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Modeling and analyzing of nonlinear dynamics for linear guide slide platform considering assembly error

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Abstract

It is industry tendency to accurately predict the dynamics of the mechanical systems with full consideration of errors. Based on the Hertz contact theory and general bearing modeling methods, this study proposed a more practical model using numerical method to investigate the influence of assembly error on the dynamics of linear guide slide platform. First, the modeling methods of five types of assembly errors are established, based on which, a nonlinear dynamic model is developed to investigate the influence of assembly error. In consideration of assembly error, the modeling method enables the sum of restoring forces and restoring moments equal to zero when no external load applied to the platform. Second, the simulation results indicate that the assembly error can cause uneven load distribution, change the dynamics of the system. In addition, different from previous research results, the stability of the system cannot be improved by simply increasing the preload. Last, in order to validate the proposed method, the proposed model is compared with previous fewer degrees-of-freedom model, and a series of experiments are conducted on a specialized platform to estimate the parameters of the system and verify the proposed model.

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Abbreviations

\(\Delta\delta_{\rm{x}}^\varepsilon \), \(\Delta\delta_{\rm{y}}^\varepsilon \) :

Straightness assembly errors of rail II in x and y directions

\(\Delta \varepsilon_{x} ,\Delta \varepsilon_{y} ,\Delta \varepsilon_{z}\) :

Rotation assembly errors of rail II about x, y and z axis

\(\Delta{\rm{x}}_{\rm{ijk}}^{\rm{a}}\), \(\Delta{\rm{y}}_{\rm{ijk}}^{\rm{a}}\) :

Type a assembly error induced contact deformation along x and y axis

\(\Delta{\rm{x}}_{\rm{ijk}}^{\rm{b}}\), \(\Delta{\rm{y}}_{\rm{ijk}}^{\rm{b}}\) :

Type b assembly error induced contact deformation along x and y axis

\(\Delta{\rm{x}}_{\rm{ijk}}^{\rm{c}}\), \(\Delta{\rm{y}}_{\rm{ijk}}^{\rm{c}}\) :

Type c assembly error induced contact deformation along x and y axis

\(\Delta{\rm{x}}_{\rm{ijk}}^{\rm{d}}\), \(\Delta{\rm{y}}_{\rm{ijk}}^{\rm{d}}\) :

Type d assembly error induced contact deformation along x and y axis

\(\Delta{\rm{x}}_{\rm{ijk}}^{\rm{e}}\), \(\Delta{\rm{y}}_{\rm{ijk}}^{\rm{e}}\) :

Type e assembly error induced contact deformation along x and y axis

l k :

Distance between the kth and the first loaded ball in the grooves of each carriage

\({\rm{S}}_{0}^\varepsilon \) :

Initial distance between the groove curvature centers with consideration of assembly error

r cg, r rg :

The groove radius of carriage and rail

\(\alpha_{0}^{\varepsilon }\) :

The initial contact angle with consideration of assembly error

δ ijk :

The contact deformation of the kth ball in the jth groove of the ith carriage

Q ijk :

The contact force of the kth ball in the jth groove of the ith carriage

α ijk :

The contact angle of the kth ball in the jth groove of the ith carriage

\(\Delta x_{ijk}\) , \(\Delta y_{ijk}\) :

The contact deformation of the kth ball along x and y directions under external load

x, y :

The displacement of the platform along x and y axis under external load

L x , L y :

The distance between grooves along x and y directions

l x , l y :

The distance between carriages along x and y directions

φ x , φ y , φ z :

The angular displacement about x, y and z axis

F x , F y :

Total restoring force of platform along x, and y axis

M x , M y , M z :

Total restoring moment of platform about x, y and z axis

I x , I y , I z :

Moment of inertia of platform about x, y and z axis

\({\rm{w}}_{x}\) \({\rm{w}}_{y}\) \({\rm{w}}_{z}\) :

Width, length, and height of platform

i :

ith rail

j :

jth carriage

k :

kth ball

x :

x Axis

y :

y Axis

z :

z Axis

a :

Type a assembly error

b :

Type b assembly error

c :

Type c assembly error

d :

Type d assembly error

e :

Type e assembly error

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Funding

The work was supported by National Natural Science Foundation of China (Grant No. 52075087), the Fundamental Research Funds for the Central Universities (Grant No. N2003006, and N2103003), and the Natural Science Foundation of Heilongjiang Province (HL2020A017).

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Zhendong Liu, Mengtao Xu, Hongzhuang Zhang, Changyou Li, Guo Yao, Zhenyuan Li, HuiHui Miao & Chenyu Wang

  2. College of Mechanical and Automotive Engineering, Zhaoqing University, Zhaoqing, 526061, China

    Yimin Zhang

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  1. Zhendong Liu
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Contributions

ZL contributed to methodology, investigation, experimental, writing—Original Draft, writing—review & editing. MX contributed to resources and supervision. HZ contributed to resources, writing—reviewing and editing, supervision, writing—review & editing. CL conceived the presented idea. GY conceived the presented idea. ZL carried out the experiment. HM contributed to coding. CW carried out the experiment. YZ contributed to resources and supervision.

Corresponding authors

Correspondence to Changyou Li or Guo Yao.

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Appendix

Appendix

According to [45], the moment of inertia of the platform about x, y and z axis can be calculated by

$$ I_{x} = \frac{1}{12}m\left( {w^{2} + d^{2} } \right) $$
$$ I_{y} = \frac{1}{12}m\left( {w^{2} + h^{2} } \right) $$
$$ I_{z} = \frac{1}{12}m\left( {h^{2} + d^{2} } \right) $$

where m is the mass of the platform, h, w, and d represent the height, width, and length, respectively.

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Liu, Z., Xu, M., Zhang, H. et al. Modeling and analyzing of nonlinear dynamics for linear guide slide platform considering assembly error. Nonlinear Dyn 108, 2193–2221 (2022). https://doi.org/10.1007/s11071-022-07345-2

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