A study on vibration of Stewart platform-based machine tool table

Abstract

In this paper, an analytical study on the vibrations of a parallel manipulator is addressed. In the vibration equation of the moving platform, the damping and stiffness of the pods are taken into account. The eigenvalue problem of the moving platform is solved to obtain the natural frequencies. Considering the role of different factors effective on the mass and stiffness matrices of the platform, natural frequencies for different configurations are investigated. The results obtained by analytical approach are further verified through FEM simulation. The effect of variation in position and orientation of the moving platform on the change in stiffness of its supporting chain, inertia tensor and natural frequencies and mode shapes of the platform as well as the effects of different payloads are studied. The vibration of the platform in different configurations is studied in different cutting conditions. The ranges of resonance frequencies and vibration amplitudes are then investigated. Finally, proper configurations of the moving platform are determined to avoid dynamic instability in different machining conditions. It also will be illustrated in this paper that some specific features embodied in the mechanism are appropriate for high-speed milling.

Appendices

Appendix 1

The physical specifications of the test manipulator are as follows:

• Radius of the moving platform = 175 mm

• Radius of the base = 400 mm

• Angular distance between two adjacent spherical joints = 30 °;

• Angular distance between two adjacent universal joints = 14 °

• Minimum length of each pod ₌ 760.2 mm

• Maximum length of each pod =968.9 mm

• Minimum mass of moving platform together with payload =40.6 kg

• Maximum mass of moving platform together with payload =90.6 kg

• Maximum course in X axis = ±130 mm, in Y axis = ±130 mm and in Z axis = 220 mm

Appendix 2

Considering ^P I _p as the inertia tensor of the moving platform and the payload in frame {P}, the inertia tensor of the moving platform and the payload in base frame, I _p , can be obtained using the parallel axes theorem [11], which yields:

$$ {{{\bf I}}_p} = {{\bf R}}\left( {{}^{\text{P}}{{{\bf I}}_p} + {m_p}\left[ {\matrix{ {r_y^2 + r_z^2} &{ - {r_x}{r_y}} &{ - {r_x}{r_z}} \\ { - {r_x}{r_y}} &{r_x^2 + r_z^2} &{ - {r_y}{r_z}} \\ { - {r_x}{r_z}} &{ - {r_y}{r_z}} &{r_x^2 + r_y^2} \\ }<!end array> } \right]} \right){{{\bf R}}^{\text{T}}} $$

where R is the rotation 3 × 3 matrix, representing the rotation of frame {P} related to frame {W}, and can be obtained as:

$$ {{\bf R}} = \left[ {\matrix{ {C{\theta_z}C{\theta_y}} &{ - S{\theta_z}C{\theta_x} + C{\theta_z}S{\theta_y}S{\theta_x}} &{S{\theta_z}S{\theta_x} + C{\theta_z}S{\theta_y}C{\theta_x}} \\ {S{\theta_z}C{\theta_y}} &{C{\theta_z}C{\theta_x} + S{\theta_z}S{\theta_y}S{\theta_x}} &{ - C{\theta_z}S{\theta_x} + S{\theta_z}S{\theta_y}C{\theta_x}} \\ { - S{\theta_y}} &{C{\theta_y}S{\theta_x}} &{C{\theta_y}C{\theta_x}} \\ }<!end array> } \right] $$

in which

$$ C{\theta_x} = \cos \left( {{\theta_x}} \right) {\text{and}}\,S{\theta_x} = { \sin }\left( {{\theta_x}} \right) $$

The vector ${{\bf r}} = {\left[ {\matrix{ {{r_x}} &{{r_y}} &{{r_z}} \\ }<!end array> } \right]^{\text{T}}}$ is the position vector of the mass centre of the moving platform and the payload in frame {W} and can be obtained as:

$$ {{\bf r}} = {{\bf R}}\,{{{\bf r}}_o} $$

in which r _o is the position vector of the mass centre of the moving platform and the payload in frame {P}.

The total damping and stiffness coefficient of the upper joints of the platform, C _Ti and K _Ti , can be obtained with series combination of damper’s and spring’s rules [8], respectively, which yields:

$$ {1}/{C_{{{{T}}i}}} = {1}/{C_{{{{s}}i}}} + {1}/{C_{{{{u}}i}}} + {1}/{C_{{{{a}}i}}} + {1}/{C_{{{{d}}i}}} + {1}/{C_{{uni}}} + 6/{C_{{b}}} $$

$$ {1}/{K_{{{{T}}i}}} = {1}/{K_{{{{s}}i}}} + {1}/{K_{{{{u}}i}}} + {1}/{K_{{{{a}}i}}} + {1}/{K_{{{{d}}i}}} + {1}/{K_{{uni}}} + 6/{K_{{b}}} $$

where C _si , C _ui , C _ai , C _di , C _uni and C _b are the damping coefficients of the spherical joints, the upper part of pods, the sliding joints, the lower part of pods, the universal joints and the stationary platform, respectively. K _si , K _ui , K _ai , K _di , K _uni and K _b are the stiffness coefficients. In the above relations, it is implicitly assumed that the stationary platform can be modelled as six concentrated vibratory elements, each situated under one pod and combined in parallel.

Each part of the pod can be considered as the homogeneous beam with uniform cross-section; therefore, the stiffness coefficients can be determined using:

$$ K = E\,A/l $$

where E is the modulus of elasticity; A and l are the area of the cross-section and the length of the beam, respectively.

Considering n _i as the unit vector along the direction of the ith pod and q _i as the position vector of the ith platform connection point on the moving platform in the base frame, the inverse Jacobian matrix can be obtained [10], which yields:

$$ {{{\bf J}}^{{ - 1}}} = \left[ {\matrix{ {{{\bf n}}_1^{\text{T}}} &{{{\left( {{{{\bf q}}_1} \times {{{\bf n}}_1}} \right)}^{\text{T}}}} \\ \vdots & \vdots \\ {{{\bf n}}_6^{\text{T}}} &{{{\left( {{{{\bf q}}_6} \times {{{\bf n}}_6}} \right)}^{\text{T}}}} \\ }<!end array> } \right] $$