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

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 [

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 [

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

The schematic diagram of the 6-dof parallel structure seismic simulator is shown in Figure

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

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

The local coordinate system of the

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 [

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 [

The nodal elastic displacement of the element is shown in Figure

Nodal elastic displacement of the element.

The elastic displacement vector of an arbitrary point

The polynomial of the nodal displacement of the element is chosen to formulate the displacement of the point

Considering the node

Considering the knowledge of material mechanics, the strain of the point

Substituting (

According to the knowledge of material mechanics, the strain energy of the element can be expressed as

Based on the presented assumption, there are kinematics relationships as follows:

It can be proved that (see [

So the kinetic energy of the element can be expressed as

Substituting (

For the strut within

The oscillation of the slider along the axial direction of the lead-screw can be expressed as

The compatibility of the deformations between the rigid moving platform and the flexible strut can be expressed as

The compatibility of the deformations between the rigid slider and the flexible strut can be expressed as

The oscillation equation of the rigid moving platform substructure in the coordinate system

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

Gathering the dynamic equations of the substructures and employing the deformation compatibility conditions between the rigid moving platform and the flexible strut yields

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

The parameters of the base platform (m).

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

0.400000 | 0.000000 | −0.400000 | 0.400000 | −0.400000 | −2.000000 | |

−0.400000 | 0.400000 | −0.400000 | −2.000000 | −2.000000 | 0.000000 | |

0.000000 | 0.000000 | 0.000000 | 1.500000 | 1.500000 | 1.500000 |

The parameters of the moving platform which are measured in the coordinate frame

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

0.400000 | 0.000000 | −0.400000 | 0.400000 | −0.400000 | −0.681000 | |

−0.400000 | 0.400000 | −0.400000 | −0.681000 | −0.681000 | 0.000000 | |

−0.166000 | −0.166000 | −0.166000 | −0.037500 | −0.037500 | −0.037500 |

The length of the strut

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 | 1.000000 |

The mass parameters of the manipulator (kg).

1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|

20 | 20 | 20 | 20 | 20 | 20 | |

50 | 100 | 50 | 50 | 50 | 100 |

The mass of the moving platform is

^{2}, ^{3},

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 (

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

It is shown in Figure

The sensitivity analysis is usually used to evaluate the effect of the structural design variables on the performance of the manipulator. From (

Sensitivities of the first-order natural frequency to the structure parameters. (a) Radius of the struts. (b) Radius of the lead screws.

The computation of the energy ratio is usually used to evaluate the allocation of the stiffness and the mass of the manipulator. Suppose that

So the energy ratio of the substructure can be achieved as

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.

The displacement response analysis will be carried out by solving (

Since the damping in the structure is a very complex subject [

From (

Substituting (

Substituting (

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 [

Let

Assuming that the investigated trajectory of the moving platform used in the simulation is expressed as

Displacement responses of the moving platform.(a)

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.

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.

From the sensitivity analysis, the first-order natural frequency is sensitive to the radius of the strut and the radius of the lead screw.

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.

For the investigated trajectory, the displacement response of the moving platform along the

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.