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Journal of Robotics
Volume 2011 (2011), Article ID 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: Yangmin Li

Copyright © 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 𝑥 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 𝑥-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 [14]. The parallel manipulator is a closed-loop kinematic chain mechanism whose end effector is linked to the base by several independent kinematic chains [57]. 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 [1023]. 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 𝐵1,𝐵2, and 𝐵3 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 𝑂𝑥𝑦𝑧 is attached to the fixed base; another moving coordinate frame 𝑂𝑢𝑣𝑤 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 matrix𝑜𝐑𝑜. Let the rotation matrix be defined by the roll, pitch, and yaw angles, namely, a rotation of 𝜙𝑥 about the fixed 𝑥 axis, followed by a rotation of 𝜙𝑦 about the fixed 𝑦 axis, and a rotation of 𝜙𝑧 about the fixed 𝑧 axis. Thus, the rotation matrix is𝑜𝐑𝑜=Rot𝑧,𝜙𝑧Rot𝑦,𝜙𝑦Rot𝑥,𝜙𝑥,(1) where s𝜙 denotes the sine of angle 𝜙, and c𝜙 denotes the cosine of angle 𝜙. In the hypothesis of small rotations, the angular velocity of the moving platform is given by [24, 25]̇𝜙𝝎=𝑥̇𝜙𝑦̇𝜙𝑧𝑇.(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 𝜙𝑖 about the 𝑧 axis resulting in a 𝐶𝑖𝑥𝑖𝑦𝑖𝑧𝑖 system followed by another rotation of 𝜑𝑖 about the rotated 𝑦𝑖-axis. So the rotation matrix of the 𝑖th strut can be written as𝑜𝐑𝑖=Rot𝑧,𝜙𝑖𝑦Rot𝑖,𝜑𝑖=c𝜙𝑖c𝜑𝑖s𝜙𝑖c𝜙𝑖s𝜑𝑖s𝜙𝑖c𝜑𝑖c𝜙𝑖s𝜙𝑖s𝜑𝑖s𝜑𝑖0c𝜑𝑖,𝑖=1,2,,6.(3) The unit vector along the strut in the coordinate system 𝑂𝑥𝑦𝑧 is𝐰𝑖=𝑜𝐑𝑖𝑖𝐰𝑖=𝑜𝐑𝑖001=c𝜙𝑖s𝜑𝑖s𝜙𝑖s𝜑𝑖c𝜑𝑖.(4) So the Euler angles 𝜙𝑖 and 𝜑𝑖 can be computed as follows:c𝜑𝑖=𝑤𝑖𝑧,s𝜑𝑖=𝑤2𝑖𝑥+𝑤2𝑖𝑦,0𝜑𝑖,<𝜋s𝜙𝑖=𝑤𝑖𝑦s𝜑𝑖,𝜑𝑖,0c𝜙𝑖=𝑤𝑖𝑥s𝜑𝑖,𝜑𝑖,0if𝜑𝑖=0,then𝜙𝑖=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=𝐉𝑇𝐟𝑒𝐧𝑒𝐉𝑇𝑚𝑝𝐠𝟎+6𝑖=1𝐉𝑇𝑖𝑣𝜔𝑚𝑖𝑖𝐑𝑜𝐠𝟎+𝐉𝑇(𝑚𝑐1𝐠)𝑇𝐞1𝑚𝑐2𝐠𝑇𝐞2𝑚𝑐3𝐠𝑇𝐞3𝑚𝑐4𝐠𝑇𝐞4𝑚𝑐5𝐠𝑇𝐞5𝑚𝑐6𝐠𝑇𝐞6𝑇+𝐉𝑇𝑚𝑝̇𝐯𝑜𝐈𝑝̇𝝎+6𝑖=1𝐉𝑇𝑖𝑣𝜔𝑚𝑖𝑖̇𝐯𝑖𝑖𝐈𝑖𝑖̇𝝎𝑖+𝐉𝑇𝑚𝑐1̈𝑞1𝑚𝑐2̈𝑞2𝑚𝑐3̈𝑞3𝑚𝑐4̈𝑞4𝑚𝑐5̈𝑞5𝑚𝑐6̈𝑞6𝑇+𝐉𝑇𝟎𝝎×𝑜𝐈𝑝𝝎+6𝑖=1𝐉𝑇𝑖𝑣𝜔𝟎𝑖𝝎𝑖×𝑖𝐈𝑖𝑖𝝎𝑖,(6)where 𝟎=000𝑇,𝐉𝑖𝑣𝜔 is the link Jacobian matrix which maps the velocity of the moving platform into the velocity of the 𝑖th strut in the 𝐶𝑖𝑥𝑖𝑦𝑖𝑧𝑖 coordinate system, 𝑚𝑝, 𝑚𝑐𝑖, and 𝑚𝑖 denote the mass of the moving platform, the mass of the slider, and the mass of the 𝑖th strut, respectively, 𝑜𝐈𝑝 is the inertia matrix of the moving platform taken about the center of mass expressed in the 𝑂𝑥𝑦𝑧 coordinate system, 𝑖𝐈𝑖 is the inertia matrix of the 𝑖th cylindrical strut about their respective centers of mass expressed in the 𝐶𝑖𝑥𝑖𝑦𝑖𝑧𝑖 coordinate system, 𝐟𝑒 and 𝐧𝑒 are the external force and moment exerted at the center of mass of the moving platform, ̇𝐯 and ̇𝝎 are the linear and angular acceleration of the moving platform,𝑖̇𝐯𝑖,𝑖𝝎𝑖, and 𝑖̇𝝎𝑖 are the linear acceleration, the angular velocity, and acceleration of the ith strut expressed in the 𝐶𝑖𝑥𝑖𝑦𝑖𝑧𝑖 coordinate system, respectively. ̈𝑞𝑖 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. And1𝐉=diag𝐰𝑇1𝐞11𝐰𝑇2𝐞21𝐰𝑇3𝐞31𝐰𝑇4𝐞41𝐰𝑇5𝐞51𝐰𝑇6𝐞6×𝐰1𝐰2𝐰3𝐰4𝐰5𝐰6𝐚1×𝐰1𝐚2×𝐰2𝐚3×𝐰3𝐚4×𝐰4𝐚5×𝐰5𝐚6×𝐰6𝑇,𝐉𝑖𝑣𝜔=𝑖𝐑𝑜𝑆𝑖𝐚𝑖𝑖𝐑𝑜+𝑙𝑖2𝑆𝑖𝐰𝑖𝐉𝑖𝜔1𝑙𝑖𝑆𝑖𝐰𝑖𝑖𝐑𝑜𝑆𝑖𝐰𝑖𝑆𝑖𝐚𝑖𝑖𝐑𝑜𝑖𝐰𝑖×𝑖𝐞𝑖𝐰𝑇𝑖𝐰𝑇𝑖𝐞𝑖𝐚𝑖×𝐰𝑖𝑇𝐰𝑇𝑖𝐞𝑖=𝐉𝑖𝑣𝐉𝑖𝜔,𝑆𝑖𝐰𝑖=0𝑖𝑤𝑖𝑖𝑧𝑤𝑖𝑖𝑦𝑤𝑖𝑧0𝑖𝑤𝑖𝑥𝑖𝑤𝑖𝑖𝑦𝑤𝑖𝑥0,𝑆𝑖𝐚𝑖=0𝑖𝑎𝑖𝑖𝑧𝑎𝑖𝑖𝑦𝑎𝑖𝑧0𝑖𝑎𝑖𝑥𝑖𝑎𝑖𝑖𝑦𝑎𝑖𝑥0,𝑖𝝎𝑖=𝐉𝑖𝜔𝐯𝝎,̈𝑞𝑖=1𝐰𝑇𝑖𝐞𝑖𝐰𝑇𝑖̇𝐚𝐯+𝑖×𝐰𝑖𝑇̇𝝎+𝐰𝑇𝑖𝝎×𝝎×𝐚𝑖𝐰𝑇𝑖𝝎𝑖×𝝎𝑖×𝑙𝑖𝐰𝑖,𝑖̇𝝎𝑖=𝐉𝑖𝜔̇𝐯̇𝝎+1𝑙𝑖𝚫1+𝚫2,𝚫1=𝑖𝐰𝑖×𝑖𝐞𝑖𝐰𝑇𝑖𝐞𝑖𝐰𝑇𝑖𝝎𝐚𝑇𝑖𝝎𝐰𝑇𝑖𝐚𝑖𝝎𝑇𝝎+𝑙𝑖||𝝎𝑖×𝐰𝑖||2,𝚫2=𝑖𝝎𝑇𝑖𝑖𝐚𝑖𝑖𝐰𝑖×𝑖𝝎𝑖𝑖𝝎𝑇𝑖𝝎𝑖𝐰𝑖×𝑖𝐚𝑖,𝑖̇𝐯𝑖=𝐉𝑖𝑣̇𝐯̇𝝎+𝑆𝑖𝝎𝑖𝑆𝑖𝝎𝑖𝑖𝐚𝑖+12𝑆𝑖𝐰𝑖𝚫1+𝚫2𝑙𝑖2𝑆𝑖𝝎𝑖𝑆𝑖𝝎𝑖𝑖𝐰𝑖.(7)𝐞𝑖, 𝐚𝑖, and 𝐰𝑖 are shown in Figure 2; they are the unit vector along the lead screw, the vector 𝐎𝐀𝐢, and the unit vector along strut 𝐶𝑖𝐴𝑖, 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 𝜹𝜹=𝑖𝜹𝑗𝑇,(8) where𝜹𝑖=𝑢𝑖𝑣𝑖𝑤𝑖𝜃𝑖𝑥𝜃𝑖𝑦𝜃𝑖𝑧𝑇,𝜹𝑗=𝑢𝑗𝑣𝑗𝑤𝑗𝜃𝑗𝑥𝜃𝑗𝑦𝜃𝑗𝑧𝑇.(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=𝑢𝑣𝑤𝜃𝑇=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 as𝑢=𝑎0+𝑎1𝑥,𝜃=𝑑0+𝑑1𝑥.(11) The transverse displacements of the point 𝑃 can be expressed as 𝑣=𝑏0+𝑏1𝑥+𝑏2𝑥2+𝑏3𝑥3,𝑤=𝑐0+𝑐1𝑥+𝑐2𝑥2+𝑐3𝑥3.(12) Substituting (11) and (12) into (10) yields𝑢𝑣𝑤𝜃=𝐇𝑝=𝑢𝐇(𝑥)𝐚𝑣(𝐇𝑥)𝐛𝑤𝐇(𝑥)𝐜𝜃(𝑥)𝐝,(13) where𝑎𝐚=0𝑎1𝑇,𝑏𝐛=0𝑏1𝑏2𝑏3𝑇,𝑐𝐜=0𝑐1𝑐2𝑐3𝑇,𝑑𝐝=0𝑑1𝑇,𝐇𝑢(𝑥)=𝐇𝜃(,𝐇𝑥)=1𝑥𝑣(𝑥)=𝐇𝑤(𝑥)=1𝑥𝑥2𝑥3.(14)

Considering the node 𝑖 and the node 𝑗, where 𝑥=0 and 𝑥=𝑙, respectively, yields𝐚=𝐀1𝑢𝜃𝜹𝑢,𝐛=𝐀1𝑣𝑤𝜹𝑣,𝐜=𝐀1𝑣𝑤𝜹𝑤,𝐝=𝐀1𝑢𝜃𝜹𝜃,(15) where𝜹𝑢=𝑢𝑖𝑢𝑗𝑇,𝜹𝑣=𝑣𝑖𝜃𝑖𝑧𝑣𝑗𝜃𝑗𝑧𝑇,𝜹𝑤=𝑤𝑖𝜃𝑖𝑦𝑤𝑗𝜃𝑗𝑦𝑇,𝜹𝜃=𝜃𝑖𝑥𝜃𝑗𝑥𝑇,𝐀1𝑢𝜃=110𝑙1𝑙,𝐀1𝑣𝑤=310000100𝑙22𝑙3𝑙21𝑙2𝑙31𝑙22𝑙31𝑙2.(16) Substituting (15) into (13) yields 𝑢𝑣𝑤𝜃=𝐡𝐩=𝑢𝐡(𝑥)𝑣(𝐡𝑥)𝑤𝐡(𝑥)𝜃(𝑥)𝐀𝜹=𝐍𝜹,(17) where𝐡𝑢,𝐡(𝑥)=100000𝑥00000𝑣(𝑥)=01000𝑥0𝑥2000𝑥3,𝐡𝑤(𝑥)=0010𝑥000𝑥20𝑥30,𝐡𝜃,1(𝑥)=000100000𝑥00𝐀=100000000000010000000000001000000000000100000000000010000000000001000000𝑙100000𝑙3000000𝑙22000𝑙03𝑙21000𝑙300𝑙220𝑙3000𝑙210𝑙01000𝑙100000𝑙20000𝑙301𝑙22000𝑙301𝑙2002𝑙31000𝑙220𝑙31000𝑙2.(18)

Considering the knowledge of material mechanics, the strain of the point 𝑃 is 𝜀𝜺=0𝜀𝑏𝑦𝜀𝑏𝑧𝜀𝑟=𝑑𝑢𝑑𝑑𝑥𝑦2𝑣𝑑𝑥2𝑑𝑧2𝑤𝑑𝑥2𝐺𝐸16𝐽𝑘𝜋𝑑3𝑑𝜃𝑑𝑥,(19) where 𝜀0 is the axial strain, 𝜀𝑏𝑦 and 𝜀𝑏𝑧 are the flexural strain in the plane 𝑂𝑥𝑦 and 𝑂𝑥𝑧, respectively, 𝜀𝑟 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, 𝐽𝑘 is the polar moment of inertia of cross-section, 𝑑 is the diameter of the strut.

Substituting (17) into (19) yields𝜺=𝑢(𝑥)𝑦𝑣(𝑥)𝑧𝑤𝐺(𝑥)𝐸16𝐽𝑘𝜋𝑑3𝜃(𝑥)𝐀𝜹=𝐁𝜹,(20) where𝑢,(𝑥)=000000100000𝑣,(𝑥)=000000020006𝑥𝑤,(𝑥)=00000000206𝑥0𝜃.(𝑥)=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 as1𝑈=2𝜺𝑇𝝈𝑑𝑣.(23) Substituting (20) and (22) into (23) yields 1𝑈=2𝐸𝜹𝑇𝐁𝑇1𝐁𝜹𝑑𝑣=2𝜹𝑇𝐸𝐁𝑇1𝐁𝑑𝑣𝜹=2𝜹𝑇𝐤𝜹,(24) where1𝐤=2𝐸𝐁𝑇𝐁𝑑𝑣(25) is the element stiffness matrix. Substituting (20) into (25) yields𝐤=𝐸𝐴𝑙00000𝐸𝐴𝑙00000012𝐸𝐼𝑧𝑙30006𝐸𝐼𝑧𝑙2012𝐸𝐼𝑧𝑙30006𝐸𝐼𝑧𝑙20012𝐸𝐼𝑦𝑙306𝐸𝐼𝑦𝑙200012𝐸𝐼𝑦𝑙306𝐸𝐼𝑦𝑙20000𝐺𝐽𝑘𝑙00000𝐺𝐽𝑘𝑙00006𝐸𝐼𝑦𝑙204𝐸𝐼𝑦𝑙0006𝐸𝐼𝑦𝑙202𝐸𝐼𝑦𝑙006𝐸𝐼𝑧𝑙20004𝐸𝐼𝑧𝑙06𝐸𝐼𝑧𝑙20002𝐸𝐼𝑧𝑙𝐸𝐴𝑙00000𝐸𝐴𝑙00000012𝐸𝐼𝑧𝑙30006𝐸𝐼𝑧𝑙2012𝐸𝐼𝑧𝑙30006𝐸𝐼𝑧𝑙20012𝐸𝐼𝑦𝑙306𝐸𝐼𝑦𝑙200012𝐸𝐼𝑦𝑙306𝐸𝐼𝑦𝑙20000𝐺𝐽𝑘𝑙00000𝐺𝐽𝑘𝑙00006𝐸𝐼𝑦𝑙202𝐸𝐼𝑦𝑙0006𝐸𝐼𝑦𝑙204𝐸𝐼𝑦𝑙006𝐸𝐼𝑧𝑙20002𝐸𝐼𝑧𝑙06𝐸𝐼𝑧𝑙20004𝐸𝐼𝑧𝑙,(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:̇𝐩𝑎=̇𝐩𝑟+̇̈𝐩𝐩,𝑎=̈𝐩𝑟+̈𝐩,(27) where ̇𝐩𝑎 and ̈𝐩𝑎 are the absolute velocity and the absolute acceleration of a certain point within the element, ̇𝐩𝑟 and ̈𝐩𝑟 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])̇𝐩𝑟=𝐍𝑟̇𝜹𝑟̇𝜹=𝐍𝑟,̈𝐩𝑟=𝐍𝑟̈𝜹𝑟̈𝜹=𝐍𝑟,(28) where ̇𝜹𝑟 and ̈𝜹𝑟 are the nodal velocity and the acceleration of the rigid-body motion.

So the kinetic energy of the element can be expressed as 1𝑇=2𝑙0̇𝐩𝑚(𝑥)𝑇𝑎̇𝐩𝑎=1𝑑𝑥2𝑙0𝑚̇𝜹(𝑥)𝑇𝑎𝐍𝑇𝐍̇𝜹𝑎=1𝑑𝑥2̇𝜹𝑇𝑎𝑙0𝑚(𝑥)𝐍𝑇̇𝜹𝐍𝑑𝑥𝑎=12̇𝜹𝑇𝑎𝐦̇𝜹𝑎,(29) where ̇𝜹𝑎 is the absolute velocity of the node. And 1𝐦=2𝑙0𝑚(𝑥)𝐍𝑇𝐍𝑑𝑥(30) is the element mass matrix. For the uniform beam, the mass function is𝑚(𝑥)=𝜌𝐴,(31) where 𝜌 is the density of the beam. So the element mass matrix is𝐦=𝜌𝐴𝑙420140000007000000015600022𝑙05400013𝑙00156022𝑙00054013𝑙0000140𝐽𝑘𝐴0000070𝐽𝑘𝐴000022𝑙04𝑙200013𝑙03𝑙20022𝑙0004𝑙2013𝑙0003𝑙270000001400000005400013𝑙015600022𝑙0054013𝑙000156022𝑙000070𝐽𝑘𝐴00000140𝐽𝑘𝐴000013𝑙03𝑙200022𝑙04𝑙20013𝑙0003𝑙2022𝑙0004𝑙2.(32)

Substituting (23) and (29) into Lagrange equation yields 𝑑𝑑𝑡𝜕𝑇𝜕̇𝜹𝜕𝑇+𝜕𝜹𝜕𝑈𝜕𝜹=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+𝐤𝜹=𝐟𝐦𝑟.(34) Equation (34) can be expressed in the coordinate system 𝑂𝑥𝑦𝑧 as𝐌𝑒𝑖̈𝐔𝑒𝑖+𝐊𝑒𝑖𝐔𝑒𝑖=𝐅𝑒𝑖𝐌𝑒𝑖̈𝐔𝑒𝑟𝑖,𝑖=1,2,,6(35) where𝐔𝑒𝑖=𝐑𝑇𝑖̈𝐔𝜹,𝑒𝑖=𝐑𝑇𝑖̈̈𝐔𝜹,𝑒𝑟𝑖=𝐑𝑇𝑖̈𝜹𝑟,𝐅𝑒𝑖=𝐑𝑇𝑖𝐌𝐟,𝑒𝑖=𝐑𝑇𝑖𝐦𝐑𝑖,𝐊𝑒𝑖=𝐑𝑇𝑖𝐤𝐑𝑖,𝐑𝑖=diag𝑜𝐑𝑖𝜋𝐑𝐨𝐭𝑦,2𝑇𝑜𝐑𝑖𝜋𝐑𝐨𝐭𝑦,2𝑇𝑜𝐑𝑖𝜋𝐑𝐨𝐭𝑦,2𝑇𝑜𝐑𝑖𝜋𝐑𝐨𝐭𝑦,2𝑇(36)

3.1.2. Strut Dynamic Equation

For the strut within 𝑖th kinematic chain, the dynamic motion in the coordinate system 𝑂𝑥𝑦𝑧 can be assembled as𝐌𝑠𝑖̈𝐔𝑠𝑖+𝐊𝑠𝑖𝐔𝑠𝑖=𝐅𝑠𝑖𝐌𝑠𝑖̈𝐔𝑠𝑟𝑖,(37) where 𝐌𝑠𝑖=3𝑗=1𝐀𝑇𝑗𝐌𝑒𝑖𝐀𝑗,𝐊𝑠𝑖=3𝑗=1𝐀𝑇𝑗𝐊𝑒𝑖𝐀𝑗,𝐅𝑠𝑖=3𝑗=1𝐀𝑇𝑗𝐅𝑒𝑖=𝐅𝑐2𝑖𝟎18×1𝐅𝑐1𝑖,𝐀1=𝐄12𝟎12×12,𝐀2=𝟎12×6𝐄𝟏𝟐𝟎12×6,𝐀3=𝟎12×12𝐄12,(38) where 𝐀𝑗 is the connectivity matrix which maps the total nodal coordinates 𝐔𝑠𝑖 of the strut within the 𝑖th kinematics chain into the 𝑗th nodal coordinate within the strut. ̈𝐔𝑠𝑖 and ̈𝐔𝑠𝑟𝑖 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. 𝐅𝑐1𝑖 is the internal force between the strut and the slider. 𝐅𝑐2𝑖 is the internal force between the rigid moving platform and the strut. All the above coordinates are measured in the coordinate system 𝑂𝑥𝑦𝑧.

3.2. Slider Dynamic Equation

The oscillation of the slider along the axial direction of the lead-screw can be expressed as𝑚𝑐𝑖̈𝑈𝑐𝑖+𝑘𝑐𝑖𝑈𝑐𝑖=𝑓𝑖𝑚𝑐𝑖̈𝑞𝑖𝐞𝑇𝑖𝐅𝑐1𝑖,(39) where 𝑈𝑐𝑖 is the elastic displacement of the slider along the axial direction of the lead-screw. 𝑘𝑐𝑖 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 𝑘𝑐𝑠𝑖,𝑘𝑐𝑛𝑖, and 𝑘𝑐𝑏𝑖 denote their respective axial stiffness then the equivalent axial stiffness can be expressed as1𝑘𝑐𝑖=1𝑘𝑐𝑠𝑖+1𝑘𝑐𝑛𝑖+1𝑘𝑐𝑏𝑖,𝑘𝑐𝑠𝑖=𝐴𝑠𝐸𝑠𝐿1𝑖+𝐿2𝑖𝐿1𝑖𝐿2𝑖,(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=𝐄3𝐚𝑆𝑖𝐔𝑝,(41) where 𝐔𝑝 denotes the generalized coordinates of vibration motion of the moving platform, 𝐔𝑐2𝑖=𝐔𝑠𝑖(13,11) denotes the elastic displacement of the corresponding node within the flexible strut, 𝑆(𝐚𝑖) is the screw matrix of 𝐚𝑖, 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 𝐞𝑇𝑖𝐔𝑐1𝑖=𝑈𝑐𝑖,(42) where 𝐔𝑐1𝑖=𝐔𝑠𝑖(1821,11) 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 𝑂𝑥𝑦𝑧 is𝑚𝑝𝐄3𝟎3×3𝟎𝑜3×3𝐈𝑝̈𝐔𝑝𝐟𝑒𝑚𝑝̇𝐯6𝑖=1𝐅𝑐2𝑖𝐧𝑒𝑜𝐈𝑝̇𝝎𝝎×𝑜𝐈𝑝𝝎6𝑖=1𝐚𝑖×𝐅𝑐2𝑖=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 as𝐌𝑘𝑖̈𝐔𝑘𝑖+𝐊𝑘𝑖𝐔𝑘𝑖=𝐅𝑘𝑐𝑖+𝐅𝑘𝑑𝑖,(44) where 𝐔𝑘𝑖=𝐔𝑠𝑖𝑈(118,11)𝑐𝑖,𝐌𝑘𝑖=𝐌𝑠𝑖(118,118)𝐌𝑠𝑖𝐌(118,2121)𝑠𝑖(2121,118)𝐌𝑠𝑖(2121,2121)+𝑚𝑐𝑖𝐌𝑖=1,2,3,𝑘𝑖=𝐌𝑠𝑖(118,118)𝐌𝑠𝑖(𝐌118,2020)𝑠𝑖(2020,118)𝐌𝑠𝑖(2020,2020)+𝑚𝑐𝑖𝐌𝑖=4,5,𝑘𝑖=𝐌𝑠𝑖(118,118)𝐌𝑠𝑖𝐌(118,1919)𝑠𝑖(1919,118)𝐌𝑠𝑖(1919,1919)+𝑚𝑐𝑖𝐊𝑖=6,𝑘𝑖=𝐊𝑠𝑖(118,118)𝐊𝑠𝑖𝐊(118,2121)𝑠𝑖(2121,118)𝐊𝑠𝑖(2121,2121)+𝑘𝑐𝑖𝐊𝑖=1,2,3,𝑘𝑖=𝐊𝑠𝑖(118,118)𝐊𝑠𝑖𝐊(118,2020)𝑠𝑖(2020,118)𝐊𝑠𝑖(2020,2020)+𝑘𝑐𝑖𝐊𝑖=4,5,𝑘𝑖=𝐊𝑠𝑖(118,118)𝐊𝑠𝑖𝐊(118,1919)𝑠𝑖(1919,118)𝐊𝑠𝑖(1919,1919)+𝑘𝑐𝑖𝐅𝑖=6,𝑘𝑐𝑖=𝐅𝑐2𝑖𝟎15×1𝟎,𝐅𝑘𝑑𝑖=𝟎18×1𝑓𝑖𝐌𝑘𝑖̈𝐔𝑘𝑟𝑖,𝐔𝑘𝑟𝑖=𝐔𝑠𝑟𝑖(118,11)̈𝑞𝑖(45)𝐅𝑘𝑐𝑖 is the internal forces between the elements within the strut, 𝐅𝑘𝑑𝑖 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 yields𝐃𝑇𝐌𝐃̈𝐔+𝐃𝑇𝐊𝐃𝐔=𝐃𝑇𝐅𝑐+𝐃𝑇𝐅𝑑,(46) where𝐄𝐃=6𝟎6×96𝐄3𝐚𝑆1𝟎3×96𝟎16×6𝐄16𝟎16×80𝐄3𝐚𝑆2𝟎3×96𝟎16×22𝐄16𝟎16×64𝐄3𝐚𝑆3𝟎3×96𝟎16×38𝐄16𝟎16×48𝐄3𝐚𝑆4𝟎3×96𝟎16×54𝐄16𝟎16×32𝐄3𝐚𝑆5𝟎3×96𝟎16×70𝐄16𝟎16×16𝐄3𝐚𝑆6𝟎3×96𝟎16×86𝐄16,𝐔𝐔=𝑝𝐔𝑘1𝐔(419,1)𝑘2𝐔(419,1)𝑘3𝐔(419,1)𝑘4𝐔(419,1)𝑘5𝐔(419,1)𝑘6,𝐅(419,1)𝑐=6𝑖=1𝐅𝑐2𝑖𝑇6𝑖=1𝐚𝑖×𝐅𝑐2𝑖𝑇𝐅𝑇𝑘𝑐1𝐅𝑇𝑘𝑐2𝐅𝑇𝑘𝑐3𝐅𝑇𝑘𝑐4𝐅𝑇𝑘𝑐5𝐅𝑇𝑘𝑐6𝑇,𝐅𝑑=𝐟𝑒𝑚𝑝̇𝐯𝑇𝐧𝑒𝑜𝐈𝑝̇𝝎𝝎×(𝑜𝐈𝑝𝝎)𝑇𝐅𝑇𝑘𝑑1𝐅𝑇𝑘𝑑2𝐅𝑇𝑘𝑑3𝐅𝑇𝑘𝑑4𝐅𝑇𝑘𝑑5𝐅𝑇𝑘𝑑6𝑇(47) Simplifying (46) yields the kineto-elasticdynamic model of the manipulator𝐌̈𝐔+𝐊𝐔=𝐅𝑑,(48) where𝐌=𝐃𝑇𝐌𝐃,(49)𝐊=𝐃𝑇𝐊𝐃,(50)𝐅𝑑=𝐃𝑇𝐅𝑑.(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 𝑂𝑢𝑣𝑤 (m).
tab3
Table 3: The length of the strut 𝐶𝑖𝐴𝑖 (m).
tab4
Table 4: The mass parameters of the manipulator (kg).

The mass of the moving platform is 𝑚𝑝=200 kg. The inertia parameters used in the simulation are given as𝑜𝐈𝑝=17.33300017.33300033.333kgm2,𝑖𝐈𝑖=1.2790001.2790000.005kgm2.(52) Other parameters used in the simulation are given as

𝐸=2.06×1011 Pa,  𝐺=79.38×109 Pa, 𝐸𝑠=2.06×1011 Pa, 𝐴𝑠=1.96×103 m2, 𝐿1𝑖+𝐿2𝑖=1.1 m, 𝜌=7800 kg/m3, 𝑑𝑖=0.244 m, 𝑑=0.05 m, 1=2 m, 2=1.5 m, =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 getdet𝜔2𝐌+𝐊=0,(53) where 𝜔 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 𝜙𝑥=𝜙𝑦=𝜙𝑧=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 get𝜔2𝑟𝝋𝐌+𝐊𝑟=0,(54) where 𝝋𝑟 and 𝜔𝑟 are the mode shape value and the natural frequency of the vibration in the 𝑟th mode. Taking the derivative of (54) with respect to the structural design value 𝑝𝑚 such as the radius of the strut and the radius of the lead screw yields 2𝜔𝑟𝜕𝜔𝑟𝜕𝑝𝑚𝐌𝜔2𝑟𝜕𝐌𝜕𝑝𝑚+𝜕𝐊𝜕𝑝𝑚𝝋𝑟+𝜔2𝑟𝐌+𝐊𝜕𝝋𝑟𝜕𝑝𝑚=0.(55) Taking dot product of 𝝋𝑟 on both sides of the equation yields𝝋𝑇𝑟2𝜔𝑟𝜕𝜔𝑟𝜕𝑝𝑚𝐌𝜔2𝑟𝜕𝐌𝜕𝑝𝑚+𝜕𝐊𝜕𝑝𝑚𝝋𝑟+𝝋𝑇𝑟𝜔2𝑟𝐌+𝐊𝜕𝝋𝑟𝜕𝑝𝑚=0.(56) Since 𝝋𝑇𝑟𝜔2𝑟=𝐌+𝐊𝜔2𝑟𝝋𝐌+𝐊𝑟𝑇𝝋=0,𝑇𝑟𝐌𝝋𝑟=𝐄.(57) give2𝜔𝑟𝜕𝜔𝑟𝜕𝑝𝑚𝜔2𝑟𝝋𝑇𝑟𝜕𝐌𝜕𝑝𝑚𝝋𝑟+𝝋𝑇𝑟𝜕𝐊𝜕𝑝𝑚𝝋𝑟=0.(58) so 𝜕𝜔𝑟𝜕𝑝𝑚1=2𝜔𝑟𝜔2𝑟𝝋𝑇𝑟𝜕𝐌𝜕𝑝𝑚𝝋𝑟𝝋𝑇𝑟𝜕𝐊𝜕𝑝𝑚𝝋𝑟.(59) Figure 6 shows the sensitivity distribution of the manipulator when the pose of the moving platform is given as 𝜙𝑥=𝜙𝑦=𝜙𝑧=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 𝑇𝑠𝑟 and 𝑉𝑠𝑟 are the maximum kinetic energy and elastic potential energy of the substructures vibrating in its 𝑟th mode. 𝑇𝐴𝑟 and 𝑉𝐴𝑟 denote the maximum kinetic energy and elastic potential energy of the system vibrating in the 𝑟th mode. Thus,𝑇𝐴𝑟=𝑁𝑠=1𝑇𝑠𝑟,𝑉𝐴𝑟=𝑁𝑠=1𝑉𝑠𝑟,(60) where𝑇𝑠𝑟=12𝜔2𝑟𝐀𝑠𝑟𝑇𝐦𝑠𝐀𝑟𝑠,𝑉𝑠𝑟=12𝐀𝑠𝑟𝑇𝐤𝑠𝐀𝑟𝑠.(61)𝐀𝑟𝑠 is the oscillating amplitude array of the substructure vibrating in the 𝑟th mode. 𝐦𝑠 and 𝐤𝑠 denote the mass matrix and the stiffness matrix of the substructure, respectively.

So the energy ratio of the substructure can be achieved as𝑇𝑠𝑟𝑇𝐴𝑟=𝛾𝑠𝑟,𝑁𝑠=1𝛾𝑠𝑟𝑉=1,𝑠𝑟𝑉𝐴𝑟=𝜇𝑠𝑟,𝑁𝑠=1𝜇𝑠𝑟=1,(62) where 𝛾𝑠𝑟 and 𝜇𝑠𝑟 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 𝜙𝑥=𝜙𝑦=𝜙𝑧=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 𝜍𝑟=0.1% are added to the Kineto-elastodynamic model of the manipulator.

From (48), we get𝜔2𝐌+𝐊𝝋=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=𝝋𝜼,(65) where 𝝋𝝋=1𝝋2𝝋3𝝋4 is the modal matrix.

Substituting (65) into (50) and adding the modal damping ratio yields𝝋𝑇̈𝐌𝝋𝜼+𝝋𝑇̇𝐂𝝋𝜼+𝝋𝑇𝐊𝝋𝜼=𝝋𝑇𝐅𝑑,(66) where𝝋𝑇𝐌𝝋=𝐄4,𝝋𝑇𝐂𝝋=𝐂=diag2𝜍1𝜔12𝜍2𝜔22𝜍3𝜔32𝜍4𝜔4,𝝋𝑇𝐊𝝋=𝛀2𝜔=diag21𝜔22𝜔23𝜔24(67)𝐄4 is the unit matrix of order four.

Substituting (67) into (66) yields̈𝜼+𝐂̇𝜼+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 intervals𝑇Δ𝑡=𝑛.(70) In the 𝑖th time interval (𝑡𝑖1<𝑡<𝑡𝑖), the motion equation of the manipulator is ̈𝜂𝑟+2𝜍𝑟𝜔𝑟(𝑖)̇𝜂𝑟+𝜔𝑟(𝑖)2𝜂𝑟=𝑁𝑟,(𝑟=1,2,,𝑁).(71) So the contribution of the 𝑟th mode to the displacement response is 𝜂𝑟(1𝑡)=𝜔(𝑖)𝑑𝑟𝑡𝑡𝑖1𝑁𝑟(𝜏)𝑒𝜍𝑟𝜔𝑟(𝑖)(𝑡𝜏)sin𝜔(𝑖)𝑑𝑟(+𝜂𝑡𝜏)𝑑𝜏𝑟𝑡𝑖11𝜍2𝑟1/2𝑒𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑡𝑖1)𝜔cos(𝑖)𝑑𝑟𝑡𝑡𝑖1𝜓𝑟+̇𝜂𝑟𝑡𝑖1𝜔(𝑖)𝑑𝑟𝑒𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑡𝑖1)sin𝜔(𝑖)𝑑𝑟𝑡𝑡𝑖1,(𝑟=1,2,,𝑁),(72) where𝜔(𝑖)𝑑𝑟=1𝜍2𝑟1/2𝜔𝑟(𝑖),(73)𝜓𝑟𝜍=arctan𝑟1𝜍2𝑟1/2.(74) Substituting 𝑡=𝑡𝑖 into (73) yields𝜂𝑟𝑡𝑖=1𝜔(𝑖)𝑑𝑟𝑡𝑖𝑡𝑖1𝑁𝑟(𝜏)𝑒𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑖𝜏)sin𝜔(𝑖)𝑑𝑟𝑡𝑖+𝜂𝜏𝑑𝜏𝑟𝑡𝑖11𝜍2𝑟1/2𝑒𝜍𝑟𝜔𝑟(𝑖)Δ𝑡𝜔cos(𝑖)𝑑𝑟Δ𝑡𝜓𝑟+̇𝜂𝑟𝑡𝑖1𝜔(𝑖)𝑑𝑟𝑒𝜍𝑟𝜔𝑟(𝑖)Δ𝑡sin𝜔(𝑖)𝑑𝑟Δ𝑡,(𝑟=1,2,,𝑁).(75) Taking the derivative of (72) with respect to time and substituting 𝑡=𝑡𝑖 into it yields ̇𝜂𝑟𝑡𝑖=1𝜔(𝑖)𝑑𝑟𝑡𝑖𝑡𝑖1𝑁𝑟(𝜏)𝜍𝑟𝜔𝑟(𝑖)𝑒𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑖𝜏)sin𝜔(𝑖)𝑑𝑟𝑡𝑖𝜏+𝜔(𝑖)𝑑𝑟𝑒𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑖𝜏)cos𝜔(𝑖)𝑑𝑟𝑡𝑖𝜏𝑑𝜏𝜂𝑟𝑡𝑖1𝜔(𝑖)𝑑𝑟1𝜍2𝑟1/2𝑒𝜍𝑟𝜔𝑟(𝑖)Δ𝑡𝜔sin(𝑖)𝑑𝑟Δ𝑡𝜓𝑟+𝜍𝑟𝜔𝑟(𝑖)1𝜍2𝑟1/2𝑒𝜍𝑟𝜔𝑟(𝑖)Δ𝑡𝜔cos(𝑖)𝑑𝑟Δ𝑡𝜓𝑟̇𝜂𝑟𝑡𝑖1𝜍𝑟𝜔𝑟(𝑖)𝜔(𝑖)𝑑𝑟𝑒𝜍𝑟𝜔𝑟(𝑖)Δ𝑡sin𝜔(𝑖)𝑑𝑟Δ𝑡𝑒𝜍𝑟𝜔𝑟(𝑖)Δ𝑡cos𝜔(𝑖)𝑑𝑟(Δ𝑡𝑟=1,2,,𝑁).(76) It is shown from (75) and (76) that 𝜂𝑟(𝑡𝑖) and ̇𝜂𝑟(𝑡𝑖) can be achieved when 𝜂𝑟(𝑡𝑖1) and ̇𝜂𝑟(𝑡𝑖1) are given. As for 𝑡𝑖=0,𝜂𝑟(𝝋0)=(𝑟)𝑇𝐌𝐔(0),̇𝜂𝑟𝝋(0)=(𝑟)𝑇𝐌̇𝐔(0).(77) So the total displacement response can be achieved by combining these modal contributions 𝐔𝑡𝑖=𝑁𝑟=1𝜂𝑟𝑡𝑖𝝋𝑟𝑡𝑖,(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 𝑎𝑥=0.1+max𝑇212𝜋𝜏,𝑎2𝜋sin(2𝜋𝜏)𝑦=0.1+max𝑇212𝜋𝜏,𝑎2𝜋sin(2𝜋𝜏)𝑧=1.644+max𝑇212𝜋𝜏,𝜙2𝜋sin(2𝜋𝜏)𝑥𝑎=0.1+max𝑇212𝜋𝜏,𝜙2𝜋sin(2𝜋𝜏)𝑦𝑎=0.1+max𝑇212𝜋𝜏,𝜙2𝜋sin(2𝜋𝜏)𝑧𝑎=0.1+max𝑇212𝜋𝜏,2𝜋sin(2𝜋𝜏)(79) where 𝑎max=9.8m/s2, 𝜏=𝑡/𝑇, 𝑇=2𝜋𝑆/𝑎max 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 𝑥 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 𝑥 axis and about the 𝑦 axis.

fig8
Figure 8: Displacement responses of the moving platform.(a) 𝑥 direction, (b) 𝑦 direction, (c) 𝑧 direction, (d) 𝜙𝑥 direction, (e) 𝜙𝑦 direction, and (f) 𝜙𝑧 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 𝑥-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 𝑥-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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