The Asian Congress of Structural and Multidisciplinary Optimization, May 21-24, 2018, Dalian, China
Shape optimization realizing given vibration mode and its application to fish robot Wares Chancharoen1∗ , Hideyuki Azegami2 1,2
Graduate School of Information Science, Nagoya University, A4-2 (780) Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan. 1∗
[email protected], 2
[email protected] Abstract Shape optimization method make it possible to design the shape of linear elastic body which vibration mode approaches to a given mode. In this study, this method is applied to design the fish-like linear elastic body which vibration mode becomes a well-known swimming function. If it is successive, this method can be applied to design a fish robot swimming with the vibration mode. According to the research [1], they studied about a fish robot swimming that mimics the real fish swimming. They used the equations of fish swimming by biological experiment of Lighthill [2] as √ uR (x1 , t) = 2πx1 (c1 + c2 x1 ) {cos (ωt) + i sin (kt)} , (1) where uR is the lateral displacement, x1 and t are the longitudinal location and time, c1 and c2 are constants determined from experiment, ω is the driving frequency, and k is the wave number. In this study, Eq. (1) is used as a given mode. On the other hand, the motion equation of three-dimensional linear elastic body is given by −ω 2 ρuT − (1 + ig) ∇T S (u) = χΩb0 bT (1 + ig) S (u) ν = 0Rd
in Ω (ϕ) ,
(2)
on ΓN (ϕ) ,
(3)
where S (u) denotes the stress tensor, g is the damping ratio, ρ is the density, b is the body force generated by an actuator located in Ωb0 and χΩb0 = 1 if x ∈ Ωb0 and = 0 elsewhere. Object function is defined as ∫ c (u2 − αuR ) (u2 − αuR ) ddx, (4) f0 (ϕ, u, α) = ΩR0
where u2 is the lateral component of displacement u, and α ∈ R is an arbitrary constant. In order to solve the optimization problem, we applied an iteration algorithm based on the H 1 gradient method for reshaping. Finally, it is found that the optimized shape as shown in Fig. 1 (b) has large shape gradient around the actuator, and the position of actuator should be located at the highest point of traveling wave by Eq. (1) in order to make high vibration response. Figure 2 shows the iteration history of the cost function.
Cost function
1.0 0.8 0.6 0.4 0.2
(a) Initial shape Fig. 1. Vibration modes.
(b) Optimized shape
f0/f0 init
0
2
4 6 Step number k
8
10
Fig. 2. Iteration history.
References 1. David Scott Barrett (1996) Propulsive Efficiency of Flexible Hull Underwater Vehicle. Doctor of Philosophy in Ocean Engineering Massachusetts Institute of Technology. 2. M.J. Lighthill, Large amplitude elongated body theory of fish locomotion. Proc. R. Soc. London B179 (1971) 125-138.
