Improved modeling of rods with periodic arrays of shunted piezoelectric patches

Abstract

This work improves the modeling of rods with periodic arrays of shunted piezoelectric patches. The modeling method abandons the previous incorrect assumptions and leads to a more precise result. A numerical investigation is based on an epoxy host rod with a periodic array of resistive–inductive shunted piezoelectric patches. The bandgaps are predicted by the application of Bloch theorem and transfer-matrix method. The results show that the boundary frequencies and attenuation extents both have some difference between the predications of the two models. In particular, the maximum attenuation in the locally resonant gap is much smaller in the result of the improved model.

Keywords

piezoelectric shunting phononic bandgaps periodicity locally resonant

Introduction

Over the last decades, extensive efforts have been exerted to investigate the application of piezoelectric shunting technique in structure-borne vibration control (Forward, 1979; Hagood and Von Flotow, 1991; Hollkamp, 1994; Oliver, 2008). Piezoelectric-shunt damping is an attractive technique for control of vibrating structures, which use damping of shunting circuits to dissipate vibration energy in structures. A passive shunt damper consists of an impedance network made from passive elements, for example, capacitors, inductors, and resistors. Contrary to active strategies, the electrical device of the system is a passive electrical network directly connected to the electrodes of piezoelectric elements. Therefore, the stability of the coupled system can be guaranteed. The effectiveness of the shunting system also stems from its lightweight, easy use, and good performance.

The piezoelectric shunting technique was first introduced by Forward (1979) who used piezoelectric elements with inductive shunts to significantly reduce the mechanical response of a membrane mirror. An elegant formulation of passive shunting was first proposed by Hagood and Von Flotow (1991), which is still most commonly used. The study demonstrated how a piezoelectric patch shunted through a resistive–inductive (RL) circuit, that is, a resonant-shunt, acted as a vibration absorber tuned at the resonance frequency of the circuit. Since then, more complex shunting circuits have been investigated to extend the effectiveness over broader frequency bands (Hollkamp, 1994).

Elastic-wave propagation in periodic structures has been researched for many years (Mead, 1970, 1986; Mead and Markus, 1983). The vibration response of periodic structures has been applied primarily to pass-band and stop-band analyses. Recently, the propagation of elastic or acoustic waves in periodic composite materials called phononic crystals (PCs) has received considerable attention (Goffaux et al., 2002; Kushwaha et al., 1994; Liu et al., 2000; Sigalas and Economou, 1992). There are two kinds of gap formation mechanism for PCs, Bragg scattering mechanism (Kushwaha et al., 1994; Sigalas and Economou, 1992) and locally resonant (LR) mechanism (Goffaux et al., 2002; Liu et al., 2000). The studies have shown that the existence of the Bragg gaps is strongly connected with a large acoustic impedance ratio between the scatters and the matrix. The center frequencies are always given by Bragg’s condition f = n(v/2a) (n = 1, 2, 3, …), where v is the elastic velocity of the matrix material and a is the lattice constant (Goffaux et al., 2002). The pioneering work of Liu et al. (2000) has opened additional field of PCs. The authors studied three-dimensional PCs consisting of cubic arrays of coated lead spheres immersed in an epoxy matrix and proposed a new kind of gap formation mechanism, that is, LR. The resonances of scatter units are the dominant factor in the formation of LR gaps.

Similarly, the structures periodically shunted by resonant-shunts not only produce Bragg gaps but also generate LR gaps. Therefore, a rather different approach to broadband vibration attenuation using simple RL shunts was proposed by Thorp et al. (2001, 2005). They used a periodic array of RL-shunted piezos mounted on the structure to control the propagation of elastic waves. The elastic waves in the structure will be attenuated when the corresponding frequencies are in the Bragg gaps or LR gaps. More recently, this strategy was extended to flat plates hosting periodic arrays of RL-shunted piezo-patches (Casadei et al., 2010; Spadoni et al., 2009).

The modeling method of the periodically shunted structures still commonly use the formulation (Thorp et al., 2001) proposed by Hagood and Von Flotow. This mathematical model is generally accepted and well used in modeling of piezoelectric shunting systems, but it is not suitable for modeling of periodically shunted structures. The foregoing formulation is based on the postulate that the electric displacement on the electrode surfaces of each piezo-patch is uniform, that is, the strain in the patch is uniform. This assumption is not satisfied well in periodically shunted structures because of the above-mentioned Bragg’s condition. In fact, the assumption is unsuitable not only for Bragg gaps but also for LR gaps. So, the mathematical model should be modified when we calculate the bandgaps of periodically shunted structures.

In this article, we abandon the above-mentioned assumption and propose a more suitable formulation when modeling rods with periodic arrays of shunted piezo-patches. The results based on the two mathematical models are both calculated by transfer-matrix method.

Models and transfer-matrix method

Piezoelectric patches with RL shunting circuits are periodically bonded to the surfaces of a host rod as shown in Figure 1. The parts with and without piezoelectric patches are, respectively, denoted as A and B.

The polarizing direction of the piezoelectric patches is along the z-axis, and all surfaces of the patch are free of constraint except along the x-axis as shown in Figure 2. In the piezoelectric equation, the directions x, y, and z in the coordinate system are denoted as 1, 2, and 3, respectively. Therefore, the piezoelectric equations can be reduced to (Thorp et al., 2001)

[\begin{matrix} S_{1} \\ D_{3} \end{matrix}] = [\begin{matrix} s_{11}^{E} & d_{31} \\ d_{31} & ε_{33}^{T} \end{matrix}] [\begin{matrix} T_{1} \\ E_{3} \end{matrix}]

(1)

where S₁ and T₁ are the mechanical strain and stress in the x-axis direction, respectively. $s_{11}^{E}$ is the piezoelectric material’s compliance coefficient at constant electric field intensity. d₃₁ is the piezoelectric constant that couples the mechanical and electrical properties of the piezoelectric material. D₃ is the electric displacement on the electrodes. $ε_{33}^{T}$ is the dielectric constant of the material, and E₃ is the electric field in the patch.

Equation (1) can be solved as

D_{3} = \frac{d_{31} S_{1}}{s_{11}^{E}} + ε_{33}^{s} E_{3}

(2)

where $ε_{33}^{s}$ is given by

ε_{33}^{s} = (ε_{33}^{T} - \frac{d_{31}^{2}}{s_{11}^{E}})

(3)

Because the electrical potential on each electrode is identical in different locations, $E_{3}$ , will be uniform in the patch. As a result, the distribution of $D_{3}$ is determined by the distribution of strain.

Previous modeling strategy

Under the assumptions of the previous model, that the strain and electric displacement are uniform, the charges on the electrodes can be written as

Q = D_{3} A_{s} = (\frac{d_{31} S_{1}}{s_{11}^{E}} + ε_{33}^{s} E_{3}) A_{s}

(4)

where $A_{s}$ is the area of electrodes, which is given by $A_{s} = lw$ .

The complex impedance of shunting circuits can be written as

Z = Ls + R

(5)

where s is the Laplace operator, L is the inductance of the shunting inductor, and R is the resistance of the shunting resistor.

Applying the definition of current and its relation with voltage, it can be written as

I = - sQ = \frac{V}{Z}

(6)

where V is the voltage on the electrodes of piezo-patches.

Substituting equations (4) and (5) into equation (6), the voltage can be solved as

V = - (w \frac{d_{31}}{s_{11}^{E}} S_{1} + ε_{33}^{s} E_{3}) A_{s} (L s^{2} + Rs)

(7)

The voltage also can be expressed as

V = E_{3} h

(8)

where h is the thickness of piezoelectric patches.

Combining equations (7) and (8), the electric field can be solved as

E_{3} = \frac{- d_{31} A_{s} (L s^{2} + Rs)}{s_{11}^{E} h [C_{p}^{s} (L s^{2} + Rs) + 1]} S_{1}

(9)

where

C_{p}^{s} = \frac{ε_{33}^{s} A_{s}}{h}

(10)

Substituting equation (9) into equation (1), the relation between stress and strain can be simplified as

T_{1} = {\frac{1}{s_{11}^{E}} + \frac{d_{31}^{2} A_{s} (L s^{2} + Rs)}{h {(s_{11}^{E})}^{2} [C_{p}^{s} (L s^{2} + Rs) + 1]}} S_{1}

(11)

Therefore, under the assumptions of the previous model, the effect of shunting circuits to the piezo-patch can be described as an equivalent modulus

E_{p} = \frac{T_{1}}{S_{1}} = \frac{1}{s_{11}^{E}} + \frac{d_{31}^{2} A_{s} (L s^{2} + Rs)}{h {(s_{11}^{E})}^{2} [C_{p}^{s} (L s^{2} + Rs) + 1]}

(12)

However, it was a contradiction that the assumption of uniform strain was relaxed, when the bandgaps were calculated by transfer-matrix method with the previous modeling strategy (Thorp et al., 2001). In other words, the formula of equivalent modulus was used in the transfer-matrix method, while the constraint of uniform stain was left without consideration.

The transfer matrix can be derived from the governing differential equations and periodic boundary conditions, which is shown as follows.

The governing differential equations of longitudinal vibration in A and B of cell n, as shown in Figure 3, can be respectively given by

(ρ_{b} A_{b} + 2 ρ_{p} A_{p}) \frac{\partial^{2} ϕ_{n, A} (x, t)}{\partial t^{2}} = \frac{\partial}{\partial x} [(E_{b} A_{b} + 2 E_{p} A_{p}) \frac{\partial ϕ_{n, A} (x, t)}{\partial x}]

(13a)

ρ_{b} A_{b} \frac{\partial^{2} ϕ_{n, B} (x, t)}{\partial t^{2}} = E_{b} A_{b} \frac{\partial^{2} ϕ_{n, B} (x, t)}{\partial x^{2}}

(13b)

where $ρ_{b}$ , $A_{b}$ , $E_{b}$ and $ρ_{p}$ , $A_{p}$ , $E_{p}$ are the densities, cross-section areas, and elastic modules of the host rod and piezo-patches, respectively. $ϕ_{n, A} (x, t)$ and $ϕ_{n, B} (x, t)$ are the displacements of the rod at coordinate x and time t.

The harmonic solution to equation (13) can be expressed as

ϕ_{n, A} (x, t) = Φ_{n, A} (x) e^{i ω t}

(14a)

ϕ_{n, B} (x, t) = Φ_{n, B} (x) e^{i ω t}

(14b)

where $Φ_{A} (x)$ and $Φ_{B} (x)$ are the vibration amplitudes. These can be written as

Φ_{n, A} (x) = A_{n, 1} \cos (λ_{1} x) + A_{n, 2} \sin (λ_{1} x)

(15a)

Φ_{n, B} (x) = B_{n, 1} \cos (λ_{2} x) + B_{n, 2} \sin (λ_{2} x)

(15b)

where $A_{n, 1}$ , $A_{n, 2}$ , $B_{n, 1}$ , and $B_{n, 2}$ are undetermined coefficients, $λ_{1}$ and $λ_{2}$ are wave numbers, which are given by

λ_{1} = \sqrt{\frac{ρ_{b} A_{b} + 2 ρ_{p} A_{p}}{E_{b} A_{b} + 2 E_{p} A_{p}}} ω

(16a)

λ_{2} = \sqrt{\frac{ρ_{b}}{E_{b}} ω}

(16b)

The continuities of displacement and force at the interfaces between cell n−1 and n give

Φ_{n, A} (0) = Φ_{n - 1, B} (b)

(17a)

(E_{b} A_{b} + 2 E_{p} A_{p}) Φ_{n, A}' (0) = E_{b} A_{b} Φ_{n - 1, B}' (b)

(17b)

The matrix form of equation (17) can be given by

K_{p 1} A_{n} = H_{p 1} B_{n - 1}

(18)

where $A_{n} = [\begin{matrix} A_{n, 1} A_{n, 2} \end{matrix}]$ , $B_{n - 1} = [\begin{matrix} B_{n - 1, 1} B_{n - 1, 2} \end{matrix}]$ and

K_{p 1} = [\begin{matrix} 1 & 0 \\ 0 & λ_{1} (E_{b} A_{b} + 2 E_{p} A_{p}) \end{matrix}]

(19a)

H_{p 1} = [\begin{matrix} \cos (λ_{2} b) & \sin (λ_{2} b) \\ - λ_{2} E_{b} A_{b} \sin (λ_{2} b) & λ_{2} E_{b} A_{b} \cos (λ_{2} b) \end{matrix}]

(19b)

Similarly, the continuities of displacement and force at the interfaces between A and B in cell n give

Φ_{n, A} (a) = Φ_{n, B} (0)

(20a)

(E_{b} A_{b} + 2 E_{p} A_{p}) Φ_{n, A}' (a) = E_{b} A_{b} Φ_{n, B}' (0)

(20b)

The matrix form of equation (20) can be given by

K_{p 2} A_{n} = H_{p 2} B_{n}

(21)

where

K_{p 2} = [\begin{matrix} \cos (λ_{1} a) & \sin (λ_{1} a) \\ - λ_{1} (E_{b} A_{b} + 2 E_{p} A_{p}) \sin (λ_{1} a) & λ_{1} (E_{b} A_{b} + 2 E_{p} A_{p}) \cos (λ_{1} a) \end{matrix}]

(22a)

H_{p 2} = [\begin{matrix} 1 & 0 \\ 0 & λ_{2} E_{b} A_{b} \end{matrix}]

(22b)

Based on equations (18) and (21), the relation between (n−1)th cell and nth cell is given by

B_{n} = T_{p} B_{n - 1}

(23)

where $T_{p} = H_{p 2}^{- 1} K_{p 2} K_{p 1}^{- 1} H_{p 1}$ is the transfer matrix got from the previous modeling strategy.

Improved modeling strategy

If the assumption of uniform strain is discarded, the more general formula of electrical charges on the electrodes can be described as

Q = \int \int_{A_{s}} D_{3} ds

(24)

Substituting equation (2) into equation (24), it can be rewritten as

Q = \int_{0}^{w} \int_{0}^{l} (\frac{d_{31} S_{1}}{s_{11}^{E}} + ε_{33}^{s} E_{3}) dxdy

(25)

If the displacement of piezo-patch in the x direction is denoted as $u (x)$ , the strain in the piezo-patch can be expressed as

S_{1} = \frac{du}{dx}

(26)

So equation (25) can be solved as

Q = \int_{0}^{w} \int_{0}^{l} \frac{d_{31}}{s_{11}^{E}} \frac{du}{dx} dxdy + ε_{33}^{s} E_{3} A_{s} = \frac{d_{31} w}{s_{11}^{E}} [u (l) - u (0)] + ε_{33}^{s} E_{3} A_{s}

(27)

Substituting equations (27) and (5) into equation (6), the voltage can be solved as

V = - {\frac{d_{31} w}{s_{11}^{E}} [u (l) - u (0)] + ε_{33}^{s} E_{3} A_{s}} (L s^{2} + Rs)

(28)

Combining equations (8) and (28), the electric field can be solved as

E_{3} = \frac{- w d_{31} [u (l) - u (0)] (L s^{2} + Rs)}{s_{11}^{E} h [C_{p}^{s} (L s^{2} + Rs) + 1]}

(29)

Substituting equation (29) into equation (1), the relation between stress and strain can be simplified as

T_{1} = \frac{S_{1}}{s_{11}^{E}} + \frac{w {d_{31}}^{2} [u (l) - u (0)] (L s^{2} + Rs)}{{(s_{11}^{E})}^{2} h [C_{p}^{s} (L s^{2} + Rs) + 1]}

(30)

Comparing equations (11) and (30), one can easily find that the two models generate big differences when modeling the shunted piezo-patch. The shunting effect is modeled as an appended elastic modulus with the previous modeling strategy, while the improved modeling strategy reveals that the shunting effect should be modeled as an appended stress. So an error will occur when the deformation of piezo-patch is non-uniform with the previous modeling strategy. Moreover, the error increases with the increase of vibration frequency, where the corresponding wavelength is short when compared with the size of piezo-patches.

Obviously, the derivation of transfer matrix also becomes different. The governing differential equations of longitudinal vibration in A and B of cell n, as shown in Figure 3, can be respectively given by

(ρ_{b} A_{b} + 2 ρ_{p} A_{p}) \frac{\partial^{2} ϕ_{n, A} (x, t)}{\partial t^{2}} = \frac{\partial}{\partial x} [E_{b} A_{b} \frac{\partial ϕ_{n, A} (x, t)}{\partial x} + 2 T_{1} A_{p}]

(31a)

ρ_{b} A_{b} \frac{\partial^{2} ϕ_{n, B} (x, t)}{\partial t^{2}} = E_{b} A_{b} \frac{\partial^{2} ϕ_{n, B} (x, t)}{\partial x^{2}}

(31b)

Substituting equation (30) into equation (31a), it can be rewritten as

(ρ_{b} A_{b} + 2 ρ_{p} A_{p}) \frac{\partial^{2} ϕ_{n, A} (x, t)}{\partial t^{2}} = (E_{b} A_{b} + \frac{2 A_{p}}{s_{11}^{E}}) \frac{\partial^{2} ϕ_{n, A} (x, t)}{\partial x^{2}}

(32)

The following derivation is similar to that of the previous modeling method except for the boundary conditions.

The continuities of displacement and force at the interfaces between cells n−1 and n give

Φ_{n, A} (0) = Φ_{n - 1, B} (b)

(33a)

E_{b} A_{b} Φ_{n, A}' (0) + 2 T_{1} A_{p} = E_{b} A_{b} Φ_{n - 1, B}' (b)

(33b)

Substituting $s = i ω$ and equation (30) into equation (33b), it can be rewritten as

(E_{b} A_{b} + \frac{2 A_{p}}{s_{11}^{E}}) Φ_{n, A}' (0) + \frac{2 A_{p} {wd}_{31}^{2} [Φ_{n, A} (a) - Φ_{n, A} (0)] (- L ω^{2} + iR ω)}{{(s_{11}^{E})}^{2} h [C_{p}^{s} (- L ω^{2} + iR ω) - 1]} = E_{b} A_{b} Φ_{n - 1, B}' (b)

(33c)

One can obtain the matrix form of equations (33a) and (33c)

K_{i 1} A_{n} = H_{i 1} B_{n - 1}

(34)

where

K_{i 1} = [\begin{matrix} 1 & 0 \\ ψ [\cos (λ_{1} a) - 1] & λ_{1} (E_{b} A_{b} + \frac{2 A_{p}}{s_{11}^{E}}) + ψ \sin (λ_{1} a) \end{matrix}]

(35a)

ψ = \frac{2 A_{p} {wd}_{31}^{2} (- L ω^{2} + iR ω) [\cos (λ_{1} a) - 1]}{{(s_{11}^{E})}^{2} h [C_{p}^{s} (- L ω^{2} + iR ω) - 1]}

(35b)

H_{i 1} = [\begin{matrix} \cos (λ_{2} b) & \sin (λ_{2} b) \\ - λ_{2} E_{b} A_{b} \sin (λ_{2} b) & λ_{2} E_{b} A_{b} \cos (λ_{2} b) \end{matrix}]

(35c)

Similarly, the continuities of displacement and force at the interfaces between A and B in cell n give

Φ_{n, A} (a) = Φ_{n, B} (0)

(36a)

E_{b} A_{b} Φ_{n, A}' (a) + 2 T_{1} A_{p} = E_{b} A_{b} Φ_{n, B}' (0)

(36b)

The matrix form of equation (36) can be given by

K_{i 2} A_{n} = H_{i 2} B_{n}

(37)

where

K_{i 2} = [\begin{matrix} \cos (λ_{1} a) & \sin (λ_{1} a) \\ - λ_{1} (E_{b} A_{b} + \frac{2 A_{p}}{s_{11}^{E}}) \sin (λ_{1} a) + ψ [\cos (λ_{1} a) - 1] & λ_{1} (E_{b} A_{b} + \frac{2 A_{p}}{s_{11}^{E}}) \cos (λ_{1} a) + ψ \sin (λ_{1} a) \end{matrix}]

(38a)

H_{i 2} = [\begin{matrix} 1 & 0 \\ 0 & λ_{2} E_{b} A_{b} \end{matrix}]

(38b)

Based on equations (34) and (37), the relation between (n−1)th cell and nth cell is given by

B_{n} = T_{i} B_{n - 1}

(39)

where $T_{i} = H_{i 2}^{- 1} K_{i 2} K_{i 1}^{- 1} H_{i 1}$ is the transfer matrix.

Comparing equations (19) and (22) with equations (35) and (38), one can find that the transfer matrices extracted with the two modeling methods are quite different.

Bandgaps description

Due to the periodicity of the infinite structure in the x direction, the vector $B_{n}$ must satisfy the Bloch theorem (Kittle, 1986)

B_{n} = e^{μ} B_{n - 1}

(40)

where $μ$ is the propagation constant.

Combining equation (40) with equation (23) or (39), the propagation constant can be determined by the eigenvalues of the transfer matrix

| T_{ζ} - e^{μ} I | = 0 (ζ = p or i)

(41)

where I is unit matrix.

The bandgaps of the periodically shunted rod can be described by the propagation constant. The real part of the propagation constant, namely, the attenuation constant, indicates that amplitude attenuation occurs as the elastic wave propagates from one cell to the next.

Results and discussion

As an example, we choose epoxy as the material for the host rod and lead zirconate titanate (PZT)-5H as the material for piezo-patches. The geometric properties and material parameters employed in the calculations are listed in Tables 1 and 2.

Table 1.

Geometric and material properties of the host rod

Parameter	Value
Density	1180 kg/m³
Elastic modulus	4.35 × 10⁹ Pa
Width	0.02 m
Thickness	0.01 m

Table 2.

Geometric and material properties of the piezo-patch

Parameter	Value	Parameter	Value
Density	7500 kg/m³	Width	0.02 m
$s_{11}^{E}$	1.65 × 10⁻¹¹ m³/N	Thickness	5.0 × 10⁻⁴ m
$d_{31}$	−2.74 × 10⁻¹⁰ C/m²	Length	0.05 m
$ε_{33}^{T}$	3.01 × 10⁻⁸ F/m	Lattice constant	0.08 m

The bandgaps are, respectively, calculated on the two mathematical models. Figure 4 shows the variation of the attenuation constant over the 0–20 kHz range when the shunting inductance and resistance are L = 4.5 mH and $R = 1 Ω$ , respectively. The first gap is induced by resonances of RL shunts, of which the frequency is strongly related to the eigenfrequency, that is, it is a LR gap. The second gap is a Bragg gap, which is mainly determined by the lattice constant, material properties of components, and so on.

Comparing the two curves, one can find that the bandgaps obtained by the two modeling methods are different. The LR gap calculated from the previous model has a much bigger attenuation. From equation (27), one can find that the charges on the electrodes of a piezo-patch are only determined by the relative difference between boundary displacements, that is, $u (l) - u (0)$ . Therefore, the strain energy stored in the piezo-patch and host rod are smaller than the actual case, if one assumes the strain in the piezo-patch is uniform. In other words, the previous model exaggerates the shunting capability of the circuits, that is, the percentage of shunted energy is larger than actual value. The result shows that the Bragg gaps are also different in the two cases. The lower boundary of the Bragg gap is much lower when one uses the improved model to calculate the gap.

If $R = 0$ , the shunting circuits become undamped, so the boundaries of the gaps can be easily discriminated as shown in Figure 5. The lower and upper boundaries of the LR gap are denoted as A and B, and the maximum attenuation point is denoted as C. The lower and upper boundaries of the Bragg gap are denoted as D and E.

Figure 6 shows the variations of characteristic frequencies with shunting inductance, which are calculated based on the two different mathematical models. The lower boundary frequency calculated from the improved model is much lower than that calculated from the previous model as shown in Figure 6(a). The frequency at the maximum attenuation is much higher when one uses the improved modeling method. However, the upper boundary frequencies from the two models almost completely overlap. Figure 6(b) shows the variations of boundary frequencies of the Bragg gap. One can find that a difference mainly emerges at the lower boundary of the Bragg gap.

The vibration mode corresponding to the lower bounding frequency of the Bragg gap is shown in Figure 7. Comparing the two subfigures, one can find that the mode shapes obtained from the two modeling strategies are quite similar, and both show that the difference of the displacements between the two ends of each piezo-patch is 0, that is, $u (l) - u (0) = 0$ . Therefore, the positive and negative charges on the electrodes will completely counteract each other, leading to vanishing of the shunting effect at this frequency. However, when referring to Figure 6(b), one can observe that the lower bounding frequency got from the previous model varies with the inductance L, which reveals that the shunting effect still exists here. Obviously, there is contradiction here, which shows that the previous modeling strategy is not self-consistent. Furthermore, we get the variation of attenuation constants with shunting resistance R as shown in Figure 8. From the foregoing discussion of Figure 7, one can find that the result of the improved model (Figure 8(a)) is more reasonable, where the damping effect will vanish at the lower bounding frequency of the Bragg gap. On the other hand, a noticeable discrepancy of attenuation constant will be produced at this frequency.

Figure 9 shows the variations of bandgaps calculated from the two mathematical models. The value of color spectrum represents the magnitude of the attenuation constant. Comparing the distribution of the attenuation constant in the two maps, one can find that the results of the two models are different, especially at the location where the LR gap and Bragg gap start overlapping. Moreover, the maximum attenuation in the LR gap is always much bigger when using the previous model.

In order to verify the theoretical results, the transmission property of the shunted rod with six periods is simulated using ANSYS. Each of the components of the rod are, respectively, meshed with the elements listed in Table 3.

Table 3.

Element types used to mesh the shunting system

Component	Element type
Host rod	SOLID45
Piezo-patch	SOLID5
Inductor	CIRCUIT94
Resistor	CIRCUIT94

The finite element model is shown in Figure 10. Exciting the rod with unit displacement on one end, the response at the other end is shown in Figure 11(a). In order to compare the theoretical results of the two modeling strategies with the simulated results, the transmission properties of theoretical calculation are shown in Figure 11(b).

Comparing Figure 11(a) with (b), one can find that the results of improved model are in better agreement with the ANSYS simulation. Specifically, the attenuation in the LR gap predicted by the previous model reveals a considerable error.

Conclusion

In conclusion, the results calculated from the two models are quite different. The maximum attenuation in the LR gap is much smaller when using the improved modeling method proposed in this article, while the corresponding frequency is higher. There is a big difference between the results of the two models with regard to the boundary frequencies. The modeling method proposed in this article is more suitable for modeling periodically shunted structures than that proposed in the study by Thorp et al. (2001) because it discards the unsuitable assumptions and leads to a more precise result.

Footnotes

Funding

This work was funded by the National Natural Science Foundation of China (grant numbers: 50905182 and 50875255).

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