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Abstract

This paper presents an experimental study of the vibration attenuation achieved by the combined action of Constrained Layer Damping (CLD) and piezoelectric patches. The test structure is a cantilever beam excited at the clamped end, and the objective is to reduce the vibration of the free end. It will be shown that the most effective combination is to clad 25% of the length of the beam with CLD on just one side, and adhere a piezoelectric patch on the constraining layer rather than on the beam itself. It will also be shown that adding CLD and patches on both sides does not further improve attenuation. To guide the experimental analysis, and predict beam behaviour prior to testing, a model has been developed to include CLD cladding and piezoelectric patches. The model is based on the simplified Euler–Bernoulli beam but taking into account the stepped or segmented nature of the beams under consideration. Mode shapes are obtained for each segment and assembled imposing continuity. Experimental results show that the model can be used as a valid first approximation.

Keywords

lightweight structure passive damping viscoelastic constrained layer damping piezoelectric patch shunt circuit

Introduction

Structural vibrations are very often undesirable in a wide range of industries and applications. Metallic structures usually lack sufficient damping to cancel or mitigate vibrations. According to Johnson,¹ around 85% of passive damping systems used viscoelastic materials. They are still widely used today in sectors such as the automotive,^2,3 railway⁴ and aerospace^5,6 industries, but they are mostly focused on noise reduction rather than on reducing the vibrations transmitted to passengers. Viscoelasticity is exhibited by materials with long molecular chains suitable for converting mechanical deformation into heat. The phenomenon is very sensitive to temperature and frequency. Viscoelastic materials show a glassy region, usually at high frequencies and/or low temperatures, where the mobility of the polymer chains is hindered; an elastic region, at low frequencies and/or high temperatures, with less restricted chain mobility that allows high and reversible deformations; and an intermediate transition region suitable for energy dissipation.

The most widespread viscoelastic technique is Constrained Layer Damping (CLD). It consists of applying a viscoelastic layer sandwiched between a rigid constricting layer and the structure itself in order to create shear strain in the layer rather than axial deformation.⁷ Under these conditions the viscoelastic layer is able to dissipate more energy in the form of heat than in configurations without a constricting layer.^8–10

In addition to viscoelastic damping, piezoelectric materials are another widely adopted solution to increase structural damping by offering high, fast, linear and robust electro–mechanical coupling. Direct piezoelectricity is the generation of electrical charges when a mechanical stress is applied. The stresses on the material cause the rotation of the electric dipoles, creating an apparent charge flow between the electrodes and thus an electric displacement. Many works can be found in the literature where inverse piezoelectricity is used in active control systems.^11–15 Conversely, the direct piezoelectric effect can be used to harvest^16,17 or to passively dissipate the electrical energy generated by deformation by connecting the patch to shunt circuits.¹⁸

The most robust, stable and energy–efficient dissipative shunt circuits are purely resistive (R–shunt).^19–23 The dynamic behaviour of these shunt circuits is similar to that of viscoelastic materials: it adds a frequency–dependent loss factor to the structural dynamics but, unlike viscoelastic materials, the loss factor fortunately remains relatively insensitive to temperature variations. Alternatively, resonant shunt circuits (RL–shunt)^{20,21,23–25} are the electrical analog of a dynamic vibration absorber tuned to the natural frequency of the host structure.¹⁸ However, RL circuits designed to operate in the low frequency range (as is common in railway structures), require high inductances which can only be achieved via synthetic electronic circuits such as the one proposed by Antoniou²⁶ or more recently by Dekemele for high–voltage synthetic inductors.²⁷ This technique turns piezoelectric systems with RL shunt circuits into semi–passive systems because of the need to power the operational amplifiers.¹⁸ Vibration cancellation of flexible structures with RL shunt circuits has been studied by several authors: dell’Isola et al.²⁸ applied it to a cantilever beam; Hassan et al.²⁹ tried series and parallel RL shunt circuits; and later Lossouarn et al.³⁰ proposed new synthetic inductance designs. Recent studies have proposed advanced shunt circuits, such as optimal elementary shunt circuits to replace multi–branch shunt circuits,³¹ LRLC–shunt circuits,³² self–tuning RL–shunt circuits,³³ or multi–frequency adaptive tuned mass dampers based on shunt circuits with negative capacitances.³⁴

Despite the rich literature using and comparing³⁵ CLDs and piezoelectrics with shunt circuits, hybrid systems passively combining both techniques have received less attention. A few studies can be found where the CLD is optimised to cancel vibrations in the high–frequency range and the piezoelectric shunt circuit is designed to target vibrations in the low–frequency range. For example, the works of Chul et al.,³⁶ those from Larbi’s research group,^37–42 and more recently that of Araújo and Aguilar.⁴³ The integration of both techniques by constraining the viscoelastic layer with the piezoelectric material itself was proposed by Ghoneim⁴⁴ under the name “Electromechanical Surface Damping” (EMSD). It was used, again, to extend the range of actuation, optimising the viscoelastic for the first mode of vibration and the shunt circuit for the second. Ghoneim later extended the studies to a plate–type structure by applying two EMSD elements.^45,46

The work presented here is an experimental study that, starting from the separate optimisation of both, the CLD and the piezoelectric patches, combines them to study hybrid systems. To the best of our knowledge, hybrid techniques have only been studied to extend the frequency range, but have not been combined to maximise damping of a single mode of interest. On the other hand, a model is presented to study any combination of layers including piezoelectric materials. In contrast to similar models found in the literature, mode shapes have been tailored to the stepped or segmented nature of the beams under consideration.

Model

The notation in this section tries to avoid conflicts between usual electrical symbols and standard mechanical notation. Thus, for example, $E$ stands for electric field rather than elastic modulus, which will hereafter be referred to as $Y$ , in honour of Thomas Young. Likewise, as another example, $ϵ$ refers to electric permitivity, forcing us to use $S$ for strain. The meaning of each symbol will be described as they appear.

Modal parameters for a beam of variable section

The equation of motion for a uniform Euler–Bernoulli beam in free transverse vibration is

YI \frac{\partial^{4} w (x, t)}{\partial x^{4}} + μ I \frac{\partial^{5} w (x, t)}{\partial t \partial x^{4}} + ρ \frac{\partial^{2} w (x, t)}{\partial t^{2}} = 0

(1)

where $w (x, t)$ is the transversal displacement as a function of the longitudinal coordinate $x$ and time $t$ , $ρ$ is the mass per unit length, $I$ is the cross section moment of inertia, $Y$ is the Young’s modulus and $μ$ the internal viscous damping coefficient.

By testing harmonic solutions with separated variables, $ϕ (x) e^{i ω t}$ , one obtains, for the undamped case, discrete natural frequencies $ω_{k}$ and their corresponding mode shapes $ϕ_{k} (x)$ . Truncating the number of modes, the general beam deflection can be written as

w (x, t) = \sum_{k = 1}^{n} ϕ_{k} (x) r_{k} e^{i ω_{k} t}

(2)

where $r_{k}$ is the amplitude of each contributing mode.

The mode shapes for a uniform beam are of the type

\begin{matrix} ϕ_{k} (x) = C_{1} \cos (β_{k} x) + C_{2} \sin (β_{k} x) + C_{3} \cosh (β_{k} x) \\ + C_{4} \sinh (β_{k} x) \end{matrix}

(3)

where constants $C_{1} - C_{4}$ depend on the boundary conditions. For the case of a non–uniform beam composed of different layers that may not extend the entire length of the beam (see Figure 1), an equivalent cross–section stiffness ( $\bar{YI}$ ) can be determined for each segment, and the mode shape can be assembled from expressions like those in equation (3), imposing continuity of deflection, slope, bending moment and shear force between segments.

Figure 1.

Beam segments and layers.

Consider the beam in Figure 1 with $N$ segments of lengths ${\bar{L}}_{i}$ ( $i = 1, \dots N$ ). Each segment has constant thickness resulting from the addition of $N_{i}$ layers. A generic layer $j$ has $ρ_{j}$ mass per unit length, cross–section area $A_{j}$ , thickness $t_{j}$ , cross section moment of inertia $I_{j}$ , Young’s modulus $Y_{j}$ and internal viscous damping coefficient $μ_{j}$ . Some layers are shared by several segments, and it is assumed that layer $j = 1$ is the “host” beam.

Let the origin of coordinate $z^{'}$ (Figure 1) be at the centroid of layer 1. The distance from the origin to the centroid of layer $j$ is

d_{j} = \frac{1}{A_{j}} \int_{A_{j}} z' d A_{j} = \frac{1}{2 t_{j}} (z_{j, 2}^{' 2} - z_{j, 1}^{' 2})

(4)

where $z_{j, 1}'$ and $z_{j, 2}'$ indicate the beginning and end of the layer, respectively. With these centroids, the neutral axis of segment $i$ is located at a distance

{\bar{d}}_{i} = \frac{\sum_{j = 1}^{N_{i}} Y_{j} A_{j} d_{j}}{\sum_{j = 1}^{N_{i}} Y_{j} A_{j}}

(5)

from the origin.

The equivalent parameters of each segment can now be calculated as

{\bar{ρ}}_{i} = \sum_{j = 1}^{N_{i}} ρ_{j}

(6)

{\bar{YI}}_{i} = \sum_{j = 1}^{N_{i}} Y_{j} (I_{j} + A_{j} {(d_{j} - {\bar{d}}_{i})}^{2}) = \sum_{j = 1}^{N_{i}} Y_{j} I_{ij}'

(7)

{\bar{μ I}}_{i} = \sum_{j = 1}^{N_{i}} μ_{j} (I_{j} + A_{j} {(d_{j} - {\bar{d}}_{i})}^{2}) = \sum_{j = 1}^{N_{i}} μ_{j} I_{ij}'

(8)

where $I_{j}$ is the inertia of layer $j$ with respect to its own centroid, whereas $I_{ij}'$ is the inertia of layer $j$ with respect to the neutral axis in segment $i$ .

As mentioned, equivalent parameters are used in equation (3) to obtain the following expressions:

\begin{matrix} {\bar{ϕ}}_{i, k} (x) = {\bar{C}}_{i, 1} \cos ({\bar{β}}_{i, k} x) + {\bar{C}}_{i, 2} \sin ({\bar{β}}_{i, k} x) + {\bar{C}}_{i, 3} \cosh ({\bar{β}}_{i, k} x) \\ + {\bar{C}}_{4} \sinh ({\bar{β}}_{i, k} x) \end{matrix}

(9)

where ${\bar{ϕ}}_{i, k} (x)$ is the $k^{th}$ modal shape (constants pending) for the $i^{th}$ segment, and ${\bar{β}}_{i, k}^{4} = ω_{k}^{2} {\bar{ρ}}_{i} / {\bar{EI}}_{i}$ .

Four boundary conditions must be imposed at each segment transition, as well as two at each end of the beam. A system of $4 N$ algebraic equations (N being the number of segments) is thus solved to determine the $4 N$ constants ( ${\bar{C}}_{i, 1} - {\bar{C}}_{i, 4} for i = 1 . . . N$ ). Solutions exists when the coefficient matrix (be it $A$ ) is singular, and this condition ( $\det (A) = 0$ ) allows us to determine $ω_{k}$ .⁴⁷ In our case, the boundary conditions mentioned above are the clamped origin ( $w (0, t) = 0$ and $w' (0, t) = 0$ ); the free end ( $w^{″} (L, t) = 0$ and $w^{″'} (L, t) = 0$ ); displacement and rotation continuity at segment transitions ( ${\bar{ϕ}}_{i, k} (L_{i}) = {\bar{ϕ}}_{i + 1, k} (L_{i})$ and ${\bar{ϕ}}_{i, k}' (L_{i}) = {\bar{ϕ}}_{i + 1, k}' (L_{i})$ ); as well as bending moment and shear force continuity ( ${\bar{YI}}_{i} {\bar{ϕ^{″}}}_{i, k} (L_{i}) = {\bar{YI}}_{i + 1} {\bar{ϕ^{″}}}_{i + 1, k} (L_{i})$ and ${\bar{YI}}_{i} {\bar{ϕ^{″'}}}_{i, k} (L_{i}) = {\bar{YI}}_{i + 1} {\bar{ϕ^{″'}}}_{i + 1, k} (L_{i})$ ). The natural frequencies and the corresponding sets of constants describe the piecewise modal shapes⁴⁷ which are then assembled as

ϕ_{k} (x) = {\begin{matrix} {\bar{ϕ}}_{1, k} (x) & if & x \leq L_{1} \\ ⋮ \\ {\bar{ϕ}}_{i, k} (x) & if & L_{i - 1} \leq x \leq L_{i} \\ ⋮ \\ {\bar{ϕ}}_{N, k} (x) & if & L_{N - 1} \leq x \leq L_{N} \end{matrix}

(10)

and normalised so that the integral along the beam of the mode shape squared is one.

Piezoelectric layers

Some of the layers in the model above are piezoelectric layers in which mechanical and electrical responses are intertwined. Strain is not only caused by mechanical stress but by electric fields as well. Likewise, electric displacement is not only due to electric field, but to mechanical stress as well. Let us assume that stress is unidirectional ( $T = T_{1}$ , $T_{2} = T_{3} = T_{12} = 0$ ), and that the electric field exists only in the transversal direction ( $E = E_{3}$ , $E_{1} = E_{2} = 0$ ). In this case

{\begin{matrix} S \\ D \end{matrix}} = [\begin{matrix} 1 / Y^{E} & d \\ d & ϵ^{T} \end{matrix}] {\begin{matrix} T \\ E \end{matrix}}

(11)

where $S$ and $D$ are strain and electric displacement, respectively; $Y^{E}$ is the Young’s modulus for the piezoelectric layer in short circuit ( $E = 0$ ), and $ϵ^{T}$ is the permittivity when the material is free to deform ( $T = 0$ ). As seen, superscripts are used to indicate a variable that is to remain null when measuring the corresponding material property. The piezoelectric coefficient $d$ couples electric field and strain, as well as stress and electric displacement.

Equation (11) may be inverted to obtain

{\begin{matrix} T \\ E \end{matrix}} = [\begin{matrix} Y^{D} & - h \\ - h & β^{S} \end{matrix}] {\begin{matrix} S \\ D \end{matrix}}

(12)

where $h = k^{2} / ((1 - k^{2}) d)$ , $k$ being the piezoelectric coupling given by $k^{2} = d^{2} Y^{E} / ϵ^{T}$ , and $β^{S} = {(ϵ^{T} (1 - k^{2}))}^{- 1}$ .

The potential energy per unit volume, stored as mechanical strain and electric displacement, is

e_{p} = \int_{0}^{S} T dS + \int_{0}^{D} E dD = \int_{(0, 0)}^{(S, D)} (T, E) {\begin{matrix} dS \\ dD \end{matrix}}

(13)

Since $T$ and $E$ depend on $S$ and $D$ (see equation (12)), the integral is

e_{p} = \frac{1}{2} Y^{D} S^{2} - hSD + \frac{1}{2} β^{S} D^{2}

(14)

In an Euler–Bernoulli beam, the strain at a distance $z$ from the neutral axis of the segment (see Figure 1) where the piezoelectric layer is attached is given by

S = - z w^{″}

(15)

The assumption will now be made that the shape functions obtained in Section “Modal parameters for a beam of variable section” hold in the case of some layers being piezoelectric. Moreover, it will be assumed that the first mode suffices to account for beam vibration, in which case

w (x, t) = ϕ (x) r (t)

(16)

where $ϕ (x)$ is the first mode shape function, and $r (t)$ the mode amplitude as a function of time which will serve as generalised coordinate in what follows. Therefore, the strain as a function of $z$ (and time) may be expressed as

S = - z ϕ^{″} r

(17)

On the other hand, the electric displacement ( $D$ ) can be assumed uniform in the piezoelectric patch area ( $A_{p}$ ), and thus

D = \frac{q}{A_{p}}

(18)

where $q$ is the electric charge. Introducing equations (17) and (18) in equation (14), and integrating in the volume of the piezoelectric layer, the total potential energy of said layer is⁴⁸

\begin{matrix} E_{p} = \frac{1}{2} \underset{K^{D}}{\underset{︸}{[\int_{V_{p}} Y^{D} z^{2} {ϕ^{″}}^{2} d V_{p}]}} r^{2} - \underset{Θ}{\underset{︸}{[\int_{V_{p}} \frac{h}{A_{p}} (- z) ϕ^{″} d V_{p}]}} rq \\ + \frac{1}{2} \underset{1 / (C_{p}^{S})}{\underset{︸}{[\int_{V_{p}} \frac{β^{S}}{A_{p}^{2}} d V_{p}]}} q^{2} \end{matrix}

(19)

where $C_{p}^{S}$ is the piezoelectric capacitance with strain impeded ( $S = 0$ ). This expression may be used in a Lagrange formulation with $r$ and $q$ as generalised coordinates. This approach allows us to identify $K^{D}$ as a stiffness, and $Θ$ as a beam–level piezoelectric coupling term linking both generalised coordinates. Stiffness $K^{D}$ is just another expression for the contribution of the open circuit piezoelectric layer to the cross–section equivalent $\bar{YI}$ already considered in equation (7). The piezoelectric coupling $Θ$ can be further expressed as

Θ = \int_{V_{p}} \frac{h}{A_{p}} (- z) ϕ^{″} d V_{p} = \frac{- {ht}_{p}^{2}}{2 L_{p} L} (τ + 1) \int_{L_{p} / L} φ^{″} d ζ

(20)

where $t_{p}$ is the piezoelectric layer thickness, $L_{p}$ its length and $L$ the total length of the beam. Note that the non–dimensional longitudinal coordinate ζ = x/L has been used, and that the non–dimensional shape function has been named $φ (ζ)$ . Parameter $τ$ is the ratio between twice the distance from the piezoelectric layer to the segment neutral axis (be it $t_{s}$ ) and the piezoelectric layer thickness ( $τ = t_{s} / t_{p}$ τ = t_s/t_p).

Dynamic equations

Let us now consider the dynamics of a segmented cantilever beam, with possible piezoelectric layers and CLDs, subjected to the motion of its clamped end with acceleration $a_{b}$ . The Langrange–d’Alembert principle may be used to obtain the governing equations from potential and kinetic energy expressions in terms of generalised coordinates. Equation (19) represents the contribution of a piezoelectric layer to the potential energy. The rest of the layers contribute to the stiffness term with their own Young’s modulus and inertia. On the other hand, the structural kinetic energy is given by

E_{c} = \int_{V_{s}} \frac{1}{2} ρ_{v} {\overset{\cdot}{w}}^{2} d V_{s} = \frac{1}{2} [\int_{V_{s}} ρ_{v} ϕ^{2} d V_{s}] {\overset{\cdot}{r}}^{2} = \frac{1}{2} M {\overset{\cdot}{r}}^{2}

(21)

where $ρ_{v}$ is mass per unit volume at each point in the solid of volume $V_{s}$ . With this, the governing equations in terms of generalised coordinates $r$ and $q$ are

\begin{matrix} M \overset{\cdot\cdot}{r} + C \overset{\cdot}{r} + K^{D} r - Θ q & = - a_{b} \sum_{i = 1}^{N} {\bar{ρ}}_{i} L \int_{ζ_{i - 1}}^{ζ_{i}} φ d ζ \end{matrix}

(22)

- Θ r + \frac{1}{C_{p}^{S}} q = v

(23)

where $M$ , the modal mass of the overall beam, is obtained from the integral in equation (21) using the piecewise mode, to yield

M = \sum_{i = 1}^{N} {\bar{ρ}}_{i} L \int_{ζ_{i - 1}}^{ζ_{i}} φ^{2} d ζ

(24)

The damping coefficient $C$ is the modal damping of the overall beam given by

C = \sum_{i = 1}^{N} \frac{{\bar{μ I}}_{i}}{L^{3}} \int_{ζ_{i - 1}}^{ζ_{i}} {(φ ″)}^{2} d ζ

(25)

Likewise, $K^{D}$ is now the modal stiffness of the overall beam with all piezoelectric layers in open circuit

K^{D} = \sum_{i = 1}^{N} \frac{{\bar{YI}}_{i}}{L^{3}} \int_{ζ_{i - 1}}^{ζ_{i}} {(φ ″)}^{2} d ζ

(26)

and $Θ$ is the piezoelectric coupling redefined to now comprise all $N_{p}$ piezoelectric layers in the beam

Θ = \sum_{p = 0}^{N_{p}} \frac{- {ht}_{p}^{2}}{2 L_{p} L} (τ_{p} + 1) \int_{L_{p} / L} φ ″ d ζ

(27)

Voltage in equation (23) may be supplied by a voltage source or may be related to current (rate of change of electric charge) via an external shunt circuit. In the case of a resistor–inductor (RL) shunt circuit, the relation is $- v = R \overset{\cdot}{q} + L \overset{\cdot\cdot}{q}$ , and the three governing equations may be grouped as

\begin{matrix} \overset{\cdot\cdot}{r} + 2 ξ^{D} ω_{n}^{D} \overset{\cdot}{r} + {(ω_{n}^{D})}^{2} r - \frac{Θ}{M} q = \frac{- a_{b}}{M} \sum_{i = 1}^{N} {\bar{ρ}}_{i} L \int_{ζ_{i - 1}}^{ζ_{i}} φ d ζ \\ - Θ r + \frac{1}{C_{p}^{S}} q = v \\ \overset{\cdot\cdot}{q} + \frac{R}{L} \overset{\cdot}{q} = \frac{- v}{L} \end{matrix}

(28)

where $ω_{n}^{D}$ and $ξ^{D}$ are the beam natural frequency and damping coefficient, respectively, with all the piezoelectric layers in open circuit.

It is often convenient to rewrite these equations in a state–space form in which

\begin{matrix} \overset{\cdot}{x} & = Ax + Bu \\ y & = Cx + Du \end{matrix}

where vector $x$ describes the state of the system; $u$ and $y$ are the input and output vector, respectively; and system matrices $A$ , $B$ , $C$ and $D$ are easily determined when rearranging the dynamic equations.

For the RL shunt circuit, the system state comprises modal coordinate $r$ and its derivative $\overset{\cdot}{r}$ , along with electric charge $q$ and its derivative $\overset{\cdot}{q}$ ; whereas the input is the acceleration ( $a_{b}$ ) of the clamped end. As output we select the acceleration at the free tip of the beam $\overset{\cdot\cdot}{w} (1, t)$ and the voltage in the piezoelectric patch $v$ . That is

\begin{matrix} x = {\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}} = {\begin{matrix} r \\ \overset{\cdot}{r} \\ q \\ \overset{\cdot}{q} \end{matrix}} \\ u = {a_{b}} \\ y = {\begin{matrix} y_{1} \\ y_{2} \end{matrix}} = {\begin{matrix} \overset{\cdot\cdot}{w} (1, t) \\ v \end{matrix}} \end{matrix}

(30)

It can readily be seen that, with these definitions and equation (28), the system matrices are

A = [\begin{matrix} 0 & 1 & 0 & 0 \\ - {(ω_{n}^{D})}^{2} & - 2 ξ^{D} ω_{n}^{D} & Θ / M & 0 \\ 0 & 0 & 0 & 1 \\ Θ / L & 0 & - 1 / L C_{P}^{S} & - R / L \end{matrix}] B = [\begin{matrix} 0 \\ - m_{I} \\ 0 \\ 0 \end{matrix}]

(31)

and

\begin{matrix} C = [\begin{matrix} - {(ω_{n}^{D})}^{2} φ (1) & - 2 ξ^{D} ω_{n}^{D} φ (1) & \frac{Θ}{M} φ (1) & 0 \\ - Θ & 0 & \frac{- 1}{C_{P}^{S}} & 0 \end{matrix}] & D = [\begin{matrix} - φ (1) m_{I} \\ 0 \end{matrix}] \end{matrix}

(32)

where

m_{I} = \frac{1}{M} \sum_{i = 1}^{N} {\bar{ρ}}_{i} L \int_{ζ_{i - 1}}^{ζ_{i}} φ d ζ

(33)

These state–space equations may be simplified for specific cases such as short circuit, open circuit or resistive shunt circuit (without inductance). The simplified systems are shown next.

Short circuit

When the electrodes of the piezoelectric layer are short–circuited ( $E = 0$ , and hence $v = 0$ ), charge can be written as $q = Θ C_{p}^{S} r$ . This allows the dynamic equations to be rewritten with the beam stiffness in short circuit given by $K^{E} = K^{D} - Θ^{2} C_{p}^{S}$ and its corresponding natural frequency in short circuit $ω_{n}^{E}$ . The state space is now represented by just the modal coordinate and its derivative, $x = {x_{1}, x_{2}}^{T} = {r, \overset{\cdot}{r}}^{T}$ , the output is just $y = {y_{1}}$ since $y_{2} = v = 0$ , and the state–space equations become

{\begin{matrix} {\overset{\cdot}{x}}_{1} \\ {\overset{\cdot}{x}}_{2} \end{matrix}} = [\begin{matrix} 0 & 1 \\ - {(ω_{n}^{E})}^{2} & - 2 ξ^{E} ω_{n}^{E} \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \end{matrix}} + [\begin{matrix} 0 \\ - m_{I} \end{matrix}] {a_{b}}

(34)

and

{\begin{matrix} y_{1} \end{matrix}} = [\begin{matrix} - {(ω_{n}^{E})}^{2} φ (1) & - 2 ξ^{E} ω_{n}^{E} φ (1) \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \end{matrix}} + [\begin{matrix} - φ (1) m_{I} \end{matrix}] {a_{b}}

(35)

Open circuit

When the piezoelectric patch is left open circuit ( $D = 0$ and hence $q = 0$ ), the dynamic equations are written in terms of the natural frequency in open circuit $ω_{n}^{D}$ . The state–space variables are $x = {x_{1}, x_{2}}^{T} = {r, \overset{\cdot}{r}}^{T}$ with output $y = {\overset{\cdot\cdot}{w} (1, t), v}^{T}$ , and the system can be written as

{\begin{matrix} {\overset{\cdot}{x}}_{1} \\ {\overset{\cdot}{x}}_{2} \end{matrix}} = [\begin{matrix} 0 & 1 \\ - {(ω_{n}^{D})}^{2} & - 2 ξ^{D} ω_{n}^{D} \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \end{matrix}} + [\begin{matrix} 0 \\ - m_{I} \end{matrix}] {a_{b}}

(36)

and

\begin{matrix} {\begin{matrix} y_{1} \\ y_{2} \end{matrix}} = [\begin{matrix} - {(ω_{n}^{D})}^{2} φ (1) & - 2 ξ^{D} ω_{n}^{D} φ (1) \\ - Θ & 0 \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \end{matrix}} \\ + [\begin{matrix} - φ (1) m_{I} \\ 0 \end{matrix}] {a_{b}} \end{matrix}

(37)

Resistive R–shunt circuit

If the piezoelectric patch is shunted with an ideal resistive R–circuit, charge must be retained as a state–space variable, but its second derivative is null. In this case $x = {x_{1}, x_{2}, x_{3}}^{T} = {r, \overset{\cdot}{r}, q}^{T}$ and $y = {y_{1}, y_{2}}^{T} = {\overset{\cdot\cdot}{w} (1, t), v}^{T}$ , the state–space system being

{\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \\ {\dot{x}}_{3} \end{matrix}} = [\begin{matrix} 0 & 1 & 0 \\ - {(ω_{n}^{D})}^{2} & - {(ω_{n}^{D})}^{2} & \frac{Θ}{M} \\ \frac{Θ}{R} & 0 & \frac{- 1}{R C_{P}^{S}} \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}} + [\begin{matrix} 0 \\ - m_{I} \\ 0 \end{matrix}] {a_{b}}

(38)

and

\begin{array}{l} {\begin{matrix} y_{1} \\ y_{2} \end{matrix}} = [\begin{matrix} - {(ω_{n}^{D})}^{2} φ (1) & - 2 ξ^{D} ω_{n}^{D} φ (1) & 0 \\ - Θ & 0 & \frac{1}{C_{P}^{S}} \end{matrix}] {\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}} \\ + [\begin{matrix} - φ (1) m_{I} \\ 0 \end{matrix}] {a_{b}} \end{array}

(39)

Methodology

Specimens

A laboratory–scale (292 mm long, 35 mm wide and 3 mm thick) cantilever aluminium beam is the “host” structure to test several configurations of constrained layer damping (CLD) attachments and piezoelectric layers. The goal is to reduce the vibration response when the clamped end is shaken at different frequencies.

The CLD patches in our study are composed of an aluminium sheet that acts as a constricting layer, and a very thin film of viscoelastic material. The viscoelastic material is commercially manufactured by Heathcote Industrial Plastics (HIP). Its mechanical properties (shear modulus and loss factor) depend, not only on temperature, but on excitation frequency as well. Constant temperature conditions are considered in this study. Two types of CLD have been studied: a 0.13 mm thick viscoelastic polymer layer constrained by a 0.5 mm thick aluminium sheet (labelled 2005); and the same viscoelastic layer constrained by a 1.0 mm thick aluminium sheet (labelled 4005).

The piezoelectric patches in our study have a nominal maximum operating voltage between 100 and 1000 V, depending on the active layer thickness. The patches are laminated structures consisting of a piezoceramic plate, electrodes and polymer materials, manufactured with bubble–free injection methods. The polymer coating simultaneously serves as electrical insulation and mechanical preload (to allow bending). Three types of piezoelectric patches manufactured by PI Ceramics have been selected, namely models P–876.A11, P–876.A12 and P–876.A15 (just A11, A12 and A15 hereafter). Their main properties are summarised in Table 1. In all three patches the piezoelectric material is PI Ceramics’ PIC255, which is a modified lead zirconate titanate.

Table 1.

Piezoelectric patch characteristics.

Piezoelectric patch type	A11	A12	A15
Piezoceramic thickness ( $μ$ m)	100	200	500
Electrical capacitance (nF)	150	90	45
Blocking force (N)	90	265	775

As mentioned, several configurations of constrained layer damping attachments and piezoelectric layers have been studied. Figure 2 shows all configurations. $W$ denotes the reference bare aluminium beam. Specimens with CLD are denoted by the type of constraining layer (C = 2005 or C = 4005), a superscript indicating the percentage ( $p$ ) of beam length covered with CLD ( $C^{p %}$ ), and a subscript indicating whether the CLD is on one side of the beam ( $C_{x 1}^{p %}$ ) or on both sides ( $C_{x 2}^{p %}$ ). Therefore, for example, specimen $4005_{x 2}^{25 %}$ refers to a beam with a pair of 4005 CLDs, one on each side of the beam, covering 25% of its length.

Figure 2.

Specimen types and labelling convention.

Specimens with piezoelectric patches are referred to by the patch type (P = A11, P = A12 or P = A15), a subscript which specifies the number of patches ( $x 1$ or $x 2$ ), and, in the case of two patches, a second subscript to indicate whether they are on the same side of the beam ( $a$ ) or on opposite sides ( $b$ ). Thus, for example, specimen $A 15_{x 2 a}$ refers to a beam with a pair of A15 piezoelectric patches aligned on the same side of the beam.

Hybrid cases with CLD as well as piezoelectric patches have also been studied. Hybrid specimens are labelled by juxtaposing the CLD and patch descriptions. In this case, however, the second patch subscript is allowed to take two new values to indicate whether the single patch is adhered to the beam ( $c$ ) or to the CLD ( $d$ ). For example, specimen $4005_{x 2}^{25 %} A 15_{x 1 d}$ refers to a beam with a pair of 4005 CLDs covering 25% of the beam and an A15 piezoelectric patch bonded on the CLD (See Figure 2). In total, 19 different specimens have been analysed in this work.

Test bench

Figure 3(a) shows a schematics of the excitation and measurement systems. The excitation system is an electrodynamic shaker model V5344 from Data Physics, connected to a slip table. Figure 3(b) shows the test bench and a ready to test specimen with its connectors and sensors.

Figure 3.

(a) Schematics of the excitation and measurement system and (b) actual test bench.

The clamped end of the beam is firmly screwed to the slip table, and it is at this clamp where the control accelerometer is placed (see Figure 3(b)). This sensor is a Kistler model 8772A50 with a 50 g measuring range and 100 mV/g sensitivity. The response at the free end of the aluminium beam is measured by means of a miniature accelerometer from PCB Piezotronics model 352C22, with a 0.5 g mass, 500 g measuring range and 10 mV/g sensitivity.

These acceleration signals are recorded by means of a National Instruments data acquisition equipment (cDAQ–9174) and its NI–9234 sound and vibration input acquisition modules of 51.2 kS/s. Acceleration signals are post–processed to obtain frequency response functions between tip and excitation accelerations.

Numerical procedure

The state-space differential equations developed in Section “Model” have been solved in Matlab via the ode45 function, which uses Runge-Kutta’s embedded RK5(4) formulation implemented by Dormand-Prince. Simulations were obtained with a 0.5 ms time step.

Frequency analysis was also performed in Matlab via the cpsd function, which estimates the cross (or auto) power spectral density using a 20-s Hanning window with 50% overlap. This allows a frequency resolution of 0.05 Hz, sufficient to detect minimal changes in resonance peaks.

Results and discussion

CLDs

Figure 4 shows the FRF of beams with a single 2005 CLD (0.5 mm thin constricting layer) and different lengths. Following the nomenclature described above, these are specimens $2005_{x 1}^{p %}$ , with $p \in [0, 100]$ . The added CLD increases the modal mass, stiffness and damping as described by equations (24)–(26), but these variations depend on the length of the CLD with respect to the beam length. When this relative length is bellow 25%, the CLD provides high damping and modal stiffness without barely increasing modal mass, which translates into higher natural frequencies and damping coefficients. Beyond 25%, the CLD extends too far from the zone of maximum strain (the clamped end) contributing more to modal mass than to damping or stiffness, so modal mass dominates and the evolution of both natural frequency and damping change trends. The experimental optimum for 2005–CLDs was obtained for the $2005_{x 1}^{25 %}$ case with an $84.0$ % FRF peak reduction with respect to the bare beam.

Figure 4.

Influence of cladding percentage for 2005 CLDs ( $2005_{x 1}^{p %}$ ).

Figure 5 replicates the experimental results from Figure 4 by means of simulations using the model developed in Section “Model.” It can be seen that, even with the many assumptions adopted, a fairly accurate natural frequency evolution with CLD length is obtained. Note that the frequency axis is fairly narrow and that the errors are lower than 4%. Modal damping is also accurately predicted, but only after adjusting material damping parameters with experimental data. Comparison of Figures 5 and 4 allow us to consider the model validated and ready to be used in any beam–CLD combination.

Figure 5.

Influence of cladding percentage for 2005 CLDs ( $2005_{x 1}^{p %}$ ). Model predictions.

Figure 6 compares the FRFs for specimens with 25% CLD cladding and different constraining layers (types 2005 and 4005) as well as single or double–sided cladding configurations. The addition of viscoelastic material via two–sided cladding translates in significant reductions of the FRF peak because it is applied exclusively in the region of large strain. In addition, it is always better to use a constricting layer with higher inertia to maximise shear strain and thus energy dissipation. The experimental optimum among all possible CLD configurations was obtained for the $4005_{x 2}^{25 %}$ case with an FRF peak reduction of $93.7$ %.

Figure 6.

Influence of CLD type and configuration.

Piezoelectric patches

Figure 7 shows the FRF of the beam with a single A15 piezoelectric patch ( $A 15_{x 1}$ ). The plot shows results for the patch in short circuit ( $E = 0$ ), in open circuit ( $D = 0$ ), as well as connected to different R–shunt circuits. The short–circuited patch behaves as a non–piezoelectric layer, whereas the open–circuited patch shifts the resonance frequency as a result of the electromechanical coupling factor. The R–shunt circuits dissipate electric energy forcing the FRF peak to decrease. The experimental optimum, close to the analytical optimum given in Hagood and Flotow²⁰ and Heuss et al.,²¹ provides a $56.4$ % peak reduction for $R = 143.8$ Ohm.

Figure 7.

Influence of R–shunt circuit resistance for $A 15_{x 1}$ .

Figure 8 shows the influence of RL–shunt circuit parameters for the same specimen ( $A 15_{x 1}$ ). Setting a $5.1$ kΩ constant resistance, the mechanical resonance may be fine–tuned with the circuit inductance.²⁷ Low inductance for low mechanical resonance frequency and viceversa. The experimental optimal inductance was found to be $809$ H after searching near the analytical optimal inductance.^20,21 By fixing that value and modifying the circuit resistance one can modulate the resonance peak. High resistance to increase the peak and low resistance to split the peak into two higher ones. The experimental optimum RL–shunt was achieved for $L = 809$ H and $R = 10$ kΩ, leading to a $91.5$ % reduction, $35.1$ % larger reduction than without inductance (R–shunt circuit).

Figure 8.

Influence of RL–shunt circuit resistance and inductance for $A 15_{x 1}$ .

Figure 9 shows the simulated results (using the model in Section “Model”) for the open and short circuit cases, as well as for a particular RL–shunt circuit. Modal frequency and modal damping for the short and open circuit cases are similar to those presented in Figure 7 (differences lower than 1%). A generalised electromechanical coupling coefficient $K_{31}$ to measure the fraction of total modal strain energy converted into electrical energy, may be defined as²⁰

K_{31}^{2} = \frac{{(ω_{n}^{D})}^{2} - {(ω_{n}^{E})}^{2}}{{(ω_{n}^{E})}^{2}}

(40)

Figure 9.

Model predictions in open, short and RL–shunt circuit for $A 15_{x 1}$ .

$ω_{n}^{D}$ and $ω_{n}^{E}$ being the open and short circuit resonance frequency, respectively. Experimental results yield $K_{31}^{2} = 1.90$ %, whereas the model predicts $K_{31}^{2} = 1.65$ %. Optimal peak attenuation (around $90$ %) is correctly predicted, albeit with differences in circuit parameters, $R = 20$ kΩ and $L = 705$ H for the numerical optimum, versus $R = 10$ kΩ and $L = 809$ H experimentally (see Figure 8). This is not only due to model simplifying assumptions, but also to the parasitic impedances introduced by the electronic circuits that generate the synthetic inductance.²⁷

Figure 10 compares three different piezoelectric patches connected to an optimised RL–shunt. The highest attenuation ( $91.5$ %) is obtained for the A15 patch (the one with the highest blocking force), but it is also the one that increases the resonant frequency the most. Conversely, the A11 patch (the one with the lowest blocking force) cannot take full advantage of the RL–shunt. The A12 patch is close to the optimum peak reduction ( $89.8$ %) with a small increase in resonant frequency with respect to the bare beam. This type of patch would be an interesting option if resonant frequency were a critical design feature.

Figure 10.

Influence of the type of piezoelectric patch.

Figure 11 compares the use of one or several piezoelectric patches (A15 with optimal RL–shunt) and their relative position in the case of bonding more than one. Differences among the three cases considered are subtle. The use of two patches is slightly better than the use of just one. And, indeed, when the two are on opposite faces of the beam also leads to slightly better peak reductions than when they are aligned on the same side. This is explained by the dependence of electromechanical coupling on modal deflection (see equation (27)), which is higher the closer to the clamp. Nevertheless, if forced to choose between very similar outputs, the choice would be the $A 15_{x 2 b}$ with an FRF peak reduction of 92.5%.

Figure 11.

Influence of the configuration of piezoelectric patches.

Hybrid systems

Figure 12 compares different combinations of CLD cladding and piezoelectric patches. The particular CLD and patch where found optimal when used independently: 25% of length clad in 4005–CLD, and A15 patch connected to a shunt circuit. It can be seen in the figure that combination $4005_{x 1}^{25 %} A 15_{x 1 c}$ achieves a $93.7$ % FRF peak reduction, and that bonding the piezoelectric patch on the CLD itself, as in $4005_{x 1}^{25 %} A 15_{x 1 d}$ , further improves peak reduction up to $95.0$ %. This is so because the patch at this location increases the inertia of the constraining layer. It is interesting to highlight that, contrary to what might have been expected, no further improvement seems to be achieved by using two CLDs and two piezoelectric patches. This seems to indicate that the combined action of both techniques saturates at levels close to full cancellation ( $95$ %) of the vibration mode.

Figure 12.

Influence of the hybrid configuration.

Figure 13 shows the attenuation predicted by the model for the shunted optimal case ( $4005_{x 1}^{25 %} A 15_{x 1 d}$ ), as well as its open circuit and short circuit cases. There are some differences in resonance frequency, although no more than 8% for a complex beam case with three segments of up to three different layers. The attenuation, on the other hand, is correctly predicted. Again, differences in circuit parameters ( $R = 17$ kΩ and $L = 500$ H for the numerical optimum, vs $R = 1$ kΩ and $L = 710$ H experimentally), are due to resonance frequency discrepancies and parasitic impedances of the synthetic inductance.

Figure 13.

Model predictions in open, short and RL–shunt circuit for hybrid $4005_{x 1}^{25 %} A 15_{x 1 d}$ .

Table 2 shows FRF peak values and attenuation with respect to the reference beam. The optimal configuration is $4005_{x 1}^{25 %} A 15_{x 1 d}$ , as it achieves the maximum attenuation ( $95$ %) with the least number of elements (just one CLD and one piezoelectric patch). However, there are less invasive configurations that achieve almost identical attenuations (around $90$ %), and that might be preferable depending on other design factors such as cost, feasibility of implementation, availability of electronic circuits, accessibility of the structural component, etc. To elaborate on this point, Figure 14 gathers experimental and simulated results for the best hybrid system found in this study alongside with the best cases found with just CLD and just piezoeletric patches. The attenuation and frequency shift of each technique is clealy seen in this figure.

Table 2.

Summary of results.

Passive damping technique	Configuration	FRF peak	Reduction (%)
Reference beam	$W$	215.07	–
CLD	$2005_{x 1}^{10 %}$	67.76	68.5
	$2005_{x 1}^{25 %}$	34.32	84.0
	$2005_{x 1}^{50 %}$	43.46	79.8
	$2005_{x 1}^{75 %}$	67.12	68.8
	$2005_{x 1}^{100 %}$	66.50	69.1
	$2005_{x 2}^{25 %}$	23.69	89.0
	$4005_{x 1}^{25 %}$	24.00	88.8
	$4005_{x 2}^{25 %}$	13.64	93.7
PZT	$A 15_{x 1}$ (R–shunt)	93.69	56.4
	$A 15_{x 1}$ (RL–shunt)	18.21	91.5
	$A 12_{x 1}$	21.95	89.8
	$A 11_{x 1}$	36.58	83.0
	$A 15_{x 2 a}$	16.19	92.5
	$A 15_{x 2 b}$	16.19	92.5
Hybrid	$4005_{x 1}^{25 %} A 15_{x 1 c}$	13.61	93.7
	$4005_{x 1}^{25 %} A 15_{x 1 d}$	10.69	95.0
	$4005_{x 2}^{25 %} A 15_{x 1}$	10.78	95.0
	$4005_{x 2}^{25 %} A 15_{x 2}$	11.33	94.7

Figure 14.

Experimental (a) and simulated (b) results for the best hybrid ( $4005_{x 1}^{25 %} A 15_{x 1 d}$ ), CLD ( $4005_{x 2}^{25 %}$ ) and piezoelectric patch ( $A 15_{x 2 b}$ ) configurations.

Conclusions

The main conclusion is that there exists an optimum combination of CLD and piezoelectric patch that nearly cancels the vibration at the tip of the cantilever beam. The combination is the one labelled $4005_{x 1}^{25 %} A 15_{x 1 d}$ , and the attenuation is 95%. The performance of the hybrid system improves when the piezoelectric patch is adhered to the CLD constraining layer, rather than to the beam itself. Moreover, defying intuition, no further improvements are achieved by using t CLDs and two piezoelectric patches.

It may also be concluded that one can get close to the attenuation level achieved by the hybrid optimum using just CLD cladding or just piezoelectric patches. However, both non–hybrid approaches have some drawbacks that need to be considered when choosing the damping technique. Attenuation with just CLD cladding significantly increases the resonant frequency, and using just piezoelectric patches may penalise energy consumption to synthesise the shunt circuit inductance.

The model developed here, using piecewise modal shapes, represents a fairly good first approximation to experimental results and can be used to predict the behaviour of any CLD–piezoelectric patch combination prior to testing.

Footnotes

Acknowledgements

The authors would like to show their gratitude to the manufacturer Heathcote Industrial Plastics for providing the viscoelastic CLD coatings.

Handling Editor: Sharmili Pandian

ORCID iDs

Miguel Melero

Javier Jiménez

Eduardo Palomares

Antonio J. Nieto

Angel L. Morales

Publio Pintado

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by grant PID2020–113747RB–I00, funded by MCIU/AEI/10.13039/501100011033 and “ERDF A way of making Europe,” and grant SBPLY/23/180225/000172, funded by JCCM–ERDF.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Experimental study of hybrid viscoelastic and piezoelectric systems for vibration cancellation in lightweight structures

Abstract

Keywords

Introduction

Model

Modal parameters for a beam of variable section

Piezoelectric layers

Dynamic equations

Short circuit

Open circuit

Resistive R–shunt circuit

Methodology

Specimens

Test bench

Numerical procedure

Results and discussion

CLDs

Piezoelectric patches

Hybrid systems

Conclusions

Footnotes

Acknowledgements

ORCID iDs

Funding

Declaration of conflicting interests

References