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Machining vibration suppression of cantilever parts of aerospace structure using robot-assisted clamping system

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Abstract

Cantilever parts are extensively used in aerospace applications. However, machining vibrations readily arise under the action of cutting forces due to the poor fixed stiffness of cantilever parts mounted on large structures, compromising working accuracy and efficiency. To address the shortcomings of traditional special vibration suppression fixtures including lengthy development cycles, high costs and limited universality, a mobile robot-assisted clamping system was proposed to fulfill the vibration suppression requirements of diverse cantilever parts on large structures. Furthermore, to analyze the vibration responses under different clamping states, a workpiece-fixture dynamics model incorporating the mobile robot-assisted clamping system is established using the transfer matrix method for multibody systems (MSTMM). Then, the effects of different clamping states on cantilevered part vibration are explored using acceleration responses as the metric. Finally, the effectiveness of the vibration suppression method and the accuracy of the dynamics model are validated through experiment and simulation. This work provides a solution for vibration suppression and optimal clamping states selection of different cantilever parts.

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Funding

This work was co-supported by the National Natural Science Foundation of China (Nos. 52005254, 52075256 and U22A20204), and the Fundamental Research Funds for the Central Universities (No. NT2022016).

Author information

Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Pinzhang Wang, Wei Tian & Bo Li

  2. Jiangsu Key Laboratory of Precision and Micro-Manufacturing Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Bo Li

Authors
  1. Pinzhang Wang
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  2. Wei Tian
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  3. Bo Li
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Contributions

Pinzhang Wang: experiments design, dynamics modeling, data analysis, writing—original draft. Wei Tian: experiments design, data analysis, writing—reviewing and editing. Bo Li: investigation, supervision, reviewing and editing.

Corresponding authors

Correspondence to Wei Tian or Bo Li.

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Appendices

Appendix 1

The transfer equation of the state vector \({\boldsymbol{\rm Z}}_{r,O}\) with respect to \({{\varvec{Z}}}_{r,I}\) is defined as

$${{\varvec{Z}}}_{r,O}={{\varvec{U}}}_{r}{{\varvec{Z}}}_{r,I}(r=\mathrm{2,5},\mathrm{8,11,15,17,19,21,23,25,27,29})$$
(12)

where \({{\varvec{U}}}_{r}\) is the transfer matrices of the rigid body, it is expressed as

$${{\varvec{U}}}_{r}=\left[\begin{array}{cccc}{{\varvec{I}}}_{3}& -{\widetilde{{\varvec{l}}}}_{IO}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ m{\omega }^{2}{\widetilde{{\varvec{l}}}}_{CO}& -{\omega }^{2}\left(m{\widetilde{{\varvec{l}}}}_{IO}{\widetilde{{\varvec{l}}}}_{IC}+{{\varvec{J}}}_{I}\right)& {{\varvec{I}}}_{3}& {\widetilde{{\varvec{l}}}}_{IO}\\ m{\omega }^{2}{{\varvec{I}}}_{3}& -m{\omega }^{2}{\widetilde{{\varvec{l}}}}_{IC}& {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}\end{array}\right]$$
(13)

where

$${\widetilde{{\varvec{l}}}}_{IO}=\left[\begin{array}{ccc}0& -{z}_{O}& {y}_{O}\\ {z}_{O}& 0& -{x}_{O}\\ -{y}_{O}& {x}_{O}& 0\end{array}\right],{\widetilde{{\varvec{l}}}}_{IC}=\left[\begin{array}{ccc}0& -{z}_{C}& {y}_{C}\\ {z}_{C}& 0& -{x}_{C}\\ -{y}_{C}& {x}_{C}& 0\end{array}\right],{{\varvec{J}}}_{I}=\left[\begin{array}{ccc}{J}_{x}& -{J}_{xy}& -{J}_{xz}\\ -{J}_{xy}& {J}_{y}& -{J}_{yz}\\ -{J}_{xz}& -{J}_{yz}& {J}_{z}\end{array}\right]$$

where \({\widetilde{{\varvec{l}}}}_{CO}={\widetilde{{\varvec{l}}}}_{IO}-{\widetilde{{\varvec{l}}}}_{IC}\), I and O are the input and output end respectively, C is the center of mass, and \({\widetilde{{\varvec{l}}}}_{PQ}\) is the coordinate matrices of point Q relative to P. \({{\varvec{J}}}_{I}\) is the moment of inertia with respect to I, m is the mass of the body, \(\omega\) is the system natural frequency.

The transfer equation and the geometric relationship between multiple input ends are

$$\left\{\begin{array}{c}{{\varvec{Z}}}_{13,O}=\sum_{m=1}^{4}{{\varvec{U}}}_{13,m}{{\varvec{Z}}}_{13,m}\\ {{\varvec{H}}}_{13,n}{{\varvec{Z}}}_{13,n}={{\varvec{H}}}_{13,1}{{\varvec{Z}}}_{13,1}\end{array}\right.\left(n=2,3,4\right)$$
(14)

where the transfer matrix \({{\varvec{U}}}_{\mathrm{13,1}}\) is consistent with the form of \({{\varvec{U}}}_{r}\) in Eq. (13), the transfer matrices of the output end with respect to input ends n (\(n=\mathrm{2,3},4\)) are represented as

$${{\varvec{U}}}_{13,n}=\left[\begin{array}{cccc}{{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {\widetilde{{\varvec{l}}}}_{{I}_{r}O}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}\end{array}\right]$$
(15)

The geometrical relationship of the first end with respect to itself is

$${{\varvec{H}}}_{13,1}=\left[\begin{array}{cccc}{{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\end{array}\right]$$
(16)

The additional geometrical equations describing geometrical relationships between the first and nth \(\left(n=2,3,4\right)\) input ends can be expressed as

$${{\varvec{H}}}_{13,n}=\left[\begin{array}{cccc}{{\varvec{I}}}_{3}& {\widetilde{{\varvec{l}}}}_{{I}_{1}{I}_{n}}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\end{array}\right]$$
(17)

The transfer matrix of workpiece 31 can be obtained according to the above method.

The transfer matrix of spatial elastic hinge is

$${{\varvec{U}}}_{v}=\left[\begin{array}{cccc}{{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{K}}}_{v}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {{\varvec{K}}}_{v}{\prime}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}\end{array}\right]$$
(18)

where v = 1, 3, 4, 6, 7, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, and

$${{\varvec{K}}}_{v}=\left[\begin{array}{ccc}-1/{K}_{v,x}& 0& 0\\ 0& -1/{K}_{v,y}& 0\\ 0& 0& -1/{K}_{v,z}\end{array}\right],{{\varvec{K}}}_{v}{\prime}=\left[\begin{array}{ccc}1/{K}_{v, x}{\prime}& 0& 0\\ 0& 1/{K}_{v, y}{\prime}& 0\\ 0& 0& 1/{K}_{v, z}{\prime}\end{array}\right]$$

in which \({K}_{v,x},{K}_{v,y}\), and \({K}_{v,z}\) donate the stiffnesses of springs in the \(x,y\), and \(z\) directions, respectively. Moreover, \({K}_{v,x}{\prime},{K}_{v,y}{\prime}\), and \({K}_{v,z}{\prime}\) are the torsional stiffnesses of rotary springs about the \(x,y\), and \(z\) directions, respectively.

According to the automatic deduction, the main transfer equation is established as follows

$${{\varvec{T}}}_{31-32}{{\varvec{Z}}}_{\mathrm{32,0}}+{{\varvec{T}}}_{31-1}{{\varvec{Z}}}_{\mathrm{1,0}}+{{\varvec{T}}}_{31-4}{{\varvec{Z}}}_{\mathrm{4,0}}+{{\varvec{T}}}_{31-7}{{\varvec{Z}}}_{\mathrm{7,0}}+{{\varvec{T}}}_{31-10}{{\varvec{Z}}}_{\mathrm{10,0}}={{\varvec{Z}}}_{\mathrm{31,0}}$$
(19)

where \({{\varvec{Z}}}_{k,0}\) reprintings the state vector of the output end of element k, and

$$\begin{array}{c}{{\varvec{T}}}_{31-32}={{\varvec{U}}}_{31,32}{{\varvec{U}}}_{32}\\ {{\varvec{T}}}_{31-1}={{\varvec{U}}}_{31,30}{{\varvec{U}}}_{30}{{\varvec{U}}}_{29}{{\varvec{U}}}_{28}{{\varvec{U}}}_{27}{{\varvec{H}}}_{X}\left({\theta }_{6}\right){{\varvec{U}}}_{26}{{\varvec{U}}}_{25}{{\varvec{H}}}_{Y}\left({\theta }_{5}\right){{\varvec{U}}}_{24}{{\varvec{U}}}_{23}{{\varvec{H}}}_{X}\left({\theta }_{4}\right){{\varvec{U}}}_{22}{{\varvec{U}}}_{21}{{\varvec{H}}}_{Y}\left({\theta }_{3}\right)\\ {{\varvec{U}}}_{20}{{\varvec{U}}}_{19}{{\varvec{H}}}_{Y}\left({\theta }_{2}\right){{\varvec{U}}}_{18}{{\varvec{U}}}_{17}{{\varvec{H}}}_{Z}\left({\theta }_{1}\right){{\varvec{U}}}_{16}{{\varvec{U}}}_{15}{{\varvec{U}}}_{14}{{\varvec{U}}}_{13,3}{{\varvec{U}}}_{3}{{\varvec{U}}}_{2}{{\varvec{U}}}_{1}\\ {{\varvec{T}}}_{31-4}={{\varvec{U}}}_{31,30}{{\varvec{U}}}_{30}{{\varvec{U}}}_{29}{{\varvec{U}}}_{28}{{\varvec{U}}}_{27}{{\varvec{H}}}_{X}\left({\theta }_{6}\right){{\varvec{U}}}_{26}{{\varvec{U}}}_{25}{{\varvec{H}}}_{Y}\left({\theta }_{5}\right){{\varvec{U}}}_{24}{{\varvec{U}}}_{23}{{\varvec{H}}}_{X}\left({\theta }_{4}\right){{\varvec{U}}}_{22}{{\varvec{U}}}_{21}{{\varvec{H}}}_{Y}\left({\theta }_{3}\right)\\ {{\varvec{U}}}_{20}{{\varvec{U}}}_{19}{{\varvec{H}}}_{Y}\left({\theta }_{2}\right){{\varvec{U}}}_{18}{{\varvec{U}}}_{17}{{\varvec{H}}}_{Z}\left({\theta }_{1}\right){{\varvec{U}}}_{16}{{\varvec{U}}}_{15}{{\varvec{U}}}_{14}{{\varvec{U}}}_{\mathrm{13,6}}{{\varvec{U}}}_{6}{{\varvec{U}}}_{5}{{\varvec{U}}}_{4}\\ {{\varvec{T}}}_{31-7}={{\varvec{U}}}_{31,30}{{\varvec{U}}}_{30}{{\varvec{U}}}_{29}{{\varvec{U}}}_{28}{{\varvec{U}}}_{27}{{\varvec{H}}}_{X}\left({\theta }_{6}\right){{\varvec{U}}}_{26}{{\varvec{U}}}_{25}{{\varvec{H}}}_{Y}\left({\theta }_{5}\right){{\varvec{U}}}_{24}{{\varvec{U}}}_{23}{{\varvec{H}}}_{X}\left({\theta }_{4}\right){{\varvec{U}}}_{22}{{\varvec{U}}}_{21}{{\varvec{H}}}_{Y}\left({\theta }_{3}\right)\\ {{\varvec{U}}}_{20}{{\varvec{U}}}_{19}{{\varvec{H}}}_{Y}\left({\theta }_{2}\right){{\varvec{U}}}_{18}{{\varvec{U}}}_{17}{{\varvec{H}}}_{Z}\left({\theta }_{1}\right){{\varvec{U}}}_{16}{{\varvec{U}}}_{15}{{\varvec{U}}}_{14}{{\varvec{U}}}_{\mathrm{13,9}}{{\varvec{U}}}_{9}{{\varvec{U}}}_{8}{{\varvec{U}}}_{7}\\ {{\varvec{T}}}_{31-10}={{\varvec{U}}}_{31,30}{{\varvec{U}}}_{30}{{\varvec{U}}}_{29}{{\varvec{U}}}_{28}{{\varvec{U}}}_{27}{{\varvec{H}}}_{X}\left({\theta }_{6}\right){{\varvec{U}}}_{26}{{\varvec{U}}}_{25}{{\varvec{H}}}_{Y}\left({\theta }_{5}\right){{\varvec{U}}}_{24}{{\varvec{U}}}_{23}{{\varvec{H}}}_{X}\left({\theta }_{4}\right){{\varvec{U}}}_{22}{{\varvec{U}}}_{21}{{\varvec{H}}}_{Y}\left({\theta }_{3}\right)\\ {{\varvec{U}}}_{20}{{\varvec{U}}}_{19}{{\varvec{H}}}_{Y}\left({\theta }_{2}\right){{\varvec{U}}}_{18}{{\varvec{U}}}_{17}{{\varvec{H}}}_{Z}\left({\theta }_{1}\right){{\varvec{U}}}_{16}{{\varvec{U}}}_{15}{{\varvec{U}}}_{14}{{\varvec{U}}}_{\mathrm{13,12}}{{\varvec{U}}}_{12}{{\varvec{U}}}_{11}{{\varvec{U}}}_{10}\end{array}$$
(20)

where \({{\varvec{U}}}_{i}\) is the transfer matrix corresponding to the element i, \({{\varvec{U}}}_{31,30}\) and \({{\varvec{U}}}_{31,32}\) are the transfer matrices of workpiece 31 with respect to input ends 30 and 32, \({{\varvec{U}}}_{13,3}\),\({{\varvec{U}}}_{13,6}\),\({{\varvec{U}}}_{13,9}\) and \({{\varvec{U}}}_{13,12}\) are the transfer matrices of AGV 13 with respect to input ends 3, 6, 9, and 12.

Furthermore, the coordinate transformation matrices are

$${{\varvec{H}}}_{X}\left({\theta }_{k}\right)=\left[\begin{array}{cccc}{{\varvec{h}}}_{X}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{X}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{X}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{X}\left({\theta }_{k}\right)\end{array}\right],k=1,2,{3,4},\mathrm{5,6}$$
(21)
$${{\varvec{H}}}_{Y}\left({\theta }_{k}\right)=\left[\begin{array}{cccc}{{\varvec{h}}}_{Y}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{Y}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{Y}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{Y}\left({\theta }_{k}\right)\end{array}\right], k=1,2,\mathrm{3,4},\mathrm{5,6}$$
(22)
$${{\varvec{H}}}_{Z}\left({\theta }_{k}\right)=\left[\begin{array}{cccc}{{\varvec{h}}}_{Z}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{Z}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{Z}\left({\theta }_{k}\right)& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{h}}}_{Z}\left({\theta }_{k}\right)\end{array}\right],k=1,2,\mathrm{3,4},\mathrm{5,6}$$
(23)

where \({\theta }_{k}\) is the angle of the joint k, and

$${{\varvec{h}}}_{X}(\theta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\theta & \mathrm{sin}\theta \\ 0& -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right],{{\varvec{h}}}_{Y}(\theta )=\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right],{{\varvec{h}}}_{Z}(\theta )=\left[\begin{array}{ccc}\mathrm{cos}\theta & \mathrm{sin}\theta & 0\\ -\mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]$$
(24)

The elements with multiple input ends in the ARGW system include workpiece 31 and vehicle body 13, so system geometric equations can be found as

$$\begin{array}{c}{{\varvec{G}}}_{31-32}{{\varvec{Z}}}_{\mathrm{32,0}}+{{\varvec{G}}}_{31-1}{{\varvec{Z}}}_{\mathrm{1,0}}+{{\varvec{G}}}_{31-4}{{\varvec{Z}}}_{\mathrm{4,0}}+{{\varvec{G}}}_{31-7}{{\varvec{Z}}}_{\mathrm{7,0}}+{{\varvec{G}}}_{31-10}{{\varvec{Z}}}_{\mathrm{10,0}}={{\varvec{O}}}_{6\times 1}\\ {{\varvec{G}}}_{13-1}{{\varvec{Z}}}_{\mathrm{1,0}}+{{\varvec{G}}}_{13-4}{{\varvec{Z}}}_{\mathrm{4,0}}={0}_{6\times 1}\\ {{\varvec{G}}}_{13-1}{{\varvec{Z}}}_{\mathrm{1,0}}+{{\varvec{G}}}_{13-7}{{\varvec{Z}}}_{\mathrm{7,0}}={0}_{6\times 1}\\ {{\varvec{G}}}_{13-1}{{\varvec{Z}}}_{\mathrm{1,0}}+{{\varvec{G}}}_{13-10}{{\varvec{Z}}}_{\mathrm{10,0}}={0}_{6\times 1}\end{array}$$
(25)

where \({{\varvec{O}}}_{6\times 1}\) denotes a 6 × 1 zero column matrix, and the parameters in Eq. (25) can be expressed as.

$$\begin{array}{c}{{\varvec{G}}}_{13-1}=-{{\varvec{H}}}_{13,3}{{\varvec{U}}}_{3}{{\varvec{U}}}_{2}{{\varvec{U}}}_{1}\\ {{\varvec{G}}}_{13-4}={{\varvec{H}}}_{13,6}{{\varvec{U}}}_{6}{{\varvec{U}}}_{5}{{\varvec{U}}}_{4}\\ {{\varvec{G}}}_{13-7}={{\varvec{H}}}_{13,9}{{\varvec{U}}}_{9}{{\varvec{U}}}_{8}{{\varvec{U}}}_{7}\\ {{\varvec{G}}}_{13-10}={{\varvec{H}}}_{13,12}{{\varvec{U}}}_{12}{{\varvec{U}}}_{11}{{\varvec{U}}}_{10}\\ {{\varvec{G}}}_{31-1}={{\varvec{H}}}_{31,30}{{\varvec{U}}}_{31,30}^{-1}{{\varvec{T}}}_{31-1}\\ {{\varvec{G}}}_{31-4}={{\varvec{H}}}_{31,30}{{\varvec{U}}}_{31,30}^{-1}{{\varvec{T}}}_{31-4}\\ {{\varvec{G}}}_{31-7}={{\varvec{H}}}_{31,30}{{\varvec{U}}}_{31,30}^{-1}{{\varvec{T}}}_{31-7}\\ {{\varvec{G}}}_{31-10}={{\varvec{H}}}_{31,30}{{\varvec{U}}}_{31,30}^{-1}{{\varvec{T}}}_{31-10}\\ {G}_{31-32}=-{{\varvec{H}}}_{31,32}{{\varvec{U}}}_{32}\end{array}$$
(26)

where

$${{\varvec{H}}}_{i,o}=\left[\begin{array}{cccc}{{\varvec{I}}}_{3}& {\widetilde{{\varvec{l}}}}_{io}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\\ {{\varvec{O}}}_{3\times 3}& {{\varvec{I}}}_{3}& {{\varvec{O}}}_{3\times 3}& {{\varvec{O}}}_{3\times 3}\end{array}\right]$$
(27)

where \({l}_{io}\) represents the displacement matrix of the input point end o relative to first input end i, \({\widetilde{{\varvec{l}}}}_{io}\) refers to the skew-symmetric matrix of \({{\varvec{l}}}_{io}\). In this paper, i includes 13, and 31, o is the corresponding 3, 6, 9, 12, and 30, 32.

Combing the system main transfer equation Eq. (18) and the system geometric equations Eq. (25), the system overall transfer equation can be obtained as

$$\left[\begin{array}{cccccc}-{{\varvec{I}}}_{12}& {{\varvec{T}}}_{31-32}& {{\varvec{T}}}_{31-1}& {{\varvec{T}}}_{31-4}& {{\varvec{T}}}_{31-7}& {{\varvec{T}}}_{31-10}\\ {{\varvec{O}}}_{6\times 12}& {{\varvec{G}}}_{31-32}& {{\varvec{G}}}_{31-1}& {{\varvec{G}}}_{31-4}& {{\varvec{G}}}_{31-7}& {{\varvec{G}}}_{31-10}\\ {{\varvec{O}}}_{6\times 12}& {{\varvec{O}}}_{6\times 12}& {{\varvec{G}}}_{13-1}& {{\varvec{G}}}_{13-4}& {{\varvec{O}}}_{6\times 12}& {{\varvec{O}}}_{6\times 12}\\ {{\varvec{O}}}_{6\times 12}& {{\varvec{O}}}_{6\times 12}& {{\varvec{G}}}_{13-1}& {{\varvec{O}}}_{6\times 12}& {{\varvec{G}}}_{13-7}& {{\varvec{O}}}_{6\times 12}\\ {{\varvec{O}}}_{6\times 12}& {{\varvec{O}}}_{6\times 12}& {{\varvec{G}}}_{13-1}& {{\varvec{O}}}_{6\times 12}& {{\varvec{O}}}_{6\times 12}& {{\varvec{G}}}_{13-10}\end{array}\right]\left[\begin{array}{c}{{\varvec{Z}}}_{\mathrm{31,0}}\\ {{\varvec{Z}}}_{\mathrm{32,0}}\\ {{\varvec{Z}}}_{\mathrm{1,0}}\\ {{\varvec{Z}}}_{\mathrm{4,0}}\\ \begin{array}{c}{{\varvec{Z}}}_{\mathrm{7,0}}\\ {{\varvec{Z}}}_{\mathrm{10,0}}\end{array}\end{array}\right]=0$$
(28)

Furthermore

$${{\varvec{U}}}_{\text{all}}{{\varvec{Z}}}_{\text{all}}={{\varvec{O}}}_{72\times 1}$$
(29)

Appendix 2. Calculation of the dynamic equation

The form of the coefficients in the overall dynamic equations are

$$\begin{array}{c}M={\text{d}}{\text{i}}{\text{a}}{\text{g}}\left({{\varvec{M}}}_{2},{{\varvec{M}}}_{5},\cdots ,{{\varvec{M}}}_{31}\right)\\ K={\text{d}}{\text{i}}{\text{a}}{\text{g}}\left({{\varvec{K}}}_{2},{{\varvec{K}}}_{5},\cdots ,{{\varvec{K}}}_{31}\right)\\ v={\left[{{\varvec{v}}}_{2}^{\text{T}},{{\varvec{v}}}_{5}^{\text{T}},\cdots ,{{\varvec{v}}}_{31}^{\text{T}}\right]}^{\text{T}}\\ f={\left[{{\varvec{f}}}_{2}^{\text{T}},{{\varvec{f}}}_{5}^{\text{T}},\cdots ,{{\varvec{f}}}_{31}^{\text{T}}\right]}^{\text{T}}\end{array}$$
(30)

And the augmented eigenvector of the kth model is defined as

$$\left\{\begin{array}{c}{{\varvec{V}}}_{i}^{k}={{\varvec{H}}{\varvec{Z}}}_{i}^{k}, \, \, i=\mathrm{2,5},\cdots 30,H={\left[{{\varvec{I}}}_{6*1 }, {{\varvec{O}}}_{6*1 }\right]}^{\mathrm{T}}\\ {{\varvec{V}}}^{k}={\left[{\left({{\varvec{V}}}_{2}^{k}\right)}^{\text{T}},{\left({{\varvec{V}}}_{5}^{k}\right)}^{\text{T}},\cdots ,{\left({{\varvec{V}}}_{30}^{k}\right)}^{\text{T}}\right]}^{\text{T}}\end{array}\right.$$
(31)

The damping \({\varvec{C}}=2\zeta \omega {\varvec{M}}\), where \(\zeta\) is the modal damping ratio. According to the orthogonality of the augmented eigenvectors, the relation between M, K, and the augmented eigenvector \({{\varvec{V}}}^{j}\) is satisfied

$$\begin{array}{c}\langle {{\varvec{M}}{\varvec{V}}}^{k},{{\varvec{V}}}^{j}\rangle ={\delta }_{k,j}{M}^{j}A\\ \langle {{\varvec{K}}{\varvec{V}}}^{k},{{\varvec{V}}}^{j}\rangle ={\delta }_{k,j}{({\omega }^{j})}^{2}{M}^{j}\end{array}$$
(32)

where \({M}^{j}\) and \({K}^{j}={({\omega }^{j})}^{2}{M}^{j}\) are the jth-order modal mass and stiffness, respectively. Only when k = j, \({\delta }_{k,j}=1\), in other situations, \({\delta }_{k,j}=0\).

Appendix 3. Parameters in the simulation

Tables 8 and 9

Table 8 The physical parameters
Full size table
Table 9 The physical parameters
Full size table

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Wang, P., Tian, W. & Li, B. Machining vibration suppression of cantilever parts of aerospace structure using robot-assisted clamping system. Int J Adv Manuf Technol 128, 4103–4121 (2023). https://doi.org/10.1007/s00170-023-12090-w

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