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Abstract

In order to study the elastodynamic behaviour of 4- universal joints- prismatic pairs- spherical joints / universal joints- prismatic pairs- universal joints 4-UPS-UPU high-speed spatial PCMMs(parallel coordinate measuring machines), the nonlinear time-varying dynamics model, which comprehensively considers geometric nonlinearity and the rigid-flexible coupling effect, is derived by using Lagrange equations and finite element methods. Based on the Newmark method, the kinematics output response of 4-UPS-UPU PCMMs is illustrated through numerical simulation. The results of the simulation show that the flexibility of the links is demonstrated to have a significant impact on the system dynamics response. This research can provide the important theoretical base of the optimization design and vibration control for 4-UPS-UPU PCMMs.

Keywords

Parallel coordinate measuring machine nonlinearity dynamics model flexible link

1. Introduction

The PCMM, which has advantages of high stiffness, good dynamic performance, no accumulated precision errors, tight construction and strong modularization, has gained a lot of attention and developed rapidly^1–2. In high-speed and high-acceleration conditions, due to the greater external and internal loads, the deformations of the links might induce noise and strong vibration and lead to reducing the motion precision and the stability of the PCMM. Therefore, the dynamic behaviours of high-speed PCMMs must be studied.

In recent years, many researchers, such as Wang³, Du⁴, Zhang⁵, and Zhou⁶, have studied the elastodynamic behaviour of planar mechanisms, and a number of approaches have been developed to predict the elastic dynamic behaviour of flexible planar mechanisms. However, there have been few efforts made oriented to spatial parallel machines and the links' flexibility and the coupling effects of the flexible links' elastic displacement have not been sufficiently considered.

This paper takes the 4-UPS-UPU PCMM(see Figure 1), which consists of a stationary platform, a moving platform, four UPS(Universal joint-Prismatic pair-Spherical joint) actuating limbs, a UPU(Universal joint-prismatic pair-Universal joints) actuating limb and a working head, as an example. The nonlinear elastodynamic model is established and then the numerical simulation analysis of dynamic response for 4-UPS-UPU PCMM, including displacement error output response, velocity error output response, and acceleration error output response can be calculated.

Figure 1.

Structural diagram of a parallel mechanism.

2. Dynamic Modeling of 4-Ups-Upu High-Speed Spatial Pcmms

2.1 Model of the flexible beam element

The flexible driving limbs of the 4-UPS-UPU PCMM can be divided into several parts which are composed of equal cross-section spatial beam elements.

As shown in Figure 2, the displacements of a point in the element which include axial, transverse, rotary and curvature displacements are expressed as

Figure 2.

Model of a flexible beam element

u (x_{i j}, t) = N_{i j 1}^{​} δ_{i j}

(1)

υ (x_{i j}, t) = N_{i j 2}^{​} δ_{i j}

(2)

w (x_{i j}, t) = N_{i j 3}^{​} δ_{i j}

(3)

ψ_{x} (x_{i j}, t) = N_{i j 4}^{​} δ_{i j}

(4)

ψ_{y} (x_{i j}, t) = (\frac{\partial N_{i j 3}}{\partial x}) δ_{i j} = {\dot{N}}_{i j 3}^{​} δ_{i j}

(5)

ψ_{z} (x_{i j}, t) = (\frac{\partial N_{i j 2}}{\partial x}) δ_{i j} = {\dot{N}}_{i j 2}^{​} δ_{i j}

(6)

Where N _ij1, N _ij2, N _ij3 and N _ij4 are the vectors of quintic Hermite polynomials and a linear polynomial, respectively.

2.2 Kinetic energy of the beam element

The kinetic energy of the beam element consists of translating the kinetic energy and rotating kinetic energy. The translating kinetic energy of the beam element can be described as:

T_{i j 1} = \frac{1}{2} \int_{0}^{l_{i j}} ρ_{i j} a_{i j} {\dot{r}}^{T} \dot{r} d x_{i j}

(7)

The rotating kinetic energy of the beam element can be denoted as:

T_{i j 2} = \frac{1}{2} \int_{0}^{l_{i j}} J_{i j} {\dot{ψ}}_{A x}^{T} {\dot{ψ}}_{A x} d x_{i j}

(8)

So the kinetic energy of the beam element is given by:

T_{i j} = T_{i j 1} + T_{i j 2} = \frac{1}{2} \int_{0}^{l_{i j}} ρ_{i j} a_{i j} {\dot{r}}^{T} \dot{r} d x_{i j} + \frac{1}{2} \int_{0}^{l_{i j}} J_{i j} {\dot{ψ}}_{A x}^{T} {\dot{ψ}}_{A x} d x_{i j}

(9)

After arranged, the kinetic energy of the beam element is obtained as follows:

T_{i j} = \frac{1}{2} {\dot{δ}}_{i j}^{T} P_{i j} {\dot{δ}}_{i j} + δ_{i j}^{T} {\tilde{B}}_{i j} {\dot{δ}}_{i j} + {\bar{B}}_{i j} {\dot{δ}}_{i j} + \frac{1}{2} δ_{i j}^{T} {\hat{B}}_{i j} δ_{i j} + {\bar{D}}_{i j} δ_{i j} + \frac{1}{2} h_{i 0}

(10)

Where $δ_{i j}^{T} {\tilde{B}}_{i j} {\dot{δ}}_{i j}$ , ${\bar{B}}_{i j} {\dot{δ}}_{i j}$ , $\frac{1}{2} δ_{i j}^{T} {\hat{B}}_{i j} δ_{i j}$ and ${\bar{D}}_{i j} δ_{i j}$ reflect the coupling effects of elastic motion and rigid motion

2.3 Potential energy of the beam element

The potential energy of the beam element consists of gravitational potential energy and elastic potential energy. The gravitational potential energy of the beam element can be described as:

V_{i j}^{g} = \int_{0}^{l_{i j}} ρ_{i j} g a_{i j} x d x_{i j} = ρ_{i j} g a_{i j} l_{i j} (Ω_{i j} + Θ_{i j} δ_{i j})

(11)

The elastic potential energy of the beam element can be denoted as:

V_{i j}^{s} = \frac{1}{2} δ_{i j}^{T} K_{i j} δ_{i j}

(12)

Where

\begin{array}{l} K_{i j} = K_{i j 1} + K_{i j 2} \\ K_{i j 1} = E_{i j} I_{i j y} \int_{0}^{l_{i j}} {(\frac{\partial^{2} N_{i j 2}}{\partial x_{i j}^{2}})}^{T} (\frac{\partial^{2} N_{i j 2}}{\partial x_{i j}^{2}}) d x_{i j} \\ + E_{i j} I_{i j z} \int_{0}^{l_{i j}} {(\frac{\partial^{2} N_{i j 3}}{\partial x_{i j}^{2}})}^{T} (\frac{\partial^{2} N_{i j 3}}{\partial x_{i j}^{2}}) d x_{i j} \\ + G I_{i j p} \int_{0}^{l_{i j}} {(\frac{\partial^{2} N_{i j 4}}{\partial x_{i j}^{2}})}^{T} (\frac{\partial^{2} N_{i j 4}}{\partial x_{i j}^{2}}) d x_{i j} \\ K_{i j 2} = E_{i j} a_{i j} \int_{0}^{l_{i j}} [{N^{'}}_{i j 1}^{T} + \frac{1}{2} ({N^{'}}_{i j 1}^{T} {N^{'}}_{i j 1} + {N^{'}}_{i j 2}^{T} {N^{'}}_{i j 2} + {N^{'}}_{i j 3}^{T} {N^{'}}_{i j 3}) δ_{i j}] \\ [{N^{'}}_{i j 1} + \frac{1}{2} δ_{i j}^{T} ({N^{'}}_{i j 1}^{T} {N^{'}}_{i j 1} + {N^{'}}_{i j 2}^{T} {N^{'}}_{i j 2} + {N^{'}}_{i j 3}^{T} {N^{'}}_{i j 3})] d x_{i j} \end{array}

So the potential energy of the beam element is given by:

V_{i j} = V_{i j}^{g} + V_{i j}^{s} = ρ_{i j} g a_{i j} l_{i j} (Ω_{i j} + Θ_{i j} δ_{i j}) + \frac{1}{2} δ_{i j}^{T} K_{i j} δ_{i j}

(13)

2.4 Dynamic equations of the element

The dynamic equation of the element, which is derived by using the Lagrange equation, is given by:

{\tilde{M}}_{i j} {\ddot{\bar{δ}}}_{i j} + {\tilde{C}}_{i j} {\dot{\bar{δ}}}_{i j} + {\tilde{K}}_{i j} {\bar{δ}}_{i j} = {\tilde{Q}}_{i j}

(14)

Where

\begin{array}{l} δ_{i j} = {\bar{R}}_{i j} {\bar{δ}}_{i j} \\ {\bar{R}}_{i j} = [\begin{matrix} R_{i j} & 0 & 0 & 0 & 0 & 0 \\ 0 & R_{i j} & 0 & 0 & 0 & 0 \\ 0 & 0 & R_{i j} & 0 & 0 & 0 \\ 0 & 0 & 0 & R_{i j} & 0 & 0 \\ 0 & 0 & 0 & 0 & R_{i j} & 0 \\ 0 & 0 & 0 & 0 & 0 & R_{i j} \end{matrix}] \end{array}

2.5 Dynamic equations of the limbs

The kinematic constraint conditions of elements are express as follows:

{\bar{δ}}_{i 11} = {\bar{δ}}_{i 12} = {\bar{δ}}_{i 13} = 0

(15)

{\bar{δ}}_{i 14} = {\bar{δ}}_{i 15} = {\bar{δ}}_{i 16} = 0

(16)

{\begin{cases} {\bar{δ}}_{i j 10} = {\bar{δ}}_{i (j + 1) 1} \\ {\bar{δ}}_{i j 11} = {\bar{δ}}_{i (j + 1) 2} \\ {\bar{δ}}_{i j 12} = {\bar{δ}}_{i (j + 1) 3} \\ {\bar{δ}}_{i j 13} = {\bar{δ}}_{i (j + 1) 4} \\ {\bar{δ}}_{i j 14} = {\bar{δ}}_{i (j + 1) 5} \\ {\bar{δ}}_{i j 15} = {\bar{δ}}_{i (j + 1) 6} \\ {\bar{δ}}_{i j 16} = {\bar{δ}}_{i (j + 1) 7} \\ {\bar{δ}}_{i j 17} = {\bar{δ}}_{i (j + 1) 8} \\ {\bar{δ}}_{i j 18} = {\bar{δ}}_{i (j + 1) 9} \end{cases}

(17)

When i = 1, we can get ${\bar{δ}}_{i (m_{1} + m_{2}) 17} = {\bar{δ}}_{i (m_{1} + m_{2}) 18} = 0$

When i = 2,3,4,5, we can get ${\bar{δ}}_{i (m_{1} + m_{2}) 16} = {\bar{δ}}_{i (m_{1} + m_{2}) 17} = {\bar{δ}}_{i (m_{1} + m_{2}) 18} = 0$

The relation equation between q _i and ${\bar{δ}}_{i j}$ is given by

{\bar{δ}}_{i j} = {\bar{A}}_{i j} q_{i}

(18)

The transformation matrix Ā_ij is varied correspondingly with the different limbs and different connection elements.

The dynamic equations of limbs oriented to element j are given by:

{\hat{M}}_{i j} {\ddot{q}}_{i} + {\hat{C}}_{i j} {\dot{q}}_{i} + {\hat{K}}_{i j} q_{i} = {\hat{Q}}_{i j}

(19)

When fabricating all the dynamic equations of limb oriented to elements, the dynamic equation of limb oriented to limb i is given by:

{\bar{M}}_{i} {\ddot{q}}_{i} + {\bar{C}}_{i} {\dot{q}}_{i} + {\bar{K}}_{i} q_{i} = {\bar{Q}}_{i}

(20)

2.6 System dynamic equations

The kinematic constraint conditions can be expressed by:

q_{S i} = [\begin{matrix} 1 & 0 & 0 & 0 & ^{A} Z_{S i} & -^{A} Y_{S i} \\ 0 & 1 & 0 & -^{A} Z_{S i} & 0 & ^{A} X_{S i} \\ 0 & 0 & 1 & ^{A} Y_{S i} & -^{A} X_{S i} & 0 \end{matrix}] q_{0}

(21)

Where $q_{S i} = {[\begin{matrix} {\bar{δ}}_{i (m_{1} + m_{2}) 10} & {\bar{δ}}_{i (m_{1} + m_{2}) 11} & {\bar{δ}}_{i (m_{1} + m_{2}) 12} \end{matrix}]}^{T}$

q_{0} = {[q_{1}, q_{2}, q_{3}, q_{4}, q_{5}, q_{6}]}^{T}

The dynamic constraint conditions can be expressed by:

\begin{array}{c} [\begin{matrix} m_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{x x} & I_{x y} & I_{x z} \\ 0 & 0 & 0 & I_{y x} & I_{y y} & I_{y z} \\ 0 & 0 & 0 & I_{z x} & I_{z y} & I_{z z} \end{matrix}] [\begin{matrix} {\ddot{q}}_{1} \\ {\ddot{q}}_{2} \\ {\ddot{q}}_{3} \\ {\ddot{q}}_{4} \\ {\ddot{q}}_{5} \\ {\ddot{q}}_{6} \end{matrix}] = \\ [\begin{matrix} \sum F_{i x} \\ \sum F_{i y} \\ \sum F_{i z} \\ \sum M_{i x} \\ \sum M_{i y} \\ \sum M_{i z} \end{matrix}] + [\begin{matrix} \sum F_{o x} \\ \sum F_{o y} \\ \sum F_{o z} \\ \sum M_{o x} \\ \sum M_{o y} \\ \sum M_{o z} \end{matrix}] - [\begin{matrix} m_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & m_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{x x} & I_{x y} & I_{x z} \\ 0 & 0 & 0 & I_{y x} & I_{y y} & I_{y z} \\ 0 & 0 & 0 & I_{z x} & I_{z y} & I_{z z} \end{matrix}] [\begin{matrix} {\ddot{X}}_{B} \\ {\ddot{Y}}_{B} \\ {\ddot{Z}}_{B} \\ \ddot{γ} \\ \ddot{β} \\ \ddot{α} \end{matrix}] \end{array}

(22)

When simplifying the dynamic constraint conditions can be expressed by:

M_{0} {\ddot{q}}_{0} = Q_{0}

(23)

The relation between q _i , which are generalized coordinates on the element, and q _i ^*, which are generalized coordinates on limb, is given by:

q_{i} = R_{i}^{*} q_{i}^{*} \begin{matrix} ​ & ​ & ​ \end{matrix} (i = 1, 2, \dots, 5)

(24)

Then we can get

M_{i} {\ddot{q}}_{i}^{*} + C_{i} {\dot{q}}_{i}^{*} + K_{i} q_{i}^{*} = Q_{i} \begin{matrix} ​ & ​ \end{matrix} (i = 1, 2, \dots, 5)

(25)

The relation between q , which is the generalized coordinates of the system, and q _i ^*, which is the generalized coordinates on the limb, is given by:

q_{i}^{*} = R_{i} q

(26)

Where

q = {[\begin{matrix} q_{01}^{T} & q_{02}^{T} & q_{03}^{T} & q_{04}^{T} & q_{05}^{T} & q_{0}^{T} \end{matrix}]}^{T}

The dynamic constraint conditions and the dynamic equations of the limb expressed by q _i ^* are assembled and then the system dynamic equation is given by:

M \ddot{q} + \hat{C} \dot{q} + K q = Q

(27)

When considering the actual damping, the system dynamic equation of a 4-UPS-UPU PCMM is given by:

M \ddot{q} + C \dot{q} + K q = Q

(28)

q_{S i} = J_{S i} q_{O_{B}} \begin{matrix} ​ & ​ & ​ \end{matrix} (i = 1, 2, 3, 4, 5)

(29)

Where $C = \hat{C} + ζ_{1} M + ζ_{2} K$ is the gross damping matrix of the system, ζ₁ and ζ₂ are the damping factors.

3. Numerical Simulation and Analysis

The primary structural parameters of the 4-UPS-UPU PCMM are illustrated as follows. The distance between the furthest universal joint and the centre point on the stationary platform is 780mm, the distribution radius of the universal joints on the stationary platform is 720mm, the distribution radius of the spherical joints on the moving platform is 200mm, the distribution position angle of the universal joints on the stationary platform is π/2, the distribution position angle between the furthest universal joint and the neighbouring universal joint on the stationary platform is π/4, the distribution position angle of the spherical joints on the moving platform is 2π/5.

The system parameters of the 4-UPS-UPU PCMM is illustrated as follows: Each driving limbs is made of steel with a density of 7800kg/m³, an elastic modulus of 2.1×1011Pa and a Poisson's ratio of 0.3. The mass of the moving platform is 105.85kg.

The motion of the working head for a 4-UPS-UPU PCMM is described as:

{\begin{cases} X_{B} = 0.97 + 0.01 \cos (\frac{π}{6} - \frac{π}{3} t) \\ Y_{B} = 0.01 \sin (\frac{π}{6} - \frac{π}{3} t) \\ Z_{B} = 0 \\ α = \frac{π}{6} - \frac{π}{3} t \\ β = 0 \end{cases} (0 \leq t \leq 1)

(30)

The dynamic responses of a 4-UPS-UPU PCMM are realized by the Newmark method and thus the displacement error output response, velocity error output response and acceleration error output response, are obtained.

Figure 3.

Displacement error output in X axis.

Figure 4.

Displacement error output in Y axis.

Figure 5.

Displacement error output in Z axis.

Figure 6.

Velocity error output in X axis

Figure 7.

Velocity error output in Y axis.

Figure 8.

Velocity error output in Z axis

Figure 9.

Acceleration error output in X axis.

Figure 10.

Acceleration error output in Y axis

Figure 11.

Acceleration error output in Z axis.

4. Conclusion

The nonlinear elastodynamic model of high-speed 4-UPS-UPU PCMMs is derived by the application of the Lagrange equation and the finite element method. The dynamic responses including displacement error output, velocity error output, and acceleration error output are obtained. The simulation results show that the links' flexibility and the coupling effects of elastic motion and rigid motion have a significant impact on a 4-UPS-UPU PCMM system, and provide an important theoretical base for the optimization of design in controlling and reducing vibration for these parallel mechanisms.

Footnotes

5. Acknowledgments

This research is supported by the National Natural Science Foundation of China[Grant No. 51005138], Shandong Young Scientists Award Fund[Grant No. BS2012ZZ008], the Natural Science Foundation of Shandong Education Department of China[Grant No. J09LD54], the Science Foundation of SUST[Grant No. 2011KYJQ102].

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Nonlinear Elastodynamic Behaviour Analysis of High-Speed Spatial Parallel Coordinate Measuring Machines

Abstract

Keywords

1. Introduction

2. Dynamic Modeling of 4-Ups-Upu High-Speed Spatial Pcmms

2.1 Model of the flexible beam element

2.2 Kinetic energy of the beam element

2.3 Potential energy of the beam element

2.4 Dynamic equations of the element

2.5 Dynamic equations of the limbs

2.6 System dynamic equations

3. Numerical Simulation and Analysis

4. Conclusion

Footnotes

5. Acknowledgments

References