Optimal configurations of circular bars under free torsional and longitudinal vibration based on Pontryagin’s maximum principle

Abstract

Optimal configurations of circular bars under free torsional and longitudinal vibration are investigated using the Pontryagin’s maximum principle (PMP). The necessary optimality condition of PMP is analysed by considering the cross-sectional area of bars as control variable and by using Maier objective functional to control the final state of the objective functional. Optimal configurations associated to different orders of eigen frequencies, eigen vectors, and specific boundary conditions are demonstrated. Their relations are also explained qualitatively. Numerical results show the equivalent about optimal configurations and eigen frequencies under specific boundary conditions.

Appendix: Proof of the necessary optimality condition (11b) cited in the Sect. 2.2

The natural frequency ω _i is here considered as a state variable. It means that the role of ω _i is equivalent to those of u and N in Eq. (2b). The total weight W is also a state variable. So, the state Eq. (2b) of the bar can be rewritten in the form:

$$\frac{du}{dx} = \frac{N}{EA}$$

(15a)

$$\frac{dN}{dx} = - \rho A\omega_{i}^{2} u$$

(15b)

$$\frac{{d\omega_{i} }}{dx} = 0$$

(15c)

$$\frac{dW}{dx} = \rho A$$

(15d)

The objective function, Eq. (10), can be rewritten in term of the Maier objective functional as follows:

$$F = (1 - k_{\omega W} )\frac{{c_{\omega i} \omega_{i} (L)}}{{\omega_{0i} }} + k_{\omega W} \frac{{c_{W} W(L)}}{{W_{0} }} \to \hbox{min} .$$

(16)

The Hamiltonian function H can be established as follows:

$$H = p_{u} \frac{N}{EA} - p_{N} \rho A\omega_{i}^{2} u + p_{\omega } \omega_{i}^{\prime} + p_{W} \rho A,(\omega_{i}^{{\prime }} = 0)$$

(17)

The adjoint equations can be expressed under the following forms:

$$\frac{{dp_{u} }}{dx} = - \frac{\partial H}{\partial u} = \rho A\omega_{i}^{2} p_{N}$$

(18a)

$$\frac{{dp_{N} }}{dx} = - \frac{\partial H}{\partial N} = - \frac{1}{EA}p_{u}$$

(18b)

$$\frac{{dp_{\omega } }}{dx} = - \frac{\partial H}{{\partial \omega_{i} }} = 2\rho_{i} A_{i} \omega_{i} up_{N}$$

(18c)

$$\frac{{dp_{W} }}{dx} = - \frac{\partial H}{\partial W} = 0$$

(18d)

The adjoint variables $p_{\varphi } ,p_{M} ,p_{\omega } ,p_{w}$ are determined as below:

$$\sum\limits_{j = 1}^{m} {p_{j} (L)\delta y_{j} (L)} - \sum\limits_{j = 1}^{m} {p_{j} (0)\delta y_{j} (0)} + \delta F = 0$$

(19)

where m = 4; y ₁ = u, y ₂ = N, y ₃ = ω _i and y ₄ = W are state variables and p ₁ = p _u , p ₂ = p _N , p ₃ = p _ω and p ₄ = p _W are adjoint variables. Thus,

$$\begin{aligned} p_{u} (L)\delta u(L) + p_{N} (L)\delta N(L) + p_{\omega } (L)\delta \omega_{i} (L) + p_{W} (L)\delta W(L){-}p_{u} (0)\delta u(0){-}p_{N} (0)\delta N(0) \hfill \\ {-}\,p_{\omega } (0)\delta \omega_{i} (0){-}p_{W} (0)\delta W(0)(1 - k_{\omega W} )\frac{{c_{\omega i} \delta \omega_{i} (L)}}{{\omega_{0i} }} + k_{\omega W} \frac{{c_{W} \delta W(L)}}{{W_{0} }} = \, 0 \hfill \\ \end{aligned}$$

(20)

Or,

$$\begin{aligned} p_{u} (L)\delta u(L) + p_{N} (L)\delta N(L) + \left[ {p_{\omega } (L) + \frac{{c_{\omega i} (1 - k_{\omega W} )}}{{\omega_{0i} }}\delta \omega_{i} (L)} \right] + \left[ {p_{W} (L) + \frac{{c_{W} k_{\omega W} }}{{W_{0} }}} \right]\delta W(L) \hfill \\ {-}\,p_{u} (0)\delta u(0){-}p_{N} (0)\delta N(0) - p_{\omega } (0)\delta \omega_{i} (0){-}\,p_{W} (0)\delta W(0) = 0 \hfill \\ \end{aligned}$$

(21)

We obtain:

$$\left\{ {\begin{array}{*{20}c} {p_{\omega } (L) = - \frac{{c_{\omega i} (1 - k_{\omega W} )}}{{\omega_{0i} }}} \\ {p_{W} (L) = - \frac{{c_{W} k_{\omega W} }}{{W_{0} }}} \\ {p_{\omega } (0) = 0} \\ {p_{W} (0) = 0} \\ \end{array} } \right.$$

(22)

and other expressions which are depended on the boundary conditions of the bar system as:

• (a)
  $${\text{free}}{\text{-}}{\text{free:}}\;N(0) = N(L) = 0\;(3{\text{b}}) \Rightarrow p_{u} (L) = 0;p_{u} (0) = 0$$

  (23a)
• (b)
  $${\text{fixed}}{\text{-}}{\text{free:}}\;u(0) = 0,N(L) = 0\; (4{\text{b)}} \Rightarrow p_{u} (L) = 0;p_{N} (0) = 0$$

  (23b)
• (c)
  $${\text{free}}{\text{-}}{\text{fixed:}}\;N(0) = 0,u(L) = 0\; (5{\rm b})\Rightarrow p_{N} (L) = 0;p_{u} (0) = 0$$

  (23c)
• (d)
  $${\text{fixed}}{\text{-}}{\text{fixed:}}\;u(0) = 0,u(L) = 0\; (6{\text{b)}} \Rightarrow p_{N} (L) = 0;p_{N} (0) = 0$$

  (23d)

Assigning:

$$\left\{ {\begin{array}{*{20}c} {p_{u} = - N_{H} } \\ {p_{N} = u_{H} } \\ \end{array} } \right.$$

(24)

Combining Eqs. (18a), (18b) and (24) yields:

$$\left\{ {\begin{array}{*{20}c} {\frac{{du_{H}^{{}} }}{dx} = \frac{{N_{H} }}{EA}} \\ {\frac{{dN_{H}^{{}} }}{dx} = - \rho A\omega_{i}^{2} u_{H} } \\ \end{array} } \right.$$

(25)

Equations (23a, 23b, 23c, 23d) can be rewritten as follows:

• (a)
  $$p_{u} (L) = 0;p_{u} (0) = 0\;(23{\rm a}) \Rightarrow N_{H} (0) = 0;N_{H} (L) = 0$$

  (26a)
• (b)
  $$p_{u} (L) = 0;p_{N} (0) = 0\; (23{\text{b)}} \Rightarrow u_{H} (0) = 0;N_{H} (L) = 0$$

  (26b)
• (c)
  $$p_{N} (L) = 0;p_{u} (0) = 0\;(23{\text{c}}) \Rightarrow N_{H} (0) = 0;u_{H} (L) = 0$$

  (26c)
• (d)
  $$p_{N} (L) = 0;p_{N} (0) = 0\;(23{\text{d}}) \Rightarrow u_{H} (0) = 0;u_{H} (L) = 0$$

  (26d)

It should be emphasized that Eqs. (25) exhibit a similar functional form as Eqs. (2b). Equations (3b, 4b, 5b, 6b) and (26a, 26b, 26c, 26d) take also the same functional forms, describing the boundary conditions of the adjoint system.

Therefore, it can be concluded that an analogy between the adjoint variables and the original variables is obtained. It allows yielding the following equations:

$$\left\{ {\begin{array}{*{20}c} {kN_{H} = N \, } \\ {ku_{H} = u} \\ \end{array} } \right.$$

(27)

Explicit expression of k can be determined by integrating the Eq. (18c) with appropriate conditions in Eq. (22):

$$\int\limits_{0}^{L} {p_{\omega }^{\prime} dx = p_{\omega }^{{}} } (L) - p_{\omega } (0) = - \frac{{c_{\omega i} (1 - k_{\omega W} )}}{{\omega_{0i} }} = \frac{{2\rho \omega_{i} }}{k}\int\limits_{0}^{L} {Au^{2} dx}$$

(28)

Hence,

$$k = \frac{{2\rho \omega_{i} \omega_{0i} }}{{c_{\omega i} (1 - k_{\omega W} )}}\int\limits_{0}^{L} {Au^{2} dx}$$

(29)

Thus, the analogy coefficient k is positive/negative for the case of maximizing/minimizing ω _i . It was demonstrated by considering the natural frequency ω _i as a state variable. The Hamiltonian function, Eq. (11b), will be maximized if:

$$H = \frac{1}{k}\left[ { - \frac{{N^{2} }}{EA} - \rho A\omega_{i}^{2} u^{2} } \right] - \frac{{c_{W} k_{\omega W} }}{{W_{0} }}\rho A \to \hbox{max} \,({\text{in}}\;A)$$

(11b)

$${\text{Or}}\;H = - \frac{{N^{2} }}{EA} - \rho A\omega^{2} u^{2} - k^{*} \rho A \to \hbox{max}$$

(30)

where, $k^{*} = k.$ $\frac{{c_{W} k_{\omega W} }}{{W_{0} }}$ is a constant, if we have no objective of weight, $k^{*} = 0.$