Vibration reduction by natural frequency optimization for manipulation of a variable geometry truss

Abstract

This study investigates the optimization of motion plans for the two dimensional variable geometry trusses (VGTs) in order to semi-actively reduce the vibration amplitude while the VGTs move a target payload from one point to another. We assume that the payload such as a trouble satellites have a control moment gyro or reaction wheel; corresponding sinusoidal disturbance is generated by the payload and it probably cause the resonance vibration of the VGTs during the manipulation. The natural frequency during the manipulation of the VGTs is given as a function of the geometry; hence, the motion plan for the VGTs can be optimized by considering difference between the frequency of the disturbance applied by the payload and the natural frequency of the VGTs. In order to study the effect of the geometry changes on the structural vibration during the manipulation, a numerical, two-dimensional VGT model that has three variable length members in the trusses is constructed by finite element method. The numerical results show that resonant vibration can occur for the specific trajectory of the VGT to move the target payload. We assumed that the VGT changes their geometry as quasi-static way. We optimize the trajectory and the three variable length members of the VGT in order to prevent the resonance by using a sequential quadratic programming. As a result, the optimized manipulation effectively reduces the amplitude of vibration. Furthermore, the stress of the truss members and the total number of the motion steps for the manipulation are evaluated.

Appendix

Appendix This study uses the SQP method to find the optimal set of three angles $\alpha _{i}$ for $i=1,2,3$. The SQP algorithms is briefly described here. Interested readers may consult with reference (Gill et al. 1981, 1991) for the details of the method.

The SQP method is known to be one of the reliable optimization method for constrained function of multi design variables. For the SQP problem, the search direction d for the design parameters $\textbf {x}_{k}$ for kth iteration is found by using the quasi-Newton method. This direction finding process is usually called “QP subproblem” and is summarized as follows.

$$\mathrm{minimize} : \frac{1}{2} \textbf{d}^{T}B_{k} \textbf{d}+ \nabla f(\textbf{x}_{k})^{T} \textbf{d} \ \mathrm{such \ that} \ \textbf{d}\in \mathcal{R}^{n} $$

(22)

$$ \mathrm{subject \ to} : g_{i}(\textbf{x}_{k})+\nabla g_{i}(\textbf{x}_{k})^{T}\mathbf{d} \leq 0 \ \ (i=1,2,\dots ,m) $$

(23)

$$ h_{j}(\textbf{x}_{k})+\nabla h_{j}(\textbf{x}_{k})^{T} \textbf{d} =0 \ \ (j=1,2,\dots ,l) $$

(24)

where f is an objective function, $\nabla$ is a first order gradient, $g_{i}$ is inequality constraints, and $h_{j}$ is equality constraints. Thus, the SQP method uses second order information to obtain search direction vector.

Although the solution for (22)–(24) is obtained by using a penalty function approach, it is now considered relatively inefficient. Thus, (22)–(34) are then replaced by the Kunh–Tucker (KT) equations for the QP subproblem defined as follows.

$$ B_{k}\textbf{d}+\nabla f(\textbf{x}_{k})+\sum\limits_{i=1}^{m} \lambda _{i}\nabla g_{i}(\textbf{x}_{k})+\sum\limits_{j=1}^{l} \mu _{j} \nabla h_{j}(\textbf{x}_{k}) = \mathbf{0} $$

(25)

$$ g_{i}(\textbf{x}_{k})+\nabla g_{i}(\textbf{x}_{k})^{T} \textbf{d} \leq 0 \ \ (i=1,2,\dots ,m) $$

(26)

$$ h_{j}(\textbf{x})+\nabla h_{j}(\textbf{x}_{k})^{T} \textbf{d}=0 \ \ (j=1,2,\dots ,l) $$

(27)

$$ \lambda _{i} \left\{ g_{i}(\textbf{x}_{k})+\nabla g_{i}(\textbf{x}_{k})\right\}^{T}\textbf{d}=0 \ \ (i=1,2,\dots ,m) $$

(28)

$$ \boldsymbol{\lambda}_{i} \geq 0 \ \ (i=1,2,\dots ,m) $$

(29)

The quasi-Newton method is then applied to find d for KT equations. For the quasi-Newton method, the approximation of Hessian matrix by a symmetric and positive definite matrix $B_{k}$ is used. In each iteration, the matrix $B_{k}$ should be updated for next iteration. Then, $B_{k}$ in (22) is updated by the Broydon-Fletcher-Goldfarb-Shanno (BFGS) method (Fletcher 1987).

$$ B_{k+1} = B_{k}-\frac{B_{k}\textbf{s}_{k}(B_{k}\textbf{s}_{k})^{T}}{(\textbf{s}_{k})^{T}B_{k}\textbf{s}_{k}}+\frac{\textbf{z}_{k}(\textbf{z}_{k})^{T}}{(\textbf{s}_{k})^{T}\textbf{z}_{k}} $$

(30)

$$ \textbf{s}_{k} = \textbf{x}_{k+1}-\textbf{x}_{k} $$

(31)

$$ \textbf{y}_{k} = \nabla_{\textbf{x}} L(\textbf{x}_{k+1},\boldsymbol{\lambda}_{k+1},\boldsymbol{\mu}_{k+1})- \nabla_{\textbf{x}} L(\textbf{x}_{k},\boldsymbol{\lambda}_{k},\boldsymbol{\mu}_{k}) $$

(32)

$$ \textbf{z}_{k} = \phi _{k}\textbf{y}_{k}+(1-\phi _{k})B_{k}\textbf{s}_{k} $$

(33)

$$\phi_{k} = \left\{\begin{array}{c}1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{if} \ \ (\textbf{s}_{k})^{T} \textbf{y}_{k} \geq \omega (\textbf{s}_{k})^{T}B_{k} \textbf{s}_{k} \\[5pt]\frac{(1-\omega)(\textbf{s}_{k})^{T} B_{k} \textbf{s}_{k}}{(\textbf{s}_{k})^{T}(B_{k}\textbf{s}_{k}-\textbf{y}_{k})} \ \ \mathrm{if} \ \ (\textbf{s}_{k})^{T} \textbf{y}_{k} < \omega (\textbf{s}_{k})^{T}B_{k} \textbf{s}_{k}\end{array}\right.$$

(34)

$$ 0 \leq \omega \leq1 $$

(35)

where L is a Lagrangian function, $\lambda _{k}$ and $\mu _{k}$ are Lagrange multipliers that contains constraints as follows.

$$ L(\textbf{x}_{k}, \boldsymbol{\lambda}_{k}, \boldsymbol{\mu}_{k}){} ={} f(\textbf{x}_{k}){}+{}\sum_{i=1}^{m}{}\lambda_{i}^{k} g_{i}(\textbf{x}_{k}){}+{}\sum_{j=1}^{l} \mu _{j}^{k} h_{j}(\textbf{x}_{k}) $$

(36)

Once the solution $\textbf {d}_{k}$ is obtained, the next search point $\textbf {x}_{k+1}$ of the design parameters is calculated as follows

$$ \textbf{x}_{k+1}=\textbf{x}_{k}+ \boldsymbol{\alpha}_{k} \textbf{d}_{k} $$

(37)

where scalar $\boldsymbol {\alpha }_{k}$ is obtained by solving a line search problem as follows.

$$ \mathrm{minimize} : \ \ L(\mathbf{x}+\alpha \textbf{d}) \ \ \mathrm{ such \ \ that} \ \ \alpha >0 $$

(38)

Thus, the design parameters are sequentially updated by (37). The iteration of the design parameters is finished until the convergence criteria is satisfied.