Active vibration control of conical shells using piezoelectric materials

Abstract

In this paper, the active vibration control of conical shells is studied using velocity feedback and linear quadratic regulator methods. Up to now, many researches on the active vibration control of beams, plates and cylindrical shells have been published, however, to our knowledge, few people have studied the active vibration control of conical shells. Normally, in the equation of motion of the conical shells, some coefficients are variables, which makes the equation of motion of the conical shells very complicated and difficult to solve analytically. In order to solve this problem, Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the complex electromechanical coupling system. This equation of motion for the conical shell and piezoelectric patch system can be easily solved and effectively used for the structural active vibration control. Based on the traditional theory of structural dynamics, this method is easy to understand and is verified by numerical simulations. The forced vibration responses of the conical shells with two piezoelectric patches are computed to study the active vibration control. The optimal design for the locations of the piezoelectric patches is also developed by the genetic algorithm. From the results it can be seen that the control gain has a significant effect on the vibration control of the conical shell, but the effect of the size of the piezoelectric patches on controlling the vibration amplitudes is not so obvious. The overall vibration of the conical shell can be effectively reduced by the velocity feedback control method. With the increase of the control gain, the active damping characteristics of the conical shell are improved. Moreover, the optimal placement scheme of the piezoelectric patches obtained by the genetic algorithm can significantly reduce the vibration amplitudes of the conical shell.

Keywords

Active vibration control assumed mode method conical shell genetic algorithm Hamilton’s principle linear quadratic regulator piezoelectric material

1. Introduction

Many people have studied the problem of active vibration control in various kinds of engineering structures, such as the beams (Sun and Tong, 2004; Vasques and Rodriques, 2008), the plates (Carra et al., 2008) and the cylindrical shells (Kumar et al., 2003; Balamurugan and Narayanan, 2008). Lin and Nien (2005) investigated the modeling and active vibration control of a smart beam using piezoelectric damping-modal actuators/sensors. They presented the theoretical formulations and numerical solutions for the analysis of a laminated composite beam. Hong et al. (2006) studied the vibration control of beams using multiobjective state-feedback control. Using Hamilton’s principle, they derived the equation of motion for a piezo/beam system and designed an active vibration suppression system using a multiobjective state-feedback controller. Gao and Liao (2005) studied the active vibration control of simply supported (S-S) beams with enhanced self-sensing active constrained layer damping (ACLD) treatments.

Hu and Ng (2005) developed an approach for active vibration control of flexible structures with integrated piezoelectric actuators using a robust vibration control method. Qiu et al. (2007) used piezoelectric ceramic patches as sensors and actuators to suppress the vibration of the smart flexible clamped plate. They derived the modal equations and piezoelectric control equations of cantilever plate, and developed an optimal placement method for the locations of piezoelectric actuators and sensors. Baz and Chen (2000) presented the distributed-parameter modeling of thin cylindrical shells treated with ACLD. They used Hamilton’s principle to develop the shell/ACLD model and the associated boundary conditions, and developed a globally stable boundary control strategy to damp out the vibration of the shell/ACLD system. Li and Yang (2007) studied the dynamic response and active control of a composite cylindrical shell with piezoelectric shear actuators.

Although the active vibration control of beams, plates and cylindrical shells has received many people’s attention, to our knowledge, few people have studied the active vibration control of conical shells. Conical shells are widely used as important structural components in many engineering fields (Lim and Liew, 1995, 1996; Sweedan and El Damatty, 2003; Senthil Kumar and Ganesan, 2008; Kayran and Yavuzbalkan, 2009; Sofiyev, 2009; Tornabene, 2009; Kerboua et al., 2010). Normally, the equation of motion of the conical shell is very complicated and some coefficients of the equation of motion are variables (Lam and Li, 1999b), which makes it difficult to analytically solve the equation of motion of the conical shell. We have presented a methodology to calculate the natural frequencies and forced vibration responses of the conical shell (Li et al., 2009), but active vibration control for the conical shell has not been studied. So, finding effective methods to study the active vibration control of conical shells has much important theoretical and practical significance.

It should be pointed out that Pinto Correia et al. (2002) presented a semi-analytical axisymmetric shell finite element model to study the active vibration control of axisymmetric shells. In the numerical simulations, they took a composite cylindrical thin shell with piezoelectric layers as an example to study the active vibration control. Fares et al. (2004) studied the problem of minimizing the dynamic response of laminated truncated conical shells with minimum control force. The equations governing the dynamic response were obtained by a first-order shear deformation shell theory. Liew et al. (2002) derived a generic static and dynamic finite element formulation for the modeling and active vibration control of doubly curved functionally graded material shell under coupled displacement, temperature and electric potential fields. Tzou et al. (2002) investigated the finite element modeling, analysis and vibration control of conical shells laminated with distributed actuators. Pradhan (2005) studied the vibration control of the functionally graded material doubly curved shells with embedded magnetostrictive layers. The first-order shear deformation shell theory was employed to obtain the equations of motion.

In this paper, on the basis of Li et al. (2008), wherein the active vibration control of beams was studied, we further investigate the active vibration control of conical shells using negative velocity feedback and linear quadratic regulator (LQR) methods. Hamilton’s principle with the assumed mode method is used to derive the equation of motion of the conical shell and piezoelectric patches system. A set of simple formulations of the principal mode shapes is employed and verified to be effective by numerical simulations. For the conical shell with piezoelectric patches, the forced vibration responses are calculated in order to study the active vibration control. The influences of several parameters on the vibration control are analyzed. The present method is based on the traditional theory of structural dynamics and is easy to understand.

2. Equation of motion

A thin, homogeneous and isotropic conical shell with constant thickness is considered. Figure 1 shows the schematic diagram of the conical shell with piezoelectric patches. Suppose n_p distributed piezoelectric patches are bonded on the surface of the conical shell. The position of the ith piezoelectric patch is located by the coordinates (ξ_i₁, ζ_i₁) and (ξ_i₂, ζ_i₂). The corresponding Cartesian coordinates o-xyz and curvilinear surface coordinates O-ξζη are also shown in Figure 1. The curvilinear surface coordinates are limited to be orthogonal ones that coincide with the lines of principal curvature of the neutral surface. For conical shells, the lines of principal curvatures are the meridians, the parallel circles of the conical shell and the lines perpendicular to the shell surface.

For the shell structures, the strain-displacement relationships are given by (Soedel, 1981)

\begin{matrix} ɛ_{11} = \frac{1}{A_{1} (1 + η / R_{1})} (\frac{\partial u_{1}}{\partial ξ} + \frac{u_{2}}{A_{2}} \frac{\partial A_{1}}{\partial ζ} + u_{3} \frac{A_{1}}{R_{1}}), \\ ɛ_{22} = \frac{1}{A_{2} (1 + η / R_{2})} (\frac{\partial u_{2}}{\partial ζ} + \frac{u_{1}}{A_{1}} \frac{\partial A_{2}}{\partial ξ} + u_{3} \frac{A_{2}}{R_{2}}), \\ ɛ_{33} = \frac{\partial u_{3}}{\partial η}, ɛ_{12} = \frac{A_{1} (1 + η / R_{1})}{A_{2} (1 + η / R_{2})} \frac{\partial}{\partial ζ} [\frac{u_{1}}{A_{1} (1 + η / R_{1})}] \\ + \frac{A_{2} (1 + η / R_{2})}{A_{1} (1 + η / R_{1})} \frac{\partial}{\partial ξ} [\frac{u_{2}}{A_{2} (1 + η / R_{2})}], \end{matrix}

\begin{matrix} ɛ_{13} = A_{1} (1 + η / R_{1}) \frac{\partial}{\partial η} [\frac{u_{1}}{A_{1} (1 + η / R_{1})}] \\ + \frac{1}{A_{1} (1 + η / R_{1})} \frac{\partial u_{3}}{\partial ξ}, \\ ɛ_{23} = A_{2} (1 + η / R_{2}) \frac{\partial}{\partial η} [\frac{u_{2}}{A_{2} (1 + η / R_{2})}] \\ + \frac{1}{A_{2} (1 + η / R_{2})} \frac{\partial u_{3}}{\partial ζ}, \end{matrix}

(1)

where ε_ij are the strains, u_i are the displacements in ξ, ζ and η directions, R₁ and R₂ are the radii of curvature of the two lines of principal curvatures in the neutral surface of the shell and A₁ and A₂ are two variables.

In the neutral surface of the conical shell, the lines of principal curvatures are the meridians and the parallel circles of the conical shell. So the radii of curvature R₁ and R₂ with regard to the meridian, that is, ξ-axis, and the circle, that is, ζ-axis, can be written as

R_{1} = + \infty, R_{2} = ξ \tan α_{0},

(2)

in which α₀ is the semi-vertex cone angle of the conical shell. For the conical shell, the two variables A₁ and A₂ are given by (Soedel, 1981)

A_{1} = 1, A_{2} = ξ \sin α_{0} .

(3)

For the case of thin shells, we can simplify the relationships between the strains and displacements (Equation (1)). If the shell is thin, one can assume that the displacements in the ξ and ζ directions vary linearly through the shell thickness and the displacement in the η direction is independent of η:

u_{1} (ξ, ζ, η) = u (ξ, ζ) + η α (ξ, ζ),

(4)

u_{2} (ξ, ζ, η) = v (ξ, ζ) + η β (ξ, ζ),

(5)

u_{3} (ξ, ζ, η) = w (ξ, ζ),

(6)

where u(ξ, ζ), v(ξ, ζ) and w(ξ, ζ) denote the displacements in the ξ, ζ and η directions, respectively, in the neutral surface of the conical shell, and α and β are the rotations of the normal to the neutral surface about the ζ and ξ-axis, respectively, and they can be expressed as (Soedel, 1981)

α = \frac{u}{R_{1}} - \frac{1}{A_{1}} \frac{\partial w}{\partial ξ}, β = \frac{v}{R_{2}} - \frac{1}{A_{2}} \frac{\partial w}{\partial ζ} .

(7)

Substituting Equations (2)–(6) into Equations (1) and (7) yields

\begin{matrix} α = - \frac{\partial w}{\partial ξ}, β = \frac{v}{ξ \tan α_{0}} - \frac{1}{ξ \sin α_{0}} \frac{\partial w}{\partial ζ}, \\ ɛ_{11} = \frac{\partial u}{\partial ξ} - η \frac{\partial^{2} w}{\partial ξ^{2}}, \\ ɛ_{22} = \frac{1}{ξ \sin α_{0}} \frac{\partial v}{\partial ζ} + \frac{u}{ξ} + \frac{w}{ξ \tan α_{0}} + η (\frac{1}{ξ^{2} \sin α_{0} \tan α_{0}} \frac{\partial v}{\partial ζ} \\ - \frac{1}{ξ^{2} \sin^{2} α_{0}} \frac{\partial^{2} w}{\partial ζ^{2}} - \frac{1}{ξ} \frac{\partial w}{\partial ξ}), ɛ_{33} = 0, \\ ɛ_{12} = \frac{1}{ξ \sin α_{0}} \frac{\partial u}{\partial ζ} + \frac{\partial v}{\partial ξ} - \frac{v}{ξ} + η (- \frac{2}{ξ \sin α_{0}} \frac{\partial^{2} w}{\partial ξ \partial ζ} \\ + \frac{1}{ξ \tan α_{0}} \frac{\partial v}{\partial ξ} - \frac{2}{ξ^{2} \tan α_{0}} v + \frac{2}{ξ^{2} \sin α_{0}} \frac{\partial w}{\partial ζ}), \\ ɛ_{13} = 0, ɛ_{23} = \frac{v}{ξ \tan α_{0}} . \end{matrix}

(8)

During the derivation of Equation (8), the approximations of η/R₁ ≪ 1 and η/R₂ ≪ 1 for thin shells are used.

The relationships between the principal stresses and strains are written as

\begin{matrix} ɛ_{11} = \frac{1}{E} [σ_{11} - μ (σ_{22} + σ_{33})], \\ ɛ_{22} = \frac{1}{E} [σ_{22} - μ (σ_{11} + σ_{33})], \\ ɛ_{33} = \frac{1}{E} [σ_{33} - μ (σ_{11} + σ_{22})], \\ ɛ_{12} = \frac{σ_{12}}{G}, ɛ_{13} = \frac{σ_{13}}{G}, ɛ_{23} = \frac{σ_{23}}{G}, \end{matrix}

(9)

where σ₁₁, σ₂₂ and σ₃₃ are the normal stresses, σ₁₂, σ₁₃ and σ₂₃ are the shear stresses, E is the modulus of elasticity of the material, μ is the Poisson’s ratio and G = E/2(1 + μ) is the shearing modulus of the material. The normal stress σ₃₃ acts in the normal direction to the neutral surface. For an unloaded outer shell surface, σ₃₃ = 0. For loaded shells, σ₃₃ is usually equivalent in magnitude to the external load on the shell, which is a relatively small value in most cases compared with the other two normal stresses (Soedel, 1981). In this analysis, the effects of normal stress σ₃₃ are neglected. From Equation (8) we see that ε₃₃ and ε₁₃ are zero. Thus solving Equation (9) for the stresses yields

\begin{matrix} σ_{11} = \frac{E (ɛ_{11} + μ ɛ_{22})}{1 - μ^{2}}, σ_{22} = \frac{E (ɛ_{22} + μ ɛ_{11})}{1 - μ^{2}}, \\ σ_{12} = G ɛ_{12}, σ_{23} = G ɛ_{23} . \end{matrix}

(10)

The piezoelectric material layers are assumed to be transversely isotropic and the polarization direction is along the z-axis. The constitutive equations of the piezoelectric layer are expressed as

\begin{matrix} {\begin{matrix} σ_{11}^{p} \\ σ_{22}^{p} \\ σ_{33}^{p} \\ σ_{23}^{p} \\ σ_{13}^{p} \\ σ_{12}^{p} \end{matrix}} = [\begin{matrix} c_{11} & c_{12} & c_{13} & 0 & 0 & 0 \\ c_{12} & c_{11} & c_{13} & 0 & 0 & 0 \\ c_{13} & c_{13} & c_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{66} \end{matrix}] {\begin{matrix} ɛ_{11} \\ ɛ_{22} \\ ɛ_{33} \\ ɛ_{23} \\ ɛ_{13} \\ ɛ_{12} \end{matrix}} \\ - [\begin{matrix} 0 & 0 & e_{31} \\ 0 & 0 & e_{31} \\ 0 & 0 & e_{33} \\ 0 & e_{15} & 0 \\ e_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{matrix} E_{ξ} \\ E_{ζ} \\ E_{η} \end{matrix}}, \end{matrix}

(11)

\begin{matrix} {\begin{matrix} D_{1} \\ D_{2} \\ D_{3} \end{matrix}} = [\begin{matrix} 0 & 0 & 0 & 0 & e_{15} & 0 \\ 0 & 0 & 0 & e_{15} & 0 & 0 \\ e_{31} & e_{31} & e_{33} & 0 & 0 & 0 \end{matrix}] {\begin{matrix} ɛ_{11} \\ ɛ_{22} \\ ɛ_{33} \\ ɛ_{23} \\ ɛ_{13} \\ ɛ_{12} \end{matrix}} \\ + [\begin{matrix} \in_{11} & 0 & 0 \\ 0 & \in_{11} & 0 \\ 0 & 0 & \in_{33} \end{matrix}] {\begin{matrix} E_{ξ} \\ E_{ζ} \\ E_{η} \end{matrix}}, \end{matrix}

(12)

where c₁₁, c₁₂, c₁₃, c₄₄ and c₆₆ are the elastic constants in which c₆₆ = (c₁₁ – c₁₂)/2, e₃₁, e₃₃ and e₁₅ are the piezoelectric constants, ∈₁₁ and ∈₃₃ are the dielectric constants, D₁, D₂ and D₃ are the electric displacements and E_ξ, E_ζ and E_η are the electric intensity along the ξ, ζ and η directions, respectively. In this analysis, the outer voltage is applied across the thickness of the piezoelectric layer. So, only E_η is considered. The electric intensity E_η in the η-direction can be written as (Choi et al., 2007; Azrar et al., 2008; Deü et al., 2008; Ramesh Kumar and Narayanan, 2008; Zhang et al., 2008):

E_{η} = V_{0} (t) / h_{p}

, where V₀(t) is the external applied voltage on the piezoelectric layer and h_p is the thickness of the piezoelectric layer. For thin shells, ε₃₃ = 0, ε₁₃ = 0 and

σ_{33}^{p}

is neglected. So Equations (11) and (12) are reduced to

\begin{matrix} σ_{11}^{p} = c_{11} ɛ_{11} + c_{12} ɛ_{22} - e_{31} E_{η}, σ_{22}^{p} = c_{12} ɛ_{11} \\ + c_{11} ɛ_{22} - e_{31} E_{η}, σ_{12}^{p} = c_{66} ɛ_{12}, σ_{23}^{p} = c_{44} ɛ_{23}, \end{matrix}

(13)

D_{2} = e_{15} ɛ_{23}, D_{3} = e_{31} (ɛ_{11} + ɛ_{22}) + \in_{33} E_{η} .

(14)

Hamilton’s principle with the assumed mode method will be used to determine the equation of motion of the conical shell and piezoelectric patches system. At first, the variational formulation for the structural system should be established. Hamilton’s principle is written by

\begin{matrix} \int_{t_{1}}^{t_{2}} δ (T - U) d t + \int_{t_{1}}^{t_{2}} δ W d t \\ = \int_{t_{1}}^{t_{2}} δ T d t - \int_{t_{1}}^{t_{2}} δ U d t + \int_{t_{1}}^{t_{2}} δ W d t = 0, \end{matrix}

(15)

where δ(·) denotes the first variation, T and U are the kinetic energy and potential energy, respectively, of the total conical shell and piezoelectric patches system, W is the work done by the external applied forces and t₁ and t₂ are the integration time limits.

The total kinetic energy of the conical shell and piezoelectric patches consists of the kinetic energies of the base shell and the piezoelectric patches. For thin shells, the influence of rotatory inertia can be neglected. So the total kinetic energy of the system can be written as

\begin{matrix} T = \frac{1}{2} \int_{V} ρ [(\frac{\partial u}{\partial t})^{2} + (\frac{\partial v}{\partial t})^{2} + (\frac{\partial w}{\partial t})^{2}] d V \\ + \sum_{i = 1}^{n_{p}} \frac{1}{2} \int_{V_{pi}} ρ_{p} [(\frac{\partial u}{\partial t})^{2} + (\frac{\partial v}{\partial t})^{2} + (\frac{\partial w}{\partial t})^{2}] d V, \end{matrix}

(16)

where ρ and ρ_p are the mass densities of the base shell and piezoelectric patches, respectively, V and V_p_i are the volumes of the conical shell and the ith piezoelectric patch, respectively, and n_p is the number of the piezoelectric patches. The infinitesimal volume dV is given by

d V = A_{1} A_{2} d ξ d ζ d η = ξ \sin α_{0} d ξ d ζ d η .

(17)

The potential energy of the total conical shell and piezoelectric patches includes the strain energy of the base conical shell and the strain energy and electric potential energy of the piezoelectric patches. The total potential energy can be written as

\begin{matrix} U = \frac{1}{2} \int_{V} (σ_{11} ɛ_{11} + σ_{22} ɛ_{22} + σ_{12} ɛ_{12} + σ_{23} ɛ_{23}) d V \\ + \sum_{i = 1}^{n_{p}} \frac{1}{2} \int_{V_{pi}} (σ_{11}^{p} ɛ_{11} + σ_{22}^{p} ɛ_{22} + σ_{12}^{p} ɛ_{12} + σ_{23}^{p} ɛ_{23}) d V \\ - \sum_{i = 1}^{n_{p}} \frac{1}{2} \int_{V_{pi}} D_{3} E_{η} d V . \end{matrix}

(18)

The work done by the external applied forces can be written as

δ W = \int_{A} (q_{1} δ u + q_{2} δ v + q_{3} δ w) d A,

(19)

where A is the surface area of the conical shell, and q₁, q₂ and q₃ are the distributed load components per unit area along the ξ, ζ, and η directions, respectively, and are assumed to act on the neutral surface of the shell. The units of q₁, q₂ and q₃ are [N/m²]. The infinitesimal area dA is given by

dA = A_{1} A_{2} d ξ d ζ = ξ \sin α_{0} d ξ d ζ .

(20)

In order to use the assumed mode method, the displacements u, v and w in the ξ, ζ and η directions, respectively, in the neutral surface of the conical shell should be expressed in terms of generalized coordinates

u (ξ, ζ, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} U_{ij} (ξ, ζ) p_{ij} (t) = U^{T} (ξ, ζ) p (t),

(21)

v (ξ, ζ, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} V_{ij} (ξ, ζ) r_{ij} (t) = V^{T} (ξ, ζ) r (t),

(22)

w (ξ, ζ, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} W_{ij} (ξ, ζ) s_{ij} (t) = W^{T} (ξ, ζ) s (t),

(23)

where p = [p₁₁, … , p₁ _n , p₂₁, … , p₂ _n , … , p_m₁, … , p_mn]^T, r = [r₁₁, … , r₁ _n , r₂₁, … , r₂ _n , … , r_m₁, … , r_mn]^T and s = [s₁₁, … , s₁ _n , s₂₁, … , s₂ _n , … , s_m₁, … , s_mn]^T are the generalized coordinates or modal coordinates of the shell and U = [U₁₁, … , U₁ _n , U₂₁, … , U₂ _n , … , U_m₁, … , U_mn]^T, V = [V₁₁, … , V₁ _n , V₂₁, … , V₂ _n , … , V_m₁, … , V_mn]^T and W = [W₁₁, … , W₁ _n , W₂₁, … , W₂ _n , … , W_m₁, … , W_mn]^T are the displacement shape functions or the principal vibration modes that must satisfy the geometric boundary conditions.

Then the kinetic energy, potential energy and work can be expressed in terms of the generalized coordinates and the displacement shape functions. Substituting Equations (17) and (21)–(23) into Equation (16) leads to the kinetic energy expression

\begin{matrix} T = \frac{1}{2} \frac{d p^{T}}{d t} M_{1} \frac{d p}{d t} + \frac{1}{2} \frac{d r^{T}}{d t} M_{2} \frac{d r}{d t} \\ + \frac{1}{2} \frac{d s^{T}}{d t} M_{3} \frac{d s}{d t}, \end{matrix}

(24)

where M ₁, M ₂ and M ₃ are the modal mass matrices of the conical shell and piezoelectric patches. The expressions of these mass matrices are listed in the Appendix.

Substituting Equations (8), (10), (13), (14), (17) and (21)–(23) into Equation (18), we can obtain the potential energy of the total electromechanical coupling system:

\begin{matrix} U = \frac{1}{2} p^{T} K_{1} p + \frac{1}{2} p^{T} K_{2} r + \frac{1}{2} p^{T} K_{3} s + \frac{1}{2} r^{T} K_{2}^{T} p \\ + \frac{1}{2} r^{T} K_{4} r + \frac{1}{2} r^{T} K_{5} s + \frac{1}{2} s^{T} K_{3}^{T} p + \frac{1}{2} s^{T} K_{5}^{T} r \\ + \frac{1}{2} s^{T} K_{6} s + V_{p}^{T} K_{7} p + V_{p}^{T} K_{8} r + V_{p}^{T} K_{9} s \\ + \frac{1}{2} \sum_{i = 1}^{n_{p}} K_{10}^{i} V_{0 i}^{2}, \end{matrix}

(25)

where

V_{p} = [V_{01}, V_{02}, \dots, V_{0 n_{p}}]^{T}

is the vector composed of the external applied voltages, K ₁, K ₂, … , K ₆ are the modal stiffness matrices of the conical shell and piezoelectric material system, K ₇, K ₈ and K ₉ are the electromechanical coupling matrices of the piezoelectric patches and

K_{10}^{i}

is the piezoelectric capacitance of the ith piezoelectric patch. These matrices are also presented in the Appendix.

Substituting Equations (20)–(23) into Equation (19), the virtual work is expressed as

δ W = q_{1} F_{q 1} δ p + q_{2} F_{q 2} δ r + q_{3} F_{q 3} δ s,

(26)

where F _q ₁, F _q ₂ and F _q ₃ are the forcing matrices of the base conical, and are also given in the Appendix.

Substituting Equations (24)–(26) into Equation (15) and performing the variation operation in terms of p (t), r (t), s (t) and V₀(t), the equations of motion of the conical shell and piezoelectric patches system are obtained as

M_{1} \frac{d^{2} p}{d t^{2}} + K_{1} p + K_{2} r + K_{3} s + K_{7}^{T} V_{p} = F_{q 1}^{T} q_{1},

(27)

M_{2} \frac{d^{2} r}{d t^{2}} + K_{2}^{T} p + K_{4} r + K_{5} s + K_{8}^{T} V_{p} = F_{q 2}^{T} q_{2},

(28)

M_{3} \frac{d^{2} s}{d t^{2}} + K_{3}^{T} p + K_{5}^{T} r + K_{6} s + K_{9}^{T} V_{p} = F_{q 3}^{T} q_{3},

(29)

K_{7} p + K_{8} r + K_{9} s = - K_{10} V_{p},

(30)

where K ₁₀ is a diagonal matrix composed of

K_{10}^{i}

(i = 1, 2, … , n_p).

Rearranging the generalized coordinates p (t), r (t) and s (t) as

X (t) = [\begin{matrix} p^{T} (t) & r^{T} (t) & s^{T} (t) \end{matrix}]^{T},

(31)

Equations (27)–(29) can be integrated as

M_{t} \frac{d^{2} X}{d t^{2}} + K_{1 t} X + K_{2 t} V_{p} = Q_{0},

(32)

where M _t , K ₁ _t , K ₂ _t and Q ₀ are the generalized mass matrix, stiffness matrix, electromechanical coupling matrix and forcing matrix of the whole system, respectively, and they can be written as

\begin{matrix} M_{t} = [\begin{matrix} M_{1} & 0 & 0 \\ 0 & M_{2} & 0 \\ 0 & 0 & M_{3} \end{matrix}], K_{1 t} = [\begin{matrix} K_{1} & K_{2} & K_{3} \\ K_{2}^{T} & K_{4} & K_{5} \\ K_{3}^{T} & K_{5}^{T} & K_{6} \end{matrix}], \\ K_{2 t} = [\begin{matrix} K_{7} & K_{8} & K_{9} \end{matrix}]^{T}, Q_{0} = [\begin{matrix} F_{q 1} q_{1} & F_{q 2} q_{2} & F_{q 3} q_{3} \end{matrix}]^{T} . \end{matrix}

(33)

From Equation (33) one can see that the matrices M _t and K ₁ _t are all symmetrical.

Equation (32) is the actuator equation that characterizes the piezoelectric actuators driven under the external applied voltages. It relates the external applied voltages to the structural deformation. Equation (30), called the sensor equation, relates the structural deformation with the voltage occurring across the piezoelectric sensor. Next we will use Equation (32) to study the active vibration control of the conical shell by negative velocity feedback and LQR methods.

3. Active vibration control schemes

3.1. Negative velocity feedback method

The piezoelectric constraining layers can be activated by appropriate external control voltage to obtain active damping and suppress structural vibration. To achieve this aim, a velocity feedback control strategy is applied. The velocity at position (ξ₀, ζ₀) of the conical shell is measured by a velocity sensor and fed back to the piezoelectric actuators as a control voltage with a control algorithm. The control voltages exerted to piezoelectric actuators are proportional to the velocity at the position (ξ₀, ζ₀). In this study, the transverse velocity of the conical shell is measured. So the control voltages of the n_p piezoelectric actuators can be expressed in terms of the transverse velocity at the position (ξ₀, ζ₀) of the conical shell as (Wang and Vaicaitis, 1998; Lee and Kim, 2001; Ray and Batra, 2007, 2008; Ray and Pradhan, 2008; Ray and Shivakumar, 2009):

V_{p} = - k \frac{d w (ξ_{0}, ζ_{0}, t)}{d t},

(34)

where

k = [k_{1}, k_{2}, \dots, k_{n_{p}}]^{T}

is the feedback control gain for the n_p piezoelectric actuators.

Substituting Equation (23) into Equation (34) leads to

V_{p} = - {kW}^{T} (ξ_{0}, ζ_{0}) \frac{d s}{d t} = - {kK}_{3 t} \frac{d X}{d t},

(35)

where the coefficient matrix K ₃ _t is written by

K_{3 t} = [\begin{matrix} 0 & 0 & W^{T} (ξ_{0}, ζ_{0}) \end{matrix}] .

(36)

Substituting Equation (35) into Equation (32) results in the following equation of motion with active damping:

M_{t} \frac{d^{2} X}{d t^{2}} + C_{p} \frac{d X}{d t} + K_{1 t} X = Q_{0},

(37)

where C _p is the active damping matrix due to the piezoelectric layers and is written as

C_{p} = - K_{2 t} {kK}_{3 t} .

(38)

It is seen from Equation (37) that the algorithm of velocity feedback control provides the active damping effect to control the structural vibration.

In order to solve Equation (37), we must present the formulations of the principal mode shapes U (ξ, ζ), V (ξ, ζ) and W (ξ, ζ) in Equations (21)–(23). Many people have studied the vibration of conical shells and some types of vibration mode shapes have been applied (Goldberg et al., 1960; Chang, 1978; Khatri and Asnani, 1995; Chai et al., 2006). For example, Liew et al. (2005) employed the kernel particle functions in hybridized form with harmonic functions to approximate the vibration modes of the conical shells. Li (2000) and Lam and Li (1999a) employed types of displacement formulations of conical shells that are similar to those of cylindrical shells (Soedel, 1981). Serpico (1963) presented the elastic buckling modes of conical shells that are the same as the vibration mode shapes of cylindrical shells. He stated that the buckling modes are satisfactory for truncated cones in which the ratio of the smallest diameter of the cone to the largest diameter is greater than one-quarter. In the present work, the vibration modes of conical shells similar to those used by Li (2000) and Lam and Li (1999a) are employed. The displacement fields are written as Equations (21)–(23).

In the curvilinear surface coordinates O-ξζη, as shown in Figure 1(a), the principal mode shapes of the conical shells with two S-S boundaries can be expressed as

U_{ij} (ξ, ζ) = cos [\frac{i π (ξ - l_{0})}{l - l_{0}}] cos (j ζ),

(39)

V_{ij} (ξ, ζ) = sin [\frac{i π (ξ - l_{0})}{l - l_{0}}] sin (j ζ),

(40)

\begin{matrix} W_{ij} (ξ, ζ) = sin [\frac{i π (ξ - l_{0})}{l - l_{0}}] cos (j ζ), \\ i = 1, 2, \dots, m; j = 1, 2, \dots, n, \end{matrix}

(41)

where i and j are the wave numbers in the meridional and circumferential directions. It should be noted that Equations (39)–(41) satisfy accurately the geometric boundary conditions but satisfy approximately the force boundary conditions for the S-S boundaries (Li et al., 2009). However, the mode shapes given by Equations (39)–(41) are simple in form and later on we can see from the numerical validation that the error is acceptable.

The distributed loads are assumed to be harmonic and are expressed as

\begin{matrix} q_{1} (ξ, ζ, t) = q_{0 1} sin ω t, q_{2} (ξ, ζ, t) = q_{0 2} sin ω t, \\ q_{3} (ξ, ζ, t) = q_{0 3} sin ω t, \end{matrix}

(42)

where q₀₁, q₀₂ and q₀₃ are the amplitudes and ω is the frequency of the dynamic loads. Under the application of the external dynamic loads, the steady-state solution of Equation (37) can be written as

X_{i} (t) = A_{i} \sin ω t + B_{i} \cos ω t, i = 1, 2, \dots, 3 mn,

(43)

where A_i and B_i are the amplitude values to be determined.

Substituting Equation (43) into Equation (37), one can obtain the following algebraic equations:

\begin{matrix} (K_{1 t} - ω^{2} M_{t}) A - ω C_{p} B = Q_{1}, \\ (K_{1 t} - ω^{2} M_{t}) B + ω C_{p} A = \underline{0}, \end{matrix}

(44)

where

A = [A_{1}, A_{2}, \dots, A_{3 mn}]^{T}

and

B = [B_{1}, B_{2}, \dots, B_{3 mn}]^{T}

are the column matrices that consist of A_i and B_i, and

Q_{1} = [\begin{matrix} F_{q 1} q_{0 1} & F_{q 2} q_{0 2} & F_{q 3} q_{0 3} \end{matrix}]^{T}

. By solving Equation (44) one can determine the unknown amplitudes A_i and B_i of X_i(t). Then, from Equations (21)–(23), (31), (39)–(41) and (43), we can finally obtain the steady-state responses of the conical shell.

3.2. LQR method

In this study, the LQR with the genetic algorithm is also used to optimally control the vibration of the conical shell. From Equation (32), the state-space equation of the structural system is written as

\overset{\cdot}{z} = Ez + {FV}_{p},

(45)

where

z = [X, \overset{\cdot}{X}]^{T}, E = [\begin{matrix} 0 & I \\ - M_{t}^{- 1} K_{1 t} & 0 \end{matrix}], F = {\begin{matrix} 0 \\ - M_{t}^{- 1} K_{2 t} \end{matrix}},

(46)

where I is the unit matrix.

For the LQR control, the feedback control is designed to minimize a cost function that is proportional to the required measure of the system’s response. The cost function is given by

J = \frac{1}{2} \int_{0}^{\infty} (z^{T} Qz + V_{p}^{T} {RV}_{p}) d t,

(47)

where Q and R are the semi-positive-definite and positive-definite weighting matrices on the outputs and control inputs. The control voltage is given by

V_{p} = - Gz = - R^{- 1} F^{T} Pz,

(48)

where G = − R ⁻¹ F ^T P is the feedback control gain, and P can be obtained by solving the Riccati equation:

\overset{\cdot}{P} + E^{T} P + PE - {PFR}^{- 1} F^{T} P + Q = 0 .

(49)

4. Numerical simulations and discussions

4.1. Validation of the present method

In order to validate this methodology, comparisons with the results available in the open literature are made. In the numerical calculations, the nondimensional frequency parameter is defined as (Irie et al., 1984; Lam and Li, 1999a; Liew et al., 2005)

f = ω_{0} a_{2} \sqrt{ρ (1 - μ^{2}) / E},

(50)

where ω₀ is the natural frequency of the conical shell in radians per second. The conical shell is aluminum whose modulus of elasticity E = 70 GPa, mass density ρ = 2710 kg/m³ and Poisson’s ratio μ = 0.3. The structural parameters are h = 0.004 m, h/a₂ = 0.01, (l – l₀)sinα₀/a₂ = 0.25. For the case of the meridional wave number m = 1 and semi-vertex cone angle α₀ = 30°, 45° and 60°, the frequency parameters calculated by Equation (50) are listed in Table 1 for different circumferential wave numbers. The corresponding results by Irie et al. (1984) and Lam and Li (1999a) are also listed in Table 1. From Table 1 we can see that the frequency parameters obtained by the present method are in good agreement with those in the open literature, which verifies the validity of the present analytical method.

Table 1.

Comparisons of frequency parameter f for the conical shell with simply supported boundaries (m = 1)

n	α₀ = 30°			α₀ = 45°			α₀ = 60
n	Irie et al. (1984)	Lam and Li (1999a)	Present	Irie et al. (1984)	Lam and Li (1999a)	Present	Irie et al. (1984)	Lam and Li (1999a)	Present
2	0.7910	0.8420	0.8431	0.6879	0.7655	0.7642	0.5722	0.6348	0.6342
3	0.7284	0.7376	0.7416	0.6973	0.7212	0.7211	0.6001	0.6238	0.6236
4	0.6352	0.6362	0.6419	0.6664	0.6739	0.6747	0.6054	0.6145	0.6146
5	0.5531	0.5528	0.5590	0.6304	0.6323	0.6336	0.6077	0.6111	0.6113
6	0.4949	0.4950	0.5008	0.6032	0.6035	0.6049	0.6159	0.6171	0.6172
7	0.4653	0.4661	0.4701	0.5918	0.5921	0.5928	0.6343	0.6350	0.6347
8	0.4645	0.4660	0.4687	0.5992	0.6001	0.6005	0.6650	0.6660	0.6653

4.2. The active vibration control with the velocity feedback method

The active vibration control for the conical shell with two S-S boundaries is studied using the velocity feedback control method. In this section, two piezoelectric patches, as shown in Figure 1, are considered. In the calculation, the feedback control gains of the two piezoelectric patches are set to be the same values. The structural passive damping is considered in the calculation by using the complex Young’s modulus E(1 + iη), where η = 1 × 10⁻⁷ is the structural loss factor. The base conical shell is aluminum whose modulus of elasticity E = 70 GPa, mass density ρ = 2710 kg/m³ and Poisson’s ratio μ = 0.3. The semi-vertex cone angle is α₀ = 30°. The position coordinates of the conical shell in the curvilinear surface coordinates O-ξζη are l = 0.8 m and l₀ = 0.6 m. So the length of the conical shell is s = l − l₀ = 0.2 m. The thickness of the conical shell is h = 0.004 m. The radii at the two ends are a₁ = 0.3 m and a₂ = 0.4 m. The piezoelectric layer is PZT-4 whose elastic constants c₁₁ = 139 GPa, c₁₂ = 77.8 GPa, c₄₄ = 25.6 GPa and c₆₆ = 30.6 GPa, mass density ρ_p = 7500 kg/m³, piezoelectric constant e₃₁ = −6.98 C/m² and dielectric constant ∈₃₃ = 54.7 × 10⁻¹⁰ F/m. The thickness of the piezoelectric layer is h_p = 0.002 m. In the analysis, the velocity sensors are located at the positions where the piezoelectric actuators are bonded. The amplitudes q₀₁, q₀₂ and q₀₃ of the external dynamic loads are set to be 1.0 MPa in the calculation.

First of all, the mode convergence is studied. Figures 2 and 3 show the uncontrolled and controlled frequency-response curves at the position P(l₀ + s/2, 0) of the shell under three different values of m and n. The two piezoelectric patches are bonded at ξ₁ = 0.64 m, ξ₂ = 0.76 m, ζ₁ = −10°, ζ₂ = 10° and ξ₁ = 0.64 m, ξ₂ = 0.76 m, ζ₁ = 170°, ζ₂ = 190°, respectively. It is observed that the two frequency-response curves for m = 3, n = 4 and m = 4, n = 4 in both the figures are almost superposed, which indicates that the mode is considered to be convergent when m and n are chosen to be 3 and 4, respectively.

In order to assess the controller performance, Figure 4 shows the kinetic energy and elastic potential energy (i.e. the strain energy) of the base conical shell with reference to the control gain and frequency. The structural size of the two piezoelectric patches expressed by the curvilinear surface coordinates O-ξζη shown in Figure 1 is ξ₁ = 0.66 m, ξ₂ = 0.74 m, ζ₁ = −7°, ζ₂ = 7°. The two piezoelectric patches are placed 180° apart from each other on the outer surface of the conical shell, as shown in Figure 1, and they are located at the middle positions of the conical shell length, that is, at the positions (l₀ + s/2, 0°) and (l₀ + s/2, 180°). It can be seen from the figure that both the total kinetic energy and strain energy of the conical shell are reduced significantly after the velocity feedback control is applied on the structural system, which indicates that the overall vibration of the conical shell can be effectively reduced by the velocity feedback control method. It is also noted that the two energies vary with the control gain following the ‘inverted bell shape’ type of graph, which is a similar phenomenon to that of the literatures of Gardonio and Elliott (2004) and Zilletti et al. (2010). That is to say, it is not a linear relationship between the energy and control gain.

In Figure 5, the kinetic and strain energies of the conical shell varying with the frequency of external dynamic load under different feedback control gains are displayed. It is clearly seen from Figure 5(a) that the overall reductions in the total kinetic energies are 17.58, 28.67, 42.92 and 32.31 dB for the first resonant frequency when the control gains are k = 3 × 10², 1 × 10³, 1 × 10⁴ and 5 × 10⁴ compared with the uncontrolled case (k = 0). For higher order resonances, better control results can also be obtained. Moreover, we can see from Figure 5(a) that when the control gain k is below a certain value (say 1 × 10⁴ in this case), the larger the control gain is, the greater the reduction in the total kinetic energy. However, when the control gain k is larger than this value, the peak values of the kinetic energy will become larger (but they are still lower than that of the uncontrolled case), which is because the kinetic energy varies with the control gain following the ‘inverted bell shape’ type of graph (Gardonio and Elliott, 2004; Zilletti et al., 2010). In Figure 5(b), similar phenomena to those in Figure 5(a) can be observed.

Figures 6 and 7 show the frequency-response curves of the first nine resonant responses at the position P(l₀+s/2, 0) of the conical shell, as shown in Figure 1, and the corresponding required voltages to control the first nine resonant vibration amplitudes. The structural sizes of the two piezoelectric patches are the same as those of Figure 4. Figure 6 shows the transverse vibration amplitudes of the first nine modes at the position P(l₀ + s/2, 0) of the conical shell as the function of frequency ω of external dynamic loads for the control gain k = 0, 2 × 10⁵ and 5 × 10⁵. From Figure 6 we see that for the uncontrolled conical shell (k = 0) there are some peak values of resonant responses. For the actively controlled cases, however, with the increase of the control gain k the amplitudes of the resonant responses of the conical shell reduce seriously. The larger the control gain is, the lower the amplitudes of the resonant responses are, which clearly reveals that the piezoelectric patches can significantly improve the active damping properties of the conical shell. It is also noted that the largest reduction of vibration for the controlled conical shell in the low- and high-frequency regions are 24.8 and 7.55 dB, respectively, from which a conclusion can be drawn that the active controls are more effective to suppress the vibration responses in low-frequency regions.

The required control voltages corresponding to Figure 6 are calculated by Equation (35) and are shown in Figure 7. In Figure 7, the control gains are k = 2 × 10⁵ and 5 × 10⁵. It is seen from Figure 7 that the maximum control voltage used for controlling the first nine resonant responses is 42.13 V, which corresponds to the fourth resonant response near frequency ω = 2127 Hz under the control gain k = 5 × 10⁵. For the first resonant response, however, the required control voltage is about 8.53 V, which is much lower than half of that for the fourth resonant response.

Figure 6 only shows the vibration control effect at the point where the piezoelectric patch is bonded. However, to assess the performance of the control scheme, the vibration control effects at other points of the conical shell should also be analyzed. Because of the symmetric structural system, that is, the symmetric boundary condition as well as the symmetric loads, the vibration control effects at other six points, that is, the positions (l₀ + s/2, 30°) and (l₀ + s/2, 60°) as well as the points (l₀ + s/2, 45°), (l₀ + s/2, 90°), (l₀ + s/4, 0°) and (l₀ + 3 s/4, 0°), which are investigated and shown in Figures 8 and 9. It can be seen from all these figures that when the velocity feedback control is applied, the amplitudes of the transverse displacement are reduced, although the control effects are not so good as those for the point where the piezoelectric patch is bonded, which also verifies that the vibration control scheme is feasible to the entire conical shell.

Figure 10 also shows the kinetic energy and strain energy of the conical shell with reference to the control gain and frequency to assess the controller performance, but the size of the two piezoelectric patches is increased. It is ξ₁ = 0.64 m, ξ₂ = 0.76 m, ζ₁ = −10°, ζ₂ = 10°, which are also expressed by the curvilinear surface coordinates O-ξζη shown in Figure 1. The two piezoelectric patches are also placed 180° apart from each other on the outer surface of the conical shell. The structural loss factor is also chosen as η = 1 × 10⁻⁷. We can draw the same conclusion as that from Figure 4, that is, the velocity feedback control method can effectively reduce the overall vibration of the conical shell.

Like Figure 5, Figure 11 shows the kinetic and strain energies of the conical shell varying with the frequency of external dynamic load under different control gains. It is noted that the overall reductions in the total kinetic energies are 22.23, 32.89, 40.14 and 28.74 dB for the first resonant frequency when the control gains are k = 3 × 10², k = 1 × 10³, k = 1 × 10⁴ and k = 5 × 10⁴, respectively, compared with the uncontrolled case (k = 0). For higher order resonances, better control results are also achieved. These facts verify that the active control scheme is efficient. Furthermore, similar conclusions to those from Figure 5 can also be drawn from Figure 11.

For Figures 12 and 13, the size of the two piezoelectric patches is also increased, and is the same as that of Figure 10. In this case, the control gain k is set to be smaller than that of Figures 6 and 7, which is k = 1.5 × 10⁵ and 4 × 10⁵. The transverse vibration amplitudes of the first nine modes at the position P(l₀ + s/2, 0) of the conical shell varying with the frequency ω of external dynamic loads are shown in Figure 12.

Comparing Figure 12 with Figure 6, it is seen that, for the larger structural size of the two piezoelectric patches, good vibration control results can be achieved even for smaller control gains. In Figure 13, the required voltages to control the resonant responses corresponding to Figure12 are shown. It is seen from Figure 13 that the required control voltages corresponding to the fifth and sixth resonant responses are much larger than those of the first and second resonant responses, which means that it is more economic to suppress the vibration level in low-frequency regions by means of the active vibration control method.

In order to evaluate the performance of the vibration control scheme to the entire conical shell with two larger piezoelectric patches, the uncontrolled and controlled frequency-response curves for other six points, that is, for the position (l₀ + s/2, 30°), (l₀ + s/2, 45°), (l₀ + s/2, 60°), (l₀ + s/2, 90°), (l₀ + s/4, 0°) and (l₀ + 3 s/4, 0°) are shown in Figure 14. It is noted from Figure 14 that the response amplitudes of all these points are reduced when the control scheme is applied even for smaller control gains, which verifies the effectiveness of the vibration control scheme on the entire conical shell.

In order to study the influences of the structural size of the piezoelectric patches on the vibration control effect, the comparisons of the controlled frequency-response curves under four different sizes of the piezoelectric patches are made and are shown in Figure 15. The four different piezoelectric patch sizes are taken as ξ₁ = 0.66 m, ξ₂ = 0.74 m, ζ₁ = −7°, ζ₂ = 7°; ξ₁ = 0.64 m, ξ₂ = 0.76 m, ζ₁ = −10°, ζ₂ = 10°; ξ₁ = 0.62 m, ξ₂ = 0.78 m, ζ₁ = −20°, ζ₂ = 20° and ξ₁ = 0.6 m, ξ₂ = 0.8 m, ζ₁ = −45°, ζ₂ = 45°. It is seen from Figure 15 that with the increase of the size of the piezoelectric patches, the resonant amplitudes of the vibration response change slightly, except that the locations of the resonant points move obviously, but the responses at other frequency regions (except for the resonant frequencies) reduce seriously. Figure 16 shows the control voltages corresponding to Figure 15. It is seen that the required voltages to control the resonant responses do not change much with the size of the two piezoelectric patches increasing.

4.3. The optimal active vibration control with the genetic algorithm and LQR method

To study the optimal placement of the piezoelectric patches, the genetic algorithm is used to make the optimal design. The conical shell is divided into 64 elements, numbered from 1 to 64, as shown in Figure 17. Each piezoelectric patch is bonded on one element and the numbers 1, 2, … , 64 are coded as a string of binary bits corresponding to the chromosomes in natural genetics. The total number of piezoelectric patches used in vibration control is n_p.

In this study, the free vibration responses of the conical shell and piezoelectric material system are calculated by using the finite difference method. The initial transverse displacement at the position P(l₀ + s/2, 0) is set to be 3.0 × 10⁻⁶ m. The geometry size of the conical shell and the material parameters of the conical shell and piezoelectric patches are the same as those of Figure 6. The structural loss factor is taken as η = 1 × 10⁻¹¹, and the number of the piezoelectric patches is set to be n_p = 8. The optimal locations of the eight piezoelectric patches obtained by the genetic algorithm are shown in Figure 18. The other two arbitrarily chosen locations of the eight piezoelectric patches are also displayed in Figure 18.

Figure 19 shows the uncontrolled and controlled free vibration responses at the position P(l₀ + s/2, 0) of the conical shell for the piezoelectric patches locating at the optimal and arbitrary positions corresponding to Figure 18. It can be seen from the figure that the LQR control algorithm can effectively reduce the amplitudes of the free vibration response for the conical shell. It is also noted that for the structural system with the optimal placements of the piezoelectric patches, the maximum response amplitude is lower than that for the nonoptimal system, which indicates that the optimal location design of the piezoelectric patches using the genetic algorithm is feasible.

5. Conclusions

The active vibration control of conical shells is investigated using a negative velocity feedback and the LQR methods. The genetic algorithm is applied to design the optimal placements of the piezoelectric patches, in order to obtain the simple equation of motion of the conical shells so as to solve the forced vibration responses analytically, Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the conical shell with the piezoelectric patches system. A set of simple expressions of the principal mode shapes is presented and numerically validated. The influences of several parameters on the active vibration control are considered. From the numerical results, the following conclusions can be drawn.

For the conical shell with two piezoelectric patches, the control gain has significant influences on the active vibration control, but the effects of the size of the piezoelectric patches on controlling the vibration amplitudes are not as obvious as the control gain. With the increase of the feedback control gain, the control results are pronouncedly improved.

It is more economic to suppress the vibration level in low-frequency regions by means of the active vibration control method.

The overall vibration reduction of the conical shell can be effectively achieved using the velocity feedback control method.

The optimal placement scheme of the piezoelectric patches obtained by the genetic algorithm is efficient in the active vibration control of the conical shell.

Footnotes

Funding

This work was supported by the National Basic Research Program of China (No. 2011CB711102) and the National Natural Science Foundation of China (No. 10672017, 11172084).

Appendix

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