Multimodal suppression of vibration in smart flexible beam using piezoelectric electrode-based switching control

Mechatronics 53 (2018) 152–167

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Multimodal suppression of vibration in smart flexible beam using piezoelectric electrode-based switching control☆

T

⁎

Lin Chi-Ying , Huang Yu-Hsi, Chen Wei-Ting Department of Mechanical Engineering, National Taiwan University of Science and Technology, 43 Keelung Road, Section 4, Taipei 10672, Taiwan

A R T I C LE I N FO

A B S T R A C T

Keywords: Piezoelectric actuator Flexible structure Multimodal vibration suppression Electrode configuration Switching control

This paper describes an active vibration control system combining a piezoelectric electrode configured with a novel switching control system to achieve multimodal vibration suppression in smart flexible structures embedded with only a single piezoelectric actuator. The fact that high-frequency vibrations attenuate more rapidly than do low frequency vibrations motivated the development of a novel scheme in which the resonant excitation of coupled bending and lateral vibrations is applied to a cantilever beam. Switching times are determined using time-frequency analysis of modal responses. A smoothing function was applied to alleviate the effects of highfrequency oscillations during the switching control transition in order to facilitate the suppression of bending mode vibrations. Algorithms for proportional derivative (PD) control and positive position feedback (PPF) control can be tailored specifically for the target vibration modes and implemented in a variety of switching control schemes. The vibration responses induced by various switching control schemes was analyzed in order to derive appropriate control parameters for the development of an efficient switching control system aimed at suppressing multimodal vibrations. Experiment results demonstrate the effectiveness of the proposed piezoelectric electrode configuration in suppressing lateral mode vibrations. The proposed switching control system also proved highly satisfactory in suppressing vibrations in a system subject to simultaneous bending mode excitation.

1. Introduction Piezoelectric sensors and actuators are widely applied in smart materials and structures for active vibration control, thanks to their compactness, responsiveness, high electromechanical efficiency, and high precision [1]. The core concept behind this type of mechatronic system is the integration of structures, actuators, and sensors as a feedback control system. Suitable control inputs are calculated using vibration control algorithms with the aim of suppressing the occurrence of unwanted vibrations in flexible structure systems [2–5]. From the perspective of mechatronics, previous research related to the active vibration control of flexible structures using piezoelectric materials falls into two general categories: (1) control algorithm development [6–13] and (2) the optimal placement of sensors and actuators [14–18]. The first category includes the proportional-derivative (PD) and positive-position-feedback (PPF) control methods in smart structures [6–8]. Advanced control methods, such as robust and adaptive controls, are used to compensate for uncertainties in modeling and high-frequency dynamics in various systems involving a flexible structure [9–13]. When a structure is suitable for the placement of ☆ ⁎

This paper was recommended for publication by Associate Editor Prof. T.H. Lee. Corresponding author. E-mail address: [email protected] (C.-Y. Lin).

https://doi.org/10.1016/j.mechatronics.2018.06.005 Received 26 February 2018; Received in revised form 9 May 2018; Accepted 12 June 2018 0957-4158/ © 2018 Elsevier Ltd. All rights reserved.

piezoelectric actuators at a fixed end, such as cantilever beams and Ibeams [6,9–13], the first category of methods can be used to control low-frequency modes. When a structure involves a complex combination of mode shapes or when the primary focus is suppressing vibrations pertaining to high frequency modes, the second category of methods can be used to determine the optimal placement of sensors and actuators using theoretical finite element modal analysis and/or through experimental investigations into the vibration characteristics of the structures [14–18]. Both of these are well-accepted approaches to promoting the efficiency of actuators and suppressing vibrations. Flexible structures are infinite-dimensional low-damped mechanical systems with vibration responses comprising multiple vibration modes, including bending modes, torsional modes, lateral modes, and longitudinal modes [19,20]. The bending modes that dominate low-frequency vibrations and most other non-bending vibration modes fall within the category of high-frequency modes. Applying an electric field to the surface of a standard piezoelectric actuator causes a contraction in the active layer and expansion in the other layer [21–23], i.e., the piezoelectric actuator bends. A piezoelectric actuator with this configuration presents the highest excitation efficiency in bending modes. The

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multimodal vibrations. (2) We derived a mathematical model for smart flexible beams integrated with a piezoelectric actuator in an electrode configuration with the aim of suppressing lateral mode vibrations. (3) We conducted a series of experiments to analyze the design parameters and evaluate the proposed switching control scheme in terms of vibration suppression performance.

inherent operating principle behind the use of a single piezoelectric actuator limits their applicability to the suppression of bending mode vibrations in flexible structures. This makes it necessary to employ algorithms to control coupled vibrations that include non-bending modes [24,25]. One common strategy for the suppression of multi mode vibrations in smart flexible structures involves applying multiple pairs of actuators and sensors to achieve active vibration control [26–28]. The core concept behind this approach is the same as that of the second category mentioned above; i.e., the proper placement of actuators to achieve optimal vibration suppression performance. Finite element method can be used to analyze the distribution of strain in structures associated with various vibration modes. It can also be used to facilitate the placement of multiple actuators/sensors in order to suppress specific vibration responses [29,30]. This is a feasible strategy when seeking to suppress vibrations in high-frequency modes; however, the need to formulate a multi-input/multi-output control system [31–35] that involves cross coupled dynamics and uncertainty modeling [36,37] can greatly hinder the development of affordable vibration suppression system. It may also be difficult to predict specific vibration modes when using multiple actuators and sensors, due to the indeterminate effects of glue on the mass of the structures and stiffness of the system. Thus, the current study focuses on the suppression of multimodal vibration in smart flexible structures using a single piezoelectric actuator. The problem of suppressing non-bending mode vibrations using a single piezoelectric actuator has been dealt with by adjusting the configuration of surface electrodes to control the efficiency of mode excitation [38]. Previous studies [39,40] have investigated the vibration characteristics of thin piezoelectric resonators, and applied the dynamic electromechanical coupling factor to evaluate the efficiency of four electrode configurations. Modal excitation experiments have revealed improvements in the efficiency of mode excitation when using welldesigned electrode configurations and revealed the potential benefits that could be garnered by increasing the mode excitation efficiency of non-bending modes using a single piezoelectric actuator. In [38], the authors first used electrode configuration to suppress lateral mode vibrations in a smart cantilever beam. Their experiment results demonstrated that an appropriate electrode configuration within an active vibration control system can improve mode excitation efficiency in the first lateral mode. They also showed that PD and PPF control algorithms could improve control over the suppression of lateral vibration. However, the vibration control method proposed in [38] is applicable only to suppressing modal vibrations corresponding to the electrode configuration (i.e., lateral mode vibrations); i.e., it is inapplicable to the suppression of bending mode vibrations and complex coupled vibrations. This paper presents an active control method for multimodal vibrations in which the electrode configurations and corresponding control algorithms are controlled using a novel switching control system. Finite element software is first used to analyze the distribution of strain in a cantilever beam in order to obtain an electrode configuration capable of controlling the second bending mode and the first lateral mode of the structure. To enable fast transient responses, the switching time is determined using time-frequency analysis of modal responses, and a smoothing function is applied to reduce the effects of high frequency oscillations during the transition. The system design includes the control parameters of the PD and PPF algorithms for the target vibration modes. The results of multimodal vibration suppression were evaluated using the proposed switching control system with numerous variations in design parameters. Experiments demonstrate the efficacy of the proposed switching control system in suppressing coupled bending and lateral vibrations in a flexible structure. The contributions of this work are summarized in the following.

The remainder of this paper is organized as follows. Section 2 details the vibration mode shapes described above as well as the placement of piezoelectric actuators and sensors based on our analysis of active vibration control in a cantilever beam. Section 3 reviews the theories used to guide the configuration of electrodes in piezoelectric resonators, and describes the electrode configurations applied in this study to control the coupled vibration modes based on finite element modal analysis. Section 4 presents the apparatus used for electrode switching control with a detailed discussion on selecting design parameters. That section also presents a performance comparison in the suppression of multimodal vibration, which was conducted using a range of experiments. Finally, concluding remarks and future directions in the formulation of electrode configurations for multimodal active vibration control designs are described at the end of the paper. 2. Smart flexible beam This section first describes the use of finite element-based modal analysis to guide the placement of piezoelectric sensors and actuators along a flexible beam fixed at one end. A dynamic model of the piezoelectric beam [41] is presented before discussing the electrode configuration in order to facilitate subsequent analysis of the proposed electrode switching control system. 2.1. Modal analysis in placement of piezoelectric sensors and actuators Modal analysis was conducted using finite element software, ANSYS, to identify the appropriate locations for the placement of piezoelectric sensors and actuators [14]. The rule of thumb here is to examine the distribution of nodal lines and strain throughout the structure. The beam in this study was aluminum, with a length of 450 mm, width of 45 mm, and thickness of 1 mm. In all theoretical analysis and finite element calculations, the material property were as follows: density of 2714 kg/m3, Young's modulus of 71.41 GPa, and Poisson's ratio of 0.359. The mode shapes of a flexible structure generally comprise various vibration modes. For illustration purposes, Fig. 1 presents the mode shapes and vibration-displacement contours of the four vibration modes: first bending mode, second bending mode, first torsional mode, and first lateral mode. The solid lines indicate the contours of displacements along the y-axis, and the regions on both sides of the solid lines represent anti-phased (convex and concave) displacements. The boldest solid lines indicate the “nodal lines” mentioned above as the region with zero displacement. In the case of the bending modes in Fig. 1(a) and (b), deformation occurred along y-axis and is denoted as wy. Fig. 1(c) presents the first torsional mode of the beam that vibrated harmonically as the twist angle ϕx. The deflection of the first lateral mode is along the z-axis as shown in Fig. 1(d) and is denoted as wz (i.e., “laterally bending mode” along the x-y plane). The principle of piezoelectric sensors [42] implies that more strain produces more electric energy as an output signal. The nodal lines in the strain field are the same as those in the displacement field on the center line of the minor axis (boldest solid lines in Fig. 1(c) and (d)). This means that placing sensors in regions distal to the nodal lines is suitable for the feedback control of these four modes in smart structures. Thus, Fig. 1(a) and (b) indicate that vibration signals of the bending modes are accessible in all regions except in areas of minimum strain, i.e., free end of the beam. In cases of non-bending modes (Fig. 1(c) and (d)), clear nodal lines are concentrated in the central section of the beam. To obtain reliable

(1) We developed a piezoelectric electrode configuration with a novel time-based switching control system for the suppression of 153

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Fig. 1. Modal analysis of flexible beam with fixed end: (a) deformation at first bending mode (4 Hz); (b) deformation at second bending mode (26 Hz); (c) deformation at first torsional mode (78 Hz); (d) deformation at first lateral mode (183 Hz).

Fig. 2. Placement of sensor and actuator enabling active vibration control in flexible beam (not to scale).

feedback pertaining to these vibration modes, a polyvinylidene fluoride (PVDF) film sensor was glued to the upper half of the beam, shown in Fig. 2. A piezoelectric patch transducer on a scale similar to the dimensions of the beam was placed at the fixed end of the beam to enable efficient actuation and facilitate the implementation of subsequent electrode configurations. Our focus in this study was on the placement of piezoelectric patches (Fig. 2) with the aim of suppressing composite vibrations, including bending mode vibrations and lateral mode vibrations. We also developed a switching control scheme using a segmented electrode configuration to provide effective control over multimodal vibrations. In the next section, we review results based on conventional models used in the suppression of bending mode vibrations. A model applicable to the suppression of lateral mode vibrations is presented in Section 3.3.

Eb Ib

∂ 4wy (x , t ) ∂x 4

+ ρAb

∂2wy (x , t ) ∂t 2

=

∂2 MT (x , t ) ∂x 2

(1)

where MT(x,t) is the bending moment acting on the beam, wy(x,t), ρ, Ab, Eb, and Ib represent the transverse deflection function, linear mass density, cross-sectional area, Young's modulus, and moment of inertia of the beam, respectively. Assume that the piezoelectric patch is thin enough and consider the homogeneous Eq. (1). Applying the procedure of standard modal analysis produces the beam deflection function wy(x,t) as an infinite series with the following form: ∞

wy (x , t ) =

∑ ϕT i (x ) qTi (t )

(2)

i=1

where ϕTi(x) and qTi(t) represent the modal shape function and modal deflection function along transverse direction. Substituting (2) into (1) and using the properties of orthogonality, the original PDE (1) transforms into a set of ordinary differential equations (ODEs) integrated with damping terms [43], as follows:

2.2. Modeling of smart flexible beam without electrode configuration According to one-dimensional Bernoulli-Euler beam theory [2], if transverse force is applied to the beam, then the equation of motion of the beam could be derived as a partial differential equation (PDE), as follows:

q¨Ti (t ) + 2ξTi ωTi q˙Ti (t ) + ωTi2 qTi (t ) =

∫0

L

∂2 MT (x , t ) ϕTi (x ) dx ∂x 2

(3)

where L is the length of the beam. The force moment generated by the piezoelectric actuator can be expressed as the product of voltage 154

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where σx, σy, σz, τxz, τyz, τxy represent components of the mechanical E E E , c12 , c13E , c33 , stress tensor; Dx, Dy, Dz represent electric displacements; c11 E c44 , represent elastic stiffness values under a constant electric field; e21, S S , ɛ 22 are dielectric permite22, e16 stand for piezoelectric constants; ɛ11 tivity values under constant strain; and ɛx, ɛy, ɛz, γxy, γxz, γyz denote components of the mechanical strain tensor. The piezoelectric ceramic (model number: APC-855) used in this study was manufactured by American Piezo Ceramic (APC) International, Ltd., the material constants of which can be found in [39]. Under steady-state vibration, electric field E can be represented using electrostatic potential φ(x, y, z), as follows:

function Va(t) and constant Kf in accordance with the properties of the beam and the attached piezoelectric patches [44]. Applying Dirac's delta function, the ODEs now become

q¨Ti (t ) + 2ξTi ωTi q˙Ti (t ) + ωTi2 qTi (t ) =

Kf =

Kf Va (t ) ρAb

[ϕ′Ti (x2) − ϕ′Ti (x1)]

6d31 Ib Eb2 Ea tb (tb + ta) Eb2 tb4 + Ea Eb (4tb3 ta + 6tb2 ta2 + 4tb ta3) + Ea2 ta4

(4)

(5)

where x1 and x2 indicate the locations on both sides of the piezoelectric patch, and tb is the thickness of the beam. d31, ta, Ea represent the piezoelectric constant, the thickness, and Young's modulus of the piezoelectric patch. The transfer function GT(s) from the applied actuator voltage Va(s) to the transverse deflection of the beam qT(s) can be expressed as follows:

GT (s ) =

qT (s ) = Va (s )

n

∑ i=1

PT s 2 + 2ξTi ωTi + ωTi2

Ex = −

∞

Kf

∫ ρAb

[ϕ′Ti (xk + 1) − ϕ′Ti (xk )] e−st dt

0

(9)

Assume that the plate is negligibly thin, such that planar stress can be used to analyze the derivation. Under this assumption, mechanical contour vibration in the plate can be simplified as two-dimensional movement. The averaged stresses and displacements of the plate can be written as follows:

(6)

where

PT =

∂φ ∂φ ∂φ , Ey = − , Ez = − ∂z ∂y ∂x

1 2h 1 {σx , σz , τxz }(x , z ) = 2h {u x , uz }(x , z ) =

(7)

where s stands for the Laplace variable. In this study, GT(s) represents a smart flexible structure without any electrode configurations and with the specific aim of suppressing bending mode vibrations.

h

∫−h {ux , uz}(x, y, z ) dy h

∫−h {σx, σz, τxz}(x, y, z ) dy

(10)

where h represents the thickness of the piezoelectric plate. Using (9) and the Cauchy strain-displacement relation, the complete integral of averaged stresses can be derived as follows:

3. Electrode configuration of piezoelectric plates This section reviews the theory underlying the electrode configurations applied to the thin rectangular piezoelectric plate [39] in this study. The procedure used to derive a suitable electrode design in accordance with its corresponding resonant mode (e.g., the first lateral mode) is illustrated using finite element-based modal analysis in the following section. Also presented is a simplified dynamic model of the smart flexible beam integrated with this electrode configuration, which is used to clarify the various dynamics involved in the switching vibration control system.

σx = c11E

∂ux ∂x

+ c12E

σz = c13E

∂ux ∂x

+ c12E

τxz =

E−cE c11 13

2

(

∂ux ∂z

u y+ − u y− 2h u y+ − u y− 2h ∂uz ∂x

+

+ c13E

∂uz ∂z

+ e21 2h

+ c11E

∂uz ∂z

+ e21 2h

Φ

Φ

)

(11)

and respectively denote where Φ(x, z) = φ(x, h, z) – φ(x, -h, z), the displacement values at the electrodes on the top and bottom of the piezoelectric plate. According to thin plate theory, stresses at free faces y = ± h are negligible; i.e., σy = 0, τyz = 0, τyx = 0. Thus, the averaged stresses in the x and z directions can be rewritten as

u y+

u y−

3.1. Theory of rectangular piezoelectric plates with electrode configurations

σx = (c11E − c13E )

+ (c13E + cc12E ) θ + (e21 + ce22) 2h

Detailed theoretical analysis of contour vibrations is presented in [39]. In that study, the mode shapes and efficiency of mode excitation for piezoelectric plate resonators with segmented electrodes was confirmed through an experimental investigation into vibration characteristics, as measured using optical interferometry and impedance analysis. Note that a verified theory governing the combination of flexible structures with segmented electrodes for piezoelectric plates has yet to be established. Thus, we adopted the dynamics of piezoelectric resonators in our electrode design to illustrate how piezoelectric electrode configurations can be manipulated to optimize vibration suppression in flexible structures. Consider a rectangular piezoelectric plate with the following parameters [39]:

σz = (c11E −

+ (c13E + cc12E ) θ + (e21 + ce22) 2h

∂ux ∂x ∂u c13E ) ∂zz

Φ

Φ

E c12

where c = − c E , θ = 22

∂ux ∂x

+

∂uz . ∂z

(12)

According to the principle of the con-

servation of momentum, the equation used to describe the motion of the plate can be obtained as follows: ∂σx ∂x ∂τxz ∂x

+ +

∂τxz ∂z ∂σz ∂z

+ ρω2u x = 0 + ρω2uz = 0

(13)

where ρ is the material density of the piezoelectric plate and ω is the resonant frequency. Suppose that the length of the plate along the lateral surface is L. The mechanically free boundary conditions at L satisfy the following:

σx = c11E ɛx + c12E ɛ y + c13E ɛ z − e21 Ey E σy = c12E (ɛx + ɛ z) + c22 ɛ y − e22 Ey

σx = 0, σz = 0, τxz = 0atL

σz = c13E ɛx + c12E ɛ y + c11E ɛ z − e21 Ey

The mechanical contour vibrations in a rectangular plate resonator are governed by the boundary-value problem (13) and (14) at Φ = 0 [39]. In this case, rewriting (13) with displacement variables u x̂ and uẑ gives us the following:

E τxy = c44 γxy − e16 Ex E τyz = c44 γyz − e16 Ez

1 E (c11 − c13E ) γxz 2 S = ɛ11 Ex + e15 γxy

τxz = Dx

∂2ux̂ ∂x 2 ∂2uẑ ∂x 2

S Dy = ɛ 22 Ey + e21 (ɛx + ɛ z) + e22 ɛ y

Dz =

S ɛ11 Ez

+ e15 γyz

(8)

+

∂2ux̂ ∂z 2

+

m ∂θ ̂ m − 2 ∂x

+

2ρ 2 E − c E ) ω u x̂ (c11 13

=0

+

∂2uẑ ∂z 2

+

m ∂θ ̂ m − 2 ∂z

+

2ρ 2 E − c E ) ω u ẑ (c11 13

=0

where m = 155

(14)

E + cc E c11 12 E + cc E c13 12

+ 1 and θ ̂ =

∂ux̂ ∂x

+

∂uẑ . ∂z

(15) The boundary conditions on

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free lateral surfaces (x = ± a, y = ± b, Φ = 0) are obtained as follows:

σx̂ = 0, σẑ = 0, τxẑ = 0

with slit electrodes. For the sake of comparsion, the displacement functions S1(x,y) and S3(x,y) of these two electrode configurations are summarized as follows:

(16)

S1 =

The superposition method can be used to obtain an analytic solution to (15) and (16), where displacements u x̂ and uẑ can be represented as sums of two ordinary Fourier series using complete sets of trigonometric functions with x and z coordinates.

FE: S3 =

∞

uẑ (x , z ) =

+

∞ b ∑n = 0

UDE:

(2n + 1) π nπ orαn = , n = 0, 1, ⋯ a 2a (2k + 1) π kπ βk = orβk = , k = 0, 1, ⋯ b 2b

(18)

When Φ = 0, the boundary conditions (16) can be written as expressions of u x̂ and uẑ :

σẑ = (c11E

E 13

1

∂uẑ ∂z

τxẑ = 2 (c11E − c13E )

(

+ +

∂ux̂ ∂z

θ̂ m−2 θ̂ m−2

+

∂uẑ ∂x

), ), (19)

∞

(σx̂ + σẑ ) = (c11E − c13E )Ω20 b ∑n = 0 (−1)nXn gn βk2 + q22 2q1

αn2 + p22 2p1

·S1 (x , z )

·S3 (x , z )

(20)

⎧ 2 ,k=0 ⎧ 2 ,n=0 ⎪ Ω22 ⎪ Ω22 fk = gn = 1 ⎨ ,k>0 ⎨ 1 ⎪α2, n > 0 ⎪ β2 ⎩ n ⎩ k where Ω1 =

ω ,Ω2 c1

=

ω , c2

Ω20 =

(24)

A vibrating piezoelectric resonator produces expanding and shrinking strain under electrode configuration. To determine which mode shape can be efficiently excited and which electrode configuration should be applied for vibration suppression control, we conducted ANSYS-based modal analysis to obtain the strain distribution of a smart flexible beam bonded with a rectangular piezoelectric plate. The influence of the sensor was not included in the analysis, as the mass was negligible. Piezoelectric vibration is induced by the stretching effects of resisting the extension from electric dipole moments through the application of an electric field to the surface of a piezoelectric material. Previous studies [39] have shown that electrode configuration produces contour vibrations different from those obtained using a full electrode configuration. Enhanced mode excitation efficiency has been justified when the nodal lines are consistent with the slit lines on electrode surfaces. These facts can be used to facilitate the design of electrode configurations aimed at suppressing coupled vibrations of bending modes and non-bending modes. For illustration, Fig. 3 presents strain modal analysis results of the first bending mode, second bending mode, first torsional mode, and first lateral mode. Similar to the modal analysis results in the displacement field (Fig. 1), the warm-colored and cold-colored regions represent the anti-phased (convex and concave) strains, whereas the transition region (green color) between those two regions indicates the strain nodal lines. Note that the stiffness of the piezoelectric transducer imposes a slight influence on the natural frequencies and mode shapes of the structure. It is clear from Fig. 3(a) that the strain associated with the first bending mode presents a distribution with uniform directionality (i.e., all positive or all negative). In this case, applying full electrode configuration to the piezoelectric actuator would generate maximal amplitudes, which could be beneficial to exciting the first bending mode and other higher bending modes

Manipulation of (17) and (19) gives the expression of the sum of normal stresses represented as the combination of two principal displacement functions S1(x,y) and S3(x,y).

∞

(23)

3.2. Determination of electrode configuration for modal vibration control

)

− (c11E − c13E )Ω20 a ∑n = 0 (−1)k Zk fk

cosh q1 x sin βk z sinh q1 a

Clearly, the difference between (22) and (23) can be attributed to the use of odd functions or even functions in S3. The even functions of S3 in the FE configuration produce motions in the same direction. This explains why the FE configuration for piezoelectric actuators is better suited to the suppression of bending mode vibrations. In contrast, the odd functions of S3 in the UDE configuration are indicative of antisymmetric properties, such that the motions caused by the piezoelectric plates differ from those obtained using the FE configuration. Refer to Krushynska et al. [39] and Huang [40] for further details on the theory of contour vibrations in a rectangular piezoelectric plate resonators. The UDE configuration could be used to suppress lateral mode vibrations and improve vibration control through the appropriate integration of the FE configuration subject to simultaneous bending mode excitation, as discussed below.

(17)

⎭

αn =

∂ux̂ ∂x

(22)

sinh p1 z cos αn x , cosh p1 b

p22 = αn2 − Ω22 , q22 = βk2 − Ω22

where U1, U3, V1, V3 are the functions that consider the two-fold symmetry of plate displacements associated with four types of contour mode, Xn and Zk are two sets of unknowns with non-negative integers n and k [39]. The notation αn and βk in (17) represent the following:

( − c )(

cos βk z

p12 = αn2 − Ω12 , q12 = βk2 − Ω12

sin αn x ⎫ (−1)nXn V3 (z ) ⎧ ⎨ cos αn x ⎬

σx̂ = (c11E − c13E )

cos αn x ,

where

sin βk z ⎫ (−1)k Zk V1 (x ) ⎧ cos βk z ⎬ ⎨ ⎩ ⎭

⎩

cosh q1 x sinh q1 a

S3 = −

sin βk z ⎫ (−1)k Zk U3 (x ) ⎧ cos βk z ⎬ ⎨ ⎩ ⎭ ∞ a ∑k = 0

sinh p1 b

S1 =

sin αn x ⎫ ∞ u x̂ (x , z ) = b ∑n = 0 (−1)nXn U1 (z ) ⎧ cos αn x ⎬ ⎨ ⎭ ⎩ + a ∑k = 0

cosh p1 z

mΩ12 m−2

(21) , c1 =

2(m − 1) c2 , (m − 2)

c2 =

E−cE c11 13

2ρ

, k,

n = 0, 1, 2, …. According to the symmetry properties of displacement, displacements u x̂ & uẑ in (17) are composed of different sets of trigonometric functions corresponding to four types of contour vibration mode associated with electrode configuration [39]. These modes include dilation modes, modes of flexure along major axes, modes of flexure along minor axes, and diagonal-type modes. For the sake of clarity, dilation modes in this paper are simply referred to as contour modes with “full electrode” (FE) configuration, i.e., no slit electrodes. The modes of flexure along minor axes and their electrode configurations were adopted for the suppression of lateral vibrations in a flexible structure, and are referred to as the “up-down electrode” (UDE) configuration in reference to the geometric shape of the piezoelectric plates 156

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Fig. 3. Modal analysis of flexible beam with fixed end and attached piezoelectric actuator: (a) strain at first bending mode (5 Hz); (b) strain at second bending mode (31 Hz); (c) strain at first torsion mode (87 Hz); (d) strain at first lateral mode (190 Hz). Fig. 4. Piezoelectric electrode configuration for the flexible beam: (a) first torsional mode; (b) first lateral mode.

vibrations. The PVDF sensor on the surface of the beam was intentionally biased to avoid the generation of an opposite-phase charge on the sensors due to opposing strain in lateral and torsional modes. The slit-electrode piezoelectric patch transducer (attached on the central line of the minor-axis in Fig. 2) was then integrated with a timebased switching control scheme to enable the suppression of multimodal vibrations by switching between FE and UDE configurations.

(Fig. 3(b)). However, this most commonly used electrode configuration has limited effects on exciting resonant modes, such as the torsional or lateral modes shown in Fig. 3(c) and (d), respectively. To obtain efficient modal excitation at these two non-bending modes we would require an electrode configuration consistent with the corresponding strain distribution. As shown in Fig. 4, the positive and minus signs indicate areas of the electrode associated with alternating phases of an applied electric field. Similar modal analysis revealed that mode excitation efficiency can be enhanced for the torsional and lateral modes in the first six vibration modes of the structure discussed in this study. Analysis indicates a reduction in the magnitude of the torsional modal response following the attachment of the piezoelectric actuator. Implementing the arc type electrode configuration shown in Fig. 4(a) is relatively difficult. Thus, to facilitate implementation and provide a clear demonstration of the proposed electrode-based switching control method, this study focused on the first lateral mode vibration associated with the second bending mode vibration by integrating the full electrode configuration with the electrode configuration, as shown in Fig. 4(b) (i.e., UDE configuration) in all subsequent experiments on multimodal vibration suppression control. It should be noted that the symbols “+” and “-” (Fig. 4) indicate the phase of motion in one instant; i.e., the effect inverses in the next moment because the vibration is in a steady state. As mentioned in Section 2, we adopted the placement of the PVDF sensor indicated in Fig. 2 to enable the measurement of multimodal

3.3. Modeling of smart flexible beam integrated with piezoelectric actuator in UDE configuration The dynamics of the described smart flexible structure vary with the electrode configuration (FE or UDE). A dynamic model composed of the dominant lateral vibrations was obtained using standard modal analysis to illustrate these differences in the switching control system. The derivation of the model representing the structure with piezoelectric actuator in UDE configuration is briefly summarized below as a reference. The equation of motion of the beam, in terms of lateral deflection function wz(x,t) can be represented as

Eb Ib

∂ 4wz (x , t ) ∂2wz (x , t ) ∂2ML (x , t ) + ρAb = . 4 2 ∂x ∂t ∂x 2

(25)

Here, ML(x,t) refers to the lateral moment acting on the beam, as indicated in Fig. 5(b). Assume that the upper stress and lower stress of the beam are symmetric with respect to the origin. The strain-stress 157

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Fig. 5. Smart piezoelectric beam under lateral deflection: (a) 3D view; (b) view of xy plane; (c) stress distribution in cross-section of the beam.

In contrast to PT, PL is the variable related to the modal function of the beam and the properties of the beam and piezoelectric actuator in a UDE configuration. In this paper, GL(s) represents the dynamic model of the smart piezoelectric beam in cases where the proposed switching control system applies the UDE configuration for lateral vibration suppression control. The resulting forced multimodal response is assumed to be expressed by superimposing the responses of systems GT(s) in Eq. (6) and GL(s) in Eq. (32).

relationship of the beam can be expressed as

σbl (z ) = Eb ɛbl (z ) = Eb α2 z.

(26)

Under the UDE configuration, the upper and lower stresses have opposite signs (shown in Fig. 5(c)), such that the strain-stress relationship of the piezoelectric actuator can be represented as

σpl = Ea ɛp, ɛp =

d31 Va (t ) ta

(27)

The strain gradient α2 can be determined by applying the moment equilibrium equation to one side as

∫0

b 2

σbl (z ) zdz −

∫0

bp 2

σpl (z ) zdz = 0.

4. Multimodal vibration suppression via switching control This paper proposes a novel switching control system to achieve multimodal vibration suppression control in smart flexible beams. To facilitate the subsequent discussion, this study considered only the suppression of multimodal vibrations associated the second bending mode and the first lateral mode of a flexible beam. As shown in Fig. 6, the proposed switching control system comprises three control schemes: (1) without controller switching & without electrode switching; (2) with controller switching & without electrode switching; (3) with controller switching & with electrode switching. Multimodal resonant excitation signals are first introduced into the flexible beam system in order to obtain vibration responses that could be used in the design of stand-alone PD and PPF controllers; i.e., the first control scheme mentioned above. The switching control system incorporates multiple controllers (i.e., the second control scheme) to deal with the issue of control saturation and improve transient performance. Switching time is based on the results of time-frequency analysis obtained from the design process of the first control scheme. The last control scheme can be activated via integration with the proposed electrode configuration and switching hardware to enhance the efficiency of multimodal vibration suppression control. Any number of vibration controllers can be used to suppress bending (or lateral) mode vibrations when the piezoelectric actuator is in FE (or UDE) configuration. In accordance with the time-switching strategy used for vibration controllers, our third control scheme permits online switching between the FE and UDE configurations. The switching mechanism and parameters are detailed in Section 4.3. In subsequent sections, the control algorithms applied in the switching control system are presented with a detailed description of the experiment setup. The design parameters associated with the

(28)

Using a manipulation similar to that in [44], the lateral moment ML(x,t) yields the following:

ML (x , t ) = Ks Va (t )[H (x − x1) − H (x − x2)]

(29)

where H(·) represents the step function and

Ks =

3Ea bp2Ib d31 ta b3

.

(30)

Applying modal analysis transforms the lateral deflection function of the beam wz(x,t) into the following expression. ∞

wz (x , t ) =

∑ ϕLi (x ) qLi (t ) i=1

(31)

where ϕLi(x) and qLi(t) respectively represent the modal shape function and modal deflection function along lateral direction. Substituting (31) into (25) and following procedures similar to those outlined in Section 2.2, the transfer function GL(s) from the applied actuator voltage Va(s) to the beam lateral deflection qL(s) can be expressed as

qL (s ) = Va (s )

n

PL s 2 + 2ξLi ωLi + ωLi 2

(32)

Ks [ϕ′Li (xk + 1) − ϕ′Li (xk )] e−st dt ∫ ρA b

(33)

GL (s ) =

∑ i=1

where ∞

PL =

0

158

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Fig. 6. Block diagram showing proposed switching control system for multimodal vibration control.

switching control (block diagram in Fig. 6) are first introduced, and then validated in experiments. The experiments on vibration control involving various switching control mechanisms are carefully analyzed to elucidate the effectiveness and practical merits of the proposed multimodal vibration control method.

Table 1 Specifications of PVDF sensor.

4.1. Vibration controller design This section introduces two typical control algorithms frequently discussed in the literature relevant to active vibration control in smart structures [6–8], PPF control, and PD control.

(34)

where ki is the tuning gain for the ith vibration mode. Damping term ζi is another design parameter typically within a range of 0.5–0.7 [7]. 4.1.2. PD control PD controllers include a damping term to dissipate resonant energy caused by low-damped dynamic systems. Unlike PPF controllers, which can be used to suppress vibrations in specified modes, the PD controller suppresses all multimodal vibrations at one time; however, the performance is limited from the perspective of the frequency domain. The transfer function of the PD controller can be represented as follows:

U (s ) = −(kp + kd s ) Y (s )

Value

Unit

Size Thickness Strain Constant, d31 Strain Constant, d33 Stress Constant, g31 Stress Constant, g33 Young's Modulus

41 × 16 52 2.3 × 10−11 −3.3 × 10−11 216 × 10−3 330 × 10−3 2–4

mm2 μm C/N C/N Vm/N Vm/N GPa

successively generated using the same piezoelectric actuator. Fig. 7 depicts the apparatus used for experiments on vibration suppression control. The vibration control algorithms were implemented using MATLAB software and the signals associated with piezoelectric sensor and actuator were acquired using a 16-bit data acquisition board (model PCI-6229 from National Instruments). The maximum voltage applied to the piezoelectric actuator through a power amplifier (model 4005 from NF electronic Co. Ltd.) was 60 V. To implement fast switching between the two electrode configurations (FE and UDE), four sets of solid state relay (SSR) circuits (model KB-20C04A from Kytech Electronics, Ltd.) with a response time of less than 2 ms were connected to the piezoelectric actuator and power amplifier. A detailed diagram of the switching circuit can be seen in Fig. 7(b). The highest frequency of the six vibration modes in the flexible beam was approximately 200 Hz [38]; therefore, a real-time sampling rate of 5 kHz (25 times of that frequency) was selected to prevent aliasing. A sensor amplifier equipped with low-pass filter circuits was used to filter out external noise with frequencies exceeding 1 kHz. An analog notch filter circuit [45] was also incorporated to eliminate 60 Hz electrical noise.

4.1.1. PPF control PPF control is meant to provide a specific quantity of energy at the controlled target mode ωi in order to suppress the corresponding vibration responses [8]. A PPF controller is a second order filter, which means that it naturally attenuates high frequency noise and undesired vibration modes. The transfer function from system output to control input can be represented as follows:

U (s ) ki = 2 Y (s ) s + 2ξi ωi s + ωi2

Parameters

4.3. Parameter selection for switching control (35) This section describes the criteria used in the selection of design parameters for the proposed switching control system in Fig. 6. The reasons for using a switching controller for multimodal vibration suppression control could be clarified by considering the standard control input of PPF control design, ustd as follows:

where kp and kd are proportional and derivative gains, respectively. 4.2. Experiment setup In this study, an aluminum cantilever beam was used for experiments on active vibration control. Based on modal analysis in Section 2.1, a pair of piezoelectric patches were attached to both sides of the cantilever beam (at the fixed end) to act as an actuator and sensor. The size of the piezoceramic patch used as an actuator was 60 × 30 × 1 mm3, and the specifications of the PVDF sensor (model LDT1-028K from Measurement Specialties, Inc.) are listed in Table 1. In the experiments, resonant excitation and vibration control signals were

ustd = k2 uppfb + k6 uppfl ustd ∈ [−10V, 10V]

(36)

where uppfb and uppfl respectively represent the control input of the PPF design with respect to the second bending mode and the first lateral mode of the flexible structure. k2 and k6 are the corresponding weighting parameters to be adjusted. Note that the subscript symbols 2 159

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Fig. 7. Experiment setup for active vibration control of smart flexible structure: (a) real-time control system; (b) zoomed-in plot of electrode configuration switching circuit.

where ti represents the time at which vibration suppression control is turned on; i.e., as resonant excitation stops. PPF control associated with the lateral mode was applied to the vibration control system from ti to ts, where ts refers to the time at which a switch is made to the other control law (i.e., PPF control associated with bending mode). This switching control scheme removes a performance constraint on the overall control system due to the voltage input. This makes it possible to manage the control energy more effectively and reduce the transient time of the vibration response. The two primary design parameters involved in this switching control mechanism are (1) switching time and (2) smooth switching control.

and 6 indicate the second and sixth vibration modes of the structure. Due to system constraints associated with the data acquisition hardware, the voltage output of ustd was limited to within ± 10 V. Unlike the case of controlling single modal vibrations, the simultaneous adjustment of two weight parameters (k2 and k6) for multimodal PPF vibration control can easily cause control saturation and undermine vibration suppression performance. To deal with this issue, we developed a time switching-based controller, which calculates individual control inputs in different time spans to enhance the efficiency of weighting parameter selection and improve multimodal vibration suppression control. The cantilever beam is a low-damped dynamic system that gradually dissipates energy following the initial excitation; therefore, the output response typically attenuates from high frequency vibrations to low frequency ones. This allowed the design of a vibration control system comprising two PPF controllers with an appropriate switching mechanism. The controller configured to suppress high frequency vibrations (lateral mode) is activated first, and then switched to the controller configured to suppress lower frequency vibrations (bending mode). The switching control law usw can be represented as follows:

4.3.1. Determining switching time The voltage input signal used for resonant excitation can be represented as follows:

u (t ) = 4 sin(ω2 t ) + 5.5sin (ω6 t )

where ω2 (the second bending mode) and ω6 (the first lateral mode) represent the second and sixth resonant frequencies in the cantilever beam, respectively. Sine-sweep excitation tests [38] revealed that ω2 = 30 Hz and ω6 = 190 Hz. In all subsequent experiments, the excitation time was set at 10 s (ti = 10) before evaluating vibration suppression performance using a variety of switching control structures. High-frequency vibrations decay extremely quick; therefore, it is

k6 uppfl , ti ≤ t < ts usw = ⎧ k u , t ≥ ts ⎨ ⎩ 2 ppfb k6 uppfl ∈ [−10V, 10V], k2 uppfb ∈ [−10V, 10V]

(38)

(37) 160

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Fig. 8. Experiment results pertaining to the selection of switching time: (a) CWT analysis: lateral mode/no control; (b) CWT analysis: lateral mode/PPF control (designed for lateral mode) under FE configuration; (c) Time responses of (a) and (b).

effects on suppressing lateral vibrations from 10 s to 10.1 s and had almost identical effects after 10.1 s. Therefore, in this study, the durations selected for controller switching and electrode switching were Δt = 0.1 s (i.e., ti = 10, ts = 10 + Δt), as shown in Eq. (37).

difficult to distinguish the transient difference and identify the switching time directly from time-domain data. To this end, time-frequency analysis, which could simultaneously clarify the relationship of time, frequency and spectrum energy in time series data, becomes an appealing solution. A continuous wavelet transform (CWT) method [46,47] was used to analyze the vibration responses to the piezoelectric actuator under full electrode configuration. As shown in Fig. 8, this results in a free response and those obtained using the PPF controller for lateral vibration suppression were also analyzed to determine the switching time. Fig. 8(a) presents the results of CWT analysis pertaining to lateral energy dissipation. Note that the depth of the color in the CWT plots indicates the strength of the energy. Without applying any control, the lateral energy in the cantilever beam decays quickly (between 10 and 10.2 s), and disappears entirely after 10.5 s. To elucidate how the lateral energy dissipates under feedback control, a PPF controller for lateral mode vibrations (abbreviated as PPFL control in Fig. 8) was implemented for vibration suppression control, the CWT analysis results of which are presented in Fig. 8(b). In this case, the energy dissipated abruptly (between 10 and 10.05 s) but presented a dissipating trend similar to the case without control after 10.1 s. The time response comparison clearly indicates that the lateral modal response (higher frequency component in Fig. 8(c)) disappeared in very little time, and the bending modal response (lower frequency component in Fig. 8(c)) continued to dominate the subsequent convergence behaviour, due to multimodal excitation. In summary, PPFL control had only positive

4.3.2. Smoothing function for switching control Based on the switching control law presented in Eq. (37), another PPF controller associated with bending mode (abbreviated as PPFB control) could be applied after ts to improve multimodal vibration suppression control. However, just like many of the switching control systems in the literature, the proposed switching action could also introduce undesired chattering associated with high frequency dynamics [48]. To deal with this issue, a time-varying function ρ(t) was adopted to allow smoother control transitions. The modified switching control law usw can be represented as follows:

k6 uppfl × (1 − ρ) + k2 uppfb × ρ 10s ≤ t < ts s usw = ⎧ ku t ≥ ts s ⎨ ⎩ 2 ppfb

(39)

The value of function ρ(t) starts at zero to ensure that PPFL control focuses on suppressing high frequency lateral vibrations during the transition. The value of ρ(t) was gradually increased, as was the percentage of the control effort of the PPFB control. When the value of ρ(t) approached 1, PPFB control was turned on fully to suppress the remaining low frequency bending vibrations. A large number of smoothing functions have been proposed in the literature. In this research, a function used to generate cycloidal motion [49] was adopted 161

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switching time for control Schemes 2 and 3. To highlight the importance of electrode switching control in improving performance, Exp. (e) was conceived of as Exp. (c) with the UDE configuration. The fast Fourier transform (FFT) results showing the magnitude values in bending mode and lateral mode are listed in Table 2 for the sake of comparison. Figs. 10–12 compare the time response of the experiments and Fig. 13 represents the corresponding FFT plots. A performance baseline for multimodal vibration suppression control was first established using PD control and PFF control in Exps. (b) and (c). The parameter values for the PD controller were kp = 3 and kd = 1 in all experiments. The parameter values for the PPF control law (36) were k2 = 0.7 and k6 = 0.3. The time response in Fig. 10 indicates that using standard PPF control reduces the transient amplitude, resulting in better suppression of lateral vibrations. This observation could be also verified by checking the peak values in the FFT results, as shown in Table 2. The high-pass filtering provided by the PD controller limited suppression performance at higher frequencies, such as lateral vibrations. However, due to the rapid dissipation of energy in highfrequency modes, the application of PPF control still made only a negligible difference with regard to the overall convergence time (approximately 12.5 s). As discussed in Section 4.3, it is important to adjust the parameters of control law (36) (k2, k6) at the same time in order to satisfy the voltage constraints imposed by the hardware. Unfortunately, this introduces a performance tradeoff between suppressing lateral mode vibrations and suppressing bending mode vibrations, thereby undermining control performance. To overcome this system constraint and thereby improve the performance of multimodal vibration suppression control, a time switching-based active vibration control system was developed to deal with coupled vibrations more effectively. The control law in (37) indicates that the controller designed for the suppression of lateral mode vibrations was activated within time span [10 s, 10 + Δt s] and then switched to the other controller designed to suppress bending mode vibrations after ts = 10 + Δt s. Note that Δt was selected as 0.1 s, as explained in the previous sections. In previous studies [38], PPF control proved more effective than PD control in controlling lateral vibrations in a smart cantilever beam. This suggests that PPFL control could be used to ensure the suppression of lateral vibrations before ts. After ts, the time response was dominated primarily by bending mode vibrations (as indicated in Fig. 8). The results obtained when using the second control scheme (Exps. (f), (g), (h) and (i)) illustrate the reasons why switching to another control law at ts is crucial to improving the overall performance. The ability to suppress specific modal vibrations makes PPFB control a straightforward option to suppress residual vibrations once the controller switch is ON, as shown in Exps. (f) and (g). PD control was used in Exps. (h) and (i) after ts to facilitate comparative analysis. A smoothing function was included in the switching control in Exps (g) and (i) to alleviate the high-frequency transition and explore other potential effects. The parameter values used for these switching control systems (Exps. (f), (g), (h) and (i)) were set at k6 = 0.4, k2 = 1.5 for control law (37). The peak values of the FFT results in Fig. 13 and Table 2 clearly show that a combination of PPFL control and PPFB control (Exps. (f) and (g)) achieve suppression performance of bending mode vibrations superior to the combination of PPFL control and PD control (Exps. (h) and (i)). This is because control law (37) allows the more aggressive use of PPF gain values to overcome performance limitations caused by the maximum voltage constraint. The time response is dominated by the bending mode; therefore, it is not surprising that applying PPFB control to improve the efficiency of energy dissipation reduced the convergence time, as shown in Fig. 11. On the other hand, switching from PPFL control to PD control shifted the suppressing effects to lateral vibrations and undermined the suppression of bending mode vibrations to below that of the baseline results in Exps. (b) and (c). It is interesting to note the remarkable suppression of lateral vibration in Exps. (h) and (i), regardless of

Fig. 9. Influence of smoothing function on switching control (switching from PPFL control to PPFB control under FE configuration of piezoelectric actuator).

for switching control, the mathematical representation of which can be expressed as

ρ (t ) =

2π (t − ti ) ⎞ ⎫ 1 ⎧ π (t − ti ) 1 ⎟ − sin ⎛⎜ π⎨ 2 ⎝ (ts − ti ) ⎠ ⎬ ⎩ (ts − ti ) ⎭

(40)

where ρ(t) satisfies the boundary conditions ρ(ti) = 0 and ρ(ts) = 1. Two sets of vibration control experiments were conducted using the switching control laws (37) and (39) in conjunction with a smoothing function (40) to highlight the necessity and effectiveness of adding this smoothing function for switching. The FE configuration of the piezoelectric actuator was adopted for both cases, and the parameter values of k2 & k6 were all the same. Fig. 9 presents the results of control voltage obtained through the implementation of these two control laws. In the case without a smoothing function, the control voltage abruptly switched from −1 V to 4.3 V at 10.1 s, and additional control energy was required to suppress bending mode vibrations. The smoothing function produced a gradual shift in the control law from uppfl to uppfb, which activated PPFB control earlier, thereby moderating the high frequency control, while enhancing the overall suppression performance of the active vibration control system. Detailed analysis of vibration control performance is presented in the next section. This switching mechanism was further integrated with the concept of switching electrode configurations and extended to a more comprehensive switching control system, as shown in Fig. 6. 4.4. Vibration control results and discussion The effectiveness of the proposed switching control algorithms in suppressing multimodal vibrations was evaluated by conducting an experiment using a series of vibration control laws based on the control block diagram in Fig. 6. The results of the experiment are summarized in Table 2. In the experiment, excitation signals were introduced into the system for 10 s before switching on the active vibration control law. In Table 2, Exp. (a) indicates the case without using any control; Exps. (a) and (b) are the cases involving the first control scheme (without controller switching & without electrode switching) using PD control and PPF control. Exps. (f), (g), (h), and (i) involve the second control scheme (with controller switching & without electrode switching), where “no smooth switching” indicates using the control law (37) and “with smooth switching” indicates using the control law (39). Exps. (j), (k), (l) and (m) involve the third control scheme (with controller switching & with electrode switching). As mentioned in the previous Section 4.3.1, Exp. (d) was conducted to determine the appropriate 162

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Table 2 Experiments evaluating the effectiveness of the proposed switching control schemes for active vibration suppression. The flexible beam was resonantly excited at the second bending mode and the first lateral mode from 0 to 10 s. The controller and electrode (if any) were switched at t = 10 + Δt (s) where the value of Δt was determined by analyzing the results of Experiment (d). Exp

Electrode switch: ON/OFF

Controller switch: ON/OFF

Controller from 10 ∼ 10 + Δt (s)

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m)

OFF OFF OFF OFF OFF OFF OFF OFF OFF ON ON ON ON

OFF OFF OFF OFF OFF ON (Without smooth switching) ON (With smooth switching) ON (Without smooth switching) ON (With smooth switching) ON (Without smooth switching) ON (With smooth switching) ON (Without smooth switching) ON (With smooth switching)

No control PD PPF (lateral + bending) PPF (lateral) PPF (lateral + bending) PPF (lateral) PPF (lateral) PPF (lateral) PPF (lateral) PPF (lateral) PPF (lateral) PPF (lateral) PPF (lateral)

(FE) (FE) (FE) (FE) (UDE) (FE) (FE) (FE) (FE)

Controller after 10 + Δt (s)

FFT Amplitude (Bending mode)

FFT Amplitude (Lateral mode)

PPF PPF PD PD PPF PPF PD PD

376.7 155.2 160 295.1 315.6 117.4 110.1 169.8 164.4 114.8 114.7 171 171.3

23 20.51 12 11 12.89 10.57 11.51 9.45 9.594 9.624 9.5 9.4 7.8

(bending) (bending)

(bending) (bending)

Fig. 10. Time response and vibration suppression in experiments using piezoelectric actuator with FE configuration: Exps. (a), (b), (c).

multimodal vibration suppression control. In this situation, the switching controller mechanism is essential for the third control scheme, due to the difficulties involved in optimizing vibration suppression using a controller with fixed parameters when system dynamics change due to a change in electrode configuration. This claim was verified in Exp. (e) in which the piezoelectric actuator was applied in the UDE configuration for vibration suppression control. Even using the same well-tuned PPF control law (36), the overall control performance was still significantly degraded, compared to the baseline results in Exp. (c) (nearly a 50% performance reduction based on FFT peak values). It is important to avoid this kind of poor control performance, while still fully exploiting the advantages of using electrode configurations to improve modal excitation efficiency and vibration suppression control [39]. Thus, in the following experiments (Exps. (j), (k), (l) and (m)) the UDE configuration was adopted for the time span [ti, ts] for lateral-mode control and then switched to the FE configuration after ts for bending-mode control. All of the design parameters were the same as those used in Exps. (f), (g), (h), and (i) except for the inclusion of electrode switching control. The time response in Fig. 12 indicates that a combination of PPFL control and PPFB control (Exps. (j) and (k)) still results in convergence speeds that are faster than the combination using PD control after ts

whether smooth switching was employed. This is because PPFB control only has positive effects on suppressing bending mode vibrations, whereas PD control is able to suppress lateral mode vibrations even after ts. The results in Table 2 and Fig. 13 reveal minor differences in suppression performance between Exps. (f) and (g). It appears that adding the smoothing function for switching control improves the suppression of bending mode vibrations but reduces the suppression of lateral mode vibrations; however, all of the effects are minor. This phenomenon may be explained by referring to Fig. 8. The earlier activation of PPFB control provides a smooth control transition; however, it also undermines the effectiveness of PPFL control earlier, thereby limiting the capacity to suppress lateral mode vibrations, and vice versa. Obviously, the performance difference depends on the switching time. This effect would be more pronounced if a longer switching transition were applied to the switching control system. Note that similar observations were made in Exps. (h) and (i). The results of experiments dealing with the second control scheme (Exps. (f), (g), (h), and (i)) demonstrate the effectiveness of the proposed switching controller. As shown in Fig. 6, the complete switching control system (third control scheme) was meant to enhance the efficiency of modal excitation while improving the effectiveness of 163

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Fig. 11. Time response and vibration suppression in experiments using piezoelectric actuator with FE configuration: Exps. (f), (g), (h), (i).

Fig. 12. Time response and vibration suppression in experiments using electrode switching control: Exps. (j), (k), (l), (m). 164

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Fig. 13. FFT results obtained in various experiments on vibration suppression control: Exps. (a), (b), (c), (f), (g), (h), (i), (j), (k), (l), (m). Plots in the left column are the results for bending mode vibrations, and the plots in the right column are for lateral mode vibrations.

revealed that Exp. (m) achieved the fastest convergence for lateral mode vibrations. Nonetheless, PD control was not as effective as PPFB control in suppressing bending mode vibrations; therefore, the dominant slow convergence for bending-mode vibrations resulted in the slowest transient behavior in Exp. (m). Compared to Exp. (g), Exp. (k) presented a 4% increase in the FFT peak values of bending mode vibrations and a 17% reduction in the FFT peak values of lateral mode vibrations. The comparison of transient performance in Table 3 also shows that Exp. (k) achieved 0.03 s faster convergence for lateral mode vibrations and 0.01 s slower transient time for bending mode vibrations. This is a clear indication that integrating the proposed switching control system with all of the necessary techniques (the setup in Exp. (k)) is the most effective design for high-performance multimodal vibration suppression control.

(Exps. (l) and (m)). This trend is similar to that observed without electrode switching control. The asymmetric and slight upward-shifting behavior in the output response may be attributed to the altered dynamics associated with electrode switching control. In contrast, the FFT results in Table 2 reveal several interesting effects that occurred after the electrode switching mechanism was added to the active vibration control system. Previous works have suggested that control over lateral vibrations could be improved by implementing piezoelectric actuators in an UDE configuration [38], regardless of whether PD control or PPF control were used. It is for this reason that Exps. (j), (k), (l), and (m) differed in the degree to which lateral mode vibrations were suppressed. Unlike the aforementioned results without electrode switching control, the inclusion of smooth transitions in switching control had positive effects on suppressing lateral mode vibrations. Furthermore, the performance of this method varied only slightly with regard to the suppression of bending mode vibrations. This clearly demonstrates the importance of the smoothing function in electrode switching control systems in guaranteeing an improvement in performance. The synergistic effects of PPFL control, PD control, and electrode switching control provided the most effective suppression of lateral-vibrations in Exp. (m), albeit at the cost of reduced performance in suppressing bending mode vibrations. Compared to the results in Exp. (i), the addition of electrode switching control further pushed the performance envelope in the suppression of lateral mode vibrations when using a combination of PPFL control and PD control for switching control. The best bending vibration suppression performance was obtained in Exp. (g); therefore, we applied time-frequency analysis to clarify the performance difference between these two extreme cases and another intermediate case, Exp. (k). Figs. 14 and 15 present the results of CWT analysis in these three cases, in terms of suppressing bending mode and lateral-mode vibrations. Table 3 summarizes the transient effects in the CWT results, in which the time of convergence is defined by the magnitude of the spectrum with a threshold value of 0.3. CWT analysis

5. Conclusion This paper presents a novel active vibration control system that uses a piezoelectric electrode configuration in conjunction with a control system based on time-switching to enable the suppression of multimodal vibrations in flexible structures. We examine the differences in performance that can be achieved using various switching control systems with a focus on bending mode vibrations and lateral mode vibrations in a cantilever beam. Experiment results demonstrate the effectiveness of the proposed switching control law in softening the control input constraints imposed by PPF control, while improving the multimodal vibration suppression performance. The application of a smoothing function to switching control was shown to alleviate the effects of high-frequency control transition and enhance control over bending mode vibrations. Integrating a mechanism in the vibration control system to enable switching between electrode configurations resulted in more efficient lateral modal excitation and more effective multimodal vibration suppression when using a single piezoelectric actuator. Future studies will include the development of more advanced 165

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Fig. 14. CWT analysis of bending mode vibrations: Exps. (g), (k), (m).

Fig. 15. CWT analysis of lateral mode vibrations: Exps. (g), (k), (m). 166

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Table 3 Transient vibration suppression performance in experiments using CWT analysis: Exps. (g), (k), (m). Exp.

Convergence time for bending mode vibrations (s)

Convergence time for lateral mode vibrations (s)

(g) (k) (m)

1.93 1.94 2.53

0.22 0.19 0.17

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