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Journal of Robotics
VolumeΒ 2011Β (2011), Article IDΒ 489695, 17 pages
http://dx.doi.org/10.1155/2011/489695
Research Article

Kineto-Elastodynamic Characteristics of the Six-Degree-of-Freedom Parallel Structure Seismic Simulator

Department of Mechatronics Engineering, Shantou University, Shantou City, Guangdong 515063, China

Received 19 January 2011; Revised 19 May 2011; Accepted 3 June 2011

Academic Editor: YangminΒ Li

Copyright Β© 2011 Yongjie Zhao. 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.

Abstract

Based on the kineto-elastodynamic assumptions, the dynamic model of the six-degree-of-freedom parallel structure seismic simulator is developed by virtue of the finite element method and the substructure synthesis technique. The kineto-elastodynamic characteristics represented by the natural frequency, the sensitivity analysis, the energy ratios, and the displacement response of the moving platform are investigated. It is shown that the second-order natural frequency is much higher than the first-order natural frequency, and the first-order natural frequency is sensitive to the radius of the strut and the radius of the lead screw. In order to improve the dynamic characteristic of the manipulator, the mass of the moving platform should be reduced or the stiffness of the strut should be increased especially for the sixth strut. For the investigated trajectory, the displacement response of the moving platform along the π‘₯ direction is smaller than these displacement responses along the 𝑦 direction and along the 𝑧 direction. The angular displacement response of the moving platform rotating about 𝑧-axis is slightly larger than those angular displacement responses rotating about the π‘₯-axis and about the 𝑦-axis.

1. Introduction

A seismic simulator is one of the most important equipments in the earthquake resistance testing.Due to the requirement of the large and variable load capability, these kinds of equipments are usually developed with the parallel structure manipulators [1–4]. The parallel manipulator is a closed-loop kinematic chain mechanism whose end effector is linked to the base by several independent kinematic chains [5–7]. For this type of manipulators, there are some potential advantages such as high accuracy, rigidity, and speed. They have been successfully used in the motion simulators, robotic end effectors, and other circumstances like fast pick-and-place operation. Many investigations have been carried out on the parallel manipulators since the concept was introduced. However, there is not many works on the flexible dynamics of the parallel manipulator [8, 9] compared with the vast of papers on the kinematics and rigid dynamics due to the following facts: (i) computational cost; (ii) geometrical complexity; (iii) unidentified mechanics property. For the 6-PSS (prismatic-spherical-spherical joint) flexible parallel manipulator under consideration in this paper, which is developed for the six-degree-of-freedom seismic simulator, the dynamics considering the structure flexibility is fundamental for the modeling, design, and control.

The demands of high speed, high load, high precision, or lightweight structure from industry make it necessary to consider the deformation, stiffness, and other dynamic characteristics for the parallel manipulator [10–23]. Mathematical modeling of a general flexible parallel manipulator is a challenging task since there is no availability of closed-form solutions to the inverse kinematic model for the flexible parallel manipulator. The nominal motion of the manipulator involves changing geometries resulting in varying system parameters. The equations of motion are usually configuration dependent and need to be computed at each configuration of the manipulator [10]. The equations of motion of a flexible five-bar manipulator were developed by means of the instantaneous structural approach, and it had been found that the mode shapes and natural frequencies of this particular manipulator are invariant throughout most of the workspace [11]. The design, dynamic modeling, and experiment validation of a three-degree-of-freedom flexible arm were presented in [12] on the assumption that all the arm mass is concentrated at the tip and at the base. So the dynamic of the arm becomes a lumped single mass model instead of the usual distributed mass model. The finite element method and the Euler-Lagrange formulation were used in [13] to model the flexible link of a three-degree-of-freedom parallel manipulator by assuming that the influence of flexible motion on rigid motion is negligible. With the piston being modeled as a mass-spring damper, a set of twelve Lagrange equations for flexible Stewart manipulator was derived by using tensor representation in [14]. The dynamic model of the 3-PRR planar parallel manipulator with flexible links was formulated by using the Lagrange equations of the first type on the assumption that the intermediate links being modeled with pinned-free boundary conditions [15]. The Lagrange finite element formulation was used to derive such a dynamic model for the flexible planar linkage with two translational and one rotational degrees of freedom, and then the dynamic model was applied to the flexible link planar parallel manipulator based on standard Kineto-elastodynamic assumptions [16]. Based on the model, strain rate feedback control using PZT transducers was used to simulate the active control of Kineto-elastodynamic responses. The dynamic finite element analysis of the flexible planar parallel manipulator was presented in [17] including the convergence analysis of the natural frequencies and the mapping of the first-order natural frequency with respect to the robot configuration. It had also been found that the geometric stiffness and the dynamic terms have a negligible effect on the response for this particular manipulator. A substructure modeling procedure was presented to develop the dynamic model for the flexible planar parallel manipulator in [18]. The Craig-Bampton method was used to reduce the model order and assemble the complete dynamic model. On the assumption that the deformations of the intermediate links are small relative to the length of the links, a procedure for the development of structural dynamic model for the 3-PRR flexible parallel manipulator was presented in [19] based on the assumed mode method. Without considering the effect of nominal motion, reference [20] provided the stationary vibration model of the sliding-leg parallel kinematic machine where the links were modeled as finite elements and the joint as virtual spring/dampers. Then, the nonstationary model of the same mechanism was developed with the elastodynamic method [21]. In the researches cited above, there is little investigation on the Kineto-elastodynamic characteristics of the six-degree-of-freedom parallel manipulator while considering the natural frequency, the sensitivity analysis, the energy ratios, and the displacement response.

This paper presents the Kineto-elastodynamic modeling and the Kineto-elastodynamic characteristics analysis of the 6-PSS parallel structure seismic simulator. It is organized as follows: in Section 2, the description of the seismic simulator and the rigid dynamic equations are presented. Section 3 gives the Kineto-elastodynamic model of the manipulator developed by virtue of the finite element method and the substructure synthesis technique. Section 4 investigates the Kineto-elastodynamic characteristics represented by the natural frequency, the sensitivity analysis, the energy ratios, and the displacement response of the moving platform through simulation. Section 5 gives the conclusions.

2. System Description and Rigid Dynamics

2.1. Description

The schematic diagram of the 6-dof parallel structure seismic simulator is shown in Figure 1. As shown in Figure 1, the parallel manipulator is composed of a moving platform and six sliders. In each kinematic chain, the platform and the slider are connected via spherical ball-bearing joints by a strut of fixed length. Each slider is driven by DC motor via a linear ball screw. The lead screws of 𝐡1,𝐡2, and 𝐡3 are vertical to the ground.

489695.fig.001
Figure 1: Schematic diagram of the 6-dof parallel structure seismic simulator.

For the purpose of analysis, the following coordinate systems are defined. As shown in Figure 2, the coordinate system π‘‚βˆ’π‘₯𝑦𝑧 is attached to the fixed base; another moving coordinate frame π‘‚β€²βˆ’π‘’π‘£π‘€ is located at the center of mass of the moving platform. The pose of the moving platform can be described by a position vector 𝐫 and a rotation matrixπ‘œπ‘π‘œξ…ž. Let the rotation matrix be defined by the roll, pitch, and yaw angles, namely, a rotation of πœ™π‘₯ about the fixed π‘₯ axis, followed by a rotation of πœ™π‘¦ about the fixed 𝑦 axis, and a rotation of πœ™π‘§ about the fixed 𝑧 axis. Thus, the rotation matrix isπ‘œπ‘π‘œξ…žξ€·=Rot𝑧,πœ™π‘§ξ€Έξ€·Rot𝑦,πœ™π‘¦ξ€Έξ€·Rotπ‘₯,πœ™π‘₯ξ€Έ,(1) where sπœ™ denotes the sine of angle πœ™, and cπœ™ denotes the cosine of angle πœ™. In the hypothesis of small rotations, the angular velocity of the moving platform is given by [24, 25]ξ‚ƒΜ‡πœ™πŽ=π‘₯Μ‡πœ™π‘¦Μ‡πœ™π‘§ξ‚„π‘‡.(2)

489695.fig.002
Figure 2: Vector diagram of a PSS kinematic.

The orientation of each kinematic strut with respect to the fixed base can be described by two Euler angles. As shown in Figure 3, the local coordinate system of the ith strut can be thought of as a rotation of πœ™π‘– about the 𝑧 axis resulting in a πΆπ‘–βˆ’π‘₯ξ…žπ‘–π‘¦ξ…žπ‘–π‘§ξ…žπ‘– system followed by another rotation of πœ‘π‘– about the rotated π‘¦ξ…žπ‘–-axis. So the rotation matrix of the 𝑖th strut can be written asπ‘œπ‘π‘–ξ€·=Rot𝑧,πœ™π‘–ξ€Έξ€·π‘¦Rotξ…žπ‘–,πœ‘π‘–ξ€Έ=⎑⎒⎒⎒⎒⎣cπœ™π‘–cπœ‘π‘–βˆ’sπœ™π‘–cπœ™π‘–sπœ‘π‘–sπœ™π‘–cπœ‘π‘–cπœ™π‘–sπœ™π‘–sπœ‘π‘–βˆ’sπœ‘π‘–0cπœ‘π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝑖=1,2,…,6.(3) The unit vector along the strut in the coordinate system π‘‚βˆ’π‘₯𝑦𝑧 is𝐰𝑖=π‘œπ‘π‘–π‘–π°π‘–=π‘œπ‘π‘–βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£001⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎒⎣cπœ™π‘–sπœ‘π‘–sπœ™π‘–sπœ‘π‘–cπœ‘π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦.(4) So the Euler angles πœ™π‘– and πœ‘π‘– can be computed as follows:cπœ‘π‘–=𝑀𝑖𝑧,sπœ‘π‘–=𝑀2𝑖π‘₯+𝑀2𝑖𝑦,ξ€·0β‰€πœ‘π‘–ξ€Έ,<πœ‹sπœ™π‘–=𝑀𝑖𝑦sπœ‘π‘–,ξ€·πœ‘π‘–ξ€Έ,β‰ 0cπœ™π‘–=𝑀𝑖π‘₯sπœ‘π‘–,ξ€·πœ‘π‘–ξ€Έ,β‰ 0ifπœ‘π‘–=0,thenπœ™π‘–=0.(5)

489695.fig.003
Figure 3: The local coordinate system of the 𝑖th strut.
2.2. Rigid Dynamics

When the seismic simulator is not at a singular configuration, the rigid dynamic model can be formulated by means of the principle of virtual work and the concept link Jacobian matrices [25]. It can be expressed as𝐅=βˆ’π‰βˆ’π‘‡βŽ‘βŽ’βŽ’βŽ£πŸπ‘’π§π‘’βŽ€βŽ₯βŽ₯βŽ¦βˆ’π‰βˆ’π‘‡βŽ§βŽͺ⎨βŽͺβŽ©βŽ‘βŽ’βŽ’βŽ£π‘šπ‘π πŸŽβŽ€βŽ₯βŽ₯⎦+6𝑖=1π‰π‘‡π‘–π‘£πœ”βŽ‘βŽ’βŽ’βŽ£π‘šπ‘–π‘–π‘π‘œπ πŸŽβŽ€βŽ₯βŽ₯⎦+𝐉𝑇(π‘šπ‘1𝐠)π‘‡πž1ξ€·π‘šπ‘2π ξ€Έπ‘‡πž2ξ€·π‘šπ‘3π ξ€Έπ‘‡πž3ξ€·π‘šπ‘4π ξ€Έπ‘‡πž4ξ€·π‘šπ‘5π ξ€Έπ‘‡πž5ξ€·π‘šπ‘6π ξ€Έπ‘‡πž6𝑇+π‰βˆ’π‘‡βŽ§βŽͺ⎨βŽͺβŽ©βŽ‘βŽ’βŽ’βŽ£π‘šπ‘Μ‡π―π‘œπˆπ‘Μ‡πŽβŽ€βŽ₯βŽ₯⎦+6𝑖=1π‰π‘‡π‘–π‘£πœ”βŽ‘βŽ’βŽ’βŽ£π‘šπ‘–π‘–Μ‡π―π‘–π‘–πˆπ‘–π‘–Μ‡πŽπ‘–βŽ€βŽ₯βŽ₯⎦+π‰π‘‡ξ‚ƒπ‘šπ‘1Μˆπ‘ž1π‘šπ‘2Μˆπ‘ž2π‘šπ‘3Μˆπ‘ž3π‘šπ‘4Μˆπ‘ž4π‘šπ‘5Μˆπ‘ž5π‘šπ‘6Μˆπ‘ž6ξ‚„π‘‡βŽ«βŽͺ⎬βŽͺ⎭+π‰βˆ’π‘‡βŽ§βŽͺ⎨βŽͺβŽ©βŽ‘βŽ’βŽ’βŽ£πŸŽξ€·πŽΓ—π‘œπˆπ‘πŽξ€ΈβŽ€βŽ₯βŽ₯⎦+6𝑖=1π‰π‘‡π‘–π‘£πœ”βŽ‘βŽ’βŽ’βŽ£πŸŽπ‘–πŽπ‘–Γ—ξ€·π‘–πˆπ‘–π‘–πŽπ‘–ξ€ΈβŽ€βŽ₯βŽ₯⎦⎫βŽͺ⎬βŽͺ⎭,(6)where ξ‚ƒξ‚„πŸŽ=000𝑇,π‰π‘–π‘£πœ” is the link Jacobian matrix which maps the velocity of the moving platform into the velocity of the 𝑖th strut in the πΆπ‘–βˆ’π‘₯𝑖𝑦𝑖𝑧𝑖 coordinate system, π‘šπ‘, π‘šπ‘π‘–, and π‘šπ‘– denote the mass of the moving platform, the mass of the slider, and the mass of the 𝑖th strut, respectively, π‘œπˆπ‘ is the inertia matrix of the moving platform taken about the center of mass expressed in the π‘‚βˆ’π‘₯𝑦𝑧 coordinate system, π‘–πˆπ‘– is the inertia matrix of the 𝑖th cylindrical strut about their respective centers of mass expressed in the πΆπ‘–βˆ’π‘₯𝑖𝑦𝑖𝑧𝑖 coordinate system, πŸπ‘’ and 𝐧𝑒 are the external force and moment exerted at the center of mass of the moving platform, ̇𝐯 and Μ‡πŽ are the linear and angular acceleration of the moving platform,𝑖̇𝐯𝑖,π‘–πŽπ‘–, and π‘–Μ‡πŽπ‘– are the linear acceleration, the angular velocity, and acceleration of the ith strut expressed in the πΆπ‘–βˆ’π‘₯𝑖𝑦𝑖𝑧𝑖 coordinate system, respectively. Μˆπ‘žπ‘– denotes the joint acceleration, 𝐠 is the gravity acceleration, 𝐉 is the Jacobian matrix which maps the velocity vector of the moving platform into the velocity vector of the actuating joint. 𝐅 is the input force vector exerted at the center of the slider. Andξ‚΅1𝐉=diag𝐰𝑇1𝐞11𝐰𝑇2𝐞21𝐰𝑇3𝐞31𝐰𝑇4𝐞41𝐰𝑇5𝐞51𝐰𝑇6𝐞6ξ‚ΆΓ—βŽ‘βŽ’βŽ’βŽ£π°1𝐰2𝐰3𝐰4𝐰5𝐰6𝐚1×𝐰1𝐚2×𝐰2𝐚3×𝐰3𝐚4×𝐰4𝐚5×𝐰5𝐚6×𝐰6⎀βŽ₯βŽ₯βŽ¦π‘‡,π‰π‘–π‘£πœ”=βŽ‘βŽ’βŽ’βŽ’βŽ£ξ‚ƒπ‘–π‘π‘œξ€·βˆ’π‘†π‘–πšπ‘–ξ€Έπ‘–π‘π‘œξ‚„+𝑙𝑖2π‘†ξ€·π‘–π°π‘–ξ€Έπ‰π‘–πœ”1π‘™π‘–ξƒ―ξ‚ƒπ‘†ξ€·π‘–π°π‘–ξ€Έπ‘–π‘π‘œξ€·βˆ’π‘†π‘–π°π‘–ξ€Έπ‘†ξ€·π‘–πšπ‘–ξ€Έπ‘–π‘π‘œξ‚„βˆ’ξ€·π‘–π°π‘–Γ—π‘–πžπ‘–ξ€Έξƒ¬π°π‘‡π‘–π°π‘‡π‘–πžπ‘–ξ€·πšπ‘–Γ—π°π‘–ξ€Έπ‘‡π°π‘‡π‘–πžπ‘–βŽ€βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£π‰ξƒ­ξƒ°π‘–π‘£π‰π‘–πœ”βŽ€βŽ₯βŽ₯⎦,𝑆𝑖𝐰𝑖=⎑⎒⎒⎒⎒⎣0βˆ’π‘–π‘€π‘–π‘–π‘§π‘€π‘–π‘–π‘¦π‘€π‘–π‘§0βˆ’π‘–π‘€π‘–π‘₯βˆ’π‘–π‘€π‘–π‘–π‘¦π‘€π‘–π‘₯0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘†ξ€·π‘–πšπ‘–ξ€Έ=⎑⎒⎒⎒⎒⎣0βˆ’π‘–π‘Žπ‘–π‘–π‘§π‘Žπ‘–π‘–π‘¦π‘Žπ‘–π‘§0βˆ’π‘–π‘Žπ‘–π‘₯βˆ’π‘–π‘Žπ‘–π‘–π‘¦π‘Žπ‘–π‘₯0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,π‘–πŽπ‘–=π‰π‘–πœ”βŽ‘βŽ’βŽ’βŽ£π―πŽβŽ€βŽ₯βŽ₯⎦,Μˆπ‘žπ‘–=1π°π‘‡π‘–πžπ‘–ξ‚€π°π‘‡π‘–Μ‡ξ€·πšπ―+π‘–Γ—π°π‘–ξ€Έπ‘‡Μ‡πŽ+π°π‘‡π‘–ξ€·ξ€·πŽΓ—πŽΓ—πšπ‘–ξ€Έξ€Έβˆ’π°π‘‡π‘–ξ€·πŽπ‘–Γ—ξ€·πŽπ‘–Γ—π‘™π‘–π°π‘–ξ‚,ξ€Έξ€Έπ‘–Μ‡πŽπ‘–=π‰π‘–πœ”βŽ‘βŽ’βŽ’βŽ£Μ‡π―Μ‡πŽβŽ€βŽ₯βŽ₯⎦+1π‘™π‘–ξ€·πš«1+𝚫2ξ€Έ,𝚫1ξ€·=βˆ’π‘–π°π‘–Γ—π‘–πžπ‘–ξ€Έπ°π‘‡π‘–πžπ‘–ξ‚€ξ€·π°π‘‡π‘–πŽπšξ€Έξ€·π‘‡π‘–πŽξ€Έβˆ’ξ€·π°π‘‡π‘–πšπ‘–πŽξ€Έξ€·π‘‡πŽξ€Έ+𝑙𝑖||πŽπ‘–Γ—π°π‘–||2,𝚫2=ξ€·π‘–πŽπ‘‡π‘–π‘–πšπ‘–ξ€Έξ€·π‘–π°π‘–Γ—π‘–πŽπ‘–ξ€Έβˆ’ξ€·π‘–πŽπ‘‡π‘–πŽξ€Έξ€·π‘–π°π‘–Γ—π‘–πšπ‘–ξ€Έ,𝑖̇𝐯𝑖=π‰π‘–π‘£βŽ‘βŽ’βŽ’βŽ£Μ‡π―Μ‡πŽβŽ€βŽ₯βŽ₯βŽ¦ξ€·+π‘†π‘–πŽπ‘–ξ€Έπ‘†ξ€·π‘–πŽπ‘–ξ€Έπ‘–πšπ‘–+12π‘†ξ€·π‘–π°π‘–πš«ξ€Έξ€·1+𝚫2ξ€Έβˆ’π‘™π‘–2π‘†ξ€·π‘–πŽπ‘–ξ€Έπ‘†ξ€·π‘–πŽπ‘–ξ€Έπ‘–π°π‘–.(7)πžπ‘–, πšπ‘–, and 𝐰𝑖 are shown in Figure 2; they are the unit vector along the lead screw, the vector πŽξ…žπ€π’, and the unit vector along strut 𝐢𝑖𝐴𝑖, respectively.

3. Kineto-Elastodynamic Model

The idea of substructure synthesis and the finite element method are adopted to develop the Kineto-elastodynamic model of the 6-PSS parallel structure seismic simulator. The finite element method used here is based on the basic assumptions [26] as follows. (1) The deflections of the links of the manipulator obey the small deflection theory. The small amplitude structural vibrations do not have a significant effect on its rigid-body motion and the coupling term between the elastic deformation and the rigid-body motion is neglected. The true motion is regarded as the sum of the rigid-body motion, and the elastic motion. (2) The instantaneous structural approach is adopted. At each instant, the manipulator is modeled as an instantaneous structure undergoing elastic deformations about its mean rigid body configuration. (3) The model is based on the Euler-Bernoulli beam theory. (4) The transverse deflections are modeled as a cubic polynomial of the nodal displacement, the longitudinal deflections and the torsional deflections are modeled as a first-order polynomial of the nodal displacement. The manipulator is divided into seven substructures, namely, one moving platform substructure and six kinematic chain substructures which are composed of the lead-screw assembly and the strut. Each strut is divided into three elements. The moving platform and the sliders are regarded as the rigid bodies since their deformations are small relative to the elastic deformations.

3.1. Strut Dynamic Equation
3.1.1. Element Model

The nodal elastic displacement of the element is shown in Figure 4. So the elastic displacement of the element can be expressed as ξ‚ƒπœΉπœΉ=π‘–πœΉπ‘—ξ‚„π‘‡,(8) whereπœΉπ‘–=ξ‚ƒπ‘’π‘–π‘£π‘–π‘€π‘–πœƒπ‘–π‘₯πœƒπ‘–π‘¦πœƒπ‘–π‘§ξ‚„π‘‡,πœΉπ‘—=ξ‚ƒπ‘’π‘—π‘£π‘—π‘€π‘—πœƒπ‘—π‘₯πœƒπ‘—π‘¦πœƒπ‘—π‘§ξ‚„π‘‡.(9)

489695.fig.004
Figure 4: Nodal elastic displacement of the element.

The elastic displacement vector of an arbitrary point 𝑃 within the element can be expressed by the nodal displacement of the element [27]𝐩=π‘’π‘£π‘€πœƒπ‘‡=𝐍𝜹,(10) where 𝐍 is relational matrix which maps the nodal elastic displacement vector of the element into that of the point 𝑃.

The polynomial of the nodal displacement of the element is chosen to formulate the displacement of the point 𝑃. The transverse displacement, the longitudinal displacement, and the torsional displacement of the point 𝑃 are modeled as a cubic polynomial and a linear function of the nodal displacement, respectively. So the longitudinal displacement and the torsional displacement of the point 𝑃 are expressed as𝑒=π‘Ž0+π‘Ž1π‘₯,πœƒ=𝑑0+𝑑1π‘₯.(11) The transverse displacements of the point 𝑃 can be expressed as 𝑣=𝑏0+𝑏1π‘₯+𝑏2π‘₯2+𝑏3π‘₯3,𝑀=𝑐0+𝑐1π‘₯+𝑐2π‘₯2+𝑐3π‘₯3.(12) Substituting (11) and (12) into (10) yieldsβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘’π‘£π‘€πœƒβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‡π‘=𝑒𝐇(π‘₯)πšπ‘£(𝐇π‘₯)𝐛𝑀𝐇(π‘₯)πœπœƒβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(π‘₯)𝐝,(13) whereξ‚ƒπ‘Žπš=0π‘Ž1𝑇,𝑏𝐛=0𝑏1𝑏2𝑏3𝑇,ξ‚ƒπ‘πœ=0𝑐1𝑐2𝑐3𝑇,𝑑𝐝=0𝑑1𝑇,𝐇𝑒(π‘₯)=π‡πœƒ(,𝐇π‘₯)=1π‘₯𝑣(π‘₯)=𝐇𝑀(π‘₯)=1π‘₯π‘₯2π‘₯3ξ‚„.(14)

Considering the node 𝑖 and the node 𝑗, where π‘₯=0 and π‘₯=𝑙, respectively, yields𝐚=π€βˆ’1π‘’πœƒπœΉπ‘’,𝐛=π€βˆ’1π‘£π‘€πœΉπ‘£,𝐜=π€βˆ’1π‘£π‘€πœΉπ‘€,𝐝=π€βˆ’1π‘’πœƒπœΉπœƒ,(15) whereπœΉπ‘’=𝑒𝑖𝑒𝑗𝑇,πœΉπ‘£=ξ‚ƒπ‘£π‘–πœƒπ‘–π‘§π‘£π‘—πœƒπ‘—π‘§ξ‚„π‘‡,πœΉπ‘€=ξ‚ƒπ‘€π‘–πœƒπ‘–π‘¦π‘€π‘—πœƒπ‘—π‘¦ξ‚„π‘‡,πœΉπœƒ=ξ‚ƒπœƒπ‘–π‘₯πœƒπ‘—π‘₯𝑇,π€βˆ’1π‘’πœƒ=βŽ‘βŽ’βŽ’βŽ£βˆ’110𝑙1π‘™βŽ€βŽ₯βŽ₯⎦,π€βˆ’1𝑣𝑀=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’310000100𝑙2βˆ’2𝑙3𝑙2βˆ’1𝑙2𝑙31𝑙2βˆ’2𝑙31𝑙2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(16) Substituting (15) into (13) yields βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘’π‘£π‘€πœƒβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎒⎒⎒⎣𝐑𝐩=𝑒𝐑(π‘₯)𝑣(𝐑π‘₯)𝑀𝐑(π‘₯)πœƒβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(π‘₯)π€πœΉ=𝐍𝜹,(17) where𝐑𝑒,𝐑(π‘₯)=100000π‘₯00000𝑣(π‘₯)=01000π‘₯0π‘₯2000π‘₯3ξ‚„,𝐑𝑀(π‘₯)=0010π‘₯000π‘₯20π‘₯30ξ‚„,π‘πœƒξ‚ƒξ‚„,βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1(π‘₯)=000100000π‘₯00𝐀=100000000000010000000000001000000000000100000000000010000000000001000000𝑙100000𝑙3000000βˆ’π‘™22000βˆ’π‘™03𝑙21000βˆ’π‘™300βˆ’π‘™220βˆ’π‘™3000𝑙210βˆ’π‘™01000βˆ’π‘™100000𝑙20000𝑙301𝑙22000βˆ’π‘™301𝑙2002𝑙31000𝑙220βˆ’π‘™31000𝑙2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(18)

Considering the knowledge of material mechanics, the strain of the point 𝑃 is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœ€πœΊ=0πœ€π‘π‘¦πœ€π‘π‘§πœ€π‘ŸβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘‘π‘’π‘‘π‘‘π‘₯βˆ’π‘¦2𝑣𝑑π‘₯2π‘‘βˆ’π‘§2𝑀𝑑π‘₯2𝐺𝐸16π½π‘˜πœ‹π‘‘3π‘‘πœƒβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‘π‘₯,(19) where πœ€0 is the axial strain, πœ€π‘π‘¦ and πœ€π‘π‘§ are the flexural strain in the plane 𝑂π‘₯𝑦 and 𝑂π‘₯𝑧, respectively, πœ€π‘Ÿ is the torsional strain, 𝑦 and 𝑧 are the distance along the 𝐲 and 𝐳 direction from the axis of the element to the point 𝑃, 𝐺 is the torsional modulus, 𝐸 is the Young’s modulus, π½π‘˜ is the polar moment of inertia of cross-section, 𝑑 is the diameter of the strut.

Substituting (17) into (19) yieldsβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£β„ŽπœΊ=ξ…žπ‘’(π‘₯)βˆ’π‘¦β„Žπ‘£ξ…žξ…ž(π‘₯)βˆ’π‘§β„Žπ‘€ξ…žξ…žπΊ(π‘₯)𝐸16π½π‘˜πœ‹π‘‘3β„Žξ…žπœƒβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(π‘₯)π€πœΉ=𝐁𝜹,(20) whereβ„Žξ…žπ‘’ξ‚ƒξ‚„,β„Ž(π‘₯)=000000100000π‘£ξ…žξ…žξ‚ƒξ‚„,β„Ž(π‘₯)=000000020006π‘₯π‘€ξ…žξ…žξ‚ƒξ‚„,β„Ž(π‘₯)=00000000206π‘₯0ξ…žπœƒξ‚ƒξ‚„.(π‘₯)=000000000100(21) So the stress is 𝝈=𝐸𝜺.(22)

According to the knowledge of material mechanics, the strain energy of the element can be expressed as1π‘ˆ=2ξ€žπœΊπ‘‡πˆπ‘‘π‘£.(23) Substituting (20) and (22) into (23) yields 1π‘ˆ=2ξ€žπΈπœΉπ‘‡ππ‘‡1ππœΉπ‘‘π‘£=2πœΉπ‘‡πΈξ€žππ‘‡1ππ‘‘π‘£πœΉ=2πœΉπ‘‡π€πœΉ,(24) where1𝐀=2πΈξ€žππ‘‡ππ‘‘π‘£(25) is the element stiffness matrix. Substituting (20) into (25) yields⎑⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎒⎣𝐀=𝐸𝐴𝑙00000βˆ’πΈπ΄π‘™00000012𝐸𝐼𝑧𝑙30006𝐸𝐼𝑧𝑙20βˆ’12𝐸𝐼𝑧𝑙30006𝐸𝐼𝑧𝑙20012𝐸𝐼𝑦𝑙30βˆ’6𝐸𝐼𝑦𝑙2000βˆ’12𝐸𝐼𝑦𝑙30βˆ’6𝐸𝐼𝑦𝑙20000πΊπ½π‘˜π‘™00000βˆ’πΊπ½π‘˜π‘™0000βˆ’6𝐸𝐼𝑦𝑙204𝐸𝐼𝑦𝑙0006𝐸𝐼𝑦𝑙202𝐸𝐼𝑦𝑙006𝐸𝐼𝑧𝑙20004𝐸𝐼𝑧𝑙0βˆ’6𝐸𝐼𝑧𝑙20002πΈπΌπ‘§π‘™βˆ’πΈπ΄π‘™00000𝐸𝐴𝑙000000βˆ’12𝐸𝐼𝑧𝑙3000βˆ’6𝐸𝐼𝑧𝑙2012𝐸𝐼𝑧𝑙3000βˆ’6𝐸𝐼𝑧𝑙200βˆ’12𝐸𝐼𝑦𝑙306𝐸𝐼𝑦𝑙200012𝐸𝐼𝑦𝑙306𝐸𝐼𝑦𝑙20000βˆ’πΊπ½π‘˜π‘™00000πΊπ½π‘˜π‘™0000βˆ’6𝐸𝐼𝑦𝑙202𝐸𝐼𝑦𝑙0006𝐸𝐼𝑦𝑙204𝐸𝐼𝑦𝑙006𝐸𝐼𝑧𝑙20002𝐸𝐼𝑧𝑙0βˆ’6𝐸𝐼𝑧𝑙20004πΈπΌπ‘§π‘™βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,(26)where 𝐼𝑦 and 𝐼𝑧 are the principal moments of inertia corresponding to the 𝐲 axis and 𝐳 axis, respectively, 𝐴 is the area of the cross-section of the uniform beam.

Based on the presented assumption, there are kinematics relationships as follows:Μ‡π©π‘Ž=Μ‡π©π‘Ÿ+Μ‡Μˆπ©π©,π‘Ž=Μˆπ©π‘Ÿ+̈𝐩,(27) where Μ‡π©π‘Ž and Μˆπ©π‘Ž are the absolute velocity and the absolute acceleration of a certain point within the element, Μ‡π©π‘Ÿ and Μˆπ©π‘Ÿ are the velocity and the acceleration of the rigid-body motion, and ̇𝐩 and ̈𝐩 are the velocity and the acceleration of the elastic motion.

It can be proved that (see [27])Μ‡π©π‘Ÿ=ππ‘ŸΜ‡πœΉπ‘ŸΜ‡πœΉ=ππ‘Ÿ,Μˆπ©π‘Ÿ=ππ‘ŸΜˆπœΉπ‘ŸΜˆπœΉ=ππ‘Ÿ,(28) where Μ‡πœΉπ‘Ÿ and ΜˆπœΉπ‘Ÿ are the nodal velocity and the acceleration of the rigid-body motion.

So the kinetic energy of the element can be expressed as 1𝑇=2ξ€œπ‘™0Μ‡π©π‘š(π‘₯)π‘‡π‘ŽΜ‡π©π‘Ž=1𝑑π‘₯2ξ€œπ‘™0π‘šΜ‡πœΉ(π‘₯)π‘‡π‘Žππ‘‡πΜ‡πœΉπ‘Ž=1𝑑π‘₯2Μ‡πœΉπ‘‡π‘Žξ€œπ‘™0π‘š(π‘₯)ππ‘‡Μ‡πœΉππ‘‘π‘₯π‘Ž=12Μ‡πœΉπ‘‡π‘Žπ¦Μ‡πœΉπ‘Ž,(29) where Μ‡πœΉπ‘Ž is the absolute velocity of the node. And 1𝐦=2ξ€œπ‘™0π‘š(π‘₯)𝐍𝑇𝐍𝑑π‘₯(30) is the element mass matrix. For the uniform beam, the mass function isπ‘š(π‘₯)=𝜌𝐴,(31) where 𝜌 is the density of the beam. So the element mass matrix is𝐦=πœŒπ΄π‘™βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£420140000007000000015600022𝑙054000βˆ’13𝑙001560βˆ’22𝑙00054013𝑙0000140π½π‘˜π΄0000070π½π‘˜π΄0000βˆ’22𝑙04𝑙2000βˆ’13𝑙0βˆ’3𝑙20022𝑙0004𝑙2013𝑙000βˆ’3𝑙270000001400000005400013𝑙0156000βˆ’22𝑙00540βˆ’13𝑙000156022𝑙000070π½π‘˜π΄00000140π½π‘˜π΄000013𝑙0βˆ’3𝑙200022𝑙04𝑙200βˆ’13𝑙000βˆ’3𝑙20βˆ’22𝑙0004𝑙2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(32)

Substituting (23) and (29) into Lagrange equation yields π‘‘ξ‚΅π‘‘π‘‘πœ•π‘‡πœ•Μ‡πœΉξ‚Άβˆ’πœ•π‘‡+πœ•πœΉπœ•π‘ˆπœ•πœΉ=𝐟,(33) where 𝐟 denotes the resultant of the applied and the internal forces exerted at the element. Thus, the kinematic differential equation of the strut element within 𝑖th kinematic chain can be achieved as follows:𝐦̈̈𝜹𝜹+𝐀𝜹=πŸβˆ’π¦π‘Ÿ.(34) Equation (34) can be expressed in the coordinate system π‘‚βˆ’π‘₯𝑦𝑧 asπŒπ‘’π‘–Μˆπ”π‘’π‘–+πŠπ‘’π‘–π”π‘’π‘–=π…π‘’π‘–βˆ’πŒπ‘’π‘–Μˆπ”π‘’π‘Ÿπ‘–,𝑖=1,2,…,6(35) where𝐔𝑒𝑖=π‘π‘‡π‘–Μˆπ”πœΉ,𝑒𝑖=π‘π‘‡π‘–ΜˆΜˆπ”πœΉ,π‘’π‘Ÿπ‘–=π‘π‘‡π‘–ΜˆπœΉπ‘Ÿ,𝐅𝑒𝑖=π‘π‘‡π‘–πŒπŸ,𝑒𝑖=𝐑𝑇𝑖𝐦𝐑𝑖,πŠπ‘’π‘–=𝐑𝑇𝑖𝐀𝐑𝑖,𝐑𝑖=diagπ‘œπ‘π‘–ξ‚€πœ‹π‘π¨π­π‘¦,βˆ’2ξ‚ξ‚π‘‡ξ‚€π‘œπ‘π‘–ξ‚€πœ‹π‘π¨π­π‘¦,βˆ’2ξ‚ξ‚π‘‡ξ‚€π‘œπ‘π‘–ξ‚€πœ‹π‘π¨π­π‘¦,βˆ’2ξ‚ξ‚π‘‡ξ‚€π‘œπ‘π‘–ξ‚€πœ‹π‘π¨π­π‘¦,βˆ’2𝑇(36)

3.1.2. Strut Dynamic Equation

For the strut within 𝑖th kinematic chain, the dynamic motion in the coordinate system π‘‚βˆ’π‘₯𝑦𝑧 can be assembled asπŒξ…žπ‘ π‘–Μˆπ”ξ…žπ‘ π‘–+πŠξ…žπ‘ π‘–π”ξ…žπ‘ π‘–=π…ξ…žπ‘ π‘–βˆ’πŒξ…žπ‘ π‘–Μˆπ”ξ…žπ‘ π‘Ÿπ‘–,(37) where πŒξ…žπ‘ π‘–=3𝑗=1π€π‘‡π‘—πŒπ‘’π‘–π€π‘—,πŠξ…žπ‘ π‘–=3𝑗=1π€π‘‡π‘—πŠπ‘’π‘–π€π‘—,π…ξ…žπ‘ π‘–=3𝑗=1𝐀𝑇𝑗𝐅𝑒𝑖=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π…βˆ—π‘2π‘–πŸŽ18Γ—1π…βˆ—π‘1π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐀1=𝐄12𝟎12Γ—12ξ‚„,𝐀2=ξ‚ƒπŸŽ12Γ—6π„πŸπŸπŸŽ12Γ—6ξ‚„,𝐀3=ξ‚ƒπŸŽ12Γ—12𝐄12ξ‚„,(38) where 𝐀𝑗 is the connectivity matrix which maps the total nodal coordinates π”ξ…žπ‘ π‘– of the strut within the 𝑖th kinematics chain into the 𝑗th nodal coordinate within the strut. Μˆπ”ξ…žπ‘ π‘– and Μˆπ”ξ…žπ‘ π‘Ÿπ‘– denote the the nodal acceleration of the elastic motion and the rigid-body motion of the strut within the 𝑖th kinematics chain. 𝐄12 is the unit matrix of order twelve. π…βˆ—π‘1𝑖 is the internal force between the strut and the slider. π…βˆ—π‘2𝑖 is the internal force between the rigid moving platform and the strut. All the above coordinates are measured in the coordinate system π‘‚βˆ’π‘₯𝑦𝑧.

3.2. Slider Dynamic Equation

The oscillation of the slider along the axial direction of the lead-screw can be expressed asπ‘šπ‘π‘–Μˆπ‘ˆπ‘π‘–+π‘˜π‘π‘–π‘ˆπ‘π‘–=π‘“π‘–βˆ’π‘šπ‘π‘–Μˆπ‘žπ‘–βˆ’πžπ‘‡π‘–π…βˆ—π‘1𝑖,(39) where π‘ˆπ‘π‘– is the elastic displacement of the slider along the axial direction of the lead-screw. π‘˜π‘π‘– is the equivalent axial stiffness of the lead screw assembly that is composed of three serially connected component, that is, lead screw, ball nut which links slider with lead-screw and the two sets of support bearings at both ends. Let π‘˜π‘π‘ π‘–,π‘˜π‘π‘›π‘–, and π‘˜π‘π‘π‘– denote their respective axial stiffness then the equivalent axial stiffness can be expressed as1π‘˜π‘π‘–=1π‘˜π‘π‘ π‘–+1π‘˜π‘π‘›π‘–+1π‘˜π‘π‘π‘–,π‘˜π‘π‘ π‘–=𝐴𝑠𝐸𝑠𝐿1𝑖+𝐿2𝑖𝐿1𝑖𝐿2𝑖,(40) where 𝐴𝑠 and 𝐸𝑠 stand for the cross-sectional area of the lead screw and its Young’s modular, and 𝐿1𝑖 and 𝐿2𝑖 are the distances between the nut and the two sets of support bearings located at each end of the lead screw.

3.3. Deformation Compatibility Condition

The compatibility of the deformations between the rigid moving platform and the flexible strut can be expressed as 𝐔𝑐2𝑖=𝐄3ξ€·πšβˆ’π‘†π‘–ξ€Έξ‚„π”π‘,(41) where 𝐔𝑝 denotes the generalized coordinates of vibration motion of the moving platform, 𝐔𝑐2𝑖=π”ξ…žπ‘ π‘–(1∢3,1∢1) denotes the elastic displacement of the corresponding node within the flexible strut, 𝑆(πšπ‘–) is the screw matrix of πšπ‘–, and 𝐄3 denotes the unit matrix of the order three.

The compatibility of the deformations between the rigid slider and the flexible strut can be expressed as πžπ‘‡π‘–π”π‘1𝑖=π‘ˆπ‘π‘–,(42) where 𝐔𝑐1𝑖=π”ξ…žπ‘ π‘–(18∢21,1∢1) is the elastic displacement of the corresponding node within the flexible strut.

3.4. Substructure Motion Equation
3.4.1. Moving Platform Substructure

The oscillation equation of the rigid moving platform substructure in the coordinate system π‘‚βˆ’π‘₯𝑦𝑧 isβŽ‘βŽ’βŽ’βŽ£π‘šπ‘π„3𝟎3Γ—3πŸŽπ‘œ3Γ—3πˆπ‘βŽ€βŽ₯βŽ₯βŽ¦Μˆπ”π‘βˆ’βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πŸπ‘’βˆ’π‘šπ‘Μ‡π―βˆ’6𝑖=1π…βˆ—π‘2π‘–π§π‘’βˆ’π‘œπˆπ‘Μ‡ξ€·πŽβˆ’πŽΓ—π‘œπˆπ‘πŽξ€Έβˆ’6𝑖=1πšπ‘–Γ—π…βˆ—π‘2π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦=0.(43)

3.4.2. Kinematic Chain Substructure

Employing the deformation compatibility conditions between the flexible strut and the rigid slider and the boundary conditions of the slider, the motion equation of the 𝑖th kinematic chain substructure can be assembled asπŒπ‘˜π‘–Μˆπ”π‘˜π‘–+πŠπ‘˜π‘–π”π‘˜π‘–=π…π‘˜π‘π‘–+π…π‘˜π‘‘π‘–,(44) where π”π‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£π”ξ…žπ‘ π‘–π‘ˆ(1∢18,1∢1)π‘π‘–βŽ€βŽ₯βŽ₯⎦,πŒπ‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£πŒξ…žπ‘ π‘–(1∢18,1∢18)πŒξ…žπ‘ π‘–πŒ(1∢18,21∢21)ξ…žπ‘ π‘–(21∢21,1∢18)πŒξ…žπ‘ π‘–(21∢21,21∢21)+π‘šπ‘π‘–βŽ€βŽ₯βŽ₯βŽ¦πŒπ‘–=1,2,3,π‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£πŒξ…žπ‘ π‘–(1∢18,1∢18)πŒξ…žπ‘ π‘–(𝐌1∢18,20∢20)ξ…žπ‘ π‘–(20∢20,1∢18)πŒξ…žπ‘ π‘–(20∢20,20∢20)+π‘šπ‘π‘–βŽ€βŽ₯βŽ₯βŽ¦πŒπ‘–=4,5,π‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£πŒξ…žπ‘ π‘–(1∢18,1∢18)πŒξ…žπ‘ π‘–πŒ(1∢18,19∢19)ξ…žπ‘ π‘–(19∢19,1∢18)πŒξ…žπ‘ π‘–(19∢19,19∢19)+π‘šπ‘π‘–βŽ€βŽ₯βŽ₯βŽ¦πŠπ‘–=6,π‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£πŠξ…žπ‘ π‘–(1∢18,1∢18)πŠξ…žπ‘ π‘–πŠ(1∢18,21∢21)ξ…žπ‘ π‘–(21∢21,1∢18)πŠξ…žπ‘ π‘–(21∢21,21∢21)+π‘˜π‘π‘–βŽ€βŽ₯βŽ₯βŽ¦πŠπ‘–=1,2,3,π‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£πŠξ…žπ‘ π‘–(1∢18,1∢18)πŠξ…žπ‘ π‘–πŠ(1∢18,20∢20)ξ…žπ‘ π‘–(20∢20,1∢18)πŠξ…žπ‘ π‘–(20∢20,20∢20)+π‘˜π‘π‘–βŽ€βŽ₯βŽ₯βŽ¦πŠπ‘–=4,5,π‘˜π‘–=βŽ‘βŽ’βŽ’βŽ£πŠξ…žπ‘ π‘–(1∢18,1∢18)πŠξ…žπ‘ π‘–πŠ(1∢18,19∢19)ξ…žπ‘ π‘–(19∢19,1∢18)πŠξ…žπ‘ π‘–(19∢19,19∢19)+π‘˜π‘π‘–βŽ€βŽ₯βŽ₯βŽ¦π…π‘–=6,π‘˜π‘π‘–=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π…βˆ—π‘2π‘–πŸŽ15Γ—1𝟎⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,π…π‘˜π‘‘π‘–=⎑⎒⎒⎣𝟎18Γ—1π‘“π‘–βŽ€βŽ₯βŽ₯βŽ¦βˆ’πŒπ‘˜π‘–Μˆπ”π‘˜π‘Ÿπ‘–,π”π‘˜π‘Ÿπ‘–=βŽ‘βŽ’βŽ’βŽ£π”ξ…žπ‘ π‘Ÿπ‘–(1∢18,1∢1)Μˆπ‘žπ‘–βŽ€βŽ₯βŽ₯⎦(45)π…π‘˜π‘π‘– is the internal forces between the elements within the strut, π…π‘˜π‘‘π‘– is the resultant force of the generalized inertial force and the outside force.

3.5. Kineto-elasticdynamic Model of the Manipulator

Gathering the dynamic equations of the substructures and employing the deformation compatibility conditions between the rigid moving platform and the flexible strut yieldsπƒπ‘‡πŒξ…žπƒΜˆπ”+πƒπ‘‡πŠξ…žπƒπ”=πƒπ‘‡π…ξ…žπ‘+πƒπ‘‡π…ξ…žπ‘‘,(46) whereβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π„πƒ=6𝟎6Γ—96𝐄3ξ€·πšβˆ’π‘†1ξ€ΈπŸŽ3Γ—96𝟎16Γ—6𝐄16𝟎16Γ—80𝐄3ξ€·πšβˆ’π‘†2ξ€ΈπŸŽ3Γ—96𝟎16Γ—22𝐄16𝟎16Γ—64𝐄3ξ€·πšβˆ’π‘†3ξ€ΈπŸŽ3Γ—96𝟎16Γ—38𝐄16𝟎16Γ—48𝐄3ξ€·πšβˆ’π‘†4ξ€ΈπŸŽ3Γ—96𝟎16Γ—54𝐄16𝟎16Γ—32𝐄3ξ€·πšβˆ’π‘†5ξ€ΈπŸŽ3Γ—96𝟎16Γ—70𝐄16𝟎16Γ—16𝐄3ξ€·πšβˆ’π‘†6ξ€ΈπŸŽ3Γ—96𝟎16Γ—86𝐄16⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π”π”=π‘π”π‘˜1𝐔(4∢19,1)π‘˜2𝐔(4∢19,1)π‘˜3𝐔(4∢19,1)π‘˜4𝐔(4∢19,1)π‘˜5𝐔(4∢19,1)π‘˜6⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐅(4∢19,1)ξ…žπ‘=βŽ‘βŽ’βŽ’βŽ£βˆ’ξƒ¬6𝑖=1π…βˆ—π‘2π‘–ξƒ­π‘‡βˆ’ξƒ¬6𝑖=1πšπ‘–Γ—π…βˆ—π‘2π‘–ξƒ­π‘‡π…π‘‡π‘˜π‘1π…π‘‡π‘˜π‘2π…π‘‡π‘˜π‘3π…π‘‡π‘˜π‘4π…π‘‡π‘˜π‘5π…π‘‡π‘˜π‘6⎀βŽ₯βŽ₯βŽ¦π‘‡,π…ξ…žπ‘‘=ξ‚ƒξ€ΊπŸπ‘’βˆ’π‘šπ‘Μ‡π―ξ€»π‘‡ξ€Ίπ§π‘’βˆ’π‘œπˆπ‘Μ‡πŽβˆ’πŽΓ—(π‘œπˆπ‘ξ€»πŽ)π‘‡π…π‘‡π‘˜π‘‘1π…π‘‡π‘˜π‘‘2π…π‘‡π‘˜π‘‘3π…π‘‡π‘˜π‘‘4π…π‘‡π‘˜π‘‘5π…π‘‡π‘˜π‘‘6𝑇(47) Simplifying (46) yields the kineto-elasticdynamic model of the manipulatorπŒΜˆπ”+πŠπ”=𝐅𝑑,(48) where𝐌=πƒπ‘‡πŒξ…žπƒ,(49)𝐊=πƒπ‘‡πŠξ…žπƒ,(50)𝐅𝑑=πƒπ‘‡π…ξ…žπ‘‘.(51)

4. Kineto-Elastodynamic Characteristics Analysis

In this section, the investigation on the Kineto-elastodynamic characteristics of the 6-PSS parallel structure seismic simulator is carried out through simulation. The program is developed by the MATLAB software. The parameters of the seismic simulator used for the simulation are given in Tables 1, 2, 3, and 4.

tab1
Table 1: The parameters of the base platform (m).
tab2
Table 2: The parameters of the moving platform which are measured in the coordinate frame π‘‚ξ…žβˆ’π‘’π‘£π‘€ (m).
tab3
Table 3: The length of the strut 𝐢𝑖𝐴𝑖 (m).
tab4
Table 4: The mass parameters of the manipulator (kg).

The mass of the moving platform is π‘šπ‘=200 kg. The inertia parameters used in the simulation are given asπ‘œξ…žπˆπ‘=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦17.33300017.33300033.333kgβ‹…m2,π‘–πˆπ‘–=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦1.2790001.2790000.005kgβ‹…m2.(52) Other parameters used in the simulation are given as

𝐸=2.06Γ—1011 Pa,  𝐺=79.38Γ—109 Pa, 𝐸𝑠=2.06Γ—1011 Pa, 𝐴𝑠=1.96Γ—10βˆ’3 m2, 𝐿1𝑖+𝐿2𝑖=1.1 m,β€ƒπœŒ=7800 kg/m3, 𝑑𝑖=0.244 m, 𝑑=0.05 m,β€ƒβ„Ž1=2 m,β€ƒβ„Ž2=1.5 m,β€ƒβ„Ž=0.01 m, 𝑧0=1.744 m.

4.1. Natural Frequency

According to the vibration theory, the rigidity of the system may be represented by the natural frequency. The seismic simulator with the higher frequency would have the higher stiffness.

From (48), we getξ€·detβˆ’πœ”2ξ€ΈπŒ+𝐊=0,(53) where πœ” denotes the natural frequency. The distribution of the natural frequency is shown in Figure 5 when the pose of the moving platform is given as πœ™π‘₯=πœ™π‘¦=πœ™π‘§=0 and 𝑧=𝑧0.

fig5
Figure 5: Distributions of the natural frequencies in the workspace. (a) First-order natural frequency. (b) Second-order natural frequency.

It is shown in Figure 5 that the second-order natural frequency is much higher than the first-order natural frequency.

4.2. Sensitivity Analysis

The sensitivity analysis is usually used to evaluate the effect of the structural design variables on the performance of the manipulator. From (48), we getξ€·βˆ’πœ”2π‘Ÿξ€Έπ‹πŒ+πŠπ‘Ÿ=0,(54) where π‹π‘Ÿ and πœ”π‘Ÿ are the mode shape value and the natural frequency of the vibration in the π‘Ÿth mode. Taking the derivative of (54) with respect to the structural design value π‘π‘š such as the radius of the strut and the radius of the lead screw yields ξ‚΅βˆ’2πœ”π‘Ÿπœ•πœ”π‘Ÿπœ•π‘π‘šπŒβˆ’πœ”2π‘Ÿπœ•πŒπœ•π‘π‘š+πœ•πŠπœ•π‘π‘šξ‚Άπ‹π‘Ÿ+ξ€·βˆ’πœ”2π‘Ÿξ€ΈπŒ+πŠπœ•π‹π‘Ÿπœ•π‘π‘š=0.(55) Taking dot product of π‹π‘Ÿ on both sides of the equation yieldsπ‹π‘‡π‘Ÿξ‚΅βˆ’2πœ”π‘Ÿπœ•πœ”π‘Ÿπœ•π‘π‘šπŒβˆ’πœ”2π‘Ÿπœ•πŒπœ•π‘π‘š+πœ•πŠπœ•π‘π‘šξ‚Άπ‹π‘Ÿ+π‹π‘‡π‘Ÿξ€·βˆ’πœ”2π‘Ÿξ€ΈπŒ+πŠπœ•π‹π‘Ÿπœ•π‘π‘š=0.(56) Since π‹π‘‡π‘Ÿξ€·βˆ’πœ”2π‘Ÿξ€Έ=𝐌+πŠξ€·ξ€·βˆ’πœ”2π‘Ÿξ€Έπ‹πŒ+πŠπ‘Ÿξ€Έπ‘‡π‹=0,π‘‡π‘ŸπŒπ‹π‘Ÿ=𝐄.(57) giveβˆ’2πœ”π‘Ÿπœ•πœ”π‘Ÿπœ•π‘π‘šβˆ’πœ”2π‘Ÿπ‹π‘‡π‘Ÿπœ•πŒπœ•π‘π‘šπ‹π‘Ÿ+π‹π‘‡π‘Ÿπœ•πŠπœ•π‘π‘šπ‹π‘Ÿ=0.(58) so πœ•πœ”π‘Ÿπœ•π‘π‘š1=βˆ’2πœ”π‘Ÿξ‚΅πœ”2π‘Ÿπ‹π‘‡π‘Ÿπœ•πŒπœ•π‘π‘šπ‹π‘Ÿβˆ’π‹π‘‡π‘Ÿπœ•πŠπœ•π‘π‘šπ‹π‘Ÿξ‚Ά.(59) Figure 6 shows the sensitivity distribution of the manipulator when the pose of the moving platform is given as πœ™π‘₯=πœ™π‘¦=πœ™π‘§=0 and 𝑧=𝑧0. It is shown that the first-order natural frequency is sensitive to the radius of the strut and the radius of the lead screw.

fig6
Figure 6: Sensitivities of the first-order natural frequency to the structure parameters. (a) Radius of the struts. (b) Radius of the lead screws.
4.3. Energy Ratio Distribution

The computation of the energy ratio is usually used to evaluate the allocation of the stiffness and the mass of the manipulator. Suppose that π‘‡π‘ π‘Ÿ and π‘‰π‘ π‘Ÿ are the maximum kinetic energy and elastic potential energy of the substructures vibrating in its π‘Ÿth mode. π‘‡π΄π‘Ÿ and π‘‰π΄π‘Ÿ denote the maximum kinetic energy and elastic potential energy of the system vibrating in the π‘Ÿth mode. Thus,π‘‡π΄π‘Ÿ=𝑁𝑠=1π‘‡π‘ π‘Ÿ,π‘‰π΄π‘Ÿ=𝑁𝑠=1π‘‰π‘ π‘Ÿ,(60) whereπ‘‡π‘ π‘Ÿ=12πœ”2π‘Ÿπ€π‘ π‘Ÿπ‘‡π¦π‘ π€π‘Ÿπ‘ ,π‘‰π‘ π‘Ÿ=12π€π‘ π‘Ÿπ‘‡π€π‘ π€π‘Ÿπ‘ .(61)π€π‘Ÿπ‘  is the oscillating amplitude array of the substructure vibrating in the π‘Ÿth mode. 𝐦𝑠 and 𝐀𝑠 denote the mass matrix and the stiffness matrix of the substructure, respectively.

So the energy ratio of the substructure can be achieved asπ‘‡π‘ π‘Ÿπ‘‡π΄π‘Ÿ=π›Ύπ‘ π‘Ÿ,𝑁𝑠=1π›Ύπ‘ π‘Ÿπ‘‰=1,π‘ π‘Ÿπ‘‰π΄π‘Ÿ=πœ‡π‘ π‘Ÿ,𝑁𝑠=1πœ‡π‘ π‘Ÿ=1,(62) where π›Ύπ‘ π‘Ÿ and πœ‡π‘ π‘Ÿ denote the kinetic energy ratio and the elastic potential energy ratio of the substructure, respectively. Figure 7 shows the distributions of the kinetic energy ratio and the elastic potential energy ratio, respectively, when the pose of the moving platform is given as πœ™π‘₯=πœ™π‘¦=πœ™π‘§=0 and 𝑧=𝑧0. It is shown that the mass of the moving platform should be reduced or the stiffness of the strut should be increased in order to improve the dynamic characteristics of the manipulator and the stiffness of the sixth strut must be increased from the energy ratios computation.

fig7
Figure 7: Distributions of the energy ratios in the workspace. (a) Kinetic energy ratios of the sliders. (b) Kinetic energy ratio of the moving platform. (c) Kinetic energy ratios of the struts. (d) Elastic potential energy ratios of the sliders. (e) Elastic potential energy ratio of the moving platform. (f) Elastic potential energy ratios of the struts.
4.4. Displacement Response Analysis

The displacement response analysis will be carried out by solving (48) subject to the initial conditions𝐔0̇𝐔=𝐔(0),0=̇𝐔(0),(63)

Since the damping in the structure is a very complex subject [28], the modal damping ratios of πœπ‘Ÿ=0.1% are added to the Kineto-elastodynamic model of the manipulator.

From (48), we getξ€·βˆ’πœ”2ξ€ΈπŒ+πŠπ‹=𝟎.(64) Neglecting higher-order terms, the displacement vector 𝐔 of a multi-degree-of-freedom system can be expressed in terms of the four dominant modal contributions. Thus, the dynamic response of the system can be expressed as𝐔=π‹πœΌ,(65) where 𝝋𝝋=1𝝋2𝝋3𝝋4ξ‚„ is the modal matrix.

Substituting (65) into (50) and adding the modal damping ratio yieldsπ‹π‘‡ΜˆπŒπ‹πœΌ+π‹π‘‡Μ‡π‚π‹πœΌ+π‹π‘‡πŠπ‹πœΌ=𝝋𝑇𝐅𝑑,(66) whereπ‹π‘‡πŒπ‹=𝐄4,𝝋𝑇𝐂𝝋=𝐂=diag2𝜍1πœ”12𝜍2πœ”22𝜍3πœ”32𝜍4πœ”4,π‹π‘‡πŠπ‹=𝛀2ξ‚€πœ”=diag21πœ”22πœ”23πœ”24(67)𝐄4 is the unit matrix of order four.

Substituting (67) into (66) yields̈𝜼+π‚Μ‡πœΌ+𝛀2𝜼=𝐍,(68) where𝐍=𝝋𝑇𝐅𝑑.(69)

The stiffness matrix and the mass matrix of the Kineto-elastodynamic model of the parallel manipulator are time varying. The common strategy of solving this kind of problem is dividing the motion period into several small time internals and regarding the stiffness matrix and the mass matrix as constant in each small time interval [26].

Let 𝑇 denote the motion period which is divided into 𝑛 intervals𝑇Δ𝑑=𝑛.(70) In the 𝑖th time interval (π‘‘π‘–βˆ’1<𝑑<𝑑𝑖), the motion equation of the manipulator is Μˆπœ‚π‘Ÿ+2πœπ‘Ÿπœ”π‘Ÿ(𝑖)Μ‡πœ‚π‘Ÿ+ξ‚€πœ”π‘Ÿ(𝑖)2πœ‚π‘Ÿ=π‘π‘Ÿ,(π‘Ÿ=1,2,…,𝑁).(71) So the contribution of the π‘Ÿth mode to the displacement response is πœ‚π‘Ÿ(1𝑑)=πœ”(𝑖)π‘‘π‘Ÿξ€œπ‘‘π‘‘π‘–βˆ’1π‘π‘Ÿ(𝜏)π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)(π‘‘βˆ’πœ)sinπœ”(𝑖)π‘‘π‘Ÿ(+πœ‚π‘‘βˆ’πœ)π‘‘πœπ‘Ÿξ€·π‘‘π‘–βˆ’1ξ€Έξ€·1βˆ’πœ2π‘Ÿξ€Έ1/2π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)(π‘‘βˆ’π‘‘π‘–βˆ’1)ξ‚ƒπœ”cos(𝑖)π‘‘π‘Ÿξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έβˆ’πœ“π‘Ÿξ‚„+Μ‡πœ‚π‘Ÿξ€·π‘‘π‘–βˆ’1ξ€Έπœ”(𝑖)π‘‘π‘Ÿπ‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)(π‘‘βˆ’π‘‘π‘–βˆ’1)sinπœ”(𝑖)π‘‘π‘Ÿξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έ,(π‘Ÿ=1,2,…,𝑁),(72) whereπœ”(𝑖)π‘‘π‘Ÿ=ξ€·1βˆ’πœ2π‘Ÿξ€Έ1/2πœ”π‘Ÿ(𝑖),(73)πœ“π‘Ÿπœ=arctanπ‘Ÿξ€·1βˆ’πœ2π‘Ÿξ€Έ1/2.(74) Substituting 𝑑=𝑑𝑖 into (73) yieldsπœ‚π‘Ÿξ€·π‘‘π‘–ξ€Έ=1πœ”(𝑖)π‘‘π‘Ÿξ€œπ‘‘π‘–π‘‘π‘–βˆ’1π‘π‘Ÿ(𝜏)π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)(π‘‘π‘–βˆ’πœ)sinπœ”(𝑖)π‘‘π‘Ÿξ€·π‘‘π‘–ξ€Έ+πœ‚βˆ’πœπ‘‘πœπ‘Ÿξ€·π‘‘π‘–βˆ’1ξ€Έξ€·1βˆ’πœ2π‘Ÿξ€Έ1/2π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)Ξ”π‘‘ξ‚ƒπœ”cos(𝑖)π‘‘π‘ŸΞ”π‘‘βˆ’πœ“π‘Ÿξ‚„+Μ‡πœ‚π‘Ÿξ€·π‘‘π‘–βˆ’1ξ€Έπœ”(𝑖)π‘‘π‘Ÿπ‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)Δ𝑑sinπœ”(𝑖)π‘‘π‘ŸΞ”π‘‘,(π‘Ÿ=1,2,…,𝑁).(75) Taking the derivative of (72) with respect to time and substituting 𝑑=𝑑𝑖 into it yields Μ‡πœ‚π‘Ÿξ€·π‘‘π‘–ξ€Έ=1πœ”(𝑖)π‘‘π‘Ÿξ€œπ‘‘π‘–π‘‘π‘–βˆ’1π‘π‘Ÿξ‚ƒ(𝜏)βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)(π‘‘π‘–βˆ’πœ)sinπœ”(𝑖)π‘‘π‘Ÿξ€·π‘‘π‘–ξ€Έβˆ’πœ+πœ”(𝑖)π‘‘π‘Ÿπ‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)(π‘‘π‘–βˆ’πœ)cosπœ”(𝑖)π‘‘π‘Ÿξ€·π‘‘π‘–ξ€Έξ‚„βˆ’πœπ‘‘πœβˆ’πœ‚π‘Ÿξ€·π‘‘π‘–βˆ’1ξ€Έξƒ¬πœ”(𝑖)π‘‘π‘Ÿξ€·1βˆ’πœ2π‘Ÿξ€Έ1/2π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)Ξ”π‘‘ξ‚€πœ”sin(𝑖)π‘‘π‘ŸΞ”π‘‘βˆ’πœ“π‘Ÿξ‚+πœπ‘Ÿπœ”π‘Ÿ(𝑖)ξ€·1βˆ’πœ2π‘Ÿξ€Έ1/2π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)Ξ”π‘‘ξ‚€πœ”cos(𝑖)π‘‘π‘ŸΞ”π‘‘βˆ’πœ“π‘Ÿξ‚ξƒ­βˆ’Μ‡πœ‚π‘Ÿξ€·π‘‘π‘–βˆ’1ξ€Έξƒ¬πœπ‘Ÿπœ”π‘Ÿ(𝑖)πœ”(𝑖)π‘‘π‘Ÿπ‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)Δ𝑑sinπœ”(𝑖)π‘‘π‘ŸΞ”π‘‘βˆ’π‘’βˆ’πœπ‘Ÿπœ”π‘Ÿ(𝑖)Δ𝑑cosπœ”(𝑖)π‘‘π‘Ÿξƒ­(Ξ”π‘‘π‘Ÿ=1,2,…,𝑁).(76) It is shown from (75) and (76) that πœ‚π‘Ÿ(𝑑𝑖) and Μ‡πœ‚π‘Ÿ(𝑑𝑖) can be achieved when πœ‚π‘Ÿ(π‘‘π‘–βˆ’1) and Μ‡πœ‚π‘Ÿ(π‘‘π‘–βˆ’1) are given. As for 𝑑𝑖=0,πœ‚π‘Ÿ(𝝋0)=(π‘Ÿ)ξ€Έπ‘‡πŒπ”(0),Μ‡πœ‚π‘Ÿξ€·π‹(0)=(π‘Ÿ)ξ€Έπ‘‡πŒΜ‡π”(0).(77) So the total displacement response can be achieved by combining these modal contributions 𝐔𝑑𝑖=π‘ξ“π‘Ÿ=1πœ‚π‘Ÿξ€·π‘‘π‘–ξ€Έπ‹π‘Ÿξ€·π‘‘π‘–ξ€Έ,(78) It is the sum of the steady-state response and the transient state response.

Assuming that the investigated trajectory of the moving platform used in the simulation is expressed as π‘Žπ‘₯=βˆ’0.1+max𝑇2ξ‚€12πœ‹πœβˆ’ξ‚,π‘Ž2πœ‹sin(2πœ‹πœ)𝑦=βˆ’0.1+max𝑇2ξ‚€12πœ‹πœβˆ’ξ‚,π‘Ž2πœ‹sin(2πœ‹πœ)𝑧=1.644+max𝑇2ξ‚€12πœ‹πœβˆ’ξ‚,πœ™2πœ‹sin(2πœ‹πœ)π‘₯π‘Ž=βˆ’0.1+max𝑇2ξ‚€12πœ‹πœβˆ’ξ‚,πœ™2πœ‹sin(2πœ‹πœ)π‘¦π‘Ž=βˆ’0.1+max𝑇2ξ‚€12πœ‹πœβˆ’ξ‚,πœ™2πœ‹sin(2πœ‹πœ)π‘§π‘Ž=βˆ’0.1+max𝑇2ξ‚€12πœ‹πœβˆ’ξ‚,2πœ‹sin(2πœ‹πœ)(79) where π‘Žmax=9.8m/s2, 𝜏=𝑑/𝑇, βˆšπ‘‡=2πœ‹π‘†/π‘Žmax s in seconds, and 𝑆=0.2 m(rad). The motion period is divided into 512 intervals. The displacement response of the moving platform is shown in Figure 8. It is shown that the displacement response of the moving platform along the π‘₯ direction is smaller than these displacement responses along the 𝑦 direction and along the 𝑧 direction. The angular displacement response of the moving platform rotating about 𝑧 axis is slightly larger than those angular displacement responses rotating about the π‘₯ axis and about the 𝑦 axis.

fig8
Figure 8: Displacement responses of the moving platform.(a) π‘₯ direction, (b) 𝑦 direction, (c) 𝑧 direction, (d) πœ™π‘₯ direction, (e) πœ™π‘¦ direction, and (f) πœ™π‘§ direction.

5. Conclusion

Based on the Kineto-elastodynamic assumption, the modeling and the Kineto-elastodynamic characteristics of the 6-PSS parallel structure seismic simulator have been systematically investigated through simulation. The conclusions are drawn from the simulation as follows.(1)The maps of the natural frequencies with respect to the manipulator configuration have been achieved. It is shown that the second-order natural frequency is much higher than the first-order natural frequency.(2)From the sensitivity analysis, the first-order natural frequency is sensitive to the radius of the strut and the radius of the lead screw.(3)The mass of the moving platform should be reduced or the stiffness of the strut should be increased in order to improve the dynamic characteristic of the manipulator, and the stiffness of the sixth strut must be increased from the energy ratios computation.(4)For the investigated trajectory, the displacement response of the moving platform along the π‘₯-direction is smaller than these displacement responses along the 𝑦 direction and along the 𝑧 direction. The angular displacement response of the moving platform rotating about 𝑧-axis is slightly larger than those angular displacement responses rotating about the π‘₯-axis and about the 𝑦-axis.

Acknowledgments

This research is jointly sponsored by the National Natural Science Foundation of China (Grant no. 50905102), the Natural Science Foundation of Guangdong Province (Grants nos. 10151503101000033 and 8351503101000001), and the Building Fund for the Academic Innovation Team of Shantou University (Grant no. ITC10003). The author would also like to thank the anonymous reviewers for their very useful comments.

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