Topology optimization of piezoelectric bi-material actuators with velocity feedback control

doi:10.1007/s11465-019-0537-y

Front. Mech. Eng.

2019, Vol. 14

Issue (2) : 190-200 https://doi.org/10.1007/s11465-019-0537-y

RESEARCH ARTICLE

Topology optimization of piezoelectric bi-material actuators with velocity feedback control

Mariana MORETTI, Emílio C. N. SILVA(

)

School of Engineering, University of São Paulo, São Paulo, SP 05508-030, Brazil

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Abstract

In recent years, the new technologies and discoveries on manufacturing materials have encouraged researchers to investigate the appearance of material properties that are not naturally available. Materials featuring a specific stiffness, or structures that combine non-structural and structural functions are applied in the aerospace, electronics and medical industry fields. Particularly, structures designed for dynamic actuation with reduced vibration response are the focus of this work. The bi-material and multifunctional concepts are considered for the design of a controlled piezoelectric actuator with vibration suppression by means of the topology optimization method (TOM). The bi-material piezoelectric actuator (BPEA) has its metallic host layer designed by the TOM, which defines the structural function, and the electric function is given by two piezo-ceramic layers that act as a sensor and an actuator coupled with a constant gain active velocity feedback control (AVFC). The AVFC, provided by the piezoelectric layers, affects the structural damping of the system through the velocity state variables readings in time domain. The dynamic equation analyzed throughout the optimization procedure is fully elaborated and implemented. The dynamic response for the rectangular four-noded finite element analysis is obtained by the Newmark’s time-integration method, which is applied to the physical and the adjoint systems, given that the adjoint formulation is needed for the sensitivity analysis. A gradient-based optimization method is applied to minimize the displacement energy output measured at a predefined degree-of-freedom of the BPEA when a transient mechanical load is applied. Results are obtained for different control gain values to evaluate their influence on the final topology.

Keywords topology optimization method bi-material piezoactuator active velocity feedback control time-domain transient analysis host structure design vibration suppression

Corresponding Author(s): Emílio C. N. SILVA

Just Accepted Date: 04 March 2019 Online First Date: 03 April 2019 Issue Date: 22 April 2019

Cite this article:

Mariana MORETTI,Emílio C. N. SILVA. Topology optimization of piezoelectric bi-material actuators with velocity feedback control[J]. Front. Mech. Eng., 2019, 14(2): 190-200.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0537-y
https://academic.hep.com.cn/fme/EN/Y2019/V14/I2/190

Fig.1 Bi-material piezoactuator

Fig.2 Transient load profile for the example proposed

Fig.3 Flowchart for the implemented optimization procedure

Parameter	Symbol	Value	Unit
Domain size (width, heigh, depth)	$(w d, h d, d d)$	(20,4,0.2)	cm
Substrate size (width, heigh, depth)	$(w s, h s, d s)$	(20,2.67,0.2)	cm
Piezoceramic size (width, heigh, depth)	$(w p, h p, d p)$	(20,0.67,0.2)	cm
Substrate mesh size	( $x s$ , $y s$ )	(120,16)	?
Piezo ceramic mesh size	( $x p$ , $y p$ )	(120,4)	?
Mechanical load amplitude	$‖ F ‖$	1	N
Initial material volume	$V ini$	30%	?
Material volume constraint	$V max$	70%	?
Proportional damping	(α,β)	(0,1×10^?⁵)	?

Tab.1 Design domain definitions for the vibration attenuation problem

Material	$E$ /GPa	$ζ$ /(kg?m⁻³)	$ν$
Fictitious (mat1)	300	1	0.24
Alumina (mat2)	392	4000	0.24

Tab.2 Material properties (mat1 and mat2 in Eqs. (36) and (37))

Fig.4 Boundaries and transient load

Fig.5 Penalization coefficients update

Fig.6 Optimized topologies for different feedback gain values. (a) Optimized topology for

K = 0

; (b) post-processed topology for

K = 0

; (c) optimized topology for K=9×10⁴; (d) Post-processed Topology for K=9×10⁴; (e) optimized topology for K=3.6×10⁵; (f) post-processed topology for K=3.6×10⁵; (g) optimized topology for K=7.2×10⁵; (h) post-processed topology for K=7.2×10⁵

Fig.7 Convergence comparison evaluated by Eq. (33) for

K = K i × 105

(i=1,2,3,4)

Fig.8 Volume constraint comparison for

K = K i × 105

(i=1,2,3,4)

K				Post-processed
K	$f (ρ)$	$V mat1$	$ω n 1$ /Hz	$f (ρ)$	$V mat1$	$ω n 1$ /Hz
0.00	3.6508×10⁻¹⁵	60.02%	1217.7312	3.6638×10⁻¹⁵	53.96%	1220.31
9.00×10⁴	3.4135×10⁻¹⁵	60.95%	1217.7312	3.4270×10⁻¹⁵	55.05%	1222.25
3.60×10⁵	3.0654×10⁻¹⁵	60.17%	1214.3580	3.0773×10⁻¹⁵	54.01%	1218.80
7.20×10⁵	2.8837×10⁻¹⁵	58.15%	1214.3580	2.8947×10⁻¹⁵	51.67%	1210.43

Tab.3 Objective function values for each optimized topology

Fig.9 Transient response for the optimized topologies

K = K i × 105

(i=1,2,3,4)

Fig.10 Transient feedback voltage for the optimized topologies

K = K i × 105

(i=1,2,3,4)

Fig.11 Analysis of the objective function values for the optimized topologies subjected to the proposed gains

K = K i × 105

(i=1,2,3,4)

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