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A study on dynamics of flexible multi-link mechanism including joints with clearance and lubrication for ultra-precision presses

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Abstract

It is essential to establish a dynamic model of flexible multi-link mechanism with clearance and lubrication for ultra-precision presses to analyze its dynamic response. Traditional dynamic models of mechanical system rarely consider the effect of flexibility, revolute and spherical clearance joints, and lubrication together, which causes lower precision analysis. In order to study the dynamic characteristics of multi-link mechanism more accurately, a novel dynamic model of flexible multi-link mechanism with clearance and lubrication for ultra-precision presses is established in the present work, which considers the effect of revolute and spherical clearance joints, lubrication, and flexibility of crank shaft and linkage. It is demonstrated that the dynamic responses of flexible multi-link mechanism with lubricated clearance joint model agree better with experimental data than those with dry clearance model and the validity of the proposed model is verified. The simulation results also show that the existence of lubrication reduces the dynamic responses of flexible multi-link mechanism with revolute and spherical clearance in a significant manner and act as a suspension for multi-link mechanism. The motion of the crank shaft center and ball center of spherical joint was mainly characterized by two phases: free flight and impact motion. Furthermore, the influences of the clearance size and input speed of crank shaft as well as blanking force on the dynamic responses of the multi-link mechanism were also investigated.

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Abbreviations

\(\delta \) :

Normal penetration at the contact point

e :

Magnitude of the eccentricity vector between the bearing and journal centers

c :

Radial clearance

\(e_x , e_y \) :

Displacement of the journal inside the bearing along X and Y directions

\(F_\mathrm{N} \) :

Normal force between the journal and bearing

\(F_\mathrm{T} \) :

Tangential force between the journal and bearing

\(\mathbf{u}_i^Q , \mathbf{u}_j^Q \) :

Global position vector of point \(Q_i\) and \(Q_j\) for revolute clearance joint

\(\mathbf{u}_i^P , \mathbf{u}_j^P \) :

Global position vector of point \(P_i\) and \(P_j\) for spherical clearance joint

\(\mathbf{A}_k \) :

Transformation matrix between global and local position for spherical clearance joint

\(\mathbf{{g}}_k^{\prime P} \) :

Local position vector of point P for spherical clearance joint

\(R_i , R_j \) :

Socket and ball radius

\(R_\mathrm{B} , R_\mathrm{J} \) :

Radius of bearing and journal

\(\mathbf{n}, \mathbf{t}\) :

Normal and tangential direction to the impacted surfaces

\(\nu _\mathrm{N} , \nu _\mathrm{T} \) :

Relative velocity in the normal and tangential direction to the surface of collision

\(\mathbf{f}_\mathrm{N} , \mathbf{f}_\mathrm{T} \) :

Normal and tangential force vectors at the contact points

\(\mathbf{M}_i , \mathbf{M}_j \) :

Moments acting on the mass center of body i and j

\(c_{\max } \) :

Normal maximum damping coefficient

\(d_{\max } \) :

Maximum normal boundary penetration

n :

Exponent of the force deformation characteristics

K :

Normal stiffness coefficient

\(C(\delta )\) :

Instantaneous normal damping coefficient

\(E_1 , E_2 \) :

Elasticity modulus of the solids in contact

\(\mu _1 , \mu _2 \) :

Possion’s ratio

\(R_1 , R_2 \) :

Radius of the solids in contact

\(\mu \) :

Friction coefficient

z :

Average bristle deflection

\(\sigma _0 \) :

Bristle stiffness

\(\sigma _1 \) :

Microscopic damping

\(\sigma _2 \) :

Viscous friction coefficient

\(\mu _\mathrm{k} \) :

Coefficient of kinetic friction

\(\mu _\mathrm{s} \) :

Coefficient of static friction

\(L_\mathrm{B} \) :

Journal–bearing length

\(\varepsilon \) :

Eccentricity ratio

\(\gamma \) :

Angle between the X and Y components of eccentricity vector

\(\omega \) :

Relative angular velocity between the journal and bearing

\(F_{\max } \) :

Maximum value of the blanking force

T :

Period of the crank shaft

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Acknowledgments

This paper was supported by the following research projects: “National Natural Science Foundation of China”, Grant # 51405238, “Natural Science Foundation of Jiangsu Province”, Grant # BK20140728, and “the Fundamental Research Funds for the Central Universities”, Grant # KJQN201558.

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Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering, Nanjing Agricultural University, Nanjing, 210031, China

    Enlai Zheng, Rui Zhu & Sihong Zhu

  2. Xuzhou Metal-Forming Machine Group Co., Ltd, Xuzhou, 221116, China

    Xinjian Lu

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  1. Enlai Zheng
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Correspondence to Enlai Zheng.

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Zheng, E., Zhu, R., Zhu, S. et al. A study on dynamics of flexible multi-link mechanism including joints with clearance and lubrication for ultra-precision presses. Nonlinear Dyn 83, 137–159 (2016). https://doi.org/10.1007/s11071-015-2315-7

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  • DOI: https://doi.org/10.1007/s11071-015-2315-7

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