Linear programming approach to design of spatial link mechanism with partially rigid joints

Abstract

A simple systematic approach is presented for designing a spatial link mechanism with partially rigid joints. A linear programming (LP) problem to find an infinitesimal mechanism that maximizes the output displacement is first formulated. The objective function of this LP problem has a penalty term to obtain a sparse solution including small numbers of hinges and members to be removed. It is shown that the dual of this LP problem can be regarded as a plastic limit analysis problem that maximizes the load factor under the equilibrium condition and upper- and lower-bound constraints on the member-end forces of a given frame structure. A heuristic approach is presented to obtain a finite mechanism by solving the LP problem after updating the nodal locations in the direction of inextensional deformation. It is shown in the numerical examples that various planar and spatial mechanisms can be easily found using the proposed method.

Appendix

We can obtain problem (11a) in Section 3.3 as a dual problem of problem (10a) in Section 3.2 from any standard duality theory of convex optimization. We here adopt the Lagrangian duality theory for explaining the derivation.

For notational convenience, rewrite problem (10) as

$$ \min\limits_{\boldsymbol{u}, \boldsymbol{\gamma}} {\quad} {-}\boldsymbol{f}_{\text{out}}^{\mathrm{T}} \boldsymbol{u} + \sum\limits_{i=1}^{n} \alpha w_{i} \gamma_{i} $$

(13a)

$$ \mathrm{s.{\,}t.} {\kern16pt}\gamma_{i} \ge |\boldsymbol{h}_{i}^{\mathrm{T}} \boldsymbol{u}| , \quad i=1,\dots,n , $$

(13b)

$$ {\kern26pt}\boldsymbol{f}_{\text{in}}^{\mathrm{T}} \boldsymbol{u} = \bar{u}_{\text{in}} . $$

(13c)

The Lagrangian of problem (13a, b, c) can be defined by

$$ L(\boldsymbol{u},\boldsymbol{\gamma},\boldsymbol{v},\boldsymbol{y},\lambda_{\text{in}}) $$

(14)

$$ =\! \left\{\begin{array}{cccc} {-}\boldsymbol{f}_{\text{out}}^{\mathrm{T}} \boldsymbol{u} + \sum_{i=1}^{n} (\alpha w_{i} \gamma_{i} - v_{i} \gamma_{i} + y_{i} \boldsymbol{h}_{i}^{\mathrm{T}} \boldsymbol{u}) \\ {}- \lambda_{\text{in}} (\boldsymbol{f}_{\text{in}}^{\mathrm{T}} \boldsymbol{u} - \bar{u}_{\text{in}}) & {} if \; v_{i} \ge |y_{i}|(i=1,\dots,n), \\ {}-\infty & {\kern-7.5pc}\text{otherwise}, \end{array}\right. $$

(15)

where $\boldsymbol {v} \in \mathbb {R}^{n}$, $\boldsymbol {y} \in \mathbb {R}^{n}$ and $\lambda _{\text {in}} \in \mathbb {R}$ are the Lagrange multipliers. Indeed, problem (13) can be expressed by using L as

$$\begin{array}{*{20}l} \min\limits_{\boldsymbol{u},\boldsymbol{\gamma}} \quad \sup \{ L(\boldsymbol{u},\boldsymbol{\gamma},\boldsymbol{v},\boldsymbol{y},\lambda_{\text{in}}) \mid (\boldsymbol{v},\boldsymbol{y},\lambda_{\text{in}}) \in \mathbb{R}^{2n+1} \} , \end{array} $$

because we have that

$$\begin{array}{@{}rcl@{}} & \sup_{v_{i},y_{i}} \{ -v_{i} \gamma_{i} + y_{i} (\boldsymbol{h}_{i}^{\mathrm{T}} \boldsymbol{u}) \mid v_{i} \ge |y_{i}| \} \\ & \ \ = \left\{\begin{array}{cccc} {}0 & {\kern1.3pc}if \gamma_{i} \ge |\boldsymbol{h}_{i}^{\mathrm{T}} \boldsymbol{u}|, \\ +\infty & \textnormal{otherwise}. \end{array}\right. \end{array} $$

Then the Lagrangian dual problem is defined by

$$\begin{array}{*{20}l} \max\limits_{\boldsymbol{v},\boldsymbol{y},\lambda_{\text{in}}} \quad \inf \{ L(\boldsymbol{u},\boldsymbol{\gamma},\boldsymbol{v},\boldsymbol{y},\lambda_{\text{in}}) \mid (\boldsymbol{u},\boldsymbol{\gamma}) \in \mathbb{R}^{d+n} \} . \end{array} $$

(16)

Observe that the relation

$$\begin{array}{@{}rcl@{}} & \inf \{ L(\boldsymbol{u},\boldsymbol{\gamma},\boldsymbol{v},\boldsymbol{y},\lambda_{\text{in}}) \mid (\boldsymbol{u},\boldsymbol{\gamma}) \in \mathbb{R}^{d+n} \}{\kern100pt}\notag\\ & = \left\{\begin{array}{cccc} {\kern-4.2pc}\bar{u}_{\text{in}} \lambda_{\text{in}} + \sum_{i=1}^{n} \inf \{ (\alpha w_{i} - v_{i}) \gamma_{i} \mid \gamma_{i} \in \mathbb{R} \} \\ \,\,\,\,\,{\kern.5pc} + \inf \Bigl\{ \sum_{i=1}^{n} y_{i}\boldsymbol{h}_{i}^{\mathrm{T}} \boldsymbol{u} - (\boldsymbol{f}_{\text{out}} + \lambda_{\text{in}} \boldsymbol{f}_{\text{in}})^{\mathrm{T}} \boldsymbol{u} \mid \boldsymbol{u}\in\mathbb{R}^{d} \Bigr\} \\ {\kern-4.2pc}\text{if} \; v_{i} \ge |y_{i}| \ \ (i=1,\dots,n), \\ {}{-}\infty \ \ \ \ \text{otherwise} \end{array}\right. \\ &{\kern-4.1pc} = \left\{\begin{array}{cccc} \bar{u}_{\text{in}} \lambda_{\text{in}} & if v_{i} = \alpha w_{i} \ge |y_{i}| \ \ (i=1,\dots,n), \\ & \sum_{i=1}^{n} y_{i}\boldsymbol{h}_{i} =\boldsymbol{f}_{\text{out}} + \lambda_{\text{in}} \boldsymbol{f}_{\text{in}}, \\ {-}\infty & {\kern-4.5pc}\text{otherwise} \end{array}\right. \end{array} $$

holds to see that problem (16) coincides with problem (11).