the distributed radius is designed as 300 m.
Influence of tower height on the minimal natural frequency is shown in Figure 10(b). Larger tower height has larger natural frequency. When the tower height is larger than 300 m, the natural frequency decreases rapidly at the border of the required workspace. Therefore, the tower height is designed as 300 m.
Figure 10(c) depicts influence of diameter of the cable on the minimal frequency. Larger diameter of the cable has larger natural frequency at the center of the required workspace but not at the border of the required workspace. When the diameter of the cable is larger than 34 mm, the natural frequency decreases rapidly at the border of the required workspace. Therefore, the diameter of the cable is designed as 34 mm.
This paper presented the vibration analysis of the six-cable-driven parallel manipulator. The weight of the long span cable results in catenary. The natural frequencies were given considering the sag of the cables. Influence of the sag of the cables on the natural frequencies was further discussed. Simplification for the first-order natural frequencies was carried out. Distribution of the first-order natural frequencies in the required workspace is provided. Finally, based on the minimal natural frequency, parameters optimization of the six-cable-driven parallel manipulator was obtained. The study in this paper provides theoretical foundations for further vibration suppression of FAST.
|:||Height of towers|
|:||Diameter of the cables|
|:||Mass of the feed cabin|
|:||Mass center of the feed cabin in|
|:||Inertia matrix of the feed cabin in|
|:||Position of the feed cabin in|
|:||Rotation matrix of the feed cabin in|
|:||Coordinate of in|
|:||Coordinate of in|
|:||Density of the cable|
|:||Cross section of the cable|
|:||Young’s modulus of the cable .|
This research is sponsored by the National Natural Science Foundation of China (nos. 11178012, 51205224).
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