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Distributed piezoelectric actuator layout-design for active vibration control of thin-walled smart structures
Thin–Walled Structures 147 (2020) 106530
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Distributed piezoelectric actuator layout-design for active vibration control of thin-walled smart structures Yisi Liu a, b, Xiaojun Wang a, *, Yunlong Li a, c a
Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100083, PR China Shenyuan Honors College, Beihang University, Beijing, 100083, PR China c Department of Mechanical Science and Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA b
A R T I C L E I N F O
A B S T R A C T
Keywords: Topology optimization Structural vibration control Piezoelectric material System performance index Sensitivity analysis Thin-walled structures
This paper proposes a scheme for distributed piezoelectric actuator layout-design to improve active vibration control performance of thin-walled smart structures. The aim of the design is to maximize the energy trans formation from actuator to structure so that better control performance will be achieved under a control strategy. The system performance index (SPI) is used to measure energy transformation. The layout-design problem is formulated by combining the topology optimization technique and the SPI. The pseudo-densities of piezoelectric materials are used as design variables and a penalty function is applied on piezoelectric materials so that a clear result can be obtained. Based on the chain rule and the adjoint method, and with the help of solving Lyapunov function, the sensitivity analysis is conducted. The optimization model is solved by nonlinear programming method. Once the optimized layout is obtained, the linear quadratic regulator (LQR) control strategy is applied and vibration suppression can be achieved. The method is load-independent. External loads and control strategy are not considered in layout-design so that a single layout can be obtained. Yet for all that, the optimized layout can achieve excellent performance in a wide range of load cases. Two numerical examples and two engineering applications demonstrate the validity of the proposed method.
1. Introduction Vibration control [1,2] is a critical problem for structural engineer ing. With the development of piezoelectric materials, smart structures which contain both traditional and piezoelectric materials show prom ising future in active control of structural vibration. Piezoelectric ma terials are compatible with thin-walled structures, which makes them suitable especially in the field of aerospace engineering and vehicle engineering. Due to all kinds of constraints (weight, cost et al.), only limited space can be used to place piezoelectric materials. Thus, the location of actuator should be optimized so that the control performance can be maximized. Extensive research has been conducted in this field [3]. Various criteria have been proposed for actuator placement. Hac and Liu [4] proposed a system performance index (SPI) for sensor and actuator placement in vibration control of flexible structures based on observ ability and controllability Gramians. Energy transmission from sensors and actuators can be maximized under the criteria. This method can balance the importance of low order and higher order modes. Yang and
Zhang [5] investigated a criterion of piezoelectric material displacement that can maximize the plate deflection of a certain model. They found that the optimal layout of actuators can be expressed by a combination of position mode functions. H∞ norm [6,7] and H2 norm [8] of structural closed-loop systems have also been researched as the optimization criteria. Possible uncertainties [9,10] in structural systems were also considered in actuator placement [11]. Heuristic optimization algo rithms such as genetic algorithms [12] are used to solve the problem. The design variables in these researches are few due to computational cost. Therefore, the design freedom is limited. Distributed piezoelectric actuators are considered in order to increase the design freedom. However, the number of design variable soars under the circumstance. Topology optimization is a strong tool dealing with large amount of design variables [13,14]. With extensive research in this field, many methods have been proposed, including homogenization method [15], solid isotropic material with penalization (SIMP) model [13], evolu tionary structural optimization (ESO) method [16], level set method [17] and moving morphable components (MMC) [18]. Multi-material topology optimization has also been investigated [19,20].
* Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.tws.2019.106530 Received 26 August 2019; Received in revised form 18 October 2019; Accepted 23 November 2019 Available online 6 December 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic plot of piezoelectric structure layout-design for vibration control.
The layout of structures obtained by topology optimization has been applied into structural vibration attenuation. Most of the studies focus on indirect ways. They extremize dynamic properties such as funda mental frequency [21], gaps between consecutive eigen-frequencies [22], magnitude of steady-state vibrations [23–25]. Dynamic proper ties are optimized to reduce structural vibration. Topology optimization has also been applied into active control of piezoelectric structures. Many researches has been conducted in the field of static deformation control [26,27]. Objective functions such as structural compliance, piezoelectric coefficients and deformation error are extremized so that a certain deformation can be achieved. Kogl and Silva [28] extend the traditional SIMP model into PEMAP-P model for piezoelectric layout design. Molter et al. [29] proposed an optimal cri terion (OC) based scheme for simultaneous structural optimization and piezoelectric actuator placement. Kang et al. applied topology optimi zation techniques into piezoelectric plate design [30,31]. Luo et al. [32] proposed a multiphase level set method of piecewise constants for piezoelectric actuator design. Yang et al. [33] investigated a multi-point constraints (MPC) based method for integrated structural optimization and piezoelectric patch placement. There are some researches involved in topology optimization of piezoelectric structures considering active vibration control [34], but is still in a preliminary stage. Zhang et al. [35,36] incorporate a constant gain velocity feedback (CGVF) control algorithm into dynamic topology optimization of piezoelectric structures to control transient response and sound radiation. Linear quadratic regulator (LQR) control is also investigated by Hu et al. [37] and Silveira et al. [38]. Control spillover effects [39] and minimum energy consumption [40] were also
considered. In these works, the frequency of external loads and the parametric value of a control strategy are all accounted in layout design. The performance of these layouts works well in design point. However, their performance deteriorate as external loads or control parameters vary. Besides, the optimized layout will be different even if the fre quency of external loads or the value of control parameters change marginally. This paper proposes a novel scheme for layout-design of distributed piezoelectric actuators aiming at vibration control of thin-walled smart structures. The schematic plot of piezoelectric structure layout-design for vibration control is shown in Fig. 1. In this paper, we combine the topology optimization technique and the SPI [4] to formulate the opti mization model of the layout-design problem. Based on the chain rule, adjoint method and solving Lyapunov equation, the sensitivity analysis of objective function is derived. The optimization model is solved by general optimization algorithm. The optimized layout obtained by the proposed method stays same regardless of the external loads and para metric value of control strategy. Then the LQR control strategy is applied to the designed layout so that vibration suppress can be achieved. The optimized layout can achieve an excellent performance in a wide range of load cases. Numerical examples show validity of the proposed method. 2. Finite element modelling of piezoelectric structure 2.1. Governing equation In this paper, the deformation of piezoelectric plate is assumed small 2
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Fig. 2. Four-nodes element for piezoelectric plate.
and linear. Thus, the constitutive equation of piezoelectric material can be expressed as follows:
where Bu denotes the strain-displacement matrix and d ¼ ½u0 ; v0 ; ⋯;
T ¼ Ce S eE D ¼ eS þ κE
Ea ¼ Bϕ ϕa
ð1Þ
ð4Þ θð4Þ x ; θy �. Substitute Eq. (5) into Eq. (3) and we can obtain:
(1)
where T and S represent stress and strain vector, respectively; D denotes electric displacement vector; E means electric field; Ce represents elastic matrix; e and κ denotes piezoelectricity and permittivity vector, respectively. In this paper, thin-walled structures are considered and piezoelectric materials are attached onto them. Thus, the whole structures can be meshed with plate elements. The finite element model of the piezo electric plate is plotted at Fig. 2. The piezoelectric layers are bounded tightly with the host layer. The strain of the piezoelectric plate can be described by Mindlin plate theory: 2 3 2 32 3 εxx u0 ∂=∂x 0 0 0 z⋅∂=∂x 6 εyy 7 6 0 6 7 ∂=∂y 0 z⋅∂=∂y 0 7 6 7 6 76 v0 7 6 γ xy 7 ¼ 6 ∂=∂y ∂=∂x 6 7 (2) 0 z⋅∂=∂x z⋅∂=∂y 7 6 7 6 76 w0 7 4 γyz 5 4 0 0 ∂=∂y 1 0 54 θx 5 γzx θy 0 0 ∂=∂x 0 1
ð1Þ
4 X
� ðiÞ ðiÞ ðiÞ � ðiÞ T Ni u0 ; v0 ; w0 ; θðiÞ x ; θx
Ni ϕðiÞ a
(8)
2.2. Reduced state-space model The degree of freedom (DOF) of Eq. (8) can be very high as elements increase, which makes it very time-consuming to obtain the controller. Therefore, model reduction is needed. The displacement can be approximated as follows:
(4)
uðtÞ � ψ qðtÞ ¼
m X
(10)
ψ i qi ðtÞ
i¼1
where ψ ¼ ½ψ 1 ; ⋯; ψ m � denotes modal shape; m represents the number of
(5)
mode chosen; qðtÞ ¼ ½q1 ðtÞ; ⋯; qm ðtÞ�T denotes the modal displacement vector. Substitute Eq. (10) into Eq. (8), and left multiply ψ T , and we can obtain:
(6)
_ þ ω2 qðtÞ ¼ ψ T f ðtÞ q€ðtÞ þ 2ξωqðtÞ
i¼1
Substitute Eq. (4) into Eq. (2) and we can obtain: S ¼ Bu d
K uϕ ϕa ðtÞ
where Ωe denotes volume of each element.
where Ni ; i ¼ 1; ⋯; 4 represent the shape function in standard form. The electric potential can also expressed as similar form: 4 X
ð4Þ T
Ωe
i¼1
ϕa ¼
ð3Þ
where M ¼ Mh þMa þ Ms represents structural mass matrix (Mh , Ma and Ms represent mass matrix of host, actuator and sensor layers, respec tively). K ¼ K h þK a þ K s denotes structural stiffness matrix. C is the damping matrix. The Rayleigh damping model is used in this paper. Only the damping effect of the host structure is considered in this paper. Therefore, the damping matrix can be expressed as C ¼ αMh þ βK h , where α and β are the coefficient of the Rayleigh damping. f ðtÞ repre sents the external loads. K uϕ denotes electro-mechanical matrix. The electro-mechanical matrix of each element can be expressed as follows: Z ðeÞ K uϕ ¼ BTu eT Bϕ dΩ (9)
where ϕa represents electric potential in actuator layer. The general displacement vector can be expressed as linear combi nation of nodal displacement: ½u0 ; v0 ; w0 ; θx ; θx �T ¼
ð2Þ
€ þ CuðtÞ _ þ KuðtÞ ¼ f ðtÞ MuðtÞ
(3)
ϕa =h
(7)
where ϕa ¼ ½ϕa ; ϕa ; ϕa ; ϕa � ; Bϕ represents the electric fieldpotential matrix. According to Eq. (1) and Eqs. (6) and (7) and the virtual work theory, we can obtain the governing equation of the piezoelectric plate at finite element form:
where u0 , v0 and w0 represent mid-plane displacement at X, Y and Z direction, respectively. θx denotes rotation around X axis and θy denotes rotation around Y axis. The height of actuator layer is h. We assume that only transverse electric fields are applied in the plate. The electric field Ea can be express as follows: Ea ¼
ð1Þ
3
ψ T K uϕ ϕa ðtÞ
(11)
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bration suppression can be maximized under a certain control input energy. The SPI proposed by Hac and Liu [4] can measure that property. The higher SPI of a system, the less energy consumption it will need in vibration control. The SPI can be expressed as follows: !s 2m 2m Y X 2m λi λi (17) SPI ¼
Table 1 Material properties of piezoelectric structure. Young’s modulus
Density
Piezoelectric constant
Poisson’s ratio
Host layer
70Gpa
–
0.30
Piezoelectric layer (PTZ)
71Gpa
2700kg/ m3 5000kg/ m3
e31 ¼ e33 ¼ 5.2C/m2
0.35
i¼1
where λi ; i ¼ 1; ⋯2m is the eigenvalues of controllability Gramian Lc of the structural system. Here we introduce the Controllability Gramian and its eigenvalue. The controllability Gramian can be expressed as follows: Z ∞ T Lc ¼ eAτ B1 BT1 eA τ dτ (18)
where ξ ¼ diagfξ1 ; ⋯; ξm g; ξi ¼
ψ Τι Cψ i ψ Τι ðαMh þ βK h Þψ i ¼ ; i ¼ 1; 2; ⋯; m ψ Τι Mψ i ψ Τι Mψ i
i¼1
(12)
and
0
ω ¼ diagfω1 ; ⋯; ωm g; ω2i ¼
ψ Τι K ψ i ; i ¼ 1; 2; ⋯; m ψ Τι Mψ i
The structural system is controllable as long as Lc is positive definite.
(13)
represents the structural model damping ratio and natural frequency. Let (14)
_ xðtÞ ¼ ½qðtÞ; qðtÞ�T be the state variable. Then Eq. (11) can be reorganized as follows:
(15)
_ ¼ AxðtÞ þ B1 ϕa ðtÞ þ B2 f ðtÞ xðtÞ where � 0 A¼
2
ω
� I ; 2ξω
� B1 ¼
� 0 ; T ψ K uϕ
� B2 ¼
0
ψ
� (16)
T
I represents the identity matrix. 3. Formulation of layout-design and sensitivity analysis 3.1. System performance index (SPI)
Fig. 4. Comparison of sensitivity analysis obtained by analytical method and FDM.
Piezoelectric materials should be allocated to the place where vi
Fig. 3. Rectangular cantilever plate.
Fig. 5. The layout obtained from different volume fraction: (a) fv ¼ 0:6; (b).fv ¼ 0:4. 4
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Table 2 Eigen-frequencies of the initial design. Mode order
Eigen-frequency
Mode order
Eigen-frequency
1 2 3 4 5 6 7 8 9 10
7.33 11.70 25.66 37.06 52.27 56.10 57.00 63.30 68.59 78.59
11 12 13 14 15 16 17 18 19 20
80.79 97.50 100.43 101.80 107.79 110.07 112.06 113.41 124.92 125.33
Table 3 Eigen-frequencies of the optimized layout. Mode order
Eigen-frequency
Mode order
Eigen-frequency
1 2 3 4 5 6 7 8 9 10
9.09 12.01 31.14 40.27 61.41 64.82 67.32 75.81 79.13 94.37
11 12 13 14 15 16 17 18 19 20
96.47 114.10 120.84 125.17 128.64 136.37 140.37 144.75 145.79 148.11
The controllability Gramian in Eq. (18) can also be obtained from the following Lyapunov equation: (19)
ALc þ Lc AT þ B1 BT1 ¼ 0
Supposes the eigenvectors of Lc are φi ; i ¼ 1; ⋯; 2m. We can easily obtain the following equation: (20)
Lc φi ¼ λi φi ; i ¼ 1; ⋯; 2m
Large SPI guarantees high energy transmission from the piezoelectric actuator to the structural system. Thus, less control input is needed to alleviate vibration. 3.2. Formulation of distributed piezoelectric actuator layout-design
Fig. 6. Iteration history of objective function: (a) fv ¼ 0:6; (b).fv ¼ 0:4.
The SPI is used as objective function in the optimization model. The volume fraction of piezoelectric materials is used as constraint. Mean while, the governing state-space function of the structural system should be satisfied. Thus, we can formulate the following topology optimization problem: find
ρe
max
gðλi Þ ¼
!s
2m X
λi i¼1
s:t:
2m Y
2m
λi i¼1
(21)
_ ¼ AxðtÞ þ B1 ϕa ðtÞ xðtÞ N N X X ρe Ve � fv Ve e¼1
e¼1
0 � ρmin � ρe � 1;
e ¼ 1; 2; ⋯; N
where ρ ¼ ½ρ1 ; ⋯; ρN �T represents the design variables; N denotes num ber of design variables; Ve represents volume of the eth element; ρmin is a small real number in case of numerical instability; fv denotes permitted volume fraction. 3.3. Sensitivity analysis
Fig. 7. Iteration history of volume fraction.
Sensitivity analysis of the objective function in Eq. (21) is critical for the topology optimization problem. Based on chain rule and adjoint 5
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vffiffiffiffiffiffiffiffiffiffi u
2m uY ∂gðλi Þ 2m 1 ¼ t λj þ 2mλi ∂λi j¼1
!s
2m X
2m Y
2m
λj j¼1
λj
(23)
j¼1
Once eigenvalue of Lc is obtained, this term can be calculated easily. Thus, the main task changes to calculate derivative of eigenvalues of ∂λi the controllability Gramian: ∂ρ ; i ¼ 1; ⋯; 2m. Differentiating the e
eigenvalue equation Eq. (20) with respect to the design variable ρe , we can obtain:
∂Lc ∂φ ∂λi ∂φ φ þ Lc i ¼ φ þ λi i ; i ¼ 1; ⋯; 2m ∂ρe i ∂ρe ∂ρe i ∂ρe
(24)
Left multiplying φTi ; i ¼ 1; ⋯; 2m at Eq. (24) and rewriting the equation, we can derive:
∂λi T ∂Lc φ φ ¼ φTi φ þ φTi Lc ∂ρe i i ∂ρe i
λi φTi
� ∂φi
∂ρe
; i ¼ 1; ⋯; 2m
(25)
Because φTi ; i ¼ 1; ⋯; 2m are normalized vectors and Lc is a sym metric positive definite matrix, substituting Eq. (20) into Eq. (25), we can obtain the following equation:
∂λi ∂Lc ¼ φTι φ ; i ¼ 1; ⋯; 2m ∂ρe ∂ρe i
(26)
Then, the task is turned into solving the derivative of controllability Gramian ∂Lc =∂ρe . Differentiating the Lyapunov equation Eq. (19), we can obtain: A
∂Lc ∂Lc T ∂A ∂AT ∂B1 T ∂BT þ A þ Lc þ Lc þ B1 þ B1 1 ¼ 0 ∂ρe ∂ρe ∂ρe ∂ρe ∂ρe ∂ρe
(27)
let H¼
∂A ∂AT ∂B1 T ∂BT Lc þ Lc þ B1 þ B1 1 ∂ρe ∂ρe ∂ρe ∂ρe
(28)
Because Lc is a positive definite symmetric matrix, H is also a positive definite symmetric matrix. Substituting Eq. (28) into Eq. (27), we can get: A
∂Lc ∂Lc T þ A þH ¼0 ∂ρe ∂ρe
(29)
Comparing Eq. (29) with Eq. (19), It can be seen that Eq. (29) is a standard Lyapunov equation. Thus, it can be solved by a well-established algorithm [41] as long as ∂A=∂ρe and ∂B1 =∂ρe is calculated. According to Eq. (16), we can derive: 3 2 0 0 ∂A 6 7 (30) ¼ 4 ∂ω2 ∂ω 5 ∂ρe 2ξ
Fig. 8. Time-domain response of initial layout and optimized layout (fv ¼ 0.4): (a) Initial state 1; (b) Initial state 2.
∂ρe
∂ρe
and 2
∂B1 6 ¼4 ∂K ∂ρe ψ T uϕ ∂ρe
3 7
(31)
5 ∂ψ T K uϕ ∂ρe
SIMP model is used here for the interpolation of mass, stiffness and coupling matrix so the results can reach either 0 (void) or 1 (piezo electric material) as nearly as possible. The interpolation scheme for piezoelectric materials can be expressed as follows:
Fig. 9. Load case for frequency response optimization.
method, and with the help of solving Lyapunov function, the sensitivity analysis is performed. The derivative of the objective function with respective design variable ρe can be expressed as: 2m ∂gðλi Þ X ∂gðλi Þ ∂λi ¼ ∂ρe ∂λi ∂ρe i¼1
0
ðeÞ
MðeÞ a ¼ ρe M a0 ; p K ðeÞ a ¼ ðρe Þ K a0 ; ðeÞ
(22)
K uϕ ¼ ðρe Þp K uϕ0 ðeÞ
The first term at the right side of Eq. (22) can be expressed as follows: 6
ðeÞ
(32)
ðeÞ
MðeÞ s ¼ ρe M s0
p K ðeÞ s ¼ ðρe Þ K s0
ðeÞ
(33) (34)
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Fig. 10. Layout obtained by frequency response optimization with different frequencies: (a) ω ¼ 8Hz; (b) ω ¼ 18Hz; (c).ω ¼ 40Hz.
Eq. (30) contains derivative of structural natural frequency. When the eigenvalue is unimodal, the derivative of ∂ω2 =∂ρe can be obtained from: � � ∂ω2i ∂K ∂M ¼ ψ Tι ω2i ψ i ; i ¼ 1; 2; ⋯; m (36)
∂ρe
∂ρe
∂ρe
When the eigenvalue is multiple (i.e. r-fold), then there are r eigen vectors (say ψ 1 ; ⋯; ψ r ) correspond to the natural frequency ωi , the de rivative can be obtained from the following equation [42]: � � � � ∂K ∂M ∂ω2i det ψ Tj (37) ω2i ψk δjk ¼ 0; j; k ¼ 1; 2; ⋯; r
∂ρe
∂ρe
∂ρe
where δjk represents the Kronecker function, which meets: � 1; j¼k δjk ¼ 0; j 6¼ k The derivative ∂ω=∂ρe can be obtained by: pffiffiffiffiffi ∂ωi ∂ ω2i 1 ∂ω2i ¼ ¼ 2ωi ∂ρe ∂ρe ∂ρe
Fig. 11. Time-domain response of different layouts. ðeÞ
ðeÞ
ðeÞ
element, and K uϕ0 is the corresponding coupling matrix; p denotes the ðeÞ
penalty coefficient. Then Eq. (31) can 2 0 ∂B1 6 ¼4 ∂ψ T p ðeÞ ∂ρe ψ T pðρe Þp 1 K ðeÞ ρe Kuϕ0 uϕ0
be rewritten as follows [25]: 3 7 5
(39)
Therefore, the sensitivity analysis of objective function can be ob tained from Eqs.(22)–(39). Once all the sensitivities are calculated, then the topology optimization model Eq. (21) can be solved by optimization algorithms such as sequential quadratic programming (SQP) [43,44] and method of moving asymptotes (MMA) [45].
where Ma0 , K a0 , Ms0 , K s0 are element mass matrix and stiffness matrix of actuator and sensor layers when piezoelectric material is full at eth ðeÞ
(38)
4. Control strategy
(35)
Once the layout has been designed, a proper control strategy is
∂ρe
Fig. 12. Voltage amplitude of the SPI layout design: (a) 0.001s; (b) 1s. 7
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Fig. 16. Iteration history of objective function. Fig. 13. Energy consumption of different layouts: (a) ω ¼ 25Hz; (b) ω ¼ 50Hz; (c).ω ¼ 100Hz.
The core purpose of LQR controller design for structural vibration control is to calculate the voltage input ϕa ðtÞ so that the following cost function can be minimized: Z � 1 ∞� T x ðtÞQxðtÞ þ ϕTa ðtÞRϕa ðtÞ dt J¼ (40) 2 0 where Q is a positive semi-definite symmetric weight coefficient matrix; R is a positive definite symmetric weight coefficient matrix. The voltage input ϕa ðtÞ can be expressed as linear function of state variables xðtÞ: ϕa ðtÞ ¼
GxðtÞ
(41)
where G ¼ R 1 BT1 P denotes feedback gain of the controller. P can be calculated from the following algebraic Riccati equation:
Fig. 14. Clamped square plate.
PA þ AT P
PB1 R 1 BT1 P þ Q ¼ 0
(42)
Once P is derived, the controller can be obtained. 5. Numerical examples and engineering application Two numerical examples and two engineering applications are used here to verify the proposed method. In all cases, the thickness of host structure is t ¼ 3mm; the thickness of actuator and sensor layer are both h ¼ 1mm. The host layer is made up of aluminum and the piezoelectric layers are constructed by piezoelectric ceramic PTZ. Material properties of the piezoelectric structure [34] are listed in Table 1. The Rayleigh damping coefficient is α ¼ 0:01 and β ¼ 0:0001, respectively. The value of penalty coefficient is p ¼ 3 in this paper. The minimum of design variables are set as ρmin ¼ 1 � 10 4 . MMA is used to solve the optimization problem Eq. (21). The algorithm converges when the difference of objective function between two sequential iterations is smaller than ε ¼ 1 � 10 8 . 5.1. Numerical example 1: rectangular cantilever plate A rectangular cantilever plate is used in this numerical example, which is plotted in Fig. 3. The left side of the plate is clamped. The length and width of the plate is a ¼ 1:6m and b ¼ 0:8m, respectively. The rectangular plate is meshed into 3200 (80 � 40) four-node Mindlin el ements, with 3321 nodes, 16605 displacement degree of freedom (DOF) and 3321 electric DOF. The top 20 modes of the rectangular plate are used to generate the reduced state-space equation. In this example, we consider the volume fraction fv ¼ 0:6 and fv ¼ 0:4. First, the sensitivities of the objective function are calculated by the proposed analytical method and finite
Fig. 15. The optimized layout.
needed to control the structural vibration. LQR is applied for vibration control in this paper. In this paper, all state variables are assumed available, although it is not true in practical engineering. However, the state variables can be obtained from state observer through limited observations. This process is a little complicated but well-established. Since this is not main the purpose of the work, we use state variables as the feedback. 8
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Fig. 17. Frequency response of initial layout and optimized layout: (a) Frequency response curve; (b) Displacement field of point A; (c) Displacement field of point B.
Fig. 18. Layout obtained by frequency response optimization with different frequencies.
difference method (FDM), respectively. In FDM, the perturbation of the design variables are 1 � 10 5 . The results are presented in Fig. 4. It show the proposed analytical method complies well with the FDM. But the efficiency of analytical method is much higher than that of FDM. The final layouts of piezoelectric materials that obtained from different constraints are plotted in Fig. 5. The piezoelectric materials are plotted with black color. It shows as the upper bound of constraint in crease (fv ¼ 0:4⇒fv ¼ 0:6), more piezoelectric materials are distributed into the plate. But the general shape of piezoelectric material stays
similar. Besides, in both cases, the results are pretty clear and no in termediate pseudo-density elements exists. The iteration history of objective function is plotted in Fig. 6. Fig. 6 shows that in both cases the curve of iteration history is comparatively smooth and converges within 110 steps. Fig. 7 is iteration history of volume fraction. It can be seen that the constraints are always satisfied through the iteration process. The eigen-frequencies before and after optimization is listed in Table 2 and Table 3. They show that the compared with the initial 9
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attenuates slowly due to existence of Rayleigh damping. (2) when LQR control is applied, the vibration suppression of optimized layout is much faster than that of initial design. 5.2. Further discussion on numerical example 1 In order to further verify the efficiency of the proposed method, the layout obtained by frequency response optimization is used. In fre quency response optimization, an external load F ¼ sinðωtÞ is applied at the tip of the plate. The load case is plotted in Fig. 9. The optimization model for frequency response analysis can be expressed as follows:
Fig. 19. Frequency response analysis of different layouts.
Fig. 20. Simplified fuselage.
design, the eigen-frequencies of optimized layout are increased. Same control strategy is applied to initial and optimized layout to verify their ability in vibration control. We choose the layout obtained by setting fv ¼ 0:4. The parameters in LQR are chosen as Q ¼ 1015 � I and R ¼ I. The initial layout without control is used here for compari son. Two initial states of the plate are considered. The initial state 1 is simple. The initial displacement field that is equal to the first order mode uðt¼0Þ ¼ ψ 1 is applied. The initial state 2 is much more complex. The displacement field that equals to combination of top 20 modes uðt¼0Þ ¼ P20 i¼1 ψ i is applied. Four order Runge-Kutta method with fixed time step t ¼ 0:001s (this time step is small enough because the maximum eigenfrequency is smaller than 150 Hz) is used. Time-domain response of initial layout and optimized layout are plotted in Fig. 8. Fig. 8 shows: (1) In the case of initial design without control, the vibration amplitude
Fig. 22. Iteration history of objective function.
Fig. 23. Load case for frequency response analysis.
Fig. 21. Layout of optimized design: (a) View 1; (b) View 2. 10
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time domain response of different layouts are plotted in Fig. 11. It shows compared with the layout obtained by frequency response optimization, the layout obtained by SPI can achieve better control performance. The voltage amplitude applied to the configuration obtained by SPI design is presented in Fig. 12. The voltage at where the ‘voids’ exist almost equal to zero and a high voltage is applied at where the piezo electric materials exist. By the way, the maximum voltage can change with the parameters of LQR. In practical engineering, the maximum voltage is determined by the property of piezoelectric materials and it is not considered in this paper. The energy consumption is also compared between these three different layouts. The energy index (EI) is defined to represent the en ergy consumption. EI is defined as the summation of voltage applied to the plate. We plot EI-time of the three layouts in Fig. 13. It shows that compared with other configurations, the total energy consumption of SPI design is minimum during the control process. 5.3. Numerical example 2: clamped square plate
Fig. 24. Frequency response of initial layout and optimized layout.
A square plate with four edges clamped is used in this example, which is plotted in Fig. 14. The length of the plate is l ¼ 1:2m. This square plate is meshed into 3600 (60 � 60) four-node Mindlin elements. There are 3721 nodes, with 18605 displacement DOF and 3721 electric DOF. The top 20 modes of the square plate are used. The permitted volume fraction is set as fv ¼ 0:3 in this example. The initial value of all design variables are set as ρe ¼ 0:3; e ¼ 1; ⋯; N. The optimization process converges within 80 steps. The final layout is plotted in Fig. 15. There is no intermediate pseudo-density element. The iteration history of objective function is plotted in Fig. 16. The iteration process is very smooth and converges quickly. The parameters in LQR are chosen as Q ¼ 1015 � I and R ¼ I. Fre quency response analysis is applied to the initial layout and optimized layout. A harmonic force whose magnitude is 1000N is applied to the center of the square plate (vertical to the plate) to perform the frequency analysis. The result is plotted in Fig. 17. Fig. 17 (a) shows: (1) The resonant frequency (eigenvalue) of optimized design is higher than that of initial design. (2) Compared with the initial design with LQR control, a 25 dB attenuation of frequency response magnitude of the first mode is achieve by the optimized layout with LQR control. (3) The frequency response curve of the optimized design is much smoother than that of initial design. The displacement field of initial design (point A in Fig. 17 (a)) and optimized design (point B in Fig. 17 (a)) under a harmonic excitation is plotted in Fig. 17 (b)–(c). It shows their modes are similar, but the maximum displacement of the optimized design is 1:1 � 10 5 m, decreased significantly from the initial design, which is 1:2 � 10 3 m. Fig. 17 demonstrates that the optimized layout obtained by the proposed method can improve the structural frequency response performance in a wide range of frequency interval.
Fig. 25. Time-domain response of initial layout and optimized layout.
find ρe min gf ¼ CTc Cc s:t: Cc ¼ FT U
�
ω2 M þ iωC þ K U ¼ F N X e¼1
ρe Ve � fv
(43)
N X
Ve e¼1
0 � ρmin � ρe � 1; e ¼ 1; 2; ⋯; N
5.4. Further discussion on numerical example 2
The detailed sensitivity analysis of the optimization model is pre sented in Appendix A. When the weight constraint fv ¼ 0:4. The final layout obtained by the above optimization model is plotted in Fig. 10. From Fig. 10 we can obtain the following conclusion. (1) The final configuration obtained from frequency response optimization changes greatly when the frequency of external load changes. (2) When fre quency of external load is comparatively higher, then there will be a lot of intermediate pseudo-density elements and the iteration is very diffi cult to converge. The reason why this happens is explained in this reference [46]. We choose the clear configuration Fig. 10 (a) to compare the result obtained by the proposed method (Fig. 5 (b)). The same control pa rameters are applied to both configurations. Initial state 2 in Fig. 8 (b) is applied to test their ability to control the complex initial response. The
Frequency response based topology optimization is also performed to compare with our proposed method. A harmonic load F ¼ sinðωtÞ is applied at the center of the square plate. The optimization model is Eq. (43) and the weight constraint is set as fv ¼ 0:3. The layout obtained by different frequency is plotted in Fig. 18. Still we can find that the layout varied as the frequency of external load changes. Besides, there will be a lot of intermediate pseudo-density elements if the frequency is comparatively high. We choose the layout obtained by ω ¼ 50Hz (Fig. 18 (b)) to compare with the proposed method. Frequency response analysis is performed to the layouts and the result is plotted in Fig. 19. Following conclusions can be obtained: (1) The resonant frequency of both SPI design and frequency design will increase. (2) The resonant frequency increase of frequency design is 11
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Thin-Walled Structures 147 (2020) 106530
Fig. 26. Center wing-box: (a) Isometric view; (b) Top view; (c) Cutaway view.
Fig. 27. Layout of the optimized design of piezoelectric materials: (a) Isometric view; (b) Top view.
higher than that of SPI design. (3) The frequency response curve of SPI design is smoother than frequency design, and the vibration control performance of SPI design is better than that of frequency design, especially in higher frequency. The above conclusion shows that compared with the frequency response based optimization, the layout obtained by SPI design is more suitable for active vibration control.
fraction. The initial value of all design variables are set as ρe ¼ 0:4; e ¼ 1; ⋯; N. The iteration converges within 50 steps. The final layout is plotted in Fig. 21. The result is very clear. The iteration history of objective function is plotted in Fig. 22. The parameters in LQR are chosen as Q ¼ 1016 � I and R ¼ I. Fre quency response analysis is applied to the simplified fuselage (see Fig. 23). The results are plotted in Fig. 24. It shows natural frequency of the optimized layout becomes higher and attenuation of the vibration is achieved. Fig. 25 shows the time-domain response of initial layout and optimized layout given a certain initial condition. The initial displace P ment field equals to uðt¼0Þ ¼ 5i¼1 ψ i . Compared with initial design with LQR control, the vibration of optimized design with LQR control de creases much faster.
5.5. Engineering application 1: simplified fuselage The fuselage of aircraft may vibrate severely due to the disturbance of airflow, which will affect fatigue property of the aircraft greatly. Therefore, vibration attenuation of fuselage is imperative. A part of simplified fuselage of aircraft is used in this example, which is plotted in Fig. 20. The simplified fuselage is cylinder shaped, with its width l ¼ 1:2m and radius r ¼ 0:2m. Both end of the simplified fuselage is clam ped. It is meshed into 3720 (62 � 60) Mindlin shell elements. There are 3782 nodes, with 22692 displacement DOF and 3782 electric DOF. The top 20 modes are used. fv ¼ 0:4 is used as the permitted volume
5.6. Engineering application 2: center wing-box A center wing-box of space shuttle is used in this engineering 12
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Thin-Walled Structures 147 (2020) 106530
Due to the severe working environment of the space shuttle (such as heat), the center wing-box may vibrate, which will reduce its stability and fatigue life. Thus, vibration attenuation is imperative. The param eters in LQR are chosen as Q ¼ 1019 � I and R ¼ I. Suppose the center wing-box vibrates due to perturbation from the environment. The initial P displacement field is complex and equals to uðt¼0Þ ¼ 5i¼1 ψ i . The timedomain response of the center wing-box of the initial design and opti mized design is plotted in Fig. 29. It shows compared with the initial design, the optimized layout can stabilize the wing-box much faster. This example shows that the proposed method can deal with more complicated engineering problems. 6. Conclusion This paper proposes a scheme for layout-design of smart structures with distributed piezoelectric actuators for vibration control, based on topology optimization technique and the system performance index (SPI). The pseudo density of piezoelectric materials are set as design variables. The optimization model is formulated to maximize SPI so that the energy transformation from actuator to structure can be maximized. The sensitivity analysis is performed based on chain rule and adjoint method, and solving Lyapunov equations. The optimization problem is solved by nonlinear programming method. Then, the linear quadratic regulator (LQR) control strategy is applied to the designed layout and vibration suppression can be achieved. External loads and control strategy is not considered in layout-design so that a single layout can be obtained. Nevertheless, the optimized layout performs well in a wide range of load cases. Numerical examples and engineering application are used to verify the proposed method. Results show that the optimization processes converge within few iterations and the constraints are not violated during the iteration. The obtained optimized layouts are very clear and in most cases there is no pseudo-density element. Compared with the initial design, the fundamental frequency of the optimized layout in creases. Moreover, although frequency of external loads are not incor porated in layout-design, the overall frequency response of optimized design are much smoother, and frequency response magnitude are attenuated a lot. In terms of the displacement perturbation, the opti mized designs can be stabilized much faster. This work provides a possible approach for conceptual design of smart structures.
Fig. 28. Iteration history of objective function.
Fig. 29. Vibration of the center wing-box.
Declaration of competing interest
application. The whole wing-box is an integral panel. The shape and size of the wing-box is plotted in Fig. 26. The left end of the wing-box is clamped. The whole model is meshed as 11376 four-node Mindlin ele ments, with 9991 nodes. The upper surface of the wing-box is applied to place piezoelectric materials for attenuating structural vibration. Thus, there are N ¼ 3120 design variables. Suppose the top 20 modes accounts for the most of vibrational energy. The volume constraint fv ¼ 0:3 is used in this example. The initial value of all design variables are set as ρe ¼ 0:3; e ¼ 1; ⋯; N. The final layout is obtained after 146 iterations. The optimized design of the piezoelectric center wing-box is plotted in Fig. 27, and the iteration history is plotted in Fig. 28.
We declared that we have no conflicts of interest. We do not have any commercial or associative interest that represents a conflict of interest in connection with the work. Acknowledgements The paper is supported by the Key Technologies Research and Development Program of China (2016YFB0200700), National Natural Science Foundation of China (11572024, 11432002, 11872089) and National Defense Basic Scientific Research Program of China (JCKY2016601B001, JCKY2017601B001).
Appendix A The sensitivity analysis of the objective of Eq. (43) can be performed as: � T � ∂gf ∂CTc ∂Cc ∂Cc ¼ Cc þ CTc ¼ 2Re Cc
∂ρe
∂ρe
∂ρe
∂ρe
where ReðÞ means the real part. Then ∂Cc =∂ρe in Eq. (A.1) can be expressed as follows: 13
(A.1)
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Thin-Walled Structures 147 (2020) 106530
�
� � 1 �T ∂ðKD Þ 1 ∂Cc ∂ FT KD 1 F ∂ðK D Þ 1 ¼ ¼ FT K D1 K D F ¼ K TD F K F ∂ρe ∂ρe ∂ρe ∂ρe D where K D ¼
∂ðKD Þ ¼ ∂ρe
(A.2)
ω2 M þ iωC þ K. Because only the damping effect of host structure is considered, then we can obtain: (A.3)
p 1 ðeÞ ω2 MðeÞ a0 þ pρe K a0 1
Let U ¼ ðK TD Þ F (U represent the conjugate vector of displacement vector U). Then Eq. (A.2) can be transformed into:
∂Cc ¼ ðUÞT ∂ρe
�
(A.4)
p 1 ðeÞ ω2 MðeÞ a0 þ pρe K a0 U
References
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Full length article
Distributed piezoelectric actuator layout-design for active vibration control of thin-walled smart structures Yisi Liu a, b, Xiaojun Wang a, *, Yunlong Li a, c a
Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing, 100083, PR China Shenyuan Honors College, Beihang University, Beijing, 100083, PR China c Department of Mechanical Science and Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA b
A R T I C L E I N F O
A B S T R A C T
Keywords: Topology optimization Structural vibration control Piezoelectric material System performance index Sensitivity analysis Thin-walled structures
This paper proposes a scheme for distributed piezoelectric actuator layout-design to improve active vibration control performance of thin-walled smart structures. The aim of the design is to maximize the energy trans formation from actuator to structure so that better control performance will be achieved under a control strategy. The system performance index (SPI) is used to measure energy transformation. The layout-design problem is formulated by combining the topology optimization technique and the SPI. The pseudo-densities of piezoelectric materials are used as design variables and a penalty function is applied on piezoelectric materials so that a clear result can be obtained. Based on the chain rule and the adjoint method, and with the help of solving Lyapunov function, the sensitivity analysis is conducted. The optimization model is solved by nonlinear programming method. Once the optimized layout is obtained, the linear quadratic regulator (LQR) control strategy is applied and vibration suppression can be achieved. The method is load-independent. External loads and control strategy are not considered in layout-design so that a single layout can be obtained. Yet for all that, the optimized layout can achieve excellent performance in a wide range of load cases. Two numerical examples and two engineering applications demonstrate the validity of the proposed method.
1. Introduction Vibration control [1,2] is a critical problem for structural engineer ing. With the development of piezoelectric materials, smart structures which contain both traditional and piezoelectric materials show prom ising future in active control of structural vibration. Piezoelectric ma terials are compatible with thin-walled structures, which makes them suitable especially in the field of aerospace engineering and vehicle engineering. Due to all kinds of constraints (weight, cost et al.), only limited space can be used to place piezoelectric materials. Thus, the location of actuator should be optimized so that the control performance can be maximized. Extensive research has been conducted in this field [3]. Various criteria have been proposed for actuator placement. Hac and Liu [4] proposed a system performance index (SPI) for sensor and actuator placement in vibration control of flexible structures based on observ ability and controllability Gramians. Energy transmission from sensors and actuators can be maximized under the criteria. This method can balance the importance of low order and higher order modes. Yang and
Zhang [5] investigated a criterion of piezoelectric material displacement that can maximize the plate deflection of a certain model. They found that the optimal layout of actuators can be expressed by a combination of position mode functions. H∞ norm [6,7] and H2 norm [8] of structural closed-loop systems have also been researched as the optimization criteria. Possible uncertainties [9,10] in structural systems were also considered in actuator placement [11]. Heuristic optimization algo rithms such as genetic algorithms [12] are used to solve the problem. The design variables in these researches are few due to computational cost. Therefore, the design freedom is limited. Distributed piezoelectric actuators are considered in order to increase the design freedom. However, the number of design variable soars under the circumstance. Topology optimization is a strong tool dealing with large amount of design variables [13,14]. With extensive research in this field, many methods have been proposed, including homogenization method [15], solid isotropic material with penalization (SIMP) model [13], evolu tionary structural optimization (ESO) method [16], level set method [17] and moving morphable components (MMC) [18]. Multi-material topology optimization has also been investigated [19,20].
* Corresponding author. E-mail address: [email protected] (X. Wang). https://doi.org/10.1016/j.tws.2019.106530 Received 26 August 2019; Received in revised form 18 October 2019; Accepted 23 November 2019 Available online 6 December 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.
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Thin-Walled Structures 147 (2020) 106530
Fig. 1. Schematic plot of piezoelectric structure layout-design for vibration control.
The layout of structures obtained by topology optimization has been applied into structural vibration attenuation. Most of the studies focus on indirect ways. They extremize dynamic properties such as funda mental frequency [21], gaps between consecutive eigen-frequencies [22], magnitude of steady-state vibrations [23–25]. Dynamic proper ties are optimized to reduce structural vibration. Topology optimization has also been applied into active control of piezoelectric structures. Many researches has been conducted in the field of static deformation control [26,27]. Objective functions such as structural compliance, piezoelectric coefficients and deformation error are extremized so that a certain deformation can be achieved. Kogl and Silva [28] extend the traditional SIMP model into PEMAP-P model for piezoelectric layout design. Molter et al. [29] proposed an optimal cri terion (OC) based scheme for simultaneous structural optimization and piezoelectric actuator placement. Kang et al. applied topology optimi zation techniques into piezoelectric plate design [30,31]. Luo et al. [32] proposed a multiphase level set method of piecewise constants for piezoelectric actuator design. Yang et al. [33] investigated a multi-point constraints (MPC) based method for integrated structural optimization and piezoelectric patch placement. There are some researches involved in topology optimization of piezoelectric structures considering active vibration control [34], but is still in a preliminary stage. Zhang et al. [35,36] incorporate a constant gain velocity feedback (CGVF) control algorithm into dynamic topology optimization of piezoelectric structures to control transient response and sound radiation. Linear quadratic regulator (LQR) control is also investigated by Hu et al. [37] and Silveira et al. [38]. Control spillover effects [39] and minimum energy consumption [40] were also
considered. In these works, the frequency of external loads and the parametric value of a control strategy are all accounted in layout design. The performance of these layouts works well in design point. However, their performance deteriorate as external loads or control parameters vary. Besides, the optimized layout will be different even if the fre quency of external loads or the value of control parameters change marginally. This paper proposes a novel scheme for layout-design of distributed piezoelectric actuators aiming at vibration control of thin-walled smart structures. The schematic plot of piezoelectric structure layout-design for vibration control is shown in Fig. 1. In this paper, we combine the topology optimization technique and the SPI [4] to formulate the opti mization model of the layout-design problem. Based on the chain rule, adjoint method and solving Lyapunov equation, the sensitivity analysis of objective function is derived. The optimization model is solved by general optimization algorithm. The optimized layout obtained by the proposed method stays same regardless of the external loads and para metric value of control strategy. Then the LQR control strategy is applied to the designed layout so that vibration suppress can be achieved. The optimized layout can achieve an excellent performance in a wide range of load cases. Numerical examples show validity of the proposed method. 2. Finite element modelling of piezoelectric structure 2.1. Governing equation In this paper, the deformation of piezoelectric plate is assumed small 2
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Thin-Walled Structures 147 (2020) 106530
Fig. 2. Four-nodes element for piezoelectric plate.
and linear. Thus, the constitutive equation of piezoelectric material can be expressed as follows:
where Bu denotes the strain-displacement matrix and d ¼ ½u0 ; v0 ; ⋯;
T ¼ Ce S eE D ¼ eS þ κE
Ea ¼ Bϕ ϕa
ð1Þ
ð4Þ θð4Þ x ; θy �. Substitute Eq. (5) into Eq. (3) and we can obtain:
(1)
where T and S represent stress and strain vector, respectively; D denotes electric displacement vector; E means electric field; Ce represents elastic matrix; e and κ denotes piezoelectricity and permittivity vector, respectively. In this paper, thin-walled structures are considered and piezoelectric materials are attached onto them. Thus, the whole structures can be meshed with plate elements. The finite element model of the piezo electric plate is plotted at Fig. 2. The piezoelectric layers are bounded tightly with the host layer. The strain of the piezoelectric plate can be described by Mindlin plate theory: 2 3 2 32 3 εxx u0 ∂=∂x 0 0 0 z⋅∂=∂x 6 εyy 7 6 0 6 7 ∂=∂y 0 z⋅∂=∂y 0 7 6 7 6 76 v0 7 6 γ xy 7 ¼ 6 ∂=∂y ∂=∂x 6 7 (2) 0 z⋅∂=∂x z⋅∂=∂y 7 6 7 6 76 w0 7 4 γyz 5 4 0 0 ∂=∂y 1 0 54 θx 5 γzx θy 0 0 ∂=∂x 0 1
ð1Þ
4 X
� ðiÞ ðiÞ ðiÞ � ðiÞ T Ni u0 ; v0 ; w0 ; θðiÞ x ; θx
Ni ϕðiÞ a
(8)
2.2. Reduced state-space model The degree of freedom (DOF) of Eq. (8) can be very high as elements increase, which makes it very time-consuming to obtain the controller. Therefore, model reduction is needed. The displacement can be approximated as follows:
(4)
uðtÞ � ψ qðtÞ ¼
m X
(10)
ψ i qi ðtÞ
i¼1
where ψ ¼ ½ψ 1 ; ⋯; ψ m � denotes modal shape; m represents the number of
(5)
mode chosen; qðtÞ ¼ ½q1 ðtÞ; ⋯; qm ðtÞ�T denotes the modal displacement vector. Substitute Eq. (10) into Eq. (8), and left multiply ψ T , and we can obtain:
(6)
_ þ ω2 qðtÞ ¼ ψ T f ðtÞ q€ðtÞ þ 2ξωqðtÞ
i¼1
Substitute Eq. (4) into Eq. (2) and we can obtain: S ¼ Bu d
K uϕ ϕa ðtÞ
where Ωe denotes volume of each element.
where Ni ; i ¼ 1; ⋯; 4 represent the shape function in standard form. The electric potential can also expressed as similar form: 4 X
ð4Þ T
Ωe
i¼1
ϕa ¼
ð3Þ
where M ¼ Mh þMa þ Ms represents structural mass matrix (Mh , Ma and Ms represent mass matrix of host, actuator and sensor layers, respec tively). K ¼ K h þK a þ K s denotes structural stiffness matrix. C is the damping matrix. The Rayleigh damping model is used in this paper. Only the damping effect of the host structure is considered in this paper. Therefore, the damping matrix can be expressed as C ¼ αMh þ βK h , where α and β are the coefficient of the Rayleigh damping. f ðtÞ repre sents the external loads. K uϕ denotes electro-mechanical matrix. The electro-mechanical matrix of each element can be expressed as follows: Z ðeÞ K uϕ ¼ BTu eT Bϕ dΩ (9)
where ϕa represents electric potential in actuator layer. The general displacement vector can be expressed as linear combi nation of nodal displacement: ½u0 ; v0 ; w0 ; θx ; θx �T ¼
ð2Þ
€ þ CuðtÞ _ þ KuðtÞ ¼ f ðtÞ MuðtÞ
(3)
ϕa =h
(7)
where ϕa ¼ ½ϕa ; ϕa ; ϕa ; ϕa � ; Bϕ represents the electric fieldpotential matrix. According to Eq. (1) and Eqs. (6) and (7) and the virtual work theory, we can obtain the governing equation of the piezoelectric plate at finite element form:
where u0 , v0 and w0 represent mid-plane displacement at X, Y and Z direction, respectively. θx denotes rotation around X axis and θy denotes rotation around Y axis. The height of actuator layer is h. We assume that only transverse electric fields are applied in the plate. The electric field Ea can be express as follows: Ea ¼
ð1Þ
3
ψ T K uϕ ϕa ðtÞ
(11)
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bration suppression can be maximized under a certain control input energy. The SPI proposed by Hac and Liu [4] can measure that property. The higher SPI of a system, the less energy consumption it will need in vibration control. The SPI can be expressed as follows: !s 2m 2m Y X 2m λi λi (17) SPI ¼
Table 1 Material properties of piezoelectric structure. Young’s modulus
Density
Piezoelectric constant
Poisson’s ratio
Host layer
70Gpa
–
0.30
Piezoelectric layer (PTZ)
71Gpa
2700kg/ m3 5000kg/ m3
e31 ¼ e33 ¼ 5.2C/m2
0.35
i¼1
where λi ; i ¼ 1; ⋯2m is the eigenvalues of controllability Gramian Lc of the structural system. Here we introduce the Controllability Gramian and its eigenvalue. The controllability Gramian can be expressed as follows: Z ∞ T Lc ¼ eAτ B1 BT1 eA τ dτ (18)
where ξ ¼ diagfξ1 ; ⋯; ξm g; ξi ¼
ψ Τι Cψ i ψ Τι ðαMh þ βK h Þψ i ¼ ; i ¼ 1; 2; ⋯; m ψ Τι Mψ i ψ Τι Mψ i
i¼1
(12)
and
0
ω ¼ diagfω1 ; ⋯; ωm g; ω2i ¼
ψ Τι K ψ i ; i ¼ 1; 2; ⋯; m ψ Τι Mψ i
The structural system is controllable as long as Lc is positive definite.
(13)
represents the structural model damping ratio and natural frequency. Let (14)
_ xðtÞ ¼ ½qðtÞ; qðtÞ�T be the state variable. Then Eq. (11) can be reorganized as follows:
(15)
_ ¼ AxðtÞ þ B1 ϕa ðtÞ þ B2 f ðtÞ xðtÞ where � 0 A¼
2
ω
� I ; 2ξω
� B1 ¼
� 0 ; T ψ K uϕ
� B2 ¼
0
ψ
� (16)
T
I represents the identity matrix. 3. Formulation of layout-design and sensitivity analysis 3.1. System performance index (SPI)
Fig. 4. Comparison of sensitivity analysis obtained by analytical method and FDM.
Piezoelectric materials should be allocated to the place where vi
Fig. 3. Rectangular cantilever plate.
Fig. 5. The layout obtained from different volume fraction: (a) fv ¼ 0:6; (b).fv ¼ 0:4. 4
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Table 2 Eigen-frequencies of the initial design. Mode order
Eigen-frequency
Mode order
Eigen-frequency
1 2 3 4 5 6 7 8 9 10
7.33 11.70 25.66 37.06 52.27 56.10 57.00 63.30 68.59 78.59
11 12 13 14 15 16 17 18 19 20
80.79 97.50 100.43 101.80 107.79 110.07 112.06 113.41 124.92 125.33
Table 3 Eigen-frequencies of the optimized layout. Mode order
Eigen-frequency
Mode order
Eigen-frequency
1 2 3 4 5 6 7 8 9 10
9.09 12.01 31.14 40.27 61.41 64.82 67.32 75.81 79.13 94.37
11 12 13 14 15 16 17 18 19 20
96.47 114.10 120.84 125.17 128.64 136.37 140.37 144.75 145.79 148.11
The controllability Gramian in Eq. (18) can also be obtained from the following Lyapunov equation: (19)
ALc þ Lc AT þ B1 BT1 ¼ 0
Supposes the eigenvectors of Lc are φi ; i ¼ 1; ⋯; 2m. We can easily obtain the following equation: (20)
Lc φi ¼ λi φi ; i ¼ 1; ⋯; 2m
Large SPI guarantees high energy transmission from the piezoelectric actuator to the structural system. Thus, less control input is needed to alleviate vibration. 3.2. Formulation of distributed piezoelectric actuator layout-design
Fig. 6. Iteration history of objective function: (a) fv ¼ 0:6; (b).fv ¼ 0:4.
The SPI is used as objective function in the optimization model. The volume fraction of piezoelectric materials is used as constraint. Mean while, the governing state-space function of the structural system should be satisfied. Thus, we can formulate the following topology optimization problem: find
ρe
max
gðλi Þ ¼
!s
2m X
λi i¼1
s:t:
2m Y
2m
λi i¼1
(21)
_ ¼ AxðtÞ þ B1 ϕa ðtÞ xðtÞ N N X X ρe Ve � fv Ve e¼1
e¼1
0 � ρmin � ρe � 1;
e ¼ 1; 2; ⋯; N
where ρ ¼ ½ρ1 ; ⋯; ρN �T represents the design variables; N denotes num ber of design variables; Ve represents volume of the eth element; ρmin is a small real number in case of numerical instability; fv denotes permitted volume fraction. 3.3. Sensitivity analysis
Fig. 7. Iteration history of volume fraction.
Sensitivity analysis of the objective function in Eq. (21) is critical for the topology optimization problem. Based on chain rule and adjoint 5
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vffiffiffiffiffiffiffiffiffiffi u
2m uY ∂gðλi Þ 2m 1 ¼ t λj þ 2mλi ∂λi j¼1
!s
2m X
2m Y
2m
λj j¼1
λj
(23)
j¼1
Once eigenvalue of Lc is obtained, this term can be calculated easily. Thus, the main task changes to calculate derivative of eigenvalues of ∂λi the controllability Gramian: ∂ρ ; i ¼ 1; ⋯; 2m. Differentiating the e
eigenvalue equation Eq. (20) with respect to the design variable ρe , we can obtain:
∂Lc ∂φ ∂λi ∂φ φ þ Lc i ¼ φ þ λi i ; i ¼ 1; ⋯; 2m ∂ρe i ∂ρe ∂ρe i ∂ρe
(24)
Left multiplying φTi ; i ¼ 1; ⋯; 2m at Eq. (24) and rewriting the equation, we can derive:
∂λi T ∂Lc φ φ ¼ φTi φ þ φTi Lc ∂ρe i i ∂ρe i
λi φTi
� ∂φi
∂ρe
; i ¼ 1; ⋯; 2m
(25)
Because φTi ; i ¼ 1; ⋯; 2m are normalized vectors and Lc is a sym metric positive definite matrix, substituting Eq. (20) into Eq. (25), we can obtain the following equation:
∂λi ∂Lc ¼ φTι φ ; i ¼ 1; ⋯; 2m ∂ρe ∂ρe i
(26)
Then, the task is turned into solving the derivative of controllability Gramian ∂Lc =∂ρe . Differentiating the Lyapunov equation Eq. (19), we can obtain: A
∂Lc ∂Lc T ∂A ∂AT ∂B1 T ∂BT þ A þ Lc þ Lc þ B1 þ B1 1 ¼ 0 ∂ρe ∂ρe ∂ρe ∂ρe ∂ρe ∂ρe
(27)
let H¼
∂A ∂AT ∂B1 T ∂BT Lc þ Lc þ B1 þ B1 1 ∂ρe ∂ρe ∂ρe ∂ρe
(28)
Because Lc is a positive definite symmetric matrix, H is also a positive definite symmetric matrix. Substituting Eq. (28) into Eq. (27), we can get: A
∂Lc ∂Lc T þ A þH ¼0 ∂ρe ∂ρe
(29)
Comparing Eq. (29) with Eq. (19), It can be seen that Eq. (29) is a standard Lyapunov equation. Thus, it can be solved by a well-established algorithm [41] as long as ∂A=∂ρe and ∂B1 =∂ρe is calculated. According to Eq. (16), we can derive: 3 2 0 0 ∂A 6 7 (30) ¼ 4 ∂ω2 ∂ω 5 ∂ρe 2ξ
Fig. 8. Time-domain response of initial layout and optimized layout (fv ¼ 0.4): (a) Initial state 1; (b) Initial state 2.
∂ρe
∂ρe
and 2
∂B1 6 ¼4 ∂K ∂ρe ψ T uϕ ∂ρe
3 7
(31)
5 ∂ψ T K uϕ ∂ρe
SIMP model is used here for the interpolation of mass, stiffness and coupling matrix so the results can reach either 0 (void) or 1 (piezo electric material) as nearly as possible. The interpolation scheme for piezoelectric materials can be expressed as follows:
Fig. 9. Load case for frequency response optimization.
method, and with the help of solving Lyapunov function, the sensitivity analysis is performed. The derivative of the objective function with respective design variable ρe can be expressed as: 2m ∂gðλi Þ X ∂gðλi Þ ∂λi ¼ ∂ρe ∂λi ∂ρe i¼1
0
ðeÞ
MðeÞ a ¼ ρe M a0 ; p K ðeÞ a ¼ ðρe Þ K a0 ; ðeÞ
(22)
K uϕ ¼ ðρe Þp K uϕ0 ðeÞ
The first term at the right side of Eq. (22) can be expressed as follows: 6
ðeÞ
(32)
ðeÞ
MðeÞ s ¼ ρe M s0
p K ðeÞ s ¼ ðρe Þ K s0
ðeÞ
(33) (34)
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Fig. 10. Layout obtained by frequency response optimization with different frequencies: (a) ω ¼ 8Hz; (b) ω ¼ 18Hz; (c).ω ¼ 40Hz.
Eq. (30) contains derivative of structural natural frequency. When the eigenvalue is unimodal, the derivative of ∂ω2 =∂ρe can be obtained from: � � ∂ω2i ∂K ∂M ¼ ψ Tι ω2i ψ i ; i ¼ 1; 2; ⋯; m (36)
∂ρe
∂ρe
∂ρe
When the eigenvalue is multiple (i.e. r-fold), then there are r eigen vectors (say ψ 1 ; ⋯; ψ r ) correspond to the natural frequency ωi , the de rivative can be obtained from the following equation [42]: � � � � ∂K ∂M ∂ω2i det ψ Tj (37) ω2i ψk δjk ¼ 0; j; k ¼ 1; 2; ⋯; r
∂ρe
∂ρe
∂ρe
where δjk represents the Kronecker function, which meets: � 1; j¼k δjk ¼ 0; j 6¼ k The derivative ∂ω=∂ρe can be obtained by: pffiffiffiffiffi ∂ωi ∂ ω2i 1 ∂ω2i ¼ ¼ 2ωi ∂ρe ∂ρe ∂ρe
Fig. 11. Time-domain response of different layouts. ðeÞ
ðeÞ
ðeÞ
element, and K uϕ0 is the corresponding coupling matrix; p denotes the ðeÞ
penalty coefficient. Then Eq. (31) can 2 0 ∂B1 6 ¼4 ∂ψ T p ðeÞ ∂ρe ψ T pðρe Þp 1 K ðeÞ ρe Kuϕ0 uϕ0
be rewritten as follows [25]: 3 7 5
(39)
Therefore, the sensitivity analysis of objective function can be ob tained from Eqs.(22)–(39). Once all the sensitivities are calculated, then the topology optimization model Eq. (21) can be solved by optimization algorithms such as sequential quadratic programming (SQP) [43,44] and method of moving asymptotes (MMA) [45].
where Ma0 , K a0 , Ms0 , K s0 are element mass matrix and stiffness matrix of actuator and sensor layers when piezoelectric material is full at eth ðeÞ
(38)
4. Control strategy
(35)
Once the layout has been designed, a proper control strategy is
∂ρe
Fig. 12. Voltage amplitude of the SPI layout design: (a) 0.001s; (b) 1s. 7
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Thin-Walled Structures 147 (2020) 106530
Fig. 16. Iteration history of objective function. Fig. 13. Energy consumption of different layouts: (a) ω ¼ 25Hz; (b) ω ¼ 50Hz; (c).ω ¼ 100Hz.
The core purpose of LQR controller design for structural vibration control is to calculate the voltage input ϕa ðtÞ so that the following cost function can be minimized: Z � 1 ∞� T x ðtÞQxðtÞ þ ϕTa ðtÞRϕa ðtÞ dt J¼ (40) 2 0 where Q is a positive semi-definite symmetric weight coefficient matrix; R is a positive definite symmetric weight coefficient matrix. The voltage input ϕa ðtÞ can be expressed as linear function of state variables xðtÞ: ϕa ðtÞ ¼
GxðtÞ
(41)
where G ¼ R 1 BT1 P denotes feedback gain of the controller. P can be calculated from the following algebraic Riccati equation:
Fig. 14. Clamped square plate.
PA þ AT P
PB1 R 1 BT1 P þ Q ¼ 0
(42)
Once P is derived, the controller can be obtained. 5. Numerical examples and engineering application Two numerical examples and two engineering applications are used here to verify the proposed method. In all cases, the thickness of host structure is t ¼ 3mm; the thickness of actuator and sensor layer are both h ¼ 1mm. The host layer is made up of aluminum and the piezoelectric layers are constructed by piezoelectric ceramic PTZ. Material properties of the piezoelectric structure [34] are listed in Table 1. The Rayleigh damping coefficient is α ¼ 0:01 and β ¼ 0:0001, respectively. The value of penalty coefficient is p ¼ 3 in this paper. The minimum of design variables are set as ρmin ¼ 1 � 10 4 . MMA is used to solve the optimization problem Eq. (21). The algorithm converges when the difference of objective function between two sequential iterations is smaller than ε ¼ 1 � 10 8 . 5.1. Numerical example 1: rectangular cantilever plate A rectangular cantilever plate is used in this numerical example, which is plotted in Fig. 3. The left side of the plate is clamped. The length and width of the plate is a ¼ 1:6m and b ¼ 0:8m, respectively. The rectangular plate is meshed into 3200 (80 � 40) four-node Mindlin el ements, with 3321 nodes, 16605 displacement degree of freedom (DOF) and 3321 electric DOF. The top 20 modes of the rectangular plate are used to generate the reduced state-space equation. In this example, we consider the volume fraction fv ¼ 0:6 and fv ¼ 0:4. First, the sensitivities of the objective function are calculated by the proposed analytical method and finite
Fig. 15. The optimized layout.
needed to control the structural vibration. LQR is applied for vibration control in this paper. In this paper, all state variables are assumed available, although it is not true in practical engineering. However, the state variables can be obtained from state observer through limited observations. This process is a little complicated but well-established. Since this is not main the purpose of the work, we use state variables as the feedback. 8
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Fig. 17. Frequency response of initial layout and optimized layout: (a) Frequency response curve; (b) Displacement field of point A; (c) Displacement field of point B.
Fig. 18. Layout obtained by frequency response optimization with different frequencies.
difference method (FDM), respectively. In FDM, the perturbation of the design variables are 1 � 10 5 . The results are presented in Fig. 4. It show the proposed analytical method complies well with the FDM. But the efficiency of analytical method is much higher than that of FDM. The final layouts of piezoelectric materials that obtained from different constraints are plotted in Fig. 5. The piezoelectric materials are plotted with black color. It shows as the upper bound of constraint in crease (fv ¼ 0:4⇒fv ¼ 0:6), more piezoelectric materials are distributed into the plate. But the general shape of piezoelectric material stays
similar. Besides, in both cases, the results are pretty clear and no in termediate pseudo-density elements exists. The iteration history of objective function is plotted in Fig. 6. Fig. 6 shows that in both cases the curve of iteration history is comparatively smooth and converges within 110 steps. Fig. 7 is iteration history of volume fraction. It can be seen that the constraints are always satisfied through the iteration process. The eigen-frequencies before and after optimization is listed in Table 2 and Table 3. They show that the compared with the initial 9
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attenuates slowly due to existence of Rayleigh damping. (2) when LQR control is applied, the vibration suppression of optimized layout is much faster than that of initial design. 5.2. Further discussion on numerical example 1 In order to further verify the efficiency of the proposed method, the layout obtained by frequency response optimization is used. In fre quency response optimization, an external load F ¼ sinðωtÞ is applied at the tip of the plate. The load case is plotted in Fig. 9. The optimization model for frequency response analysis can be expressed as follows:
Fig. 19. Frequency response analysis of different layouts.
Fig. 20. Simplified fuselage.
design, the eigen-frequencies of optimized layout are increased. Same control strategy is applied to initial and optimized layout to verify their ability in vibration control. We choose the layout obtained by setting fv ¼ 0:4. The parameters in LQR are chosen as Q ¼ 1015 � I and R ¼ I. The initial layout without control is used here for compari son. Two initial states of the plate are considered. The initial state 1 is simple. The initial displacement field that is equal to the first order mode uðt¼0Þ ¼ ψ 1 is applied. The initial state 2 is much more complex. The displacement field that equals to combination of top 20 modes uðt¼0Þ ¼ P20 i¼1 ψ i is applied. Four order Runge-Kutta method with fixed time step t ¼ 0:001s (this time step is small enough because the maximum eigenfrequency is smaller than 150 Hz) is used. Time-domain response of initial layout and optimized layout are plotted in Fig. 8. Fig. 8 shows: (1) In the case of initial design without control, the vibration amplitude
Fig. 22. Iteration history of objective function.
Fig. 23. Load case for frequency response analysis.
Fig. 21. Layout of optimized design: (a) View 1; (b) View 2. 10
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time domain response of different layouts are plotted in Fig. 11. It shows compared with the layout obtained by frequency response optimization, the layout obtained by SPI can achieve better control performance. The voltage amplitude applied to the configuration obtained by SPI design is presented in Fig. 12. The voltage at where the ‘voids’ exist almost equal to zero and a high voltage is applied at where the piezo electric materials exist. By the way, the maximum voltage can change with the parameters of LQR. In practical engineering, the maximum voltage is determined by the property of piezoelectric materials and it is not considered in this paper. The energy consumption is also compared between these three different layouts. The energy index (EI) is defined to represent the en ergy consumption. EI is defined as the summation of voltage applied to the plate. We plot EI-time of the three layouts in Fig. 13. It shows that compared with other configurations, the total energy consumption of SPI design is minimum during the control process. 5.3. Numerical example 2: clamped square plate
Fig. 24. Frequency response of initial layout and optimized layout.
A square plate with four edges clamped is used in this example, which is plotted in Fig. 14. The length of the plate is l ¼ 1:2m. This square plate is meshed into 3600 (60 � 60) four-node Mindlin elements. There are 3721 nodes, with 18605 displacement DOF and 3721 electric DOF. The top 20 modes of the square plate are used. The permitted volume fraction is set as fv ¼ 0:3 in this example. The initial value of all design variables are set as ρe ¼ 0:3; e ¼ 1; ⋯; N. The optimization process converges within 80 steps. The final layout is plotted in Fig. 15. There is no intermediate pseudo-density element. The iteration history of objective function is plotted in Fig. 16. The iteration process is very smooth and converges quickly. The parameters in LQR are chosen as Q ¼ 1015 � I and R ¼ I. Fre quency response analysis is applied to the initial layout and optimized layout. A harmonic force whose magnitude is 1000N is applied to the center of the square plate (vertical to the plate) to perform the frequency analysis. The result is plotted in Fig. 17. Fig. 17 (a) shows: (1) The resonant frequency (eigenvalue) of optimized design is higher than that of initial design. (2) Compared with the initial design with LQR control, a 25 dB attenuation of frequency response magnitude of the first mode is achieve by the optimized layout with LQR control. (3) The frequency response curve of the optimized design is much smoother than that of initial design. The displacement field of initial design (point A in Fig. 17 (a)) and optimized design (point B in Fig. 17 (a)) under a harmonic excitation is plotted in Fig. 17 (b)–(c). It shows their modes are similar, but the maximum displacement of the optimized design is 1:1 � 10 5 m, decreased significantly from the initial design, which is 1:2 � 10 3 m. Fig. 17 demonstrates that the optimized layout obtained by the proposed method can improve the structural frequency response performance in a wide range of frequency interval.
Fig. 25. Time-domain response of initial layout and optimized layout.
find ρe min gf ¼ CTc Cc s:t: Cc ¼ FT U
�
ω2 M þ iωC þ K U ¼ F N X e¼1
ρe Ve � fv
(43)
N X
Ve e¼1
0 � ρmin � ρe � 1; e ¼ 1; 2; ⋯; N
5.4. Further discussion on numerical example 2
The detailed sensitivity analysis of the optimization model is pre sented in Appendix A. When the weight constraint fv ¼ 0:4. The final layout obtained by the above optimization model is plotted in Fig. 10. From Fig. 10 we can obtain the following conclusion. (1) The final configuration obtained from frequency response optimization changes greatly when the frequency of external load changes. (2) When fre quency of external load is comparatively higher, then there will be a lot of intermediate pseudo-density elements and the iteration is very diffi cult to converge. The reason why this happens is explained in this reference [46]. We choose the clear configuration Fig. 10 (a) to compare the result obtained by the proposed method (Fig. 5 (b)). The same control pa rameters are applied to both configurations. Initial state 2 in Fig. 8 (b) is applied to test their ability to control the complex initial response. The
Frequency response based topology optimization is also performed to compare with our proposed method. A harmonic load F ¼ sinðωtÞ is applied at the center of the square plate. The optimization model is Eq. (43) and the weight constraint is set as fv ¼ 0:3. The layout obtained by different frequency is plotted in Fig. 18. Still we can find that the layout varied as the frequency of external load changes. Besides, there will be a lot of intermediate pseudo-density elements if the frequency is comparatively high. We choose the layout obtained by ω ¼ 50Hz (Fig. 18 (b)) to compare with the proposed method. Frequency response analysis is performed to the layouts and the result is plotted in Fig. 19. Following conclusions can be obtained: (1) The resonant frequency of both SPI design and frequency design will increase. (2) The resonant frequency increase of frequency design is 11
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Fig. 26. Center wing-box: (a) Isometric view; (b) Top view; (c) Cutaway view.
Fig. 27. Layout of the optimized design of piezoelectric materials: (a) Isometric view; (b) Top view.
higher than that of SPI design. (3) The frequency response curve of SPI design is smoother than frequency design, and the vibration control performance of SPI design is better than that of frequency design, especially in higher frequency. The above conclusion shows that compared with the frequency response based optimization, the layout obtained by SPI design is more suitable for active vibration control.
fraction. The initial value of all design variables are set as ρe ¼ 0:4; e ¼ 1; ⋯; N. The iteration converges within 50 steps. The final layout is plotted in Fig. 21. The result is very clear. The iteration history of objective function is plotted in Fig. 22. The parameters in LQR are chosen as Q ¼ 1016 � I and R ¼ I. Fre quency response analysis is applied to the simplified fuselage (see Fig. 23). The results are plotted in Fig. 24. It shows natural frequency of the optimized layout becomes higher and attenuation of the vibration is achieved. Fig. 25 shows the time-domain response of initial layout and optimized layout given a certain initial condition. The initial displace P ment field equals to uðt¼0Þ ¼ 5i¼1 ψ i . Compared with initial design with LQR control, the vibration of optimized design with LQR control de creases much faster.
5.5. Engineering application 1: simplified fuselage The fuselage of aircraft may vibrate severely due to the disturbance of airflow, which will affect fatigue property of the aircraft greatly. Therefore, vibration attenuation of fuselage is imperative. A part of simplified fuselage of aircraft is used in this example, which is plotted in Fig. 20. The simplified fuselage is cylinder shaped, with its width l ¼ 1:2m and radius r ¼ 0:2m. Both end of the simplified fuselage is clam ped. It is meshed into 3720 (62 � 60) Mindlin shell elements. There are 3782 nodes, with 22692 displacement DOF and 3782 electric DOF. The top 20 modes are used. fv ¼ 0:4 is used as the permitted volume
5.6. Engineering application 2: center wing-box A center wing-box of space shuttle is used in this engineering 12
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Due to the severe working environment of the space shuttle (such as heat), the center wing-box may vibrate, which will reduce its stability and fatigue life. Thus, vibration attenuation is imperative. The param eters in LQR are chosen as Q ¼ 1019 � I and R ¼ I. Suppose the center wing-box vibrates due to perturbation from the environment. The initial P displacement field is complex and equals to uðt¼0Þ ¼ 5i¼1 ψ i . The timedomain response of the center wing-box of the initial design and opti mized design is plotted in Fig. 29. It shows compared with the initial design, the optimized layout can stabilize the wing-box much faster. This example shows that the proposed method can deal with more complicated engineering problems. 6. Conclusion This paper proposes a scheme for layout-design of smart structures with distributed piezoelectric actuators for vibration control, based on topology optimization technique and the system performance index (SPI). The pseudo density of piezoelectric materials are set as design variables. The optimization model is formulated to maximize SPI so that the energy transformation from actuator to structure can be maximized. The sensitivity analysis is performed based on chain rule and adjoint method, and solving Lyapunov equations. The optimization problem is solved by nonlinear programming method. Then, the linear quadratic regulator (LQR) control strategy is applied to the designed layout and vibration suppression can be achieved. External loads and control strategy is not considered in layout-design so that a single layout can be obtained. Nevertheless, the optimized layout performs well in a wide range of load cases. Numerical examples and engineering application are used to verify the proposed method. Results show that the optimization processes converge within few iterations and the constraints are not violated during the iteration. The obtained optimized layouts are very clear and in most cases there is no pseudo-density element. Compared with the initial design, the fundamental frequency of the optimized layout in creases. Moreover, although frequency of external loads are not incor porated in layout-design, the overall frequency response of optimized design are much smoother, and frequency response magnitude are attenuated a lot. In terms of the displacement perturbation, the opti mized designs can be stabilized much faster. This work provides a possible approach for conceptual design of smart structures.
Fig. 28. Iteration history of objective function.
Fig. 29. Vibration of the center wing-box.
Declaration of competing interest
application. The whole wing-box is an integral panel. The shape and size of the wing-box is plotted in Fig. 26. The left end of the wing-box is clamped. The whole model is meshed as 11376 four-node Mindlin ele ments, with 9991 nodes. The upper surface of the wing-box is applied to place piezoelectric materials for attenuating structural vibration. Thus, there are N ¼ 3120 design variables. Suppose the top 20 modes accounts for the most of vibrational energy. The volume constraint fv ¼ 0:3 is used in this example. The initial value of all design variables are set as ρe ¼ 0:3; e ¼ 1; ⋯; N. The final layout is obtained after 146 iterations. The optimized design of the piezoelectric center wing-box is plotted in Fig. 27, and the iteration history is plotted in Fig. 28.
We declared that we have no conflicts of interest. We do not have any commercial or associative interest that represents a conflict of interest in connection with the work. Acknowledgements The paper is supported by the Key Technologies Research and Development Program of China (2016YFB0200700), National Natural Science Foundation of China (11572024, 11432002, 11872089) and National Defense Basic Scientific Research Program of China (JCKY2016601B001, JCKY2017601B001).
Appendix A The sensitivity analysis of the objective of Eq. (43) can be performed as: � T � ∂gf ∂CTc ∂Cc ∂Cc ¼ Cc þ CTc ¼ 2Re Cc
∂ρe
∂ρe
∂ρe
∂ρe
where ReðÞ means the real part. Then ∂Cc =∂ρe in Eq. (A.1) can be expressed as follows: 13
(A.1)
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�
� � 1 �T ∂ðKD Þ 1 ∂Cc ∂ FT KD 1 F ∂ðK D Þ 1 ¼ ¼ FT K D1 K D F ¼ K TD F K F ∂ρe ∂ρe ∂ρe ∂ρe D where K D ¼
∂ðKD Þ ¼ ∂ρe
(A.2)
ω2 M þ iωC þ K. Because only the damping effect of host structure is considered, then we can obtain: (A.3)
p 1 ðeÞ ω2 MðeÞ a0 þ pρe K a0 1
Let U ¼ ðK TD Þ F (U represent the conjugate vector of displacement vector U). Then Eq. (A.2) can be transformed into:
∂Cc ¼ ðUÞT ∂ρe
�
(A.4)
p 1 ðeÞ ω2 MðeÞ a0 þ pρe K a0 U
References
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