Modal analysis and demonstration of metastructure combining local resonators and an autonomous synchronized switch damping circuit

Abstract

A locally resonant metastructure is a promising approach for low-frequency vibration attenuation, whereas the attachment of many resonators results in unnecessary and multiple resonance outside the bandgap. To address this issue, we propose a damping metastructure combining local resonators and an autonomous synchronized switch damping circuit. On the basis of modal analysis, we derive an electromechanically coupled equation of the proposed metastructure. The piezo ceramics, which are attached on a small portion of the metastructure and connected to the circuit, remarkably decrease the magnitude of the resonant vibration with no extra sensors, signal processors, or power sources. The displacement at unnecessary resonance was decreased by approximately 75%. The results of the coupled analysis were similar to the experimentally observed results in terms of the location and width of the bandgap on the frequency axis and the decreased displacement for the circuit. The proposed technique can overcome the disadvantage of the metastructure.

Keywords

Metastructure local resonator bandgap synchronized switch damping vibration attenuation

Introduction

A technique for reducing a mechanical vibration or noise in vehicles,^1,2 aerospace structures, or industrial machinery³ must satisfy the requirements of being lightweight, compact, low in power consumption, and cost-effective. A technique using a piezoelectric transducer⁴ and the semi-active vibration attenuation termed as the synchronized switch damping on inductor (SSDI)⁵ can satisfy these requirements, and its development is advancing. When the switch closes at the maximum/minimum of the piezo voltage, a high electrical current flows in the circuit and the piezo voltage flips immediately through the formation of an electrical connection between the inductor and piezo intrinsic capacitor. The vibrational energy of the structure is thereby dissipated. In an early phase of development, a signal processor and a sensor activated by an external power source were required to ensure the switch closes at the precise timing of the maximum/minimum of the piezo voltage. Subsequently, Lallart et al.⁶ proposed a novel technique which allows to realize a self-sensing and self-powered system. Adopting the self-sensing and/or self-powered technique, derivative techniques such as a self-sensing (SS) synchronized switch damping on voltage (SS-SSDV) technique,⁷ a SS-synchronized switch damping on negative capacitance (SS-SSDNC) technique,⁸ and a SS-technique using a microcontroller unit with low power consumption^9,10 were developed. These techniques can also be applied to reduce the level of sound induced by a structure’s vibration that is excited by a sound source.¹¹ Recently, to predict the attenuation performance of the autonomous SSDI technique more accurately, an electromechanical model incorporating nonlinear piezoelectricity was reported.^12,13

A vibration attenuation technique of designing a metastructure is another promising approach. The metastructure is formed by deploying locally resonant structures periodically or creating a spatial variation in elasticity or mass density over a host structure. The metastructure can realize artificial properties such as negative mass and modulus, wave focusing and steering, elastic mirrors, or tunable bandgaps.¹⁴ Based on the local resonance or Bragg scattering, the metastructure can form a shielding frequency band, which is referred to as a bandgap. Compared with the Bragg scattering metastructure, the locally resonant (LR) metastructure is more suited to attenuate low-frequency vibration, which occurs in vehicles^1,2 or machinery³ in the range of several tens to several hundreds of Hertz, because it allows the absorption of a vibrational wave whose wavelength is much larger than the lattice constant of the periodic structure.¹⁵ The LR metastructure is formed by distributing mechanical or electromechanical resonators on a host structure periodically. This study briefly explains the state-of-the-art. To estimate the bandgap width of the LR beam, Liu et al.¹⁶ proposed a transfer matrix method (TMM) and validated that the frequency response function obtained using the TMM was in good agreement with that obtained using the finite element method. To create multiple bandgaps and/or widen the bandgap, techniques using a four-link-mechanism,¹⁷ interconnected resonator,^18,19 or quadruple-mode resonator²⁰ were proposed. In the TMM, the Bloch’s theorem and/or the continuity between the neighboring unit cells are employed, whereas the vibrational mode shape function of the host structure, which is determined considering the boundaries, is not considered. For a metastructure, which has a finite number of resonators and is designed for attenuating low-frequency vibration, ignoring the mode shapes may lead to inaccurate prediction of the resonance frequencies or displacement outside the bandgap. Sugino et al.^21,22 established an analytic methodology for the LR metastructure based on modal analysis and provided a closed-form solution for the bandgap width. Similarly, adopting the modal analysis, El-Borgi et al.²³ derived an equation of motion for the LR metastructure, allowing the creation of two bandgaps using two local resonators with different resonant frequencies. Unlike the aforementioned metastructure having a mechanical resonator, the metastructure having an electromechanical resonator uses piezo elements located periodically and a shunt circuit. Aghakhani et al.²⁴ developed a system-level analytic methodology for a plate-type piezo metastructure connecting to a resistive-inductive shunt circuit. Sugino et al.²⁵ proposed a methodology to achieve a wide bandgap by merging the bandgaps created by mechanical and electromechanical resonators. Chen et al.²⁶ realized an adaptive metamaterial beam by introducing a tunable electromechanical resonator, which comprises a piezo stack and shunt circuit with a negative capacitor and inductor. Compared with the use of the LR metastructure having a resistive-inductive shunt circuit, the use of the SSDI-based phononic metastructure reduces the required inductance remarkably,²⁷ allowing for a lightweight and compact metastructure. Bao et al.²⁸ proposed a unique technique of tri-interleaved SSDI topology, which uses the periodic cells of the piezo elements and the SSDI technique more efficiently. Based on the TMM, they derived a coupled equation of the complex system of the metastructure.

Given that piezo ceramics/stacks are costly, a large metastructure throughout which many of piezo ceramics/stacks are deployed²⁴⁻²⁸ may be impractical. This study considers a metastructure having mechanical resonators because of the cost-effectiveness and ease of design of such resonators. However, the addition of local resonators to the host structure creates unnecessary and multiple resonant peaks outside the bandgap. The resonant peaks on the low-frequency side pose a serious problem for practical application because the ambient vibration usually has a wide frequency band and the magnitude of displacement/acceleration at low frequency is large. ³ To address the practical problem, we propose a damping metastructure combining mechanical resonators and the autonomous SSDI technique. The piezo ceramics, which are attached on a small portion of the metastructure and connected to the circuit, decrease the magnitude of the unnecessary resonant vibration outside the bandgap with no requirement for any sensors, signal processors, or external power sources. It is important to develop an analysis technique for predicting the attenuation performance of the proposed metastructure. Two-way coupled analysis^12,29,30 combining numerical software and a circuit simulator can precisely simulate electromechanical properties of the device using the synchronized switch technique, although the analysis requires a fair amount of computational time. In this paper, we establish a simpler numerical approach by simplifying the coupled equation. We employ a modal analysis approach to derive an electromechanically coupled equation of the proposed metastructure, considering a switching delay that cannot be negligible for the autonomous system.³¹ We then solve the coupled equation and confirm that our modeling replicates experimental results.

Structure, circuit, and analysis

Metastructure

Figure 1 shows the damping metastructure embedded in the local resonators. Local resonators, each having a cantilever structure with a tip mass, were located periodically on the main composite beam with a spacing l_s of 32 mm. As shown in Figure 1(b), a stainless-steel sheet (AISI304, thickness: 0.3 mm) with 12 side beams (27 × 10 × 0.3 mm³) was formed by electro-discharging machining. The metastructure was formed manually by inserting the thin stainless-steel sheet between two straight aluminum beams (AISI5052, 192 × 20 × 2 mm³ per beam, excluding the clamp part) and then attaching a steel mass (10 × 10 × 6 mm³) on the tip of the side beams and the piezo ceramics (Fuji Ceramics Corporation, Japan, C-6, 20 × 20 × 0.2 mm³ per sheet) on both sides of and near the edge of the composite beam. An epoxy structural adhesive (3M, DP460) was used to laminate the components. We attached a costly material of piezo ceramics on only a small portion of the metastructure, thereby realizing a cost-effective attenuation technique. We connected the two piezo ceramics in series to increase the electromechanical coupling and thus the damping force. The series-connected piezo capacitance C_p was 16.1 nF. The metastructure was designed to create a bandgap at the first eigenfrequency (approximately 86 Hz) of the composite beam with no local resonators. As explained later, the metastructure adversely has two resonant peaks near the edge of the bandgap. We decreased the displacement magnitude of the unnecessary resonant peaks using the piezo ceramics connected to the SSDI circuit.

Figure 1.

(a) Oblique view of the locally resonant metastructure with the piezo ceramics, (b) top view of the stainless-steel sheet, and (c) side and (d) front views of the metastructure.

Autonomous SSDI circuit

Figure 2 shows schematics of the autonomous SSDI circuit used in this study and the waveforms of the piezo voltage v_p and displacement in the steady state. The circuit topology was originally designed for the increase in the output voltage of vibration energy harvesters³²; subsequently, it was applied to the SSDI technique.¹² The circuit in this study mainly comprises npn (T_r1 and T_r3, Nexperia, PBHV8140Z) and pnp (T_r2 and T_r4, Nexperia, PBHV9540Z) transistors, diodes (Diodes Incorporated, MUR120-T), a base resistor R_b (3.3 kΩ), an envelope capacitor C_ed (1 nF), and an inductor L₀. Compared with other SSD-based circuits,⁶⁻⁸ the circuit can increase the magnitude of the piezo voltage and thus the attenuation performance of the SSDI by decreasing C_ed required. The advantage is greater for a device having series-connected piezo elements in which the envelope capacitance is not negligible compared with the piezo capacitance, because the circuit can reduce the electric current consumed in the envelope capacitor. As shown in Figure 2(b), when the positive switch closes for a period that is much shorter than the vibrational period of the metastructure, a high electrical current flows across the inductive circuit and v_p flips immediately from v_M to −v_m. In the same manner, the negative switch activates with the opposite polarity of the current, voltage (from −v_M to v_m), and displacement. The parasitic resistor r in the inductive shunt circuit, which is the series resistance of the inductor, switch, and electrical wires, affects the inversion efficiency of the piezo voltage. The voltage ratio of v_m/v_M, which is referred to as the inversion coefficient γ, is a critical parameter that determines the attenuation performance. The piezo ceramics are open-circuited except for the switching. The autonomous SSDI circuit inherently causes a switch delay τ relative to the ideal switch timing because of the on-voltage threshold and the parasitic elements in the transistors and diodes.^7,12 This study incorporates the switch delay into the model to replicate the real performance of the circuit. The detailed workflow of the circuit can be referred to in our previous paper¹² and a long explanation is omitted for brevity here.

Figure 2.

(a) Autonomous SSDI circuit and (b) waveforms of the piezo voltage v_p and displacement in the steady state.

Coupled analysis

On the basis of modal analysis, we first derive a governing equation for the LR metastructure with the autonomous SSDI circuit and then solve the derived equation numerically.³¹ As shown in Figure 1, the electromechanical equations for the proposed metastructure are derived as

m_{c} \ddot{w} + d_{c} \dot{w} + E I w^{″ ″} - \sum_{j = 1}^{S} m_{r, j} ω_{r, j}^{2} u_{j} (t) δ (x - x_{j}) - θ [δ^{'} (x) - δ^{'} (x - l_{p})] v_{p} (t) = - m_{c} {\ddot{w}}_{b} (t)

(1)

{\ddot{u}}_{j} (t) + 2 ζ_{r, j} ω_{r, j} {\dot{u}}_{j} (t) + ω_{r, j}^{2} u_{j} (t) = - \ddot{w} (x_{j}, t) - {\ddot{w}}_{b} (t)

(2)

θ \int_{0}^{l_{p}} {\dot{w}}^{″} d x + C_{p} {\dot{v}}_{p} (t) + i = 0

(3)

where w_b is the base displacement controlled by the shaker, w(x,t) is the z-directional displacement of the composite beam relative to the base, m_c is the mass per unit length of the composite beam, d_c and EI are the damping constant and flexural rigidity of the composite beam, m_r,j, ω_r,j, u_j, x_j, and ζ_r,j are the lumped mass, first angular eigenfrequency, displacement relative to the composite beam, location along the x-axis, and damping ratio of the j-th local resonator, S (= 12) is the number of local resonators, δ is the Dirac delta function, θ is the piezo coefficient, v_p is the piezo voltage across the piezo bimorph, and i is the electrical current flowing to the circuit. θ is calculated as e₃₁b(h_p + h_c)/2, where e₃₁ is the piezo coupling constant, b is the width of the piezo ceramics and composite beam, and h_p and h_c are, respectively, the thicknesses of the piezo ceramic and composite beam. Accordingly, the absolute z-directional displacement w_abs(x,t) of the composite beam can be calculated as w(x,t) + w_b(t). The prime mark and dot above the characters represent the x-partial and time-derivative, respectively. For simplicity, m_c with/without the piezo ceramics was treated as the same value in this study. The local resonator was designed such that the bandgap expands around the first eigenfrequency of the composite beam with no local resonators. The fourth and fifth terms on the left-hand side of (1) represent the vibrational absorption by local resonators and the piezo force by the piezo ceramics connecting to the SSDI circuit, respectively. The vibrational absorption and the piezo force terms are coupled with (2) and (3), respectively. Additionally, (3) about the law of conservation of electrical currents will be solved by analyzing the SSDI circuit. The vibrational absorption term in (1) creates two resonances near the edges of the bandgap because of the addition of local resonators. The piezo force, which acts as a damping force by the SSDI circuit, reduces the peak magnitude of the two resonances.

We determine w(x,t) by solving the eigenvalue problem of the composite beam with no local resonators. Based on the discretization, w(x,t) can be expanded in series as

w (x, t) = \sum_{i = 1}^{N} ϕ_{i} (x) q_{i} (t)

(4)

where

ϕ_{i}

is the i-th mode shape function and q_i is the i-th modal coordinate, and N is the number of modes considered in the analysis. Under the condition of being undamped and short-circuited, the equation of the eigenvalue problem is written as

m_{c} \ddot{w} + E I w^{″ ″} = 0

(5)

By substituting (4) into (5), $ϕ_{i}$ must satisfy the equation $ϕ_{i}^{′'''} - m_{c} ω_{c, i}^{2} ϕ_{i} / E I = 0$ , where ω_c,i is the i-th angular eigenfrequency of the composite beam. As a general solution, the i-th mode shape function can be expressed as

ϕ_{i} (x) = A_{i 1} \cosh (\frac{λ_{i}}{l_{c}} x) + A_{i 2} \cos (\frac{λ_{i}}{l_{c}} x) + A_{i 3} \sinh (\frac{λ_{i}}{l_{c}} x) + A_{i 4} \sin (\frac{λ_{i}}{l_{c}} x)

(6)

where A_i1–A_i4 are constant coefficients of the i-th mode shape function and l_c is the length of the composite beam. λ_i is the constant coefficient and satisfies the equation

ω_{c, i} = \frac{λ_{i}^{2}}{l_{c}^{2}} \sqrt{\frac{E I}{m_{c}}}

(7)

λ_{i}

and

ω_{c, i}

can be determined by considering the boundary conditions of the tip and fixed ends of the beam. For the tip end,

ϕ_{i}

satisfies

ϕ_{i}^{″} (l_{c}) = ϕ_{i}^{′''} (l_{c}) = 0

, considering the balance of the moment and shear force. For the fixed end,

ϕ_{i}

satisfies

ϕ_{i} (0) = ϕ_{i}^{'} (0) = 0

. In this study, the mass normalization was used to determine A_i1–A_i4. Thus, the mode shape function satisfies the following orthogonal equation

m_{c} \int_{0}^{l_{c}} ϕ_{i} ϕ_{k} d x = δ_{i k}

(8)

where δ_ik is the Kronecker delta. By taking these steps, we obtain a specific function of

ϕ_{i}

in (6).

By substituting (4) into (1)−(3), multiplying by $ϕ_{k}$ on both sides of (1), and integrating from zero to l_c with respect to x, the electromechanical equation can be simplified as

{\ddot{q}}_{k} + 2 ζ_{c, k} ω_{c, k} {\dot{q}}_{k} + ω_{c, k}^{2} q_{k} - \sum_{j = 1}^{S} m_{r, j} ω_{r, j}^{2} u_{j} ϕ_{k} (x_{j}) - α_{k} v_{p} = - m_{c} \int_{0}^{l_{c}} ϕ_{k} d x {\ddot{w}}_{b}

(9)

{\ddot{u}}_{j} (t) + 2 ζ_{r, j} ω_{r, j} {\dot{u}}_{j} (t) + ω_{r, j}^{2} u_{j} (t) + \sum_{n = 1}^{N} ϕ_{n} (x_{j}) {\ddot{q}}_{n} = - {\ddot{w}}_{b} (t)

(10)

\sum_{l = 1}^{N} α_{l} {\dot{q}}_{l} + C_{p} {\dot{v}}_{p} + i = 0

(11)

where ζ_c,k is the damping ratio of the k-th mode of the composite beam and α_k is the constant coefficient of

θ ϕ_{k}^{'} (l_{p})

We now determine the piezo voltage from the waveform shown in Figure 2(b). In general, the displacement or q_l(t) can be expressed as a sine function of $\bar{q_{l}}$ sin(ωt) in the steady state, where $\bar{q_{l}}$ is the amplitude of q_l(t) and ω is the angular frequency set by the shaker. When the piezo ceramics are open-circuited and thus i in (11) is zero, the piezo voltage v_p,oc is expressed as

v_{p, o c} = - \frac{1}{C_{p}} \sum_{l = 1}^{N} α_{l} q_{l} (t) = - \frac{Q_{N} (t)}{C_{p}}

(12)

where Q_N(t) is the net charge of the piezo ceramics, which is introduced to simplify the equation. For the autonomous SSDI circuit, the piezo voltage v_p,SSDI is expressed as^12,13

v_{p, SSDI} = - \frac{Q_{N} (t)}{C_{eff}} - \frac{1}{C_{eff}} \sum_{l = 1}^{N} α_{l} h_{l} (t)

(13)

where C_eff (= C_p + C_ed) is the effective capacitance when connecting to the circuit. h_l represents a square wave that is artificially introduced to replicate the voltage inversion by the switching, and it is expressed as

h_{l} (t) = \frac{1 + γ}{1 - γ} {\bar{q}}_{l} \cos (τ) sign (\cos (ω t - τ))

(14)

where γ is the inversion coefficient and τ is the switch delay (See Figure 2(b)). By substituting v_p,SSDI into v_p in (9) and approximating (9) by considering the first harmonic term only,^12,13 the equation of the modal coordinate for the SSDI circuit is written as

\begin{array}{l} {\ddot{q}}_{k} + 2 ζ_{c, k} ω_{c, k} {\dot{q}}_{k} + \frac{4 α_{k} (1 + γ) \cos^{2} τ}{π ω C_{eff} (1 - γ)} {\dot{Q}}_{N} + ω_{c, k}^{2} q_{k} + \frac{α_{k}}{C_{eff}} (1 + \frac{4 (1 + γ) \cos τ \sin τ}{π (1 - γ)}) \\ Q_{N} - \sum_{j = 1}^{S} m_{r, j} ω_{r, j}^{2} u_{j} ϕ_{k} (x_{j}) = - m_{c} \int_{0}^{l_{c}} ϕ_{k} d x {\ddot{w}}_{b} \end{array}

(15)

As shown in (15), the term with

{\dot{Q}}_{N}

represents a damping force and the term with Q_N has a stiffness hardening effect. It is seen that the damping force can be increased by decreasing the switch delay τ and increasing the inversion coefficient γ. τ and γ depend on the voltage drops across parasitic elements such as series resistors or capacitors in the SSDI circuit.¹² The selection of the transistor and inductor with less parasitic elements or the use of negative resistance³³ are promising approach to increase the damping force. We evaluated the attenuation performance by solving the simultaneous differential equation of (10) and (15) numerically. For the short circuit, we set

{\dot{Q}}_{N}

= Q_N = 0 in (15). The frequency range used in this study is around the first eigenfrequency of the composite beam with no local resonators, and the contributions of the higher modes with an eigenfrequency exceeding several kilohertz are thus small. This study therefore considers the first three modes; that is, the number of modes N is set to three. Accordingly, λ₁, λ₂, and λ₃ are 1.875, 4.694, and 7.855, respectively, and ω_c,i is obtained from (7). ζ_r,j, ζ_c,1, ζ_c,2, and ζ_c,3, which were determined to be close to the experimental results, are 0.015, 0.003, 0.0015, and 0.0015, respectively. In this study, m_r,j, ω_r,j, and ζ_r,j are set to be constant because all resonators are made with the same mass, dimensions, and materials. The parameters are summarized in Table 1.

Table 1.

Parameters of the LR metastructure with the piezo ceramics.

Parameter	Value	Unit
l _c	0.192	(m)
m _c	0.262	(kg/m)
m _r,j	4.83×10⁻³	(kg)
EI	8.41	(Nm²)
ω_r,j	427	(rad./s)
ζ_r,j	0.015	(−)
ζ_c,1	0.003	(−)
ζ_c,2	0.0015	(−)
ζ_c,3	0.0015	(−)
θ	−5.86×10⁻⁴	(Nm/V)
C _p	1.61×10⁻⁸	(F)
C _ed	1.0×10⁻⁹	(F)
γ	0.65	(−)
τ	0.30	(rad.)
μ	1.15	(−)

We numerically investigate the transmissibility for the short and SSDI circuits to validate that the number of local resonators in this study is sufficient and the SSDI can decrease the peak magnitude of unnecessary resonance outside the bandgap. Figure 3 shows the transmissibility T_r for (a) the short-circuited and (b) SSDI circuit versus frequency and number of local resonators. When the piezo ceramics are short-circuited, the piezo ceramics act as a simple elastic material because v_p and the piezo force become zero. T_r was obtained from w_abs(x_m,t)/w_b(t), where x_m is the location of the measurement of the displacement at a distance of 5 mm from the tip edge of the composite beam and w_abs(x_m,t) is the absolute displacement at x_m. w_b(t) was calculated as −a/(2πf),² where a(t) and f are the acceleration and frequency, respectively. The magnitude of the acceleration was 0.2 G (1G = 9.8 m/s²) from 20 to 65 Hz, 2G from 70 to 95 Hz, and 1G from 100 to 180 Hz to match the experimental condition explained in the next section. For reference, the broken lines indicate the bandgap edges, which were estimated analytically based on the previous study.²¹ The estimated bandgap ranges from $ω_{r, j} / 2 π$ to $ω_{r, j} \sqrt{1 + μ} / 2 π$ , where μ is the mass ratio of local resonators to the composite beam. In the numerical study, the local resonators were evenly distributed over the composite beam while μ was held constant. As shown in Figure 3, the bandgap expands around the first eigenfrequency (approximately 86 Hz) of the composite beam with no local resonators and the 12 local resonators in this study was sufficient to yield a large bandgap width. The autonomous SSDI circuit successfully reduced the two resonant peaks outside the bandgap. In this study, we approximated the equation (13) and solved the coupled equation (15) numerically. It is noteworthy that the analysis based on the method of multiple scales can solve the equation more precisely, especially for the instability analysis under large deformation.³⁴

Figure 3.

Transmissibility T_r for (a) the short-circuited and (b) synchronized switch damping on inductor circuits versus frequency and number of local resonators. The broken lines indicate the estimated bandgap edges.²¹ The mass ratio μ of the local resonators to the composite beam is held constant in the numerical analysis.

Results and Discussion

We now investigate how the magnitude of the displacement can be decreased using the small piezo ceramics connecting to the SSDI circuit, both numerically and experimentally. We compare results for the short-circuited (SC) and autonomous SSDI circuits. Figure 4 shows the equipment for measuring the absolute displacement w_abs and piezo voltage v_p of the metastructure. A shaker applied a sinusoidal vibration by setting a frequency and acceleration. A laser displacement sensor (LK-H150, LK-G5000, Keyence) was used to measure the displacement. The signals of the displacement, piezo voltage, and acceleration were collected via an oscilloscope and analyzed on a personal computer. w_abs was measured at a distance of 5 mm from the tip edge of the composite beam. The transmissibility T_r was obtained from the absolute value of w_abs/w_b, where w_b is the base displacement applied by the shaker. w_b can be calculated as −a/(2πf)², where a and f are the acceleration and frequency applied by the shaker, respectively. Similar to the experiment, T_r in the numerical analysis was obtained from w_abs(x_m,t)/w_b(t), where x_m is the location of the measurement of the displacement. The attenuation efficiency was calculated as 20log₁₀(w_{abs, SSDI}/w_abs,SC), where w_{abs, SSDI} and w_abs,SC represent the displacement for the SSDI and short circuits, respectively. This study applied a different magnitude of the acceleration depending on the frequency, because the displacement magnitudes at the first and second resonances were greatly different at the same acceleration level and a sufficiently large signal of w_b was required to estimate the bandgap precisely, compared with the ground electrical noise. As shown in Figure 5, pulsebeat wave was observed at the first resonance when the impulsive damping force induced by the SSDI circuit was excessively strong. A different inductance was employed for the first (47 mH) and second (1 mH) resonances to avoid the complexity observed in the waveform.

Figure 4.

Equipment for evaluating the attenuation performance. PC: personal computer, SSDI: synchronized switch damping on inductor, USB: Universal Serial Bus.

Figure 5.

Pulsebeat wave induced by a strong impulsive force of the SSDI circuit at the first resonance.

Figure 6 shows the experimental and simulation results of transmissibility T_r and the frequency response curves of the displacement w_abs for the SC and SSDI circuits around the first and second resonances. The solid lines and circles respectively represent the simulation and experimental results. For reference, the estimated bandgap edges discussed in the subsection of Coupled analysis were shown in Figure 6(a) as broken lines. The magnitude of the applied acceleration was 0.2 G (1G = 9.8 m/s²) from 25 to 65 Hz, 2G from 70 to 95 Hz, and 1G from 100 to 180 Hz. The resonance frequency f_r and displacement w_abs at the first and second peaks were listed in Table 2. The relative error was calculated as |Simulation−Experiment|/Experiment. Overall, the numerical analysis showed a behavior similar to that of the experimentally observed results and replicated the location and width of the bandgap on the frequency axis and the decreased displacement for the SSDI circuit. There was a difference in the bandgap depth of the transmissibility between the experiment and simulation, because non-negligible noise remained in the displacement and acceleration signals, and a design error of the local resonators may cause a noise vibration. As shown in Figure 6(a), the bandgap formed from 73 to 100 Hz in the experiment and from 68 to 100 Hz in the simulation, when the bandgap is defined as T_r < 1. As shown in Figure 6(a), the SSDI circuit did not affect the characteristics of the bandgap because a small displacement in the bandgap cannot apply sufficiently large stress to the piezo ceramics. As shown in Figure 6(b) and (c), the piezo ceramics connected to the SSDI circuit markedly reduced the first and second resonant vibrations outside the bandgap. The attenuation efficiency at the first resonance was −11.8 dB for the experiment and −11.1 dB for the simulation, and the attenuation efficiency at the second resonance was −13.4 dB for the experiment and −11.1 dB for the simulation. The displacement for unnecessary resonance outside the bandgap thus decreased by approximately 75%.

Figure 6.

Experimental and simulation results of (a) transmissibility T_r and the frequency response curves of the displacement w_abs for the short-circuited (SC) and synchronized switch damping on inductor (SSDI) circuits around (b) the first and (c) second resonances. The broken lines indicate the estimated bandgap edges.²¹.

Table 2.

The resonance frequency f_r and displacement w_abs at the first and second peaks.

		Experiment		Simulation		Relative error
		Value	Unit	Value	Unit	Value ((%))
1st peak	f_r (SC)	47.6	(Hz)	47.7	(Hz)	0.210
	f_r (SSDI)	48.3	(Hz)	48.3	(Hz)	0
	w_abs (SC)	0.901	(mm)	1.08	(mm)	19.9
	w_abs (SSDI)	0.232	(mm)	0.301	(mm)	29.7
2nd peak	f_r (SC)	120.8	(Hz)	122.5	(Hz)	1.41
	f_r (SSDI)	120.7	(Hz)	123.8	(Hz)	2.57
	w_abs (SC)	0.905	(mm)	0.800	(mm)	11.6
	w_abs (SSDI)	0.193	(mm)	0.224	(mm)	16.1

Figure 7 shows the experimental results of waveforms of the displacement w_abs and piezo voltage v_p at the first and second resonances. As explained in the subsection of Autonomous SSDI circuit, the SSDI circuit flipped the piezo voltage at a little over the maximum/minimum displacement. The switch delay was clearly observed in the waveform. This study incorporated the non-negligible parameter in modeling and the simulation thus replicated the attenuation performance well.

Figure 7.

Waveforms of the displacement w_abs and piezo voltage v_p at the (a) first and (b) second resonances.

To investigate the potential of the proposed technique in a simple manner, this study used a linear model of the piezoelectricity and damping in modeling. However, similar to the case of previous studies,^12,13 the nonlinear piezoelectricity and damping may need to be included in modeling to predict the frequency response of the displacement and damping performance more accurately. At the second resonance, an 1.4% difference in the short-circuited resonance frequency between the experiment and simulation was observed, which may be attributed to the softening effect of the nonlinear piezoelectricity. Our future work will focus on establishing a more precise model for the LR metastructure with the autonomous SSDI circuit, considering the material and damping nonlinearities.

Owing to the sufficiently large piezo voltage, the autonomous SSDI technique in this study decreased the displacement by approximately 75%. For high frequencies beyond several kHz, the proposed technique may not work because the stress/strain imposed to the piezo material and thus the piezo voltage would be too small to activate the autonomous SSDI circuit. Re-designing the thickness ratio of piezo materials to substrate and using a microhenry inductor may be possible approach for the high frequency application. In this study, the coupled equation (15) was derived, assuming that the switch closes at every maximum/minimum displacement. For a random vibration, a switch-skipping²⁹ will occur when a sufficiently large piezo voltage is not available. Our future work will also focus on including the switch-skipping in modelling.

Conclusion

A locally resonant metastructure can attenuate low-frequency vibration by forming a bandgap, whereas the attachment of many mechanical resonators throughout the host structure leads to unnecessary and multiple resonance outside the bandgap. To address the practical issue, this study proposed a damping metastructure combining local resonators and an autonomous synchronized switch damping circuit. The piezo ceramics, which are attached on a small portion of the metastructure and connected to the autonomous SSDI circuit, can reduce the resonant vibration with no extra sensors, signal processors, or power sources. Thus, the combinational technique achieves both high attenuation performance and cost-effectiveness. This study employed the modal analysis approach and derived an electromechanical equation for the proposed metastructure, considering a non-negligible switch delay. Overall, the numerical analysis showed a behavior similar to that of the experimentally observed results and replicated the location and width of the bandgap in the frequency axis and the decreased displacement for the SSDI circuit. The use of the SSDI circuit decreased the displacement for unnecessary resonance outside the bandgap by approximately 75%.

Footnotes

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.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors received financial support from the JKA Foundation and its promotion funds from KEIRIN RACE.

ORCID iD

Haruhiko Asanuma

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