Vibrational microplatforms that exploit complex three-dimensional (3D) architectures assembled via the controlled compressive buckling technique represent promising candidates in 3D micro-electromechanical systems (MEMS), with a wide range of applications such as oscillators, actuators, energy harvesters, etc. However, the accuracy and efficiency of such 3D MEMS might be significantly reduced by the viscoelastic damping effect that arises from material viscosity. Therefore, a clear understanding and characterization of such effects are essential to progress in this area. Here, we present a study on the viscoelastic damping effect in complex 3D structures via an analytical model and finite element analysis (FEA). By adopting the Kelvin–Voigt model to characterize the material viscoelasticity, an analytical solution is derived for the vibration of a buckled ribbon. This solution then yields a scaling law for the half-band width or the quality factor of vibration that can be extended to other classes of complex 3D structures, as validated by FEA. The scaling law reveals the dependence of the half-band width on the geometries of 3D structures and the compressive strain. The results could serve as guidelines to design novel 3D vibrational microplatforms for applications in MEMS and other areas of technology.

Introduction

Micro- and nanostructures in micro-electromechanical systems (MEMS) are of significant current research interest, due partly to their relevance to wide ranging classes of applications in biomedicine [13], sensors [46], electronics and optoelectronics [79], batteries and supercapacitors [1012], robotics [1315], and others [16,17]. In MEMS, structural vibration plays a key role in the transformation from mechanical signals to electrical signals [18,19]. The utilization of three-dimensional (3D) micro- and nanostructures provide significant advantages and design flexibility compared to those based on two-dimensional (2D) structures [2023], especially in areas of energy harvesting, sensing of anisotropic mechanical properties, and simultaneous evaluation of multiple mechanical properties (density, modulus, etc.), due to their ability to support multidirectional vibrations and controlled vibration modes [24].

Diverse manufacturing techniques based on phenomena such as mechanical buckling [2528], self-folding induced by residual stress [2932], surface instabilities [3335], capillary forces [3638], and temperature changes [39,40] can provide access to 3D micro- and nanostructures. Among these methods, the compressive buckling approach is notable for its ability to construct complex 3D structures with vibration behaviors (e.g., natural frequency) that can be tuned by applying tensile strain to the soft, elastomeric assembly platform. In addition, this technique is compatible with a variety of active materials, such as poly (vinylidene fluoride), Pb(Zr,Ti)O3 (PZT), and conductive metals, allowing actuation by external stimuli based on electric and/or magnetic fields. Such options provide great potential in the applications of resonators, energy harvesters, and other systems based on continuous adaption of the resonant frequency or vibration modes [24].

One major challenge in the design of 3D MEMS arises from the effects of viscoelasticity in the constituent materials. Such effects lead to dissipation of energy, changes in stress distributions, displacements, and frequencies during vibration [4145]. For example, increases in the ambient temperature and/or humidity can increase the viscosity [4648], which, in turn, can reduce the efficiency of 3D MEMS devices. The viscoelastic properties can be characterized by dynamic mechanical analysis [49] and creep and stress relaxation methods [50] for a wide range of materials, such as poly(methyl methacrylate), SU8 epoxy resin, polypropylene, polydimethylsiloxane, etc. [5155]. The viscoelastic damping effect due to viscosity of materials in a vibration system can be characterized by the half bandwidth $ξ$ or the quality factor $Q$ ($Q=1/(2ξ)$) [5658].

Several viscoelastic models have been proposed to characterize the viscoelastic material properties including Maxwell model, Kelvin–Voigt model, and standard linear solid model [5961]. Among them, the Kelvin–Voigt model is a classical and widely used viscoelastic model. For example, Ghayesh [62] investigated the nonlinear dynamic response of a simply supported beam supported by a nonlinear spring. Mahmoodi et al. [63] reported the experimental study of nonlinear vibration and frequency responses of viscoelastic beams, with good agreement between experimental results and numerical simulations. Kolahchi [64] studied the nonlinear vibrations of viscoelastic rectangular plates. These investigations focus on the viscoelastic damping effect of 2D plates or beams. Though Tseng and Dugundji [65] investigated the vibration of a buckled beam, they only provided an exact solution for the case without viscoelastic damping. Similarly, Cottone et al. [66] studied piezoelectric buckled beams as a vibration energy harvester. The aforementioned theoretical models are restricted to a few simple geometries and cannot be directly extended to complex 3D structures. Therefore, it is important to develop a theoretical model to predict the viscoelastic damping effect of vibrations in complex 3D structures.

This paper presents a study on the viscoelastic characteristics of 3D mechanically assembled structures formed by compressive buckling techniques via an analytical model and finite element analysis (FEA). The Kelvin–Voigt model is used to capture the viscoelastic effects of conventional materials (SU8, PVDF, etc.) in these structures. An analytical model explicitly relates the half-band width of a buckled ribbon to the geometry/material parameters and the compressive strain. Then, a more general model describes the half-band width of complex structures, with two fitting parameters to account for the complexity of the vibration mode and structure. The models are validated by FEA with good agreement. They provide insights into the effect of design parameters (material, compressive strain, structure shape, etc.) on the vibration behavior and may serve as useful references in the design of 3D vibrational platforms and the potential applications of 3D MEMS.

The Viscoelastic Characteristics of Three-Dimensional Buckled Ribbon and Three-Dimensional Structures

A schematic illustration of a buckled ribbon excited by an external harmonic load is shown in Fig. 1. A straight 2D slender ribbon (length L, width b, and thickness h) is selectively bonded to a highly prestrained elastomer at two ends as shown in Fig. 1(a). Release of the prestrain induces compression and triggers buckling of the ribbon into the arch shape, as shown in Fig. 1(b), where l represents the distance between two bonding sites after compression. Considering that the viscosity of the soft elastomer may increase the half-band width of vibration, the buckled ribbon is transferred on to a rigid base using the technique reported by Yan et al. [67]. An external harmonic load (see Fig. 1(c)) then actuates the first-order vibration mode shown in Fig. 1(d). Because of the base has larger modulus than the buckled ribbon, its deformation can be neglected during the vibration analysis. Due to the ribbon thickness (h) is much smaller than its width (b) and length (L), finite-deformation beam theory with no shear deformation [6870] is adopted to establish an analytical model. In general, the deformations of a planar ribbon can be described by the displacement of the central axis u = uiEi [71] and the twist angle $ψ$ [68], where Ei is the unit vector before deformation in the Cartesian coordinates (X, Y, Z), which all calculations are based on. For the post-buckling and vibration shown in Fig. 1, only the displacement components in the XZ plane are involved.

Fig. 1
Fig. 1
Close modal

The Construction of Governing Equations.

For a moderate level of compressive strain (e.g., < 30%), the displacements of the ribbon after the post-buckling could be expressed as [24,72,73]
$u1(0)=A(0)[cos(2πLZ)+1], u3(0)=πA(0)24Lsin(4πLZ)−εcompreZ$
(1)
where $εcompre=L−l/L$ is the relative dimensional change between the two bonding sites, or called the compressive strain; $εcompre$ is related to the elastomer prestrain $εprestrain$ via $εcompre=εprestrain/(1+εprestrain)$; $εc=π2h2/(3L2)$ is the critical strain; and $A(0)=(3L2εcompre−π2h2)/3π2=L/πεcompre−εc$ is the static deflection amplitude of the buckled ribbon. Due to the rigidity of the fix stage, the boundary conditions are
$u1(±L/2)=0, u3(±L/2)=0, du1dZ(±L/2)=0$
(2)
Then a vibration displacement function $Δui(Z,t)$ is superimposed on $ui(0)(Z)$ to give the total displacement of the buckled ribbon as
$u1(Z,t)=u1(0)+Δu1(Z,t), u3(Z,t)=u3(0)+Δu3(Z,t)$
(3)
By introducing a set of series $φ(k)(Z)$ and $ϕ(k)(Z)$ into the vibration displacements, the total displacements of the buckled ribbon can be further written as
$u1(Z,t)=u1(0)+Δu1(Z,t)=u1(0)+∑k=1nΔa(k)(t)φ(k)(Z)$
(4)
$u3(Z,t)=u3(0)+Δu3(Z,t)=u3(0)+∑k=1nΔa(k)(t)ϕ(k)(Z)$
(5)
The total displacements should satisfy the boundary conditions in Eq. (2), leading to
$φ(k)(±L/2)=0, ϕ(k)(±L/2)=0 and dφkdZ(±L/2)=0$
(6)
To describe the viscoelastic characteristics of the buckled ribbon, the physical relationship for Kelvin–Voigt model is introduced as
$σ=σe+σd=Eε(X,Z,t)+ηE∂ε(X,Z,t)∂t$
(7)
where $σ$ is the stress, $ε$ is the strain, and E and $η$ are the elastic modulus and viscoelastic damping coefficient of the ribbon material, respectively. According to the beam theory
$ε=(λ−1)−Xκ$
(8)

where $λ$ is the stretch ratio and $κ$ is the curvature. Noe that X represents the distance from a point on the cross section to the central axis of the ribbon, where the origin of the coordinate system locates at.

To solve $Δa(k)(t)$ in Eqs. (4) and (5), the Lagrange's equation of motion is introduced. Specifically, the strain energy $Ws$ of the buckled ribbon is given by [74]
$Ws=12∫−L/2L/2Ebh(λ−1)2dZ+124∫−L/2L/2Ebh3κ2dZ$
(9)
By neglecting the terms of the third and higher order power of $Δa$ in Eq. (9), the potential energy can be also written as $Ws=1/2ΔaTKΔa$, in which $Δa$ is a $n×1$ vector ${Δa(1),Δa(2),…,Δa(n)}T$, and $K$ is an $n×n$ stiffness matrix. Similarly, the kinetic energy can be given as
$T(Δa˙)=12∫−L/2L/2ρbh[(∂u1(v)∂t)2+(∂u3(v)∂t)2]dZ=12Δa˙TMΔa˙$
(10)

where $Δa˙={Δa˙(1),Δa˙(2),…,Δa˙(n)}T$ is the time derivative of $Δa$, i.e., $Δa˙=d(Δa)/dt$, $ρ$ is the density of the ribbon material, and $M$ is an $n×n$ mass matrix.

Based on Eq. (7), the dissipation function of the buckled ribbon can be obtained as follows:
$D=12∫−L/2L/2ηEbh(∂λ∂t)2dZ+124∫−L/2L/2ηEbh3(∂κ∂t)2dZ$
(11)
The potential energy of the external harmonic load can be written as
$Wext=−∫−L/2L/2P(t)Δu1(Z,t)dZ$
(12)

where $P(t)$ is the external force exerted on per unit length of bucked ribbon and ω is the frequency of the external harmonic load.

The Lagrange's equation of motion requires that
$∂(T−Ws)∂Δa−ddt∂(T−Ws)∂Δa˙−∂D∂Δa˙−∂Wext∂Δa=0$
(13)
By substituting Eqs. (7)(12) into Eq. (13), the governing equation of the buckled ribbon under an external harmonic load is obtained as
$MΔä+CΔa˙+KΔa=F$
(14)

The Half-Band Width of the First-Order Vibration Mode.

For the first-order vibration mode excited by the external harmonic load, FEA results show that the vibration displacement can be well characterized by superposition of the following base functions, with two terms (i.e., n = 2) in Eqs. (4) and (5), as [24]:
$φ(1)(Z)=1+cos(2πZL), φ(2)(Z)=1−cos(4πZL)$
(15)
$ϕ(1)(Z)=πA(0)2Lsin(4πZL), ϕ(2)(Z)=πA(0)3L[6 sin(2πZL)−2 sin(6πZL)]$
(16)
The potential energy of the vibration is then derived as
$Ws=Ebh3L3(k11Δa(1)2+k22Δa(2)2)$
(17)
where
$k11=−5π6A(0)23L2+2π4A(0)2h2 and k22=28π6A(0)23L2+4π4$
(18)
The kinetic energy is then written as
$T=ρbhL(m11Δa˙(1)2+m22Δa˙(2)2+m12Δa˙(1)Δa˙(2))$
(19)
where
$m11=π2A(0)216L2+34, m22=10π2A(0)29L2+34 , m12=1$
(20)
The dissipation function is calculated as
$D=π4ηEbh33L3(c11Δa˙(1)2+c22Δa˙(2)2)$
(21)
where
$c11=−π2A(0)2L2+6A(0)2h2+1, c22=32π2A(0)2L2+16$
(22)
A harmonic load $P(t)=P0 sin(ωt)$ is exerted on the ribbon leading to the potential energy as
$Wext=−P0L(Δa(1)+Δa(2))sin(ωt)$
(23)
Substitution of Eqs. (17)(23) into Eq. (13) gives the governing equations
$[M11M12M12M22][Δä(1)Δä(2)]+[C1100C22][Δa˙(1)Δa˙(2)]+[K1100K22][Δa(1)Δa(2)]=[P0L sin(ωt)P0L sin(ωt)]$
(24)
where
$M11=2ρbhLm11,M22=2ρbhLm22,M12=ρbhLm12C11=2π4ηEbh33L3c11,C22=2π4ηEbh33L3c22K11=2Ebh3L3k11,K22=2Ebh3L3k22$
(25)
The solutions to Eq. (24) are
$[Δa(1)Δa(2)]=[B1 sin(ωt+θ1)B2 sin(ωt+θ2)]$
(26)
with the specific deducing process and the expressions of $B1$, $B2$, $θ1$, $θ2$ presented in the Appendix.
Therefore, the transverse vibrational displacement and velocity of the buckled ribbon can be written as
$Δu1(Z,t)=Δa(1)φ(1)(Z)+Δa(2)φ(2)(Z)=B(Z,ω)sin(ωt+θ)$
(27)
$v1(Z,t)=ωB(Z,ω)cos(ωt+θ)$
(28)
where
$B=[Br1(ω)φ(1)(Z)+Br2(ω)φ(2)(Z)]2+[Bi1(ω)φ(1)(Z)+Bi2(ω)φ(2)(Z)]2 tan θ=[Bi1(ω)φ(1)(Z)+Bi2(ω)φ(2)(Z)]/[Br1(ω)φ(1)(Z)+Br2(ω)φ(2)(Z)]$
(29)
with $Br1$, $Br2$, $Bi1$, and $Bi2$ presented in the Appendix. In Eq. (27), B(Z, ω) is the spatial part of the transverse vibrational displacement and the vibration amplitude A(ω) is defined as the maximum of B with respective to the position Z. At a given position, the normalized vibrational velocity as a function of the frequency is
$μ(ω)=2ωBEbh3P0L4$
(30)
The half-band width of the buckled ribbon is
$ξ=ω2−ω12ω0$
(31)
where $ω0$ is the resonant frequency, i.e., the frequency at which $μ(ω)$ reaches its maximum $μmax$; $ω1$ and $ω2$ are the two solutions to $μ(ω)=μmax/2$. We refer to Eq. (31) as the general form of the analytical model.
It is verified that $φ(1)(Z)$ and $ϕ(1)(Z)$ in Eqs. (15) and (16) could dominate the vibration mode, when the static deflection amplitude of the buckled ribbon is much smaller than the ribbon thickness, i.e., $A(0)≪h$, while $φ(2)(Z)$ and $ϕ(2)(Z)$ could dominate the vibration mode when $A(0)≫h$. Then, the vibration velocity in Eq. (30) could be degenerated to
$μ*=ω(−ω2⋅M11+K11)2+(ω⋅C11)2 (A(0)≪h)$
(32)
$μ*=ω(−ω2⋅M22+K22)2+(ω⋅C22)2 (A(0)≫h)$
(33)
which lead to the half-band width
$ξ*=ω2−ω12ω0=C112M11K11=π4ηh6L2Eρc11m11k11=π4ηω06c11k11 (A(0)≪h)$
(34)
$ξ*=ω2−ω12ω0=C222M22K22=π4ηh6L2Eρc22m22k22=π4ηω06c22k22 (A(0)≫h)$
(35)

We refer to Eqs. (34) and (35) as the simplified form of the analytical model for $A(0)≪h$ and $A(0)≫h$, respectively.

The Extension of the Analytical Model to General Three-Dimensional Structures.

Since the 3D structures formed via controlled buckling technique usually have a large static deflection amplitude, this paper focuses on the case when $A(0)≫h$. The half-band width of the buckled ribbon in Eq. (35) can be further written by using the relationship $εcompre=π2A(0)2/L2(1+h2/3A(0)2)≈π2A(0)2/L2$, as
$ξ2*=π4ηω06c22k22=ηω0ξ̂$
(36)
where $ξ̂$ is the nondimensional half band width of the buckled ribbon as
$ξ̂=π46c22k22=4εcompre+27εcompre+3$
(37)
By using the first-order Taylor expansion, the nondimensional half-band width of the buckled ribbon demonstrated in Eq. (37) can be approximately written as
$ξ̂≈23(1−13εcompre)$
(38)
Inspired by the structure of the analytical solutions in Eqs. (36) and (38), the nondimensional half-band width $ξ̂$ can be expressed in the form of scaling law by a single-variable function of the compressive strain $εcompre$ as
$ξ̂=α(1+βεcompre)$
(39)

where $α$ and $β$ can be determined by fitting the FEA results of the $ξ̂∼εcompre$ curves. As we show in Sec. 3, the scaling law Eq. (39) applies to the vibration of a variety of 3D structures formed by the compressive buckling technique with viscoelastic materials. In addition, it can be seen from Eq. (36) that this scaling law is independent of structural dimensions and material properties.

Examples and Discussions

The analytical model is validated by FEA, as presented in Fig. 2. The FEA were performed using the commercial software abaqus. The material was assumed to be a photopatternable epoxy (SU8), a typical polymer used in 3D assembly. The Young's modulus, Poisson's ratio and density of SU8 are E = 4.02 GPa, ν = 0.22, and ρ = 1.2 g/cm3 [24]. $ηω0$ is assumed to be a constant to simulate the viscoelastic damping effect of Kelvin–Voigt model based on the data obtained by dynamic mechanical analysis, creep and stress relaxation, and other methods [49,53,54,75,76]. The vibrational displacement and velocity under the harmonic external load were obtained by the steady-state analysis, after importing the shape and stress of the buckled ribbon determined from the post-buckling analysis. The ribbon was discretized by Four-node finite-strain shell elements (S4) and at least 20 elements were implemented along the width direction of the ribbon to guarantee the convergence.

Fig. 2
Fig. 2
Close modal

Under a representative compressive strain $εcompre=0.1$, the analytical results of the transverse vibrational displacement match well with the FEA results, as shown in Fig. 2(a). Figure 2(b) gives a comparison of the nondimensional half-band width $ξ̂$ calculated by the general form of the analytical model (Eq. (31)), the scaling law (Eq. (38)), and FEA. The scaling law agrees well with FEA under compressive strain $εcompre>0.05$, a typical value used in compressive buckling techniques. The general form of the analytical model agrees well with FEA in a broader range of compressive strain $εcompre$, even when $εcompre$ is close to the critical strain $εc$. Note that the nondimensional half-band width increases significantly when $εcompre$ decreases to $εc$, which is consistent with the simplified form of the analytical model for $A(0)≪h$ (Eq. (34)). Further, Fig. 2(c) plots a comparison of nondimensional velocity responses of the buckled ribbon with and without the viscoelastic damping effect. A obviously sharp peak of $μ*/μ1,max*$$ω/ω0$ curve occurs at the natural frequency ($ω/ω0=1$) for the case without the viscoelastic damping effect, where $μ1,max*$ represents the maximum of velocity response with respect to the frequency when the viscoelastic damping effect is present.

To compare the half-band width of 3D buckled structures with that of their corresponding 2D precursors, the FEA results of three pairs of structures (buckled ribbon, tent, and table) are presented in Fig. 3. The nondimensional half-band widths of these 3D structures are ∼25% larger than their 2D precursors. It is also noticed that the nondimensional half-band widths of these 2D structures are almost the same.

Fig. 3
Fig. 3
Close modal

Figure 4 shows the variations of the nondimensional half-band width $ξ̂$ of three complex structures (tent, table, and helix) with compressive strain $εcompre$ ranged from 0.05 to 0.30. The relationship of $ξ̂$ versus $εcompre$ can be well modeled by the scaling law in Eq. (39) with proper parameters $α$ and $β$. The paramters relate the nondimensional half-band width $ξ̂$ to the topology of 2D precursor and the compressive strain $εcompre$. When $εcompre$ ranges from 0.05 to 0.30, $ξ̂$ decreases by about 5%. For the same compressive strain, the nondimensional half-band widths of the three structures show about 20% difference among each other. To further illustrate the relationship between $ξ̂$ and the structure shape, we study more 3D structures as shown in Figs. 5 and 6.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal

Figure 5(a) shows the geometry of the table structures with different central circle radii r. The table structure becomes the tent structure when r equals to the ribbon width. Figures 5(b)5(g) presents the FEA results of the nondimensional half-band width $ξ̂$ of the first-order mode, when the normalized circle radius r/L ranges from 0.07 to 0.175, as well as a demonstration of r/L = 0.50, and the compressive strain $εcompre$ ranges from 0.05 to 0.30. The scaling law in Eq. (39) could remarkably characterize the FEA results for all cases with proper parameters $α$ and $β$, which change slightly with r/L as shown in Fig. 5(h).

Similarly, Fig. 6(a) shows the geometry of the helix structures with different top angles $θ$. The helix structure becomes the tent structure when $θ$ is 0 deg. The scaling law in Eq. (39) could also remarkably characterize the FEA results for all cases (Figs. 6(b)6(g)) with proper parameters $α$ and $β$, which change slightly with $θ$ as shown in Fig. 6(h).

Conclusions

This paper presents a study on the viscoelastic characteristics of 3D structures formed by compressive buckling techniques via an analytical model and the FEA. The Kelvin–Voigt model is introduced to characterize the material viscoelasticity. A validation between the analytical model and FEA is presented, and a scaling law to characterize the half-band width is obtained and extended to more complex 3D structures including tent, table, and helix. Some main conclusions can be drawn as: (1) The nondimensional half-band width of the buckled straight ribbon, tent, and table structures are about 25% larger than that of their 2D precursors. (2) The scaling law relates the half-band width to the structure shape and the compressive strain via two parameters. The model and FEA focus on the vibration when the 3D structure is transferred on to a rigid base after compressive buckling to remove the soft elastomer. Taking the underlying substrate into the model is a future direction, which may broaden the application to cases when the 3D structure is transferred on to a deformable base such as skin or remains on the soft elastomer. This paper could serve as design guidelines of MEMS for their potential applications in energy harvesting and mechanical sensors.

Acknowledgment

Y. Z. acknowledges the support from the National Natural Science Foundation of China, the National Basic Research Program of China and the Tsinghua National Laboratory for Information Science and Technology. Y. H. acknowledges the support from NSF. H. B. Li thanks the support from China Scholarship Council. This works is also financially supported from National Science Foundation of China (NSFC).

Funding Data

• National Basic Research Program of China (2015CB351900).

• National Natural Science Foundation of China (11722217 and 11732009).

• National Science Foundation (CMMI1400169, CMMI1534120, and CMMI1635443).

Appendix

Using $Δa=Im(Δã)$ and substituting $Δã=ΔÃeiωt$ into Eq. (24) leads to
$[−ω2M11+iωC11+K11−ω2M12−ω2M12−ω2M11+iωC22+K22][ΔÃ(1)ΔÃ(2)]=MΔÃ=[P0LP0L]$
(A1)
Then, by multiplying the inverse coefficients matrix $M−1$ at both sides of Eq. (A1), the solutions of $ΔÃ(1)$ and $ΔÃ(2)$ can be expressed as
$ΔÃ(1)=Br1+iBi1, ΔÃ(2)=Br2+iBi2$
(A2)
where
$Br1=−P0L{[K22+(M12−M22)ω2]T1+T2C22ω}T12+T22Bi1=−P0L{T1C22ω−T2[K22+(M12−M22)ω2]}T12+T22Br2=−P0L{[K11+(M12−M11)ω2]T1+T2C11ω}T12+T22Bi2=−P0L{T1C11ω−T2[K11+(M12−M11)ω2]}T12+T22$
(A3)
where
$T1=(M122−M11M22)ω4+(K11M22+K22M11+C11C22)ω2−K11K22T2=(C11M22+C22M11)ω3−(C11K22+C22K11)ω$
(A4)
Thereby, the solutions of Eq. (24) are obtained as
$[Δa(1)Δa(2)]=Im[Δã(1)Δã(2)]=[B1 sin(ωt+θ1)B2 sin(ωt+θ2)]$
(A5)

where, $B1=Br12+Bi12$, $B2=Br22+Bi22$, $tan θ1=Bi1/Br1$, and $tan θ2=Bi2/Br2$.

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