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

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 [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 𝐵₁,𝐵₂, and 𝐵₃ are vertical to the ground.

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 𝑂−𝑥𝑦𝑧 is attached to the fixed base; another moving coordinate frame 𝑂′−𝑢𝑣𝑤 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 matrix^𝑜𝐑_𝑜. Let the rotation matrix be defined by the roll, pitch, and yaw angles, namely, a rotation of 𝜙_𝑥 about the fixed 𝑥 axis, followed by a rotation of 𝜙_𝑦 about the fixed 𝑦 axis, and a rotation of 𝜙_𝑧 about the fixed 𝑧 axis. Thus, the rotation matrix is^𝑜𝐑_𝑜=Rot𝑧,𝜙_𝑧Rot𝑦,𝜙_𝑦Rot𝑥,𝜙_𝑥,(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]𝝎=𝜙̇_𝑥𝜙̇_𝑦𝜙̇_𝑧^𝑇.(2)

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 𝜙_𝑖 about the 𝑧 axis resulting in a 𝐶_𝑖−𝑥^𝑖𝑦^𝑖𝑧^𝑖 system followed by another rotation of 𝜑_𝑖 about the rotated 𝑦^𝑖-axis. So the rotation matrix of the 𝑖th strut can be written as^𝑜𝐑_𝑖=Rot𝑧,𝜙_𝑖Rot𝑦^𝑖,𝜑_𝑖=⎡⎢⎢⎢⎢⎣c𝜙^𝑖c𝜑_𝑖−s𝜙_𝑖c𝜙_𝑖s𝜑_𝑖s𝜙_𝑖c𝜑_𝑖c𝜙_𝑖s𝜙_𝑖s𝜑_𝑖−s𝜑_𝑖0c𝜑_𝑖𝑖=1,2,…,6.⎤⎥⎥⎥⎥⎦,(3) The unit vector along the strut in the coordinate system 𝑂−𝑥𝑦𝑧 is𝐰_𝑖=^𝑜𝐑_𝑖𝑖𝐰_𝑖=^𝑜𝐑_𝑖⎡⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣c𝜙_𝑖s𝜑_𝑖s𝜙_𝑖s𝜑_𝑖c𝜑_𝑖⎤⎥⎥⎥⎥⎦.(4) So the Euler angles 𝜙_𝑖 and 𝜑_𝑖 can be computed as follows:cs𝜑𝜑_𝑖𝑖=𝑤_𝑖𝑧,s𝜙_𝑖=𝑤^2𝑖𝑥+𝑤^2𝑖𝑦,0≤𝜑_𝑖<𝜋,=𝑤^𝑖𝑦s𝜑_𝑖c𝜙_𝑖,𝜑_𝑖≠0,=𝑤^𝑖𝑥s𝜑_𝑖if𝜑_𝑖,𝜑_𝑖≠0,=0,then𝜙_𝑖=0.(5)

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=−𝐉^−𝑇⎡⎢⎢⎣𝐟_𝑒𝐧_𝑒⎤⎥⎥⎦−𝐉_−𝑇⎧⎪⎨⎪⎩⎡⎢⎢⎣𝑚𝟎_𝑝𝐠⎤⎥⎥⎦+⁶_𝑖=1𝐉^{𝑇𝑖𝑣𝜔}+𝐉_𝑇⎡⎢⎢⎣𝑚_𝑖𝑖𝟎𝐑_𝑜𝐠⎤⎥⎥⎦+𝐉_−𝑇(𝑚_𝑐1𝐠)^𝑇𝐞₁𝑚_𝑐2𝐠^𝑇𝐞₂𝑚_𝑐3𝐠^𝑇𝐞₃𝑚_𝑐4𝐠^𝑇𝐞₄𝑚_𝑐5𝐠^𝑇𝐞₅𝑚_𝑐6𝐠^𝑇𝐞₆^𝑇⎧⎪⎨⎪⎩⎡⎢⎢⎣𝑚_𝑝𝐯_𝑜̇𝐈_𝑝𝝎̇⎤⎥⎥⎦+⁶_𝑖=1𝐉^{𝑇𝑖𝑣𝜔}⎡⎢⎢⎣𝑚_𝑖𝑖𝐯_𝑖𝑖̇𝐈_𝑖𝑖𝝎̇_𝑖⎤⎥⎥⎦+𝐉_𝑇𝑚_𝑐1𝑞₁̈𝑚_𝑐2𝑞₂̈𝑚_𝑐3𝑞₃̈𝑚_𝑐4𝑞₄̈𝑚_𝑐5𝑞₅̈𝑚_𝑐6𝑞₆̈^𝑇+𝐉_−𝑇⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩⎡⎢⎢⎣𝝎×𝟎_𝑜𝐈_𝑝𝝎⎤⎥⎥⎦+⁶_𝑖=1𝐉^{𝑇𝑖𝑣𝜔}⎡⎢⎢⎣𝟎_𝑖𝝎_𝑖×^𝑖𝐈_𝑖𝑖𝝎_𝑖⎤⎥⎥⎦⎫⎪⎬⎪⎭,(6)where 𝟎=000^𝑇,𝐉_𝑖𝑣𝜔 is the link Jacobian matrix which maps the velocity of the moving platform into the velocity of the 𝑖th strut in the 𝐶_𝑖−𝑥_𝑖𝑦_𝑖𝑧_𝑖 coordinate system, 𝑚_𝑝, 𝑚_𝑐𝑖, and 𝑚_𝑖 denote the mass of the moving platform, the mass of the slider, and the mass of the 𝑖th strut, respectively, ^𝑜𝐈_𝑝 is the inertia matrix of the moving platform taken about the center of mass expressed in the 𝑂−𝑥𝑦𝑧 coordinate system, ^𝑖𝐈_𝑖 is the inertia matrix of the 𝑖th cylindrical strut about their respective centers of mass expressed in the 𝐶_𝑖−𝑥_𝑖𝑦_𝑖𝑧_𝑖 coordinate system, 𝐟_𝑒 and 𝐧_𝑒 are the external force and moment exerted at the center of mass of the moving platform, 𝐯̇ and 𝝎̇ are the linear and angular acceleration of the moving platform,𝐯̇^𝑖_𝑖,^𝑖𝝎_𝑖, and 𝝎̇^𝑖_𝑖 are the linear acceleration, the angular velocity, and acceleration of the ith strut expressed in the 𝐶_𝑖−𝑥_𝑖𝑦_𝑖𝑧_𝑖 coordinate system, respectively. 𝑞_𝑖̈ 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. And𝐉=diag1𝐰^𝑇1𝐞₁1𝐰^𝑇2𝐞₂1𝐰^𝑇3𝐞₃1𝐰^𝑇4𝐞₄1𝐰^𝑇5𝐞₅1×⎡⎢⎢⎣𝐰₁𝐰^𝑇6𝐞₆𝐰₂𝐰₃𝐰₄𝐰₅𝐰₆𝐚₁×𝐰₁𝐚₂×𝐰₂𝐚₃×𝐰₃𝐚₄×𝐰₄𝐚₅×𝐰₅𝐚₆×𝐰₆⎤⎥⎥⎦^𝑇𝐉_𝑖𝑣𝜔,=⎡⎢⎢⎢⎣^𝑖𝐑_𝑜−𝑆^𝑖𝐚_𝑖^𝑖𝐑_𝑜+𝑙^𝑖2𝑆^𝑖𝐰_𝑖𝐉_𝑖𝜔1𝑙_𝑖𝑆^𝑖𝐰_𝑖^𝑖𝐑_𝑜−𝑆^𝑖𝐰_𝑖𝑆^𝑖𝐚_𝑖^𝑖𝐑_𝑜−^𝑖𝐰_𝑖×^𝑖𝐞_𝑖𝐰^𝑇𝑖𝐰^𝑇𝑖𝐞_𝑖𝐚_𝑖×𝐰_𝑖^𝑇𝐰^𝑇𝑖𝐞_𝑖⎤⎥⎥⎥⎦=⎡⎢⎢⎣𝐉^𝑖𝑣𝐉_𝑖𝜔𝑆_𝑖⎤⎥⎥⎦,𝐰_𝑖=⎡⎢⎢⎢⎢⎣0−^𝑖𝑤_𝑖𝑖𝑧𝑤_𝑖𝑖𝑦𝑤_𝑖𝑧0−^𝑖𝑤_𝑖𝑥𝑆_𝑖−^𝑖𝑤_𝑖𝑖𝑦𝑤_𝑖𝑥0⎤⎥⎥⎥⎥⎦,𝐚_𝑖=⎡⎢⎢⎢⎢⎣0−^𝑖𝑎_𝑖𝑖𝑧𝑎_𝑖𝑖𝑦𝑎_𝑖𝑧0−^𝑖𝑎_𝑖𝑥−^𝑖𝑎_𝑖𝑖𝑦𝑎_𝑖𝑥0⎤⎥⎥⎥⎥⎦,_𝑖𝝎_𝑖=𝐉_𝑖𝜔𝑞_𝑖̈⎡⎢⎢⎣𝝎𝐯⎤⎥⎥⎦,=1𝐰^𝑇𝑖𝐞_𝑖𝐰^𝑇𝑖𝐯̇+𝐚_𝑖×𝐰_𝑖^𝑇𝝎̇+𝐰^𝑇𝑖𝝎×𝝎×𝐚_𝑖−𝐰^𝑇𝑖𝝎_𝑖×𝝎_𝑖×𝑙_𝑖𝐰_𝑖_𝑖,𝝎̇_𝑖=𝐉_𝑖𝜔⎡⎢⎢⎣𝝎𝐯̇̇⎤⎥⎥⎦+1𝑙_𝑖𝚫₁𝚫₁+𝚫₂,=−^𝑖𝐰_𝑖×^𝑖𝐞_𝑖𝐰^𝑇𝑖𝐞_𝑖𝐰^𝑇𝑖𝚫₂𝝎𝐚^𝑇𝑖𝝎−𝐰^𝑇𝑖𝐚_𝑖𝝎^𝑇𝝎+𝑙_𝑖||𝝎_𝑖×𝐰_𝑖||²,=^𝑖𝝎^𝑇𝑖𝑖𝐚_𝑖^𝑖𝐰_𝑖×^𝑖𝝎_𝑖−^𝑖𝝎^𝑇𝑖𝝎^𝑖𝐰_𝑖×^𝑖𝐚_𝑖,_𝑖𝐯̇_𝑖=𝐉_𝑖𝑣⎡⎢⎢⎣𝝎𝐯̇̇⎤⎥⎥⎦+𝑆_𝑖𝝎_𝑖𝑆^𝑖𝝎_𝑖^𝑖𝐚_𝑖+12𝑆^𝑖𝐰_𝑖𝚫₁+𝚫₂−𝑙^𝑖2𝑆^𝑖𝝎_𝑖𝑆^𝑖𝝎_𝑖^𝑖𝐰_𝑖.(7)𝐞_𝑖, 𝐚_𝑖, and 𝐰_𝑖 are shown in Figure 2; they are the unit vector along the lead screw, the vector 𝐎^𝐀_𝐢, and the unit vector along strut 𝐶_𝑖𝐴_𝑖, 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 𝜹=𝜹_𝑖𝜹_𝑗^𝑇,(8) where𝜹𝜹_𝑖𝑗=𝑢_𝑖𝑣_𝑖𝑤_𝑖𝜃_𝑖𝑥𝜃_𝑖𝑦𝜃_𝑖𝑧^𝑇,=𝑢_𝑗𝑣_𝑗𝑤_𝑗𝜃_𝑗𝑥𝜃_𝑗𝑦𝜃_𝑗𝑧^𝑇.(9)

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=𝑢𝑣𝑤𝜃^𝑇=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 as𝜃𝑢==𝑎𝑑₀₀+𝑎₁𝑥,+𝑑₁𝑥.(11) The transverse displacements of the point 𝑃 can be expressed as 𝑤𝑣==𝑏𝑐₀₀+𝑏₁𝑥+𝑏₂𝑥²+𝑏₃𝑥³,+𝑐₁𝑥+𝑐₂𝑥²+𝑐₃𝑥³.(12) Substituting (11) and (12) into (10) yields𝑝=⎡⎢⎢⎢⎢⎢⎢⎣𝑤𝜃𝑣𝑢⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣𝐇^𝑢𝐇(𝑥)𝐚_𝑣𝐇(𝑥)𝐛_𝑤𝐇(𝑥)𝐜_𝜃(𝑥)𝐝⎤⎥⎥⎥⎥⎥⎥⎦,(13) where𝐛=𝐚=𝑏₀𝑎₀𝑎₁^𝑇,𝐜=𝑐₀𝑏₁𝑏₂𝑏₃^𝑇,𝐝=𝑑𝑐₁₀𝑐₂𝑐₃^𝑇,𝐇_𝑢𝑑₁^𝑇,𝐇(𝑥)=𝐇_𝜃(𝑥)=1𝑥_𝑣,(𝑥)=𝐇_𝑤(𝑥)=1𝑥𝑥²𝑥³.(14)

Considering the node 𝑖 and the node 𝑗, where 𝑥=0 and 𝑥=𝑙, respectively, yields𝐛𝐚==𝐀𝐀^−1𝑢𝜃_−1𝑣𝑤𝜹_𝑢,𝐜=𝐀_−1𝑣𝑤𝜹_𝑣,𝐝=𝐀_−1𝑢𝜃𝜹_𝑤,𝜹_𝜃,(15) where𝜹_𝑣𝜹_𝑢=𝑢_𝑖𝑢_𝑗^𝑇,𝜹_𝑤=𝑣_𝑖𝜃_𝑖𝑧𝑣_𝑗𝜃_𝑗𝑧^𝑇,=𝜹_𝜃𝑤_𝑖𝜃_𝑖𝑦𝑤_𝑗𝜃_𝑗𝑦^𝑇,𝐀_−1𝑢𝜃=𝜃_𝑖𝑥𝜃_𝑗𝑥^𝑇,=⎡⎢⎢⎣−110𝐀_−1𝑣𝑤𝑙1𝑙⎤⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎢⎣−103010000𝑙²−2𝑙𝑙3²−12𝑙𝑙³1𝑙²−2𝑙³1𝑙²⎤⎥⎥⎥⎥⎥⎥⎥⎦.(16) Substituting (15) into (13) yields 𝐩=⎡⎢⎢⎢⎢⎢⎢⎣𝑤𝜃𝑣𝑢⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣𝐡^𝑢𝐡(𝑥)_𝑣𝐡(𝑥)_𝑤𝐡(𝑥)_𝜃(𝑥)⎤⎥⎥⎥⎥⎥⎥⎦𝐀𝜹=𝐍𝜹,(17) where𝐡𝐡_𝑢(𝑥)=100000𝑥00000_𝑣,𝐡_𝑤(𝑥)=01000𝑥0𝑥²000𝑥³,𝐡_𝜃(𝑥)=0010𝑥000𝑥²0𝑥³0,𝐀=(𝑥⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣)−=1000001001000000001000100001000000001000000001𝑥00000000000000,0000000000000000000000000𝑙−0300001𝑙00000𝑙²000−2𝑙0𝑙3²000−100−3𝑙𝑙²0−2𝑙000𝑙3²0−1000−1𝑙0002𝑙000001𝑙00𝑙³01𝑙²000−2𝑙³01𝑙²020𝑙³0001𝑙²0−2𝑙³0001𝑙²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(18)

Considering the knowledge of material mechanics, the strain of the point 𝑃 is 𝜺=⎡⎢⎢⎢⎢⎢⎢⎣𝜀⁰𝜀_𝑏𝑦𝜀_𝑏𝑧𝜀_𝑟⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎣−𝑦𝑑𝑑𝑑𝑥𝑢₂𝑣𝑑𝑥²−𝑧𝑑²𝑤𝑑𝑥²𝐺𝐸1𝜋6𝑑𝐽^𝑘3𝑑𝜃𝑑𝑥⎤⎥⎥⎥⎥⎥⎥⎥⎦,(19) where 𝜀₀ is the axial strain, 𝜀_𝑏𝑦 and 𝜀_𝑏𝑧 are the flexural strain in the plane 𝑂𝑥𝑦 and 𝑂𝑥𝑧, respectively, 𝜀_𝑟 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, 𝐽_𝑘 is the polar moment of inertia of cross-section, 𝑑 is the diameter of the strut.

Substituting (17) into (19) yields𝜺=⎡⎢⎢⎢⎢⎢⎢⎢⎣ℎ^𝑢−𝑦ℎ(_𝑣𝑥)−𝑧ℎ_𝑤(𝑥)𝐺(𝑥)𝐸1𝜋6𝑑𝐽^𝑘3ℎ^𝜃(𝑥)⎤⎥⎥⎥⎥⎥⎥⎥⎦𝐀𝜹=𝐁𝜹,(20) whereℎℎ^𝑢(𝑥)=000000100000_𝑣,ℎ(𝑥)=000000020006𝑥_𝑤,ℎ(𝑥)=00000000206𝑥0_𝜃,(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 as𝑈=12𝜺^𝑇𝝈𝑑𝑣.(23) Substituting (20) and (22) into (23) yields 𝑈=12𝐸𝜹^𝑇𝐁^𝑇𝐁𝜹𝑑𝑣=12𝜹^𝑇𝐸𝐁^𝑇𝐁𝑑𝑣𝜹=12𝜹^𝑇𝐤𝜹,(24) where𝐤=12𝐸𝐁^𝑇𝐁𝑑𝑣(25) is the element stiffness matrix. Substituting (20) into (25) yields𝐤=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐸𝐴0𝑙12𝐸0𝐼_𝑧0000−𝐸𝑙𝐴00000𝑙³0006𝐸𝐼^𝑧𝑙²0−12𝐸𝐼^𝑧𝑙³0006𝐸𝐼^𝑧𝑙²0012𝐸𝐼^𝑦𝑙³0−6𝐸𝐼^𝑦𝑙²000−12𝐸𝐼^𝑦𝑙³0−6𝐸𝐼^𝑦𝑙²000𝐺𝐽_𝑘000−6𝐸𝐼_𝑦𝑙00000−𝐺𝑙𝐽^𝑘00𝑙²04𝐸𝐼^𝑦𝑙0006𝐸𝑙²𝐼^𝑦02𝐸𝐼^𝑦06𝐸𝐼_𝑧𝑙0𝑙²0004𝐸𝐼^𝑧𝑙0−6𝐸𝑙²𝐼^𝑧0002𝐸𝐼^𝑧−𝐸𝐴𝑙0𝑙−120𝐸𝐼_𝑧0000𝐸𝑙𝐴00000𝑙³000−6𝐸𝐼^𝑧𝑙²012𝐸𝐼^𝑧𝑙³000−6𝐸𝐼^𝑧𝑙²00−12𝐸𝐼^𝑦𝑙³06𝐸𝐼^𝑦𝑙²00012𝐸𝐼^𝑦𝑙³06𝐸𝐼^𝑦𝑙²000−𝐺𝐽_𝑘000−6𝐸𝐼_𝑦𝑙00000𝐺𝑙𝐽^𝑘00𝑙²02𝐸𝐼^𝑦𝑙0006𝐸𝑙²𝐼^𝑦04𝐸𝐼^𝑦06𝐸𝐼_𝑧𝑙0𝑙²0002𝐸𝐼^𝑧𝑙0−6𝐸𝑙²𝐼^𝑧0004𝐸𝐼^𝑧𝑙(2⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,6)where 𝐼_𝑦 and 𝐼_𝑧 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:𝐩𝐩̇̈_𝑎=𝐩̇_𝑟+𝐩̇,_𝑎=𝐩̈_𝑟+𝐩̈,(27) where 𝐩̇_𝑎 and 𝐩̈_𝑎 are the absolute velocity and the absolute acceleration of a certain point within the element, 𝐩̇_𝑟 and 𝐩̈_𝑟 are the velocity and the acceleration of the rigid-body motion, and 𝐩̇ and 𝐩̈ are the velocity and the acceleration of the elastic motion.

It can be proved that (see [27])𝐩𝐩̈̇_𝑟𝑟=𝐍_𝑟𝜹̇_𝑟=𝐍𝜹̇_𝑟,=𝐍_𝑟𝜹̈_𝑟=𝐍𝜹̈_𝑟,(28) where 𝜹̇_𝑟 and 𝜹̈_𝑟 are the nodal velocity and the acceleration of the rigid-body motion.

So the kinetic energy of the element can be expressed as 𝑇=12^𝑙0=1𝑚(𝑥)𝐩̇^𝑇𝑎𝐩̇_𝑎𝑑𝑥2^𝑙0=1𝑚(𝑥)𝜹̇^𝑇𝑎𝐍^𝑇𝐍𝜹̇_𝑎𝑑𝑥2𝜹̇^𝑇𝑎^𝑙0𝑚(𝑥)𝐍^𝑇𝐍𝑑𝑥𝜹̇_𝑎=12𝜹̇^𝑇𝑎𝐦𝜹̇_𝑎,(29) where 𝜹̇_𝑎 is the absolute velocity of the node. And 𝐦=12^𝑙0𝑚(𝑥)𝐍^𝑇𝐍𝑑𝑥(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=𝜌𝐴𝑙420⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣100040150006105006140000𝐽^𝑘−2002𝑙2002𝑙7000500450040001003𝑙−1003𝑙00−22𝑙𝐴040𝑙₂000070𝐴𝐽^𝑘00022𝑙00040𝑙₂00−13𝑙0−3𝑙²0013𝑙000−3𝑙²70000500045000470000𝐽_𝑘−1003𝑙1003𝑙1004010506100560002002𝑙−2002𝑙0013𝑙𝐴0−30𝑙₂000014𝐴0𝐽^𝑘000−13𝑙000−30𝑙₂0022𝑙04𝑙²00−22𝑙0004𝑙²⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(32)

Substituting (23) and (29) into Lagrange equation yields 𝑑𝑑𝑡𝜕𝜕𝑇𝜹̇−𝜕𝜕𝑇𝜹+𝜕𝜕𝑈𝜹=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:𝐦𝜹̈+𝐤𝜹=𝐟−𝐦𝜹̈_𝑟.(34) Equation (34) can be expressed in the coordinate system 𝑂−𝑥𝑦𝑧 as𝐌_𝑒𝑖𝐔̈_𝑒𝑖+𝐊_𝑒𝑖𝐔_𝑒𝑖=𝐅_𝑒𝑖−𝐌_𝑒𝑖𝐔̈_𝑒𝑟𝑖,𝑖=1,2,…,6(35) where𝐔̈𝐔_𝑒𝑖=𝐑^𝑇𝑖𝜹,_𝑒𝑖=𝐑^𝑇𝑖𝐅𝜹̈,_𝑒𝑖𝐔̈_𝑒𝑟𝑖=𝐑^𝑇𝑖𝜹̈_𝑟,𝐌=𝐑^𝑇𝑖𝐟,_𝑒𝑖𝐊_𝑒𝑖=𝐑^𝑇𝑖𝐦𝐑_𝑖,𝐑_𝑖=𝐑^𝑇𝑖𝐤𝐑_𝑖,=diag^𝑜𝐑_𝑖𝐑𝐨𝐭𝑦,−𝜋2^𝑇^𝑜𝐑_𝑖𝐑𝐨𝐭𝑦,−𝜋2^𝑇^𝑜𝐑_𝑖𝐑𝐨𝐭𝑦,−𝜋2^𝑇^𝑜𝐑_𝑖𝐑𝐨𝐭𝑦,−𝜋2^𝑇(36)

3.1.2. Strut Dynamic Equation

For the strut within 𝑖th kinematic chain, the dynamic motion in the coordinate system 𝑂−𝑥𝑦𝑧 can be assembled as𝐌^𝑠𝑖𝐔̈^𝑠𝑖+𝐊^𝑠𝑖𝐔^𝑠𝑖=𝐅^𝑠𝑖−𝐌^𝑠𝑖𝐔̈^{𝑠𝑟𝑖},(37) where 𝐌^𝑠𝑖=³_𝑗=1𝐊_𝑠𝑖𝐀^𝑇𝑗𝐌_𝑒𝑖𝐀_𝑗,=³_𝑗=1𝐅_𝑠𝑖𝐀^𝑇𝑗𝐊_𝑒𝑖𝐀_𝑗,=³_𝑗=1𝐀^𝑇𝑗𝐅_𝑒𝑖=⎡⎢⎢⎢⎢⎣𝟎𝐅^∗𝑐2𝑖_18×1𝐅^∗𝑐1𝑖𝐀₁⎤⎥⎥⎥⎥⎦,𝐀₂=𝐄₁₂𝟎_12×12,𝐀₃=𝟎_12×6𝐄_𝟏𝟐𝟎_12×6,=𝟎_12×12𝐄₁₂,(38) where 𝐀_𝑗 is the connectivity matrix which maps the total nodal coordinates 𝐔^𝑠𝑖 of the strut within the 𝑖th kinematics chain into the 𝑗th nodal coordinate within the strut. 𝐔̈^𝑠𝑖 and 𝐔̈^{𝑠𝑟𝑖} 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. 𝐅^∗𝑐1𝑖 is the internal force between the strut and the slider. 𝐅^∗𝑐2𝑖 is the internal force between the rigid moving platform and the strut. All the above coordinates are measured in the coordinate system 𝑂−𝑥𝑦𝑧.

3.2. Slider Dynamic Equation

The oscillation of the slider along the axial direction of the lead-screw can be expressed as𝑚_𝑐𝑖𝑈̈_𝑐𝑖+𝑘_𝑐𝑖𝑈_𝑐𝑖=𝑓_𝑖−𝑚_𝑐𝑖𝑞_𝑖̈−𝐞^𝑇𝑖𝐅^∗𝑐1𝑖,(39) where 𝑈_𝑐𝑖 is the elastic displacement of the slider along the axial direction of the lead-screw. 𝑘_𝑐𝑖 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 𝑘_𝑐𝑠𝑖,𝑘_𝑐𝑛𝑖, and 𝑘_𝑐𝑏𝑖 denote their respective axial stiffness then the equivalent axial stiffness can be expressed as1𝑘_𝑐𝑖=1𝑘_𝑐𝑠𝑖+1𝑘_𝑐𝑛𝑖+1𝑘_𝑐𝑏𝑖,𝑘_𝑐𝑠𝑖=𝐴^𝑠𝐸_𝑠𝐿_1𝑖+𝐿_2𝑖𝐿_1𝑖𝐿_2𝑖,(40) where 𝐴_𝑠 and 𝐸_𝑠 stand for the cross-sectional area of the lead screw and its Young’s modular, and 𝐿_1𝑖 and 𝐿_2𝑖 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 𝐔_𝑐2𝑖=𝐄₃−𝑆𝐚_𝑖𝐔^𝑝,(41) where 𝐔_𝑝 denotes the generalized coordinates of vibration motion of the moving platform, 𝐔_𝑐2𝑖=𝐔^𝑠𝑖(1∶3,1∶1) denotes the elastic displacement of the corresponding node within the flexible strut, 𝑆(𝐚_𝑖) is the screw matrix of 𝐚_𝑖, 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 𝐞^𝑇𝑖𝐔_𝑐1𝑖=𝑈_𝑐𝑖,(42) where 𝐔_𝑐1𝑖=𝐔^𝑠𝑖(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 𝑂−𝑥𝑦𝑧 is⎡⎢⎢⎣𝑚_𝑝𝐄₃𝟎_3×3𝟎_𝑜3×3𝐈_𝑝⎤⎥⎥⎦𝐔̈_𝑝−⎡⎢⎢⎢⎢⎢⎣𝐟^𝑒−𝑚_𝑝𝐯̇−⁶_𝑖=1𝐅^∗𝑐2𝑖𝐧_𝑒−^𝑜𝐈_𝑝𝝎̇−𝝎×^𝑜𝐈_𝑝𝝎−⁶_𝑖=1𝐚_𝑖×𝐅^∗𝑐2𝑖⎤⎥⎥⎥⎥⎥⎦=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 as𝐌_𝑘𝑖𝐔̈_𝑘𝑖+𝐊_𝑘𝑖𝐔_𝑘𝑖=𝐅_𝑘𝑐𝑖+𝐅_𝑘𝑑𝑖,(44) where 𝐔_𝑘𝑖=⎡⎢⎢⎣𝐔^𝑠𝑖𝐌_𝑘𝑖(1∶𝑈18,1∶1)_𝑐𝑖⎤⎥⎥⎦,=⎡⎢⎢⎣𝐌^𝑠𝑖𝐌(1∶18,1∶18)𝐌^𝑠𝑖(1∶18,21∶21)_𝑠𝑖(21∶21,1∶18)𝐌^𝑠𝑖(21∶21,21∶21)+𝑚_𝑐𝑖𝐌𝑖=1,2,3,_𝑘𝑖⎤⎥⎥⎦=⎡⎢⎢⎣𝐌^𝑠𝑖𝐌(1∶18,1∶18)𝐌^𝑠𝑖(1∶18,20∶20)_𝑠𝑖(20∶20,1∶18)𝐌^𝑠𝑖(20∶20,20∶20)+𝑚_𝑐𝑖𝐌𝑖=4,5,_𝑘𝑖⎤⎥⎥⎦=⎡⎢⎢⎣𝐌^𝑠𝑖𝐌(1∶18,1∶18)𝐌^𝑠𝑖(1∶18,19∶19)_𝑠𝑖(19∶19,1∶18)𝐌^𝑠𝑖(19∶19,19∶19)+𝑚_𝑐𝑖𝐊𝑖=6,_𝑘𝑖⎤⎥⎥⎦=⎡⎢⎢⎣𝐊^𝑠𝑖𝐊(1∶18,1∶18)𝐊^𝑠𝑖(1∶18,21∶21)_𝑠𝑖(21∶21,1∶18)𝐊^𝑠𝑖(21∶21,21∶21)+𝑘_𝑐𝑖𝐊𝑖=1,2,3,_𝑘𝑖⎤⎥⎥⎦=⎡⎢⎢⎣𝐊^𝑠𝑖𝐊(1∶18,1∶18)𝐊^𝑠𝑖(1∶18,20∶20)_𝑠𝑖(20∶20,1∶18)𝐊^𝑠𝑖(20∶20,20∶20)+𝑘_𝑐𝑖𝐊𝑖=4,5,_𝑘𝑖⎤⎥⎥⎦=⎡⎢⎢⎣𝐊^𝑠𝑖𝐊(1∶18,1∶18)𝐊^𝑠𝑖(1∶18,19∶19)_𝑠𝑖(19∶19,1∶18)𝐊^𝑠𝑖(19∶19,19∶19)+𝑘_𝑐𝑖𝐅𝑖=6,_𝑘𝑐𝑖⎤⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝟎𝐅^∗𝑐2𝑖_15×1𝐅_𝑘𝑑𝑖𝟎⎤⎥⎥⎥⎥⎦,=⎡⎢⎢⎣𝟎^18×1𝑓_𝑖⎤⎥⎥⎦−𝐌_𝑘𝑖𝐔_𝑘𝑟𝑖𝐔̈_𝑘𝑟𝑖,=⎡⎢⎢⎣𝐔^{𝑠𝑟𝑖}(1∶𝑞1_𝑖̈8,1∶1)⎤⎥⎥⎦(45)𝐅_𝑘𝑐𝑖 is the internal forces between the elements within the strut, 𝐅_𝑘𝑑𝑖 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 yields𝐃^𝑇𝐌^𝐃𝐔̈+𝐃^𝑇𝐊^𝐃𝐔=𝐃^𝑇𝐅^𝑐+𝐃^𝑇𝐅^𝑑,(46) where𝐃=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐄⁶𝟎_6×96𝐄₃−𝑆𝐚₁𝟎_3×96𝟎_16×6𝐄₁₆𝟎_16×80𝐄₃−𝑆𝐚₂𝟎_3×96𝟎_16×22𝐄₁₆𝟎_16×64𝐄₃−𝑆𝐚₃𝟎_3×96𝟎_16×38𝐄₁₆𝟎_16×48𝐄₃−𝑆𝐚₄𝟎_3×96𝟎_16×54𝐄₁₆𝟎_16×32𝐄₃−𝑆𝐚₅𝟎_3×96𝟎_16×70𝐄₁₆𝟎_16×16𝐄₃−𝑆𝐚₆𝟎_3×96𝟎_16×86𝐄₁₆𝐔=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐔_𝑝⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝐔𝐔_𝑘1(4∶19,1)_𝑘2𝐔(4∶19,1)_𝑘3𝐔(4∶19,1)_𝑘4𝐔(4∶19,1)_𝑘5𝐔(4∶19,1)_𝑘6𝐅(4∶19,1)_𝑐⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎣−⁶_𝑖=1𝐅^∗𝑐2𝑖^𝑇−⁶_𝑖=1𝐚_𝑖×𝐅^∗𝑐2𝑖^𝑇𝐅^{𝑇𝑘𝑐1}𝐅^{𝑇𝑘𝑐2}𝐅^{𝑇𝑘𝑐3}𝐅^{𝑇𝑘𝑐4}𝐅^{𝑇𝑘𝑐5}𝐅^{𝑇𝑘𝑐6}⎤⎥⎥⎦^𝑇𝐅_𝑑,=𝐟_𝑒−𝑚_𝑝𝐯̇^𝑇𝐧_𝑒−^𝑜𝐈_𝑝𝝎̇−𝝎×(^𝑜𝐈_𝑝𝝎)^𝑇𝐅^{𝑇𝑘𝑑1}𝐅^{𝑇𝑘𝑑2}𝐅^{𝑇𝑘𝑑3}𝐅^{𝑇𝑘𝑑4}𝐅^{𝑇𝑘𝑑5}𝐅^{𝑇𝑘𝑑6}^𝑇(47) Simplifying (46) yields the kineto-elasticdynamic model of the manipulator𝐌𝐔̈+𝐊𝐔=𝐅_𝑑,(48) where𝐌=𝐃^𝑇𝐌^𝐃,(49)𝐊=𝐃^𝑇𝐊^𝐃,(50)𝐅_𝑑=𝐃^𝑇𝐅^𝑑.(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.

Table 1 

The parameters of the base platform (m).

Table 2 

The parameters of the moving platform which are measured in the coordinate frame 𝑂^−𝑢𝑣𝑤 (m).

Table 3 

The length of the strut 𝐶_𝑖𝐴_𝑖 (m).

Table 4 

The mass parameters of the manipulator (kg).

The mass of the moving platform is 𝑚_𝑝=200 kg. The inertia parameters used in the simulation are given as^𝑜𝐈_𝑝=⎡⎢⎢⎢⎢⎣17.0033317.0033333.00333⎤⎥⎥⎥⎥⎦kg⋅m²,_𝑖𝐈_𝑖=⎡⎢⎢⎢⎢⎣1.200791.200790.00005⎤⎥⎥⎥⎥⎦kg⋅m².(52) Other parameters used in the simulation are given as

𝐸=2.06×10¹¹ Pa,  𝐺=79.38×10⁹ Pa, 𝐸_𝑠=2.06×10¹¹ Pa, 𝐴_𝑠=1.96×10⁻³ m², 𝐿_1𝑖+𝐿_2𝑖=1.1 m, 𝜌=7800 kg/m³, 𝑑_𝑖=0.244 m, 𝑑=0.05 m, ℎ₁=2 m, ℎ₂=1.5 m, ℎ=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 getdet−𝜔²𝐌+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 𝜙_𝑥=𝜙_𝑦=𝜙_𝑧=0 and 𝑧=𝑧₀.

(a)

(b)

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 get−𝜔^2𝑟𝐌+𝐊𝝋_𝑟=0,(54) where 𝝋_𝑟 and 𝜔_𝑟 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 𝑝_𝑚 such as the radius of the strut and the radius of the lead screw yields −2𝜔^𝑟𝜕𝜔_𝑟𝜕𝑝_𝑚𝐌−𝜔^2𝑟𝜕𝐌𝜕𝑝_𝑚+𝜕𝐊𝜕𝑝_𝑚𝝋^𝑟+−𝜔^2𝑟𝐌+𝐊𝜕𝝋^𝑟𝜕𝑝_𝑚=0.(55) Taking dot product of 𝝋_𝑟 on both sides of the equation yields𝝋^𝑇𝑟−2𝜔^𝑟𝜕𝜔_𝑟𝜕𝑝_𝑚𝐌−𝜔^2𝑟𝜕𝐌𝜕𝑝_𝑚+𝜕𝐊𝜕𝑝_𝑚𝝋^𝑟+𝝋^𝑇𝑟−𝜔^2𝑟𝐌+𝐊𝜕𝝋^𝑟𝜕𝑝_𝑚=0.(56) Since 𝝋^𝑇𝑟−𝜔^2𝑟𝐌+𝐊𝝋=−𝜔^2𝑟𝐌+𝐊𝝋_𝑟^𝑇=0,_𝑇𝑟𝐌𝝋_𝑟=𝐄.(57) give−2𝜔_𝑟𝜕𝜔_𝑟𝜕𝑝_𝑚−𝜔^2𝑟𝝋^𝑇𝑟𝜕𝐌𝜕𝑝_𝑚𝝋_𝑟+𝝋^𝑇𝑟𝜕𝐊𝜕𝑝_𝑚𝝋_𝑟=0.(58) so 𝜕𝜔_𝑟𝜕𝑝_𝑚=−12𝜔_𝑟𝜔^2𝑟𝝋^𝑇𝑟𝜕𝐌𝜕𝑝_𝑚𝝋_𝑟−𝝋^𝑇𝑟𝜕𝐊𝜕𝑝_𝑚𝝋_𝑟.(59) Figure 6 shows the sensitivity distribution of the manipulator when the pose of the moving platform is given as 𝜙_𝑥=𝜙_𝑦=𝜙_𝑧=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.

(a)

(b)

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 𝑇_𝑠𝑟 and 𝑉_𝑠𝑟 are the maximum kinetic energy and elastic potential energy of the substructures vibrating in its 𝑟th mode. 𝑇_𝐴𝑟 and 𝑉_𝐴𝑟 denote the maximum kinetic energy and elastic potential energy of the system vibrating in the 𝑟th mode. Thus,𝑇_𝐴𝑟=^𝑁_𝑠=1𝑉_𝐴𝑟𝑇_𝑠𝑟,=^𝑁_𝑠=1𝑉_𝑠𝑟,(60) where𝑇_𝑠𝑟=12𝜔^2𝑟𝑉_𝑠𝑟𝐀^𝑠𝑟𝑇𝐦_𝑠𝐀^𝑟𝑠,=12𝐀^𝑠𝑟𝑇𝐤_𝑠𝐀^𝑟𝑠.(61)𝐀^𝑟𝑠 is the oscillating amplitude array of the substructure vibrating in the 𝑟th mode. 𝐦_𝑠 and 𝐤_𝑠 denote the mass matrix and the stiffness matrix of the substructure, respectively.

So the energy ratio of the substructure can be achieved as𝑇_𝑠𝑟𝑇_𝐴𝑟=𝛾_𝑠𝑟,^𝑁_𝑠=1𝑉𝛾_𝑠𝑟=1,_𝑠𝑟𝑉_𝐴𝑟=𝜇_𝑠𝑟,^𝑁_𝑠=1𝜇_𝑠𝑟=1,(62) where 𝛾_𝑠𝑟 and 𝜇_𝑠𝑟 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 𝜙_𝑥=𝜙_𝑦=𝜙_𝑧=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.

(a)

(b)

(c)

(d)

(e)

(f)

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 conditions𝐔𝐔̇₀=𝐔(0),₀=𝐔̇(0),(63)

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

From (48), we get−𝜔²𝐌+𝐊𝝋=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 𝝋=𝝋₁𝝋₂𝝋₃𝝋₄ is the modal matrix.

Substituting (65) into (50) and adding the modal damping ratio yields𝝋^𝑇𝐌𝝋𝜼̈+𝝋^𝑇𝐂𝝋𝜼̇+𝝋^𝑇𝐊𝝋𝜼=𝝋^𝑇𝐅_𝑑,(66) where𝝋_𝑇𝝋^𝑇𝐌𝝋=𝐄₄,𝐂𝝋𝝋=_𝑇𝐂=diag2𝜍₁𝜔₁2𝜍₂𝜔₂2𝜍₃𝜔₃2𝜍₄𝜔₄,𝐊𝝋=𝛀²=diag𝜔²¹𝜔²²𝜔²³𝜔²⁴(67)𝐄₄ is the unit matrix of order four.

Substituting (67) into (66) yields𝜼̈+𝐂𝜼̇+𝛀²𝜼=𝐍,(68) where𝐍=𝝋^𝑇𝐅_𝑑.(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 intervalsΔ𝑡=𝑇𝑛.(70) In the 𝑖th time interval (𝑡_𝑖−1<𝑡<𝑡_𝑖), the motion equation of the manipulator is 𝜂̈_𝑟+2𝜍_𝑟𝜔^𝑟(𝑖)𝜂̇_𝑟+𝜔^𝑟(𝑖)²𝜂_𝑟=𝑁_𝑟,(𝑟=1,2,…,𝑁).(71) So the contribution of the 𝑟th mode to the displacement response is 𝜂_𝑟(𝑡)=1𝜔^{(𝑖)𝑑𝑟}^𝑡𝑡_𝑖−1+𝜂𝑁_𝑟(𝜏)𝑒^{−𝜍𝑟𝜔𝑟(𝑖)(𝑡−𝜏)}sin𝜔^{(𝑖)𝑑𝑟}(𝑡−𝜏)𝑑𝜏_𝑟𝑡_𝑖−11−𝜍^2𝑟^1/2+𝜂̇_𝑟𝑒^{−𝜍𝑟𝜔𝑟(𝑖)(𝑡−𝑡𝑖−1)}cos𝜔^{(𝑖)𝑑𝑟}𝑡−𝑡_𝑖−1−𝜓_𝑟𝑡_𝑖−1𝜔^{(𝑖)𝑑𝑟}𝑒^{−𝜍𝑟𝜔𝑟(𝑖)(𝑡−𝑡𝑖−1)}sin𝜔(𝑟^{(𝑖)𝑑𝑟}=𝑡1−,2𝑡,…_𝑖−1,,𝑁),(72) where𝜔^{(𝑖)𝑑𝑟}=1−𝜍^2𝑟^1/2𝜔^𝑟(𝑖),(73)𝜓_𝑟=arctan𝜍^𝑟1−𝜍^2𝑟^1/2.(74) Substituting 𝑡=𝑡_𝑖 into (73) yields𝜂_𝑟𝑡_𝑖=1𝜔^{(𝑖)𝑑𝑟}^𝑡𝑖_𝑡𝑖−1+𝜂𝑁_𝑟(𝜏)𝑒^{−𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑖−𝜏)}sin𝜔^{(𝑖)𝑑𝑟}𝑡_𝑖−𝜏𝑑𝜏_𝑟𝑡_𝑖−11−𝜍^2𝑟^1/2+𝜂̇_𝑟𝑒^{−𝜍𝑟𝜔𝑟(𝑖)Δ𝑡}cos𝜔^{(𝑖)𝑑𝑟}Δ𝑡−𝜓_𝑟𝑡_𝑖−1𝜔^{(𝑖)𝑑𝑟}𝑒^{−𝜍𝑟𝜔𝑟(𝑖)Δ𝑡}sin𝜔^{(𝑖)𝑑𝑟}Δ𝑡,(𝑟=1,2,…,𝑁).(75) Taking the derivative of (72) with respect to time and substituting 𝑡=𝑡_𝑖 into it yields 𝜂=̇_𝑟𝑡_𝑖1𝜔^{(𝑖)𝑑𝑟}^𝑡𝑖_𝑡𝑖−1𝑁_𝑟(𝜏+)𝜔−_{(𝑖)𝑑𝑟}𝜍_𝑟𝜔^𝑟(𝑖)𝑒^{−𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑖−𝜏)}sin𝜔^{(𝑖)𝑑𝑟}𝑡_𝑖−𝜏−𝜂_𝑟𝑒^{−𝜍𝑟𝜔𝑟(𝑖)(𝑡𝑖−𝜏)}cos𝜔^{(𝑖)𝑑𝑟}𝑡_𝑖−𝜏𝑑𝜏𝑡_𝑖−1𝜔^{(𝑖)𝑑𝑟}1−𝜍^2𝑟^1/2+𝜍_𝑟𝑒^{−𝜍𝑟𝜔𝑟(𝑖)Δ𝑡}sin𝜔^{(𝑖)𝑑𝑟}Δ𝑡−𝜓_𝑟𝜔^𝑟(𝑖)1−𝜍^2𝑟^1/2−𝜂̇_𝑟𝑒^{−𝜍𝑟𝜔𝑟(𝑖)Δ𝑡}cos𝜔^{(𝑖)𝑑𝑟}Δ𝑡−𝜓_𝑟𝑡_𝑖−1𝜍^𝑟𝜔^𝑟(𝑖)𝜔^{(𝑖)𝑑𝑟}𝑒^{−𝜍𝑟𝜔𝑟(𝑖)Δ𝑡}sin𝜔(𝑟^{(𝑖)𝑑𝑟}Δ=𝑡1−,2𝑒,…,𝑁^{−𝜍𝑟𝜔𝑟(𝑖)Δ𝑡})c.os𝜔^{(𝑖)𝑑𝑟}Δ𝑡(76) It is shown from (75) and (76) that 𝜂_𝑟(𝑡_𝑖) and 𝜂̇_𝑟(𝑡_𝑖) can be achieved when 𝜂_𝑟(𝑡_𝑖−1) and 𝜂̇_𝑟(𝑡_𝑖−1) are given. As for 𝑡_𝑖=0,𝜂𝜂̇_𝑟𝑟(0)=𝝋^(𝑟)^𝑇𝐌𝐔(0),(0)=𝝋^(𝑟)^𝑇𝐌𝐔̇(0).(77) So the total displacement response can be achieved by combining these modal contributions 𝐔𝑡_𝑖=^𝑁_𝑟=1𝜂_𝑟𝑡_𝑖𝝋^𝑟𝑡_𝑖,(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 𝑥=−0.1+𝑎^max𝑇²𝑦=−0.1+𝑎2_max𝜋𝜏−21𝜋sin(2𝜋𝜏),𝑇²𝑧=1.644+𝑎2𝜋_max𝜏−21𝜋sin(2𝜋𝜏),𝑇²𝜙2𝜋𝜏−21𝜋sin(2𝜋𝜏)_𝑥,=−0.1+𝑎^max𝑇²𝜙2𝜋𝜏−21𝜋sin(2𝜋𝜏)_𝑦,=−0.1+𝑎^max𝑇²𝜙2𝜋𝜏−21𝜋sin(2𝜋𝜏)_𝑧,=−0.1+𝑎^max𝑇²2𝜋𝜏−21𝜋sin(2𝜋𝜏),(79) where 𝑎_max=9.8m/s², 𝜏=𝑡/𝑇, 𝑇=√2𝜋𝑆/𝑎_max 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.

(a)

(b)

(c)

(d)

(e)

(f)

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.