Nonlinear dynamic behavior of a microbeam array subject to ...

Nonlinear Dyn (2012) 67:1–36 DOI 10.1007/s11071-010-9888-y

O R I G I N A L PA P E R

Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low, medium and large DC-voltages S. Gutschmidt · O. Gottlieb

Received: 2 August 2009 / Accepted: 3 November 2010 / Published online: 24 November 2010 © Springer Science+Business Media B.V. 2010

Abstract The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, most widely used theoretical approaches, although opposite extremes with respect to complexity, are nonlinear lumped-mass and finite-element models. While a lumped-mass approach is useful for a qualitative understanding of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Finite-element simulations, on the other hand, are adequate for static analysis, but inadequate for dynamic simulations. A third approach is that of a reduced-order modeling which has gained significant attention for single-element microelectromechanical systems (MEMS), yet leaves an open amount of fundamental questions when applied to MEMS arrays. In this work, we employ a nonlinear continuum-based model to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the array’s behavior in regions of its internal one-to-one, parametric, and several internal three-to-one and combination resonances, which correspond to low, medium and large

DC-voltage inputs, respectively. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weakly nonlinear system in the afore mentioned resonance regions, respectively. Analytically obtained results of a two-element system are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic behavior of the twoand three-beam systems reveal several in- and outof-phase co-existing periodic and aperiodic solutions. Stability analysis of such co-existing solutions enables construction of a detailed bifurcation structure. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearestneighbor interactions of medium- and large-size arrays. Furthermore, the results of this present work motivate future experimental work and can serve as a guideline to investigate the feasibility of new MEMS array applications.

S. Gutschmidt () Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand e-mail: [email protected]

1 Introduction

O. Gottlieb Department of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa, Israel

Keywords Internal resonance · MEMS array · Parametric actuation · Bifurcation · Chaotic dynamics

Single micro- and nano-mechanical systems (MEMS and NEMS) are small flexible electromechanical structures, such as plates, beams and wires that are excited by external electrostatic or magnetic fields. Arrays of such resonators [6, 8, 24, 49], consist of a

2

multitude of coupled elements. MEMS and NEMS arrays have successfully given birth to new technologies as in e.g. storage devices [27, 46], parallel lithography processes [22], microcantilever biosensors [1] and mass detection [24], as opto-mechanical signal processing [6] and signal mixing devices [11], for fast mapping of surfaces via atomic force microscopy [2, 32, 36], and recently for protein printing [43]. In most operational cases of these arrays the dynamic response is governed by nonlinear effects [4, 6, 10, 24, 30] which directly influence their performance. To date, the primary focus of microbeam array investigation has been experimental [6, 32, 43]. Theoretical investigations include lumped-mass derivation [4, 9, 30], which reveal the existence of in- and outof-phase periodic responses [30] and intrinsic energy localization of specific array modes [10], the widely used finite-element approach [50] and a continuumbased approach [16, 18]. While the dynamic behavior of a single-resonator MEMS has been studied extensively in literature [29, 37, 40, 44, 48], little is known so far for array devices [6, 17, 30, 41] and the collective behavior of interacting members [20, 21]. Although the scientific study of coupled oscillators goes all the way back to the seventeenth century, where Christiaan Huygens observed mutual synchronization of pendulum clocks connected by a beam [23] and recently by Bennet et al. [3], Brown et al. [5], Danzl and Moehlis [7], and Hikihara et al. [20], to the best of our knowledge, no published work accounts for systematic investigations of the dynamical behavior of a MEMS array according to distinct regions of parameter domains. Buks and Roukes (BR) [6] employed optical diffraction to study the mechanical properties of an electrically tunable array of suspended doubly-clamped beams which were parametrically excited at primary resonance. The experiments depicted complex multivalued periodic response for a bias DC-voltage range from 0 to 20 V and a very small periodic AC input of 50 mV. Motivated by their work, Lifshitz and Cross (LC) [30] proposed a set of coupled Duffingtype equations of motion for an array excited at its principal parametric resonance and were able to qualitatively explain some of the documented experimental phenomena. Their analytical steady state asymptotic analysis reveals co-existing stable and unstable periodic solutions for a large bias DC-voltage and a very small AC-voltage excitation. A recent study

S. Gutschmidt, O. Gottlieb

of Karabalin et al. [26], which is based on the LC model, also reveals the existence of what appears to be a wide-banded aperiodic response. The qualitative agreement between LC and BR includes several abrupt drops in the large-size array response as the frequency was swept upwards and downwards and reveals only periodic responses. Both works do not classify investigations according to regions of the input voltage, which, as is going to be observed in Sect. 3, reveals regions of distinct dynamical behavior and collective interactions of the array. Recently, Gutschmidt and Gottlieb investigated a similar continuum initial-boundary-value problem of a doublyclamped microbeam array parametrically excited at several DC bias and periodic AC-voltages [15–17]. For a small DC bias the natural frequencies of the array were identical and thus, the system was excited at its one-to-one internal resonance [15]. (Note that the terminology “internal resonance” refers to neighboring members of the array in this paper, and must not be confused with the elswhere used intrinsic internal resonances, or ultra-sub-harmonics, of an individual single-degree-of-freedom resonator.) For large DC-voltage inputs, near the system’s first pull-in instability, several three-to-one internal and combination resonances were identified [17]. In both cases (low and high DC bias), analytical and numerical analyses revealed multiple co-existing stable and unstable, periodic and aperiodic solutions. We note that theoretical asymptotic analysis to-date includes only derivation of periodic solutions and their consequent stability. Thus, the significance of this research is to deduce and numerically verify the analytical criteria that enable prediction of both internal and combination resonance induced quasi-periodic dynamics culminating with temporal chaos in small arrays subject to combined DC bias and AC excitation. The present work investigates the dynamical behavior of a small-size microbeam array in distinct resonance regions below the first pull-in instability threshold of the system. Corresponding regions are that of a oneto-one internal (for zero and near-zero DC-voltages), a parametric (for medium DC-voltages), and a three-toone internal and combination (for large DC-voltages) resonance, respectively. (We include some previously published results, clearly indicated as occurring, for completeness of our description which for the first time to our knowledge spans the whole DC range from small to large.) Although the analysis of the

Nonlinear dynamic behavior of a microbeam array subject to parametric actuation

present model allows no quantitative comparison to previous experimentally observed results by BR [6], it makes significant contributions to the fundamental understanding of the array behavior. The practical importance is that while actual energy transfer between in- and out-of phase co-existing periodic modes is not periodic, it can be predicted (in an asymptotic approximate form) and as such be suppressed or enhanced for either sensor, actuator or filter applications. This manuscript is organized as follows: In Sect. 2 we formulate the initial-boundary-value problem (IBVP) for the array including both localized nonlinear electrodynamic actuation and dissipation. The IBVP is reduced to a modal dynamical system via the Galerkin decomposition which then is investigated analytically and numerically in Sect. 3 (equilibrium analysis), Sect. 4 (multiple-scales asymptotics) and Sect. 5 (numerical analysis), respectively. The equilibrium analysis includes investigations of the fixed points and their stability of the single-, two- and threebeam system as well as the phenomena of the pullin of the array with respect to its neighbor beams. In Sect. 4 we employ multiple-scale asymptotics for an array of two members in the vicinity of the system’s internal and parametric resonance regions. The twoelement analysis is validated numerically and complemented by the numerical analysis of a three-element array (Sect. 5). We summarize with closing remarks in Sect. 6.

2 Model 2.1 Initial boundary value problem We consider an array of N clamped-clamped silicon beams (see Fig. 1). All microbeams (length L, width B, height H , respectively) are assumed to

Fig. 1 Definition sketch of the micro-electromechanical array; actuation and dissipative forces applied at mid-span of each beam

3

have identical material properties. We assume a linear stress-strain law and that plane sections remain plane. The equation of motion for a single clamped-clamped resonator in an array of N beams, as found in literature (e.g. [39]), is Aw˜ ntt + R˜ n (w˜ n ) + S˜n (w˜ n , w˜ nt ) ˜ n (w˜ n , w˜ n+1 , w˜ n−1 , w˜ nt , w˜ n+1t , w˜ n−1t , t), =Q for n ∈ [0, N + 1],

(1)

where variables marked with a tilde are dimensional. A in (1) is the mass per unit length (with density and cross-sectional area) of each beam n and w˜ n (x, t) are the displacements in the z-direction (in-plane motion, see Fig. 1). The restoring force R˜ n is that of a standard Euler–Bernoulli beam with immovable boundary conditions that includes the effect of residual stresses and nonlinear membrane stiffness, [38]. We consider here both, linear viscous and a Kelvin–Voigt viscoelastic damping model, [12, 45]. The elastic restoring force and the structural damping force for each beam are L 2 ˜ Rn = EI w˜ nxxxx + N0 − K (w˜ nx ) dx w˜ nxx , (2) 0

S˜n = D1 w˜ nt + D2 w˜ nxxxxt .

(3)

E and N0 in (2) are the Young’s modulus and the pre-tensional force, respectively, and I and K are given by I = H B 3 /12 and K = EA/(2L) and denoting the moment of inertia and mid-plane stretching. Unlike in BR’s experiment, the model here considers the electrodynamic interactions as concentrated loads at mid-span of each microbeam, which allows for a continued analytical description of the system’s response as the integral of the electromechanical forcing term (see Sect. 2.2) can be solved in closed form. This simplification is made to ensure that only a symmetric first mode is excited, so that the contribution of additional modes is indeed negligible. However, the emphasis of this work is laid on studying the nonlinear phenomena and dynamic behavior of a perfectly symmetric array with identical material properties of each member and thus, the radically shortened electrodes applied at mid-span of each resonator serve the aims and interests of this work. ˜ n in (1) is composed of the electroThe actuation Q ˜E dynamic actuation Q n , which is proportional to the

4


quadratic ratio between the input voltage and the relative grating of the array [44, 47], and the nonlin˜D ˜D ear electrodynamic damping force Q n . Qn is deduced from the quadratic Rayleigh dissipation function [31] 2 V (w˜ n+1 − w˜ n )2 (w˜ (n+1)t − w˜ nt )2 1 G = D3 n+1 2 (g − w˜ n + w˜ n+1 )2 V 2 (w˜ n − w˜ n−1 )2 (w˜ nt − w˜ (n−1)t )2 , − n (g − w˜ n−1 + w˜ n )2

(4)

which is motivated by experimental observations [6], which reported on a sharp increase in damping with an increase in input voltage. A complete general description of linear and nonlinear damping mechanisms in MEMS and NEMS does not exist, as nonlinear damping effects cannot be separated by a set of simple control experiments. However, it is clear from experimental evidence [33] that there is a need for consistent modeling of nonlinear electrodynamic damping as calibrated nonlinear visco-elastic damping is insufficient to make the frequency response curves be bounded. Dj for j = [1, . . . , 3] in (3) and (4) and g in (4) are linear and nonlinear damping coefficients and array grating (gap between resonators), respectively. Thus, the nonlinear damping force is L D ˜ Qn = δ x − 2 2 V (w˜ n+1 − w˜ n )2 (w˜ (n+1)t − w˜ nt ) × D3 n+1 (g − w˜ n + w˜ n+1 )2 Vn2 (w˜ n − w˜ n−1 )2 (w˜ nt − w˜ (n−1)t ) (5) − (g − w˜ n−1 + w˜ n )2 ˙˜ n assuming that nonlinear damp˜D with Q n = −∂G/∂ w ing occurs predominantly as a function of the bias voltage. In a similar way the electrodynamic actuation force can readily be deduced from the potential function to 2 Vn+1 L ˜E P Q = δ x − n 2 (g − w˜ n + w˜ n+1 )2 Vn2 − (6) (g − w˜ n−1 + w˜ n )2 for P = ε0 H /2, where ε0 is the electric constant (vacuum permittivity).

Each element of the microbeam array is clamped at both ends, i.e. w˜ n (0, t) = 0, w˜ n (0, t) = 0 and w˜ n (L, t) = 0, w˜ n (L, t) = 0. Furthermore, the first and the last beam in the array are completely fixed along their lengths (w˜ 0,N +1 (x, t) = 0). By introducing the nondimensional parameters w = w/L, ˜ τ = ωs t and ωs2 = EI /(AL4 ) the nondimensional field equations become wnτ τ = Qn (wn , wn+1 , wn−1 , wnτ , wn+1τ , wn−1τ , τ ) − Rn (wn ) − Sn (wn , wnτ ).

(7)

The nondimensionalized restoring force Rn (wn ), structural damping force Sn (wn , wnτ ), generalized dissipation force QD n (wn , wnτ ) and electrodynamic (w , τ ) of each beam are excitation QE n n Rn = wnssss

+ wnss κ1 − κ3

0

1

wn2s

ds ,

(8)

(9) Sn = μˆ 1 wnτ + μˆ 2 wnssssτ , 2 (wn+1 − wn ) (w(n+1)τ − wnτ ) 1 QD μˆ 3 n =δ s− 2 (γ + wn+1 − wn )2 (wn − wn−1 )2 (wnτ − w(n−1)τ ) , (10) − (γ + wn − wn−1 )2 1 1 E 2 ˆ ΓV Qn = δ s − 2 (γ + wn+1 − wn )2 1 , (11) − (γ + wn − wn−1 )2 respectively. The nondimensional parameters in (8)– (11) are Γˆ = 6ε0 L/(EB 3 ), μˆ 1 = D1 /(Aωs ), μˆ 2 = D2 /(Aωs L4 ), μˆ 3 = D3 /(Aωs L2 ), κ1 = N0 L2 / (EI ), κ3 = 6(L/B)2 , γ = g/L, Ωˆ = ΩAC /ωs , ˆ . whereas the voltage is V = VDC + VAC cos Ωτ The nondimensional boundary conditions (b.c.) are wn (0, τ ) = 0, wn (1, τ ) = 0 and wns (0, τ ) = 0, wns (1, τ ) = 0 while the first and last beam in the array are prevented from undergoing any motions, i.e. w0 (s, τ ) = wN +1 (s, τ ) = 0. 2.2 Modal dynamical system The dynamic response can be approximated in terms of a linear combination of a finite number of orthonormal spatial basis functions with time dependent amplitudes. The deflections of each microbeam are approximated by the series of spatial modeshapes


multiplied by time dependent amplitudes, wn (s, τ ) = (m) qn,m (τ )Φm (s). Φm (s) are the modeshapes associated to the linear undamped homogeneous system of (7). The shape functions, which satisfy the b.c. exactly [19, 28], are given by Φm (s) = Cm (S˜m sinh Sm s − Sm sin S˜m s) − Om (cosh Sm s − cos S˜m s) ,

J1 =

1

0

J3 =

(12)

1

2

(Φs ) ds ds,

ΦΦss

J4 =

ΦΦss ds, 0

1

1

J2 =

2

Φ ds,

0 1

ΦΦssss ds. 0

We rescale the resulting ordinary differential equations ¯ n /γ and t ∗ = ζ 2 τ (where ζ1 = 4.73 (for a by xn = Φq 1 clamped-clamped beam and κ1 = 0) and Φ¯ = Φ(1/2)) to yield the final modal dynamical system:

cosh Sm − cos S˜m , Sm sinh Sm + S˜m sin S˜m

2 N N0 0 Sm = + λ2m , + 2 2

2 N N0 0 ˜ + Sm = − + λ2m . 2 2

λm is the frequency parameter which relates to the eigenfrequency of the beam by λ2m = ωm /ωs and is obtained by the characteristic equation 2 S 2 − S˜m cosh Sm cos S˜m − m sinh Sm sin S˜m − 1 = 0. 2Sm S˜m (13)

Due to maintained symmetry of the parallel plate model a first-mode discretization captures the nonlinear behavior sufficiently. The separation ansatz is substituted into (7)–(11) and employing Galerkin’s method by multiplication of Φ and integration over the length of the beam (from 0 to 1) yields J1 qn,τ τ + (J4 + κ1 J2 )qn − κ3 J3 qn3 + (μˆ 1 J1 + μˆ 2 J4 )qn,τ (qn+1 − qn )2 (qn+1,τ − qn,τ ) = μˆ 3 Φ¯ 4 ¯ 2 (γ + qn+1 Φ¯ − qn Φ) (qn − qn−1 )2 (qn,τ − qn−1,τ ) − ¯ 2 (γ + qn Φ¯ − qn−1 Φ) 1 + Φ¯ Γˆ V 2 ¯ 2 (γ + qn+1 Φ¯ − qn Φ) 1 − , ¯ 2 (γ + qn Φ¯ − qn−1 Φ)

with Φ¯ = Φ(1/2) and

0

where Om = Sm S˜m

5

x¨n + αxn + βxn3 + μL x˙n (xn+1 − xn )2 (x˙n+1 − x˙n ) − μ˜ NL (1 + xn+1 − xn )2 (xn − xn−1 )2 (x˙n − x˙n−1 ) − (1 + xn − xn−1 )2 1 ∗ 2 = η˜ DC + η˜ AC cos Ωt (1 + xn+1 − xn )2 1 − , (1 + xn − xn−1 )2

(15)

whereas the parameters are defined as α = 1 − κ1 |J2 | / J1 ζ14 , β = κ3 γ 2 |J3 | / ζ14 Φ¯ 2 J1 , μL = μˆ 1 /ζ12 + (μˆ 2 J4 )/ ζ12 J1 , μ˜ NL = μˆ 3 Φ¯ 2 / ζ12 J1 , Γˆ ∗ = Γˆ Φ¯ 2 / γ 3 J1 ζ14 , η˜ k = Γˆ ∗ Vk , ˆ 12 . Ω = Ω/ζ

(14)

Derivatives in (15) are with respect to t ∗ . Note that the gap parameter γ appears in the cubic stiffness parameter β and in both, the bias ηDC and the oscillating parameter ηAC . The dynamical system in (15) readily reduces to a set of nonlinearly coupled Duffinglike equations with the peculiarity that cubic and linear stiffness parameters are a direct outcome of the linear material properties and microbeam dimensions and thus, are not independent as proposed by the lumped mass approach.

6


2.3 A note on parameters Although in this work we do not include a quantitative comparison of results to experimental observations, dimensions, material properties and field parameters are chosen according to realistic parameter domains of existing devices in literature. The parameters of the arrays which are investigated in this paper relate to dimensions and parameters of the BR-array [6]. Material and design parameters of an individual microbeam are listed in Table 3 in Appendix A, from which model parameters are deduced (see Table 4 in Appendix A).

3 Equilibrium analysis 3.1 Fixed points and their stability In this section the conservative system of (15) is investigated to obtain the fixed points and to evaluate their stability. Therefore, damping and time dependent excitation terms in (15) are set to zero. For the nth beam the equilibrium equation is (1 + xn − xn−1 )2 − (1 + xn+1 − xn )2 2 · η˜ DC (1 + xn+1 − xn )2 (1 + xn − xn−1 )2 (16) − xn α + βxn2 = 0. Note that the equilibrium equations admit multiple solutions including a trivial configuration. It can readily be shown that the trivial solution is asymptotically stable below the first pull-in threshold. 3.1.1 Single-beam system Figure 2 shows two typical fixed-point characteristics of the dynamical system x as a function of the scaled 2 /α (η DC-voltage ηDC DC = 2η˜ DC ). For ratios β/α ≤ 2 the bifurcation diagram shows two while for ratios β/α > 2 three regions, in which multiple equilibria coexist. The range of the second region (beyond pull-in) is widened with increasing ratios of β/α. The fixedpoint characteristics depend on the linear and the cubic stiffness terms as well as on the DC-voltage parameter, which, in turn, is a function of the input of DC-voltage, some geometric parameters, the fundamental eigenfrequency and the gap between the beams in the array. In region I of Fig. 2 there are three solutions that satisfy (16) for N = 1. At the transition from

Fig. 2 Fixed points of the single-beam system; solid lines: stable, dashed lines: unstable; dashed lines denoted by ppi and spi stand for primary and secondary pull-in instabilities, respectively 2 /α = 1) two additional solutions region I to II (ηDC originate from x = 0. Region II has a total of 5 solutions. In region III only the trivial solution remains. For the single-element system there exist one primary and one secondary pull-in point. We denote a primary pull-in point to be the pitchfork bifurcation point while the terminology secondary pull-in is used to denote a saddle-node bifurcation. Note, furthermore, that what we denote here and henceforth as primary pull-in point differs from the classical “pull-in” terminology or “jump-to-contact” instability, as the beam does not snap into its walls at the pitchfork bifurcation point. Instead, there is another stable solution below the secondary pull-in point. The saddle-node, thus, relates to the classical phenomena again, as the beam(s) either snap into each other, or for the single-beam case, into the wall. Stability is obtained by the investigation of the potential

V (x) =

2 1 1 1 ηDC + αx 2 + βx 4 , 2 2 (x − 1) 2 4

(17)

that corresponds to the single-beam conservative system. Two of the solutions in region I (see Fig. 2) are maxima of (17) and therefore unstable saddles. The zero solution is an elliptic fixed point (or center). The saddle points which separate the stable and the unstable solutions in region II are obtained by setting the


derivative of (16) (for N = 1) equal to zero and solving the same for x or x 2 , respectively. Equation (16) (for N = 1) is a bi-quadratic equation in x whose solutions are x1,2 = ±1 and x3,4 = ±[1/3 − (2α)/(3β)]1/2 . The first two solutions correspond to the saddle appearing at the walls. The second set provides the coordinates of the remaining set of saddle points (= secondary pullin point). 3.1.2 Two-beam system Possible equilibria for the two-element system are the trivial solution, x1 = −x2 , and x1 = x2 . Figure 3 shows the fixed points of the system as a function of

7

2 /α. In region I for each the scaled DC-voltage ηDC beam there exist four solutions, the trivial, two nontrivial and a spurious one. Spurious solutions are such branches which, although being solutions of (16) mathematically, they have no physical meaning (due to mutual penetration). The trivial as well as the spurious solutions are stable elliptic centers here while the remaining of the non-trivial solutions are unstable saddles. In region II the stable equilibrium configuration corresponds to the x1 = −x2 solution. The trivial solution is and remains unstable throughout regions II–IV. Region III depicts two co-existing equilibrium configurations, one corresponding to the former stable x1 = −x2 solution and the other to x1 ≈ x2 , which is unstable. Note that the positions of the two beams are not perfectly identical unlike in the previous case with the opposite sign. Stability of the fixed points is determined by analysis of the extreme points of the potential

2 V (x1 , x2 ) = −η˜ DC

1 1 1 + + 1 + x1 1 + x2 − x1 1 − x2

1 1 + α x12 + x22 + β x14 + x24 . 2 4

Fig. 3 Fixed points of the two-beam system, α = 1, β = 13.935; thick lines: stable, thin lines: unstable or spurious, solid black lines: x1 , dashed lines: x2 , solid grey lines: trivial solution

Fig. 4 Potential V (x1 , x2 ) 2 /α = 1/2 at ηDC

(18)

2 /α = 1/2 Figure 4 shows the potential plotted at ηDC and α = 1 and β = 13.935. In order to investigate in the stability of the zero solution the potential V (x1 , x2 ) is analyzed with respect to maxima and minima applying the following theorem [25] for an arbitrary function with two variables

1 = B 2 − AC < 0 and 2 = A + C > 0

(19)

8


with ∂ 2 V (x1 , x2 ) A= , ∂x12 (0,0) ∂ 2 V (x1 , x2 ) B= , ∂x1 ∂x2 (0,0) ∂ 2 V (x1 , x2 ) C= , ∂x22 (0,0)

(20)

where V = V (x1 , x2 ) is given in (18). For the twoelement system there exist two primary pull-in points (where the second one is meaningless as it does not show a change in stability) and one secondary pull-in, which denotes the “jump-to-contact” point of the two beams. 3.1.3 Three-beam system Equations (16) for N = 3 show immediately that the zero solution satisfies all three equations. As in the case of the system with two beams, the identical configuration x1 = x2 = x3 does not satisfy the set of equations. Also, neither combinations of x1 = −x2 = x3 or −x1 = x2 = −x3 are solutions of the set of equations above. Figure 5 shows the fixed points of the three-element system. As observed for the system with two beams, the maximum value for a non-trivial equilibrium (and no allowance for mutual penetration) is one half of the gap between beams. For completeness, spurious equilibria in Fig. 5 are included but denoted by thin lines (regardless their stability). Region I, as for previously treated systems, is marked by the stable trivial equilibrium, followed by the stable configuration of alternating signs of neighboring beams in region II (see Fig. 5b). In region III the only stable equilibrium is the spurious configuration. A new combination of a beam configuration for the array with three beams is depicted in region V. Here, the stable configuration corresponds to x1 = −x3 and x2 = 0, meaning the two outer beams are in an opposite-sign configuration while the middle beam remains at the zero position. This solution remains stable to the shown saddle points marking the transition to region VI. Region VI shows the configuration x1 ≈ x2 ≈ x3 as known from the two-beam system, which is unstable also in the three-element array. In the presented parameter domain the fixed-point analysis of the three-element system reveals three pri-

Fig. 5 (a) Fixed points of the three-beam system (b) zoom-in of region II of (a); α = 1, β = 13.935; thick lines: stable, thin lines: unstable or spurious, solid black lines: x2 (middle beam), dashed lines: x1 , dash-dotted lines: x3 , solid grey lines: trivial solution

mary and three secondary pull-in points. A more detailed study of the equilibria of small-size arrays has recently been published in [14]. 3.2 Natural frequencies and conditions for internal resonances In order to compute natural frequencies of the microbeam array, the respective stable equilibrium solutions are substituted into the Jacobian matrix of (15) for increasing DC-voltage parameters. Substitution of the stable trivial equilibrium solution yields the coefficient matrix Cω of the linearized set of equations (¨x + Cω (ηDC )x = 0)


⎡ ⎢ ⎢ Cω = ⎢ ⎢ ⎣

2 ) (α − ηDC

2 ηDC 2

2 ηDC 2

2 ) (α − ηDC

. . .

. . .

0

0

9

⎤ 0

0

...

0

2 ηDC 2

0

...

0

⎥ ⎥ ⎥. ⎥ ⎦

. . . ...

0

2 ηDC 2

2 ) (α − ηDC

(21) The natural circular frequencies are the square root of the magnitudes of the corresponding eigenvalues, either of (21) or of the Jacobian matrix for substitutions of other non-trivial equilibrium solutions. As observed in the previous section, it is the trivial equilibrium configuration that dominates the static behavior below the first system’s bifurcation point. Thus, it is convenient to express the natural circular frequencies (for smallsize arrays) in closed form: 2 , N = 1: ω1 = α − ηDC N = 2: ω1 =

Fig. 6 Natural frequency of the single-beam system

3 2 α − ηDC , 2

1 2 α − ηDC , 2

√ 2 2 ηDC , N = 3: ω1 = α − 1 + 2 2 , ω2 = α − ηDC ω2 =

ω3 =

(22)

√ 2 2 α− 1− ηDC . 2

Figures 6, 7 and 8 show the natural circular frequencies and combinations of integers of the fundamental eigenfrequencies, such as (if existing for the array) ωn , nωn , ωN − ωn , ωn − ω1 , ωN + ωn , as well as ωn + ω1 , for n = 1, . . . , N . (A conversion of parameters into dimensional quantities is not given because we do not compare results to experimental observations in this work.) Note that the two- and threeelement systems count for interesting intersections in regions generally near a pull-in instability (below as well as beyond) but especially near the first primary pull-in point, where the second natural frequency is three times of the first one (see shaded regions in Fig. 7). For the DC-voltage parameter being near zero the system’s response falls into the one-to-one internal resonance region, i.e. ωi ≈ ωj for i, j = [1, . . . , N].

Fig. 7 Natural angular frequencies and integer combinations of the fundamental frequencies of the two-beam system; thick and thin lines correspond to stable and unstable equilibrium solution, respectively, solid lines: ωi , dashed lines: 2ωi ; dash-dotted lines: 3ω1 , grey lines: ω2 + ω1 (upper) and ω2 − ω1 (lower); shaded areas: regions of one-to-one, three-to-one and combinational resonances

For values of the DC-voltage parameter near the first pull-in point, the natural frequencies reveal a region of three-to-one internal and several combination resonances. In general, the ratios of ωn /ω1 ≈ 3 occur very close to the pull-in point. Ratios with the reference frequency different from the fundamental frequency occur beyond the pull-in point. Thick and thin lines in Figs. 6–8 correspond to stable and unstable static solution branches of Figs. 2–5. Solid, dashed and dashdotted lines denote ωn , 2ωn and 3ωn , respectively, and grey lines stand for either sums (upper branches) or differences (lower branches).

10


Fig. 8 (a) Natural angular frequencies and integer combinations of the fundamental frequencies of the three-beam system, (b) zoom-in of first primary and secondary pull-in region of (a); thick and thin lines correspond to stable and unstable equilibrium solution, respectively, solid lines: ωi , dashed lines: 2ωi ; dash-dotted lines: 3ωi , grey lines: combinations of sums (upper) and differences (lower)

3.3 The pull-in threshold The first bifurcation point of an array occurs at the voltage for which each first natural frequency of the system is equal to zero. Thus, the first instabilities for the single-, two-, and three-element array occur √ √ √ at η1ppi = α = 1, η1ppi = 2α/3 = 2/3, and √ √ η1ppi = 2α/(2 + 2) = 0.5858, respectively. Figure 9 depicts the natural frequency characteristics below the system’s first pull-in threshold for a singleup to a ten-beam array. We observe a decreasing DCvoltage value for the 1ppi-bifurcation point as the members in the array increase. Figure 10 and Table 5 (in Appendix B) show the characteristic and list the values of these voltages as the numbers of beams in-

Fig. 9 Natural frequencies for arrays of one, two, three and ten beams; dashed lines: 1st ppi voltage, dash-dotted lines denote the three-to-one ratio(s) or region of frequencies below the first pull-in instability of the array

crease in the array. However, the first bifurcation voltage for a mid- and large-size array occurs to converge √ to the value η1ppi = 0.5 (see Fig. 10). This is significant information for the design of such devices


11

Fig. 10 Change of first ppi-voltage in relation to number of members in the array

(e.g. gap size between the beams) as well as for experimentalists, as it reveals knowledge about e.g. the voltages that can be safely applied without damaging the array. 3.4 A limiting bound on displacements In the following, we investigate the limiting bound on displacements of the fixed point branches of the single- and two-beam dynamical systems (see Figs. 2 and 3). The Hamiltonian form of the single-beam system is x˙ =

∂H , ∂y

y˙ = −

∂H , ∂x

Fig. 11 Hamiltonian state-space of the single-beam system; (a) region I, (b) region II; solid lines: separatrix, dashed lines: sample of inner or outer isoline, respectively

(23) The Hamiltonian form of the two beam system is

where 1 H (x, y) = y 2 + V (x), 2

x˙1 =

∂H , ∂y1

y˙1 = −

∂H , ∂x1

x˙2 =

∂H , ∂y2

y˙2 = −

∂H , ∂x2

(24)

and the potential is given by (17). In the following the branches of the equilibrium solution (see Fig. 2) for chosen voltage parameters of regions I and II are substituted into (24) and results are presented in the Hamiltonian state-space plane Fig. 11. Figure 11a depicts a standard heteroclinic structure [12, 13] for the single-beam system in region I of Fig. 2, whereas Fig. 11b reveals the structure replicated in region II of Fig. 2. In the case of two beams (see Fig. 3) the Hamiltonian function H describes the total energy T + V .

(25)

where 1 1 H (x1 , x2 , y1 , y2 ) = y12 + y22 + V (x1 , x2 ), 2 2

(26)

and the potential is given by (18). The values of the equilibrium branches are substituted into (26) at selected locations of each of the regions I–IV. Results are presented in the Hamiltonian state-space plane in Fig. 12. Figure 12a–c reveal a homoclinic structure in correspondence to the regions I, II and III of Fig. 3 for

12


voltage ηDC in Fig. 11b, whereas the opposite occurs for the two-beam system in Fig. 12.

4 Weakly nonlinear asymptotic analysis The analysis in Sect. 3.2 has revealed interesting regions of intersecting natural frequencies which sets the structure of the following analysis. In the remaining sections of this work investigations focus on the twoand three-beam array’s behavior in regions of their internal one-to-one, parametric, and internal three-toone resonances, which according to Sect. 3.2 correspond to low, medium and large DC-voltage inputs, respectively. Figure 7, for instance, depicts these three regions below the first bifurcation point of the twobeam array. For the DC-voltage parameter being near zero the system’s response falls into the one-to-one internal resonance region (region I), i.e. ωi ≈ ωj for i, j = [1, . . . , N ]. With increasing values of the DCvoltage parameter, the system’s response enters the regions of parametric resonances (region II). For values of the DC-voltage parameter near the first bifurcation point, the natural frequencies reveal a region of threeto-one internal and several combination resonances. The borders between regions I/II and II/III are set arbitrarily here in order to illustrate an existence of such regions. A proper definition can be given by defining a small parameter epsilon and its corresponding DCdomain, within which the relative differences of integers of the natural frequencies are equal or smaller than epsilon. The following analysis of the two- and three-beam array is thus carried out in all three regions below the first primary pull-in instability, while starting with DC-voltages being close to zero, which is the analysis of the system in its one-to-one internal resonance. 4.1 Single beam Fig. 12 Hamiltonian state-space of the two-beam system; (a) region I, (b) region II, (c) region III, (d) region IV; bold solid lines: separatrix, dashed lines: sample of inner or outer isoline, respectively, grey lines: spurious

the two-beam system. Figures 12c and 12d show a heteroclinic structure (similar to the structure found for the single-beam system Fig. 11b), which corresponds to the regions III and IV in Fig. 3, respectively. Note that the area contained within the heteroclinic structure in Fig. 11a decreases with the increase of the bias

Considerable work has been done for the single-beam system under the same [28] or similar [42] electrode configuration. However, for the reason of completion, we also include the analysis of the single-beam system in this work. The equations of motion of the singlebeam system, given in (15), reduce to x 2 (1 + x 2 ) x˙ (1 − x 2 )2 2 x , = ηDC + ηAC cos Ωt ∗ (1 − x 2 )2

x¨ + αx + βx 3 + μL x˙ + μNL

(27)


in which, and henceforth, the following parameters are redefined to μNL = 2μ˜ NL and ηi = 2η˜ i (i = [DC, AC]). Before carrying out the asymptotic multiple-scales method [39], (27) is pre-multiplied by the denominator of the forcing term. The dynamical response of the beam is represented by three different time scales that are distinguished by the small parameter . Note that scales and small parameters for the multiple-scales technique have to be chosen according to parameter and resonance regions as discussed in the previous section on the equilibrium analysis. For instance, considering a dynamical analysis in the region near the first bifurcation point of the system (Fig. 2 and Fig. 11), the DC-voltage parameter is rather large and from Fig. 11 the AC-voltage parameter is limited to be very small. Typical magnitudes in the near instability point region would be 101 V for the DCvoltage and 10−2 V for the AC-voltage. Thus, for this special case, the AC-voltage parameter is a candidate for the small parameter in the multiple-scales technique. Furthermore, the linear damping coefficients in MEMS devices (in general) are usually low due to large quality factors. Therefore also the linear damping parameter is treated as small parameter. In the following, the method of multiple-scales is carried out for a large DC-voltage and a small AC-voltage parameter, which corresponds to the system’s behavior in the middle of region I of Fig. 2 (below 1ppi). Due to afore mentioned low damping and excitation parameters the solution can be decomposed into fastand slow-varying components. Thus, the linear damping parameter μL as well as the AC-voltage parameter ηAC are re-scaled to μL = 2 μ¯ L and ηAC = 2 η¯ AC . The contribution of nonlinear terms in (27), which are assumed to be small, causes a deviation from the solution of the purely linear problem. This deviation is expressed by a variation of the amplitude and phase based on the slow time scale T2 = 2 t ∗ . The displacement is extended into a power series in : x = 3j =1 j xj (T0 , T1 , T2 , . . .) + O( 4 ). The coefficients of each order of are collected and form the following set of equations: O 1 : D02 x1 + ω2 x1 = 0, O 2 : D02 x2 + ω2 x2 = −2D0 D1 x1 , O 3 : D02 x3 + ω2 x3 = −2D0 D1 x2 − D12 x1 − 2D0 D2 x1

(28) (29)

13

− βx13 − μ¯ L D0 x1 − μNL x12 D0 x1 + 2x12 D02 x1 + αx1 (30) + 2ηDC η¯ AC cos Ωt ∗ x1 2 . The derivatives D with ω2 = α − ηDC m for m = ∂ dTm [0, . . . , 2] are defined as Dm = ∂Tm dt ∗ . The solution of the O( 1 ) equation is

x1 = A(T1 , T2 ) exp (iωT0 ) ¯ 1 , T2 ) exp (−iωT0 ). + A(T

(31)

Substitution of (31) into (29) leads to D1 A(T1 , T2 ) = 0 which means that A is independent of the time scale T1 . The particular solution of (29) vanishes and the homogeneous solution is identical to the solution of (28), i.e. x2 = x1 . Substitutions of solutions x1 and x2 into (30) with D1 x1 = D1 x2 = 0 yields D02 x3 + ω2 x3 = −iω 2D2 A + μ¯ L A + μNL A2 A¯ 2 2 A A¯ − 3 β − 2ηDC + ηDC η¯ AC A¯ exp (iσ T2 ) exp (iωT0 ) 3 2 A − iωμNL A3 + − β − 2ηDC + ηDC η¯ AC A exp (iσ T2 ) exp (3iωT0 ) + cc., (32) where the detuning is 2 σ = Ω − 2ω and cc. represents the conjugate complex parts. Elimination of secular terms in (32) yields iω 2D2 A + μ¯ L A + μNL A2 A¯ 2 2 A A¯ − ηDC η¯ AC A¯ exp (iσ T2 ) = 0. + 3 β − 2ηDC (33) Substituting the polar coordinates A = a exp (iθ )/2 into (33) and separating imaginary and real terms results in the following slowly varying evolution equations a = (δex sin ψ − ζL )a − ζNL a 3 ,

(34)

aψ = (σ + 2δex cos ψ)a − δ3 a 3 ,

(35)

with ψ = σ T2 − 2θ , ζL = μ¯ L /2, ζNL = μNL /8, δex = 2 )/(4ω). DerivaηDC η¯ AC /(2ω) and δ3 = 3(β − 2ηDC tives in (34) and (35) are with respect to T2 . In steady state operation any changes with respect to time vanish, i.e. a and ψ in (34) and (35) are equal to zero. One solution of (34) and (35) is the trivial

14


solution a = 0. In order to find the solutions for the non-trivial case the harmonic terms in (34) and (35) are eliminated by solving the equations for the same, respectively, then squaring each of the equation and adding them, which results in δ3 2 σ 2 2 2 2 δex = ζL + ζNL a a − + . (36) 2 2 Equation (36) is multiplied by 4 and then solved to yield the frequency response function (Ω = f ( 2 a 2 )): Ω1,2 = δ3 2 a 2

ηDC ηAC 2 μL μNL 2 2 2 + a ±2 − 2ω 2 8 + 2ω.

(37)

Note that (36) is a bi-quadratic function in the amplitude a which yields to the constraint a 2 ≥ 0. The minimum positive amplitude amin is found by differentiating (36) with respect to σ and setting da/dσ equal to zero. Substituting the extremum of σE into (36) yields ηDC ηAC 4 2 (38) − μL , amin = μNL ω which is postulated to be greater or equal zero. From (38) it becomes directly obvious that a positive amplitude squared depends mainly on the two quantities of the oscillating input voltage and the linear damping coefficient. Thus, ηAC ≥

ωμL . ηDC

(39)

For the given set of parameters α = 1, β = 13.935, ηDC = 0.5750, ηAC = 0.0028, Q = 500, and μNL = 0.1, a hardening frequency response is observed, Fig. 13. For the nonlinear damping parameter equal to zero the curves do not close and amplitudes become infinite. According to the number of stable and co-existing solutions, the bifurcation structure of the single-beam system unfolds to distinct regions. The trivial solution is stable in regions I and III (and IV for μNL = 0). The stability of the non-trivial steady state solutions is determined by analyzing the eigenvalue problem of the Jacobian of the system (see (34) and (35)). The characteristic equation is λ2 + m1 λ + m2 = 0, where m1 = 2(ζNL a 2 + ζL a + ζNL )a > 0 (∀a > 0) and m2 = a 3 [4ζNL (ζL + ζNL a 2 ) − δ3 (σ + δex − δ3 a 2 )]. Thus, the upper branch (Fig. 13), pre-

Fig. 13 Frequency response of the single-beam system, ηDC = 0.5750, ηAC = 0.0028, Q = 500, μNL = 0.1; solid lines: stable, dashed lines: unstable, triangles: numerical verification, thin lines: μNL = 0

sented by the bold solid line is stable in region II while the lower (dashed line) is unstable. 4.2 Two beams We employ the method of multiple-scales to the twomicrobeam array given by x¨1 + αx1 + βx13 + μL x˙1 x12 x˙1 (x2 − x1 )2 (x˙2 − x˙1 ) − μ˜ NL − (1 + x2 − x1 )2 (1 + x1 )2 2 = η˜ DC + η˜ AC cos Ωt ∗ 1 1 , (40) × − (1 + x2 − x1 )2 (1 + x1 )2 x¨2 + αx2 + βx23 + μL x˙2 (−x2 )2 (−x˙2 ) (x2 − x1 )2 (x˙2 − x˙1 ) − μNL − (1 − x2 )2 (1 + x2 − x1 )2 2 = η˜ DC + η˜ AC cos Ωt ∗ 1 1 , (41) × − (1 − x2 )2 (1 + x2 − x1 )2 where a three-term solution for both beams is assumed as xn =

3

j xnj (T0 , T1 , T2 , . . .) + O 4

j =1

and its respective derivatives.

(42)


Following the equilibrium analysis in Sect. 3, we examine here the cases of parametric, one-to-one, and three-to-one internal resonances. Figures 7 and 9 (N = 2) show the resonance regions in relation to the DC-voltage parameter, which reveals the following detuning: 1 = ω2 − ω1 (one-to-one internal resonance) for small ηDC (near zero), (b) ω2 = mω1 for m = 1, 2, 3, . . . (parametric resonance), and (c) 2 σ1 = ω2 − 3ω1 (three-to-one internal resonance) for large ηDC (near the first pull-in instability).

(a)

2σ

As for the single beam system, the dynamical response of the two beams is represented by three different time scales that are distinguished by the small parameter . The scales and small parameters for the multiplescales technique are chosen according to parameter and resonance regions (a)–(c). Thus, for an analysis in or near perfect one-to-one internal resonance, the DCvoltage parameter is zero or very small. On the other hand, Fig. 12 shows a limiting bound on displacements and thus defines the limit for the AC-voltage parameter to be rather large in this case. Unlike in case (a), the DC-voltage parameter for case (c) (three-to-one internal resonance) is larger and the domain of attraction reveals a limit for the AC-voltage parameter to be very small. In the case of parametric resonance yet another scenario of parameter scaling needs to be considered. Therefore, the scaling of small parameters is addressed as we carry out the multiple-scales technique in each region. (a) Parametric resonance (ω2 = mω1 ) We begin the dynamical analysis of the two-beam system in the parametric resonance region, i.e. ω2 is not a multiple integer of ω1 . The DC-voltage parameter is of two orders of magnitude larger than the AC-voltage parameter. Thus, no scaling is performed for the DCvoltage while ηAC is re-scaled to ηAC = 2 η¯ AC . The linear damping coefficient μL is very small due to relatively high Q-factors observed in MEMS in general, which justifies a rescaling to μL = 2 μ¯ L . Substitution of the solution (42) into (40) and (41), collecting terms in orders of , results in the following set of equations: (43) O 1 : D02 xn1 + bn1 x11 + b2n x21 = 0, 2 (44) O : D02 xn2 + bn1 x12 + b2n x22 = fn2 ,

O 3 : D02 xn3 + bn1 x13 + b2n x23 = fn3

15

(45)

2 ) for i = j and for n = [1, 2], where bij = (α − ηDC 1 2 bij = 2 ηDC for i = j , (i, j = [1, 2]) and fn2 and fn3 are given in Appendix C.1. The two natural frequen2 cies determined from (43) are ω1 = α − 3/2ηDC 2 . The homogeneous solution and ω2 = α − 1/2ηDC to (82) is

xn1 = (−1)n+1 A1 exp (iω1 T0 ) + A2 exp (iω2 T0 ) + cc.,

(46)

where Aj = Aj (T1 , T2 ) for j = [1, 2] and cc. denote conjugate complex terms. Substitution of (46) into (44) leads to D02 xn2 + bn1 x12 + b2n x22 = (−1)n 2iω1 D1 A1 exp (iω1 T0 ) − 2iω2 D1 A2 exp (iω2 T0 ) + NST n + cc.,

(47)

where NST n represent non-secular terms. Elimination of secular terms yields D1 Aj = 0. Thus, Aj (T1 , T2 ) are independent of the time scale T1 . The non-secular terms 3 2 2 3A1 exp(2iω1 T0 ) NST n = −(−1)n ηDC 4 − A22 exp(2iω2 T0 ) + 3|A1 |2 − |A2 |2 + (−1)n 2A1 A2 exp i(ω1 + ω2 )T0 + (−1)n 2A¯ 1 A2 exp i(ω2 − ω1 )T0

(48)

consist of quadratic and sum and difference frequency terms. Thus, an ansatz for the solutions of xn2 according to the right-hand side gives xn2 = Cn1 (T2 ) exp (2iω1 T0 ) + Cn2 (T2 ) exp i(ω1 + ω2 )T0 + Cn3 (T2 ) exp i(ω1 − ω2 )T0 + Cn4 (T2 ) exp (2iω2 T0 ) + Cn5 + cc.,

(49)

where the expressions of Cnj for j = 1, . . . , 5 are provided in Appendix C.1. The solutions, which satisfy the equations of order O() and O( 2 ), are then substituted into the equations of order O( 3 ) (45). Note that the terms containing D1n (for n = 1, 2) vanish.

16


In the following we focus our analysis on: Ω = 2ω1 : principal parametric resonance of the outof-phase mode, Ω = 2ω2 : principal parametric resonance of the inphase mode, Ω = ω1 + ω2 : combination parametric resonance of the additive type and Ω = ω2 − ω1 : combination parametric resonance of the subtractive type. Parametric excitation at Ω ≈ 2ω1 We define the detuning for the forcing frequency Ω to be 2 σ = Ω − 2ω1 .

(50)

In order to determine the solvability conditions of (45), we seek a particular solution of the form xn3 = Pn (T2 ) exp (iωn T0 ) + Qn (T2 ) exp (iωn T0 ) + cc. for n = [1, 2]. The complex evolution equations are deduced to a solution of Pn and Qn , respectively, by substitutions of the particular solution into (45) and equating the coefficients of exp(iωn T0 ) on both sides. We employ the polar ansatz and separate imaginary and real terms which yields the slowly varying dynamical system 9 a1 = − ζNL a13 − ζNL a1 a22 2 − (ζL − 3δex1 sin ψ1 )a1 ,

(51)

a1 ψ1 = (δDC11 − δ31 )a13 + (δDC12 − 2δ31 )a1 a22 + (6δex1 cos ψ1 + σ )a1 , 1 a2 = − ζNL a23 − ζNL a12 a2 − ζL a2 , 2 1 a2 ψ2 = δDC22 + δ32 a23 2 + (δDC21 + δ32 )a12 a2 ,

(52) (53)

Equation (55) is a bi-quadratic equation in a1 presenting the amplitude–frequency relationship for the out-of-phase vibration mode, which yields to the constraint a12 ≥ 0. The minimum positive amplitude a1min is found by differentiating (55) with respect to σ and setting da1 /dσ equal to zero. Substituting the ex2 yields tremum of σE into (55) and solving for a1min 2 a1min ≥

8(3ηDC ηAC − 2μL ω1 ) , 18μNL ω1

(56)

which is postulated to be greater or equal zero. As in the single-beam system, (56) reveals that a positive amplitude squared depends mainly on the two quantities of the oscillating input voltages and the linear damping coefficient. Thus, ηAC ≥

2μL ω1 . 3ηDC

(57)

Note the difference of factor 2/3 for the two-beam compared to the single-beam threshold. Parametric excitation at Ω ≈ 2ω2 For the case of principal parametric resonance of the in-phase mode the detuning is 2 σ = Ω −2ω2 . By following the same procedure as in the previous paragraph, the equations describing the slowly varying dynamical system are 9 a1 = − ζNL a13 − ζNL a1 a22 − ζL a1 , 2 1 1 δ31 − δDC11 a13 a1 ψ1 = 2 2 1 + δ31 − δDC12 a1 a22 , 2

(58)

(59)

1 a2 = − ζNL a23 − ζNL a12 a2 − [ζL − δex2 sin ψ2 ]a2 , 2 (60) a2 ψ2 = (−δ32 − 2δDC22 )a23 − (2δ32 + 2δDC21 )a12 a2

(54)

whereas the parameters are given in Appendix C.1. Note that the equations decouple for zero cubic damping (ζNL = 0). Thus, the steady state solutions are either the trivial solution (a1 = a2 = 0) or a non-trivial solution for a1 obtained from (51) and (52) where a2 = 0 2 2 2 = 2ζL + 9ζNL a12 + (δDC11 − δ31 )a12 + σ . 36δex1 (55)

+ (2δex2 cos ψ2 + σ )a2 .

(61)

The parameters are documented in Appendix C.1. Note that, also in the in-phase case, the equations decouple for zero cubic damping (ζNL = 0). The steady state solutions consist of the trivial solution (a1 = a2 = 0) or/and the non-trivial solution a2 , for a2 being obtained from (60) and (61) with a1 = 0. The characteristic equation for steady state is then 2 2 2 = 2ζL + ζNL a22 + −(2δDC22 + δ32 )a22 + σ . 4δex2 (62)


17

The solutions of (62) are solutions of the system, if all a22 are positive 2 a2min ≥

4(ηDC ηAC − 2μL ω2 ) . μNL ω2

(63)

The condition that is deduced from (63) is ηAC ≥

2μL ω2 . ηDC

(64)

Combination parametric resonance For the case of combination parametric resonance of the additive type, the following modulation equations are obtained 4iωn D1 An = n8ω1 δDCnn − 6β − ξn iωn μNL |An |2 An + n8ωn δDCnl − 12β − 2iωn μNL An |Al |2 − 2iωn μ¯ L An ,

(65)

where ξ1 = 9, ξ2 = 1, and n = 1, 2. We note that no excitation term is preserved in (65) and that there is therefore no response of the two-beam system to be expected in this case. The same occurs for the case of combination parametric resonance of the difference type, for which the modulation equations are identical to (65). Figure 14 portrays the steady state solutions for three different AC-voltage parameters. For a Q-factor of Q = 500 and a DC-voltage of ηDC = 0.5750 the AC-thresholds regarding the in-phase and out-ofphase solutions are ηAC = 0.0064 and ηAC = 0.0016, respectively. The dynamic response in Fig. 14a is plotted for an AC-voltage below the AC-threshold of the in-phase solution. The response in Fig. 14b is plotted for an AC-voltage parameter above both thresholds, where both solutions, in-phase and out-of-phase, exist. For large values of the AC-voltage the response shown in Fig. 14c exhibits an overlap of out-of-phase and in-phase solutions. Figure 14b portrays the two steady state responses for ηAC = 0.0086. The trivial solution is stable except in regions II and V. The stability of the nontrivial steady state solutions (depicted for the outof-phase case by solid lines in Fig. 14b) is determined by analyzing the eigenvalue problem of the Jacobian of the system (51)–(54). Recall that in the out-of-phase vibration mode the steady state solution exist for a2 = 0. Thus, the characteristic equa-

Fig. 14 Frequency response of the two-beam system at parametric resonance, ηDC = 0.5750, Q = 500, μNL = 1.2 (a) ηAC = 0.0029, (b) ηAC = 0.0086, (c) ηAC = 0.0575; numerical verification: triangle: out-of-phase mode, diamond: in-phase mode, asterisk: additional in-phase solution not identified by multiple-scales analysis

tion, like in the case of the single-beam system, is of second order λ2 + m1 λ + m2 = 0, with m1 = 2 +18a 2 ζ ζ − 18a12 ζNL +2ζL > 0 and m2 = 81a14 ζNL 1 NL L 2 2 + a2δ 2a14 δ31 δDC11 − a12 δ31 σ + a14 δDC11 + a14 δ31 1 DC11 σ . Upper branches in Fig. 14, presented by thick lines, are stable while lower branches (thin lines) are unstable. Figure 15 shows the stability map of the nontrivial steady state solutions of the two-beam system in parametric resonance. The thin dash-dotted lines in both plots mark the AC-value corresponding to Fig. 14b. Solid and dashed lines present the critical lines m2 = 0, respectively and thus separate stable from unstable regions.

18


Fig. 15 Stability diagram of the non-trivial solutions of the two-beam system at parametric resonance, ηDC = 0.5750, Q = 500, μNL = 1.2; (a) stability regions for the upper branches, (b) stability regions for lower branches; solid lines: m2 = 0 for out-of-phase solution, dashed lines: m2 = 0 for in-phase solutions; horizontal thin dash-dotted lines show the stability of the specific solutions (a)–(c) of Fig. 14

(b) One-to-one internal resonance (ω2 ≈ ω1 ) For ω2 ≈ ω1 the DC-voltage parameter ηDC is very small. Thus, ηDC is scaled in terms of the small parameter to ηDC = 2 η¯ DC . Previous work of the authors discuss results of the array’s response in its pure oneto-one internal resonance (for ηDC = 0) [15]. Unlike in the parametric resonance case, where the AC-voltage input occurred to be much smaller than the DC-voltage parameter, in the one-to-one internal resonance case the alternating AC-voltage can take on larger values than the DC input. However, the AC-voltage parameter is still assumed to be a small parameter of order and thus, ηAC = η¯ AC . The linear damping coefficient μL is scaled as μL = 2 μ¯ L . Substitution of the solution form (42), including the scaling of the voltage parameters and the linear damping coefficient and collecting the terms of different orders in , results in the following set of equations: O 1 : D02 xn1 + ω2 xn1 = 0, O 2 : D02 xn2 + ω2 xn2 = fn2 , O 3 : D02 xn3 + ω2 xn3 = fn3 ,

(66) (67) (68)

with ω2 = α and fn2 , fn3 given in Appendix C.2. 2 cos2 Ωt ∗ in (104) is rewritThe excitation term η¯ AC 2 ten to η¯ AC (1 + cos (2Ωt ∗ ))/2, where the harmonic part reveals the parametric excitation. Note that the occurrences of the terms containing D02 xn1 + αxn1 (n = [1, 2]), respectively, on the right-hand side of (67) and (68) have their origin in the pre-multiplication

of the denominators of the forcing terms in (40) and (41). Such terms exhibit the terms involved in the same technique but, instead of a pre-multiplication of the denominator, having expanded the forcing terms into a Taylor series. A single mode representation of the solution xn1 = An (T1 , T2 ) exp (iωT0 ) + A¯ n (T1 , T2 ) exp (−iωT0 )

(69)

is chosen, which upon substitution into (67) leads to D02 xn2 + ω2 xn2 = −2iωD1 An (T1 , T2 ) exp (iωT0 ) + cc.

(70)

The secular terms in (70) are −2iωD1 An , respectively. Other (non-secular) terms cancel out. D1 An = 0 in (70) mean that the An (T1 , T2 ) are independent of the time scale T1 and, thus, terms in (68) containing D1n vanish. Furthermore, the solution of (67) is equal to the solution of (66), and, thus, xn2 = xn1 . The principal and fundamental parametric resonance frequencies occur at Ω = ω and Ω = ω/2, respectively. Thus, the selected detuning is Ω = ω + σ . Substitutions of solutions xn1 and xn2 into (68) and elimination of secular terms yields the slowly varying complex evolution equations: 1 0 = −2iωD1 An + iωμNL A21 A¯ l 2 1 1 2 2 Al η¯ AC + η¯ DC − 2 2


1 2 + η¯ AC exp (i2σ T2 )A¯ n + μNL iωA1 A2 A¯ n 4 1 + μNL iωA22 A¯ l − μNL iωA1 A2 A¯ l 2 1 − μˆ 3 iωA2l A¯ n − (3β + μNL iω)A2n A¯ n 2 1 2 − η¯ AC exp (i2σ T2 )A¯ l ω 8 1 2 2 An , − μ¯ L i − η¯ AC − η¯ DC 2

a2 (ψ1 + ψ2 )

= −δ3 a23 − ζNL sin ψ1 a13 − a12 a2 + 2a1 a22 1 − δAC cos ψ1 + cos ψ2 + 2δDC cos ψ1 a1 2 1 + 2 δAC 1 + cos (ψ1 + ψ2 ) 2 + σ + 2δDC a2 , (75)

(71)

where, as previously defined, l = −1n+1 + n. Substituting the polar coordinates An = an exp (iθn )/2 into (71) and separating imaginary and real terms yields the following slowly varying dynamical system 1 a1 = ζNL −a13 + + cos ψ1 a12 a2 2 1 1 2 3 − 1 + cos ψ1 a1 a2 + cos ψ1 a2 2 2 1 − ζL + δAC sin (ψ1 − ψ2 ) a1 2 1 1 + δAC sin ψ1 − sin ψ2 2 2 + δDC sin ψ1 a2 , (72) a1 (ψ1

− ψ2 ) = δ3 a13 − ζNL 3 sin ψ1 2a12 a2

− a1 a22 + a23 1 − 2 δAC 1 + cos (ψ1 − ψ2 ) + σ + 2δDC a1 2 1 + δAC cos ψ1 + cos ψ2 2 + 2δDC cos ψ1 a2 , (73)

a2

19

1 1 = ζNL cos ψ1 a13 − 1 + cos ψ1 a12 a2 2 2 1 2 3 + cos ψ1 a1 a2 − a2 + 2 1 1 − δAC sin ψ1 + sin ψ2 + δDC sin ψ1 a1 2 2 1 + −ζL + δAC sin (ψ1 + ψ2 ) a2 , (74) 2

whereas parameters are presented in Appendix C.2. Fixing one of the beams in the array by setting either a1 or a2 equal to zero, results in a set of two equations which are identical to the set of equations of the single-beam system (cf. (34) and (35)). Steady state is deduced from (72)–(75) by setting an and ψn (n = 1, 2) equal to zero. There exist three non-trivial solutions for the two-beam system in addition to the trivial solution (a1 = a2 = 0). (i) The first non-trivial solution is derived from the case when both beams vibrate in-phase and with equal amplitudes, a1 = a2 . The phase angle ψ1 = θ1 − θ2 becomes 0 + 2kπ (k is an integer). a and ψ correspond to either of the beams and thus denote an and ψ2 = 2σ T2 − 2θn . a1 = a2 = a satisfies (72)–(75) along with a1 = a2 = a, sin ψ2 = 2

sin ψ1 = 0,

cos ψ1 = 1,

ζNL a 2 + 2ζL , δAC

cos ψ2 = 2

δ3 a 2 − 2σ − δAC − 2δDC . δAC

Substitutions of this solution into (72) results in the amplitude–frequency relationship for the in-phase case 2 2 δAC = 4ζL + 2ζNL a 2 2 + 2δ3 a 2 − 4σ − 2δAC − 4δDC .

(76)

(ii) The second steady state solution is deduced from the case when both beams vibrate with the same amplitude but out-of-phase. The phase angle ψ1 = θ1 − θ2 then becomes π + 2kπ . This along with a1 = a2 = a, sin ψ2 = −

sin ψ1 = 0,

2 5ζNL a 2 + 2ζL , 3 δAC

cos ψ1 = −1,

20


cos ψ2 = −

2 δ3 a 2 − 2σ − 3δAC − 6δAC 3 δAC

satisfies (72)–(75). Substitution of this solution into (72) results in the amplitude–frequency relationship for the out-of-phase case 2 2 9δAC = 4ζL + 10ζNL a 2 2 + 2δ3 a 2 − 4σ − 6δAC − 12δDC .

(77)

(iii) The third steady state solution is found by solving the algebraic set of equations (deduced from (72)– (75)) numerically. This solution incorporates unequal amplitudes (a1 = a2 ) and corresponds to an in-phase mode as shown in the next section. The in-phase (76) and out-of-phase (77) solutions are bi-quadratic equations in a, respectively. The condition for which the minimum amplitude square, a 2 , is greater than zero is determined from solutions (76) and (77), respectively, by computing the derivative of the same with respect to σ and setting da/dσ = 0, which, for the in-phase and out-of-phase cases, results in 2 amin ≥

δAC − 4ζL 2ζNL

2 amin ≥

3δAC − 4ζL . 10ζNL

and

(78) (79)

Thus, (78) and (79) yield the conditions for lower bounds on the AC-voltage parameter for in- and outof-phase response:

8 ηAC ≥ ω (in-phase) and (80) Q

8 ηAC ≥ ω (out-of-phase). (81) 3Q For a Q-factor of Q = 500, the AC-threshold obtained from (80) is ηAC = 0.2153 (in-phase) and that obtained from (81) is ηAC = 0.1243 (out-of-phase), respectively. The AC-threshold of the solution for which the two beams vibrate in-phase with unequal amplitudes is determined numerically to be ηAC = 0.1919. The influence of different AC-voltage inputs is depicted in Fig. 16. Figure 16a shows the dynamic response of the two-beam system for an AC-value above the out-of-phase threshold (ηAC > 0.1243) and below the in-phase and unequal amplitude threshold (ηAC < 0.1919). Thus, the in-phase solutions with equal and

Fig. 16 Frequency response of the two-beam system at one-to-one internal resonance [15]; dashed lines: out-of-phase mode, solid and dash-dotted lines: in-phase mode; ηDC = 0.0785, Q = 500, μNL = 1.2; (a) ηAC = 0.1831, (b) ηAC = 0.2092, (c) ηAC = 0.2171, (d) ηAC = 0.3138

unequal amplitudes do not appear. The dynamic response in Fig. 16b is plotted for 0.1919 < ηAC < 0.2153. Thus, the in-phase and equal amplitude solutions do not exist. In Fig. 16c and Fig. 16d, which are plotted for AC-values above both thresholds, all the inphase and out-of-phase solutions exist. Note that up to


Fig. 17 Frequency response of the two-beam system at one-to-one internal resonance; ηDC = 0.0785, ηAC = 0.2354, Q = 500, μNL = 0; bold black lines: stable, thin grey lines: unstable, markers denote numerical simulations, hollow markers: periodic response, solid markers: quasi-periodic response, diamond, stars: in-phase, triangles: out-of-phase; solutions: (0) trivial, (1) and (2) out-of-phase mode, (3), (4), (5) and (6) in-phase mode, H5j for j = [1, 2]: Hopf points of solution (5), BP4 : branch point of unstable solution (4); vertical grey dashed lines: region separators

another critical AC-value, the unequal amplitude solution includes two separate closed-loop branches. Only for values of ηAC = 0.2236 and above the characteristic curves join to yield continuous branches. In the following we investigate in the stability of the solutions discussed above. As previously observed, the nonlinear damping parameter holds the function of closing the response curves. The principal information of the dynamic behavior and bifurcation structure is maintained for μNL = 0, i.e. the curves show no significant change except for the region near the closing of curves for μNL = 0. Thus, for the sake of simplifying the equations in order to express stability regions in closed form the nonlinear damping parameter is set to zero. Figure 17 depicts the four steady state solutions for μNL = 0 and ηAC = 0.2354. These solutions include the trivial solution (denoted by (0)), out-ofphase (branches (1) and (2)), in-phase and equal amplitudes (branches (3) and (4)), and in-phase and unequal amplitudes (branches (5) and (6)). For stability analysis equations (72)–(75) are rewritten into the form ξ = f (ξ ) with ξ = [a1 , ψ1 , a2 , ψ2 ]T . Stability

21

analysis of the trivial steady state solution (0) is done using the cartesian form of (72)–(75), whereas stability of the non-trivial steady state solutions (1)–(6) make use of the polar form. The eigenvalue problem for the Jacobian of the system is analyzed using the quartic characteristic equation λ4 + m1 λ3 + m2 λ2 + m3 λ + m4 = 0, where m1 is always greater than zero for all solution branches and regions. Figure 18 depicts the bifurcation characteristics of the AC-voltage parameter ηAC over the scaled excitation frequency Ω/ω. For clarity only distinct lines that contribute to the stability of the system are plotted. Furthermore, reviewing (80) and (81), there exists a critical ACvoltage value for each solution for which the amplitude square, an2 , is equal to zero. This corresponding AC-threshold for each solution is represented by the bold dashed-dotted lines in Fig. 18, respectively. In region I of Fig. 17 there exists only a trivial solution which is stable. In Fig. 18-0 the m4 = 0 lines are the lines that mark the stability regions. The trivial solution in region I lies in the m4 > 0 area and is thus stable therein. In region II there are two solution branches, an unstable trivial (0) and a stable out-ofphase solution (1). Figure 18-1 shows the m4 = 0 line and the AC-threshold marked by the dash-dotted line. The solution branch (1) appears in the m4 > 0 region and is thus stable. The m4 = 0 line is identical to the condition of critical AC-values for which the amplitude square an2 is equal to zero. For values below that threshold no solutions exist. All other distinct lines like m3 = 0 or 3 = m3 (m1 m2 − m3 ) − m21 m4 = 0 (Hopf criteria) occur outside the region where solutions are valid, and are therefore not included in the figure. In region III, three solutions exist: a stable trivial (0), a stable out-of-phase (1), and an unstable outof-phase (2). For the case of no nonlinear damping the stability status of these two branches remain the same throughout all remaining regions in Fig. 17 (I–IX and III–IX, respectively). In region IV the trivial solution is unstable and, in addition to the previous three solutions, a fourth solution evolves. Figure 18-3 reveals that this solution is stable since it falls into the area of m4 > 0. In region V an additional solution, branch (4), appears, corresponding to the in-phase solution. The stability diagram (Fig. 18-4) shows that this solution is unstable here and throughout the remaining regions since m4 is less than zero. The trivial solution becomes stable again and remains stable throughout the rest of the regions. At the transition from region V

22


3 = 0 and bold black dots denote points of m4 = 0 or branch points, respectively. Within the small Hopf isle the limit cycles are expected to lose stability and to turn into simple and complex attracting tori. The amount of solution branches in region VII is the same as in the previous region VI. The frequency response of branch (5) becomes quasi-periodic as will be shown in Sect. 5.2. At the transition from region VII to VIII another bifurcation point occurs. The unstable solution branch (4) intersects with branch (6). Branch (6) is and remains unstable throughout all the rest of the regions (VIII through IX) since m4 < 0. Region IX begins with the second Hopf point at solution branch (5). It is that point in Fig. 18-5, when following the dashed line of ηAC = 0.2354 again, of exiting the Hopf isle. In region IX solution (5) is stable again and the amount of solutions is the same as in the previous region VIII. For larger input excitation (e.g. increasing ηAC above the value where there Hopf isle occurs) the number of regions in the bifurcation diagram decreases to seven. (c) Three-to-one internal resonance (ω2 ≈ 3ω1 )

Fig. 18 Stability map for the solution branches (0)–(5) of the two-beam system (cf. Fig. 17); ηDC = 0.0785, Q = 500, μNL = 0; bold solid lines and bold black dots: m4 = 0, bold dash-dotted lines: AC-threshold for existing solutions, thin dashed line in (4): m3 = 0, grey bold line in (5): Hopf criteria 3 = m3 (m1 m2 − m0 m3 ) − m21 m4 = 0, thin grey dashed lines: markers for ηAC = 0.2354

to VI a pitchfork bifurcation point occurs where the stable in-phase solution ((3) in region V) loses stability and two stable in-phase solution branches (mj > 0, 3 > 0) emerge ((5) in region VI). The transition point from region VI to VII is a Hopf bifurcation of solution branch (5). It is this point in Fig. 18-5, when following the dashed line of ηAC = 0.2354, of entering the Hopf isle, where 3 < 0 ∀mj > 0. Stability analysis of this in-phase solution is determined from Fig. 18-5, which depicts the stability regions in the domain of ηAC = [0.1; 0.3]. The bold grey line represents

In this section the two-beam array is investigated in its three-to-one internal resonance (large DC-voltages). This part has been published before and also to a more detailed level than possible in this work [17, 18]. We employ the method of multiple-scales [39] to the two-microbeam array given in (15) assuming the three-term solution (42) and its respective derivatives for both beams. The AC-voltage parameter is scaled as ηAC = 2 η¯ AC . The DC-voltage parameter ηDC is scaled in the same manner to ηDC = ηDC◦ + 2 η¯ DC , √ with ηDC◦ = 8α/13 being the DC-voltage value for which ω2 ≈ 3ω1 . The pull-in voltage for the array of √ N = 2 is ηPI = 2α/3. The linear damping coefficient is scaled to μL = 2 μ¯ L . Substitution of the solution form (42), including the scaling of the voltage parameters and the linear damping coefficient, and collecting the terms of different orders in results in (82) O 1 : D02 xn1 + bn1 x11 + b2n x21 = 0, 2 2 (83) O : D0 xn2 + bn1 x12 + b2n x22 = fn2 , 3 (84) O : D02 xn3 + bn1 x13 + b2n x23 = fn3 5 α for i = j and bij = for n = [1, 2], where bij = 13 4 13 α for i = j , (i, j = [1, 2]) and fn2 and fn3 are given


in Appendix C.3. The two natural frequencies deter√ √ mined from (82) are ω1 = α/13 and ω2 = 3 α/13. The detuning, for which the second eigenfrequency is approximately three times the first eigenfrequency, is 2 σ1 = ω2 − 3ω1 . The homogeneous solution to (82) is

which yields the slowly varying dynamical system 9 a1 = − ζNL a13 − ζNL a1 a22 2 − (ζL − 3δAC1 sin ψ1 )a1 , a1 ψ1

= δ310 a13

(85)

where Aj = Aj (T1 , T2 ) for j = [1, 2] and cc. denotes conjugate complex terms. Substitution of (85) into (83) leads to D02 xn2 + bn1 x12 + b2n x22 = (−1)n 2iω1 D1 A1 exp (iω1 T0 ) − 2iω2 D1 A2 exp (iω2 T0 ) + NST n + cc.,

(86)

where NST n represent non-secular terms. Elimination of secular terms yields D1 Aj = 0. Thus, Aj (T1 , T2 ) are independent of the time scale T1 , as in previous cases. The non-secular terms NST n consist of quadratic and sum and difference frequency terms. Thus, an ansatz for the solutions of xn2 according to the right-hand side is xn2 = (−1)n C1 exp (2iω1 T0 ) + C2 exp i(ω1 + ω2 )T0 − C3 exp i(ω1 − ω2 )T0 − (−1)n C4 exp (2iω2 T0 ) − (−1)n C5 + cc.,

(87)

whereas the expressions of Cj = Cj (T2 ) for j = [1, . . . , 5] are provided in Appendix C.3. The solutions of O() and O( 2 ) are then substituted into the equations of order O( 3 ) (84). We focus our analysis on the case where Ω is close to twice the first natural frequency ω1 . Thus, we define a detuning for the forcing frequency 2 σ2 = Ω − 2ω1 . In order to determine the solvability conditions of (84), we seek a particular solution of the form xn3 = Pn (T2 ) exp (iωn T0 ) + Qn (T2 ) exp (iωn T0 ) + cc. The complex evolution equations are deduced as a solution of Pn and Qn , respectively, by substitutions of the particular solution into (84) and equating the coefficients of exp(iωn T0 ) on both sides. We employ the polar form and separate imaginary and real terms

(88)

+ δ312 a1 a22

+ (σ2 + 12δDC1 + 6δAC1 cos ψ1 )a1 ,

xn1 = (−1)n+1 A1 exp (iω1 T0 ) + A2 exp (iω2 T0 ) + cc.,

23

(89)

1 a2 = − ζNL a23 − ζNL a12 a2 − ζL a2 , 2

(90)

a2 ψ2 = δ301 a23 + δ321 a12 a2 − 2δDC2 a2 ,

(91)

where the parameters are given in Appendix C.3. We note that the equations decouple for zero nonlinear damping. Thus, the steady state solutions are either the trivial solution (a1 = a2 = 0) or a non-trivial solution for a1 obtained from (88) and (89) where a2 = 0 2 2 = 2ζL + 9ζNL a12 36δAC1 2 + δ310 a12 + 12δDC1 + σ2 .

(92)

Equation (92) is a bi-quadratic equation in a1 presenting the amplitude–frequency relationship for the outof-phase vibration mode. As in the cases of parametric and one-to-one resonance, there exists a condition for which the minimum amplitude square a12 is greater than zero, namely a12min ≥

3ηDC◦ ηAC − 2μL ω1 . 18ζNL ω1

(93)

Thus, (93) yield the condition for a lower bound on the AC-voltage parameter: ηAC ≥

2 ω1 √ . 3 8Q

(94)

Figure 19 depicts the frequency response characteristic for the parameters ηDC = 1.3363, ηAC = 0.0013, Q = 500, and μNL = 1.2. The frequency response bifurcation structure includes four regions. Regions I and IV depict a single stable trivial solution. In region II, three solutions co-exist, a stable trivial, an unstable non-trivial (lower branch) and a stable (upper branch) solution. Region III portrays two co-existing solutions, an unstable trivial and a stable non-trivial solution. The overall frequency response behavior is softening as the contribution of the electrodynamic terms are larger than the hardening beam stiffness

24


one internal resonance) and Gutschmidt and Gottlieb [17, 18] (for three-to-one internal resonance). 5.1 Validation of periodic solutions in the single- and two-beam systems In this section the weakly nonlinear asymptotic results are validated by numerical integration of (15) for the single- and two-beam system and for the regions of one-to-one, parametric, and three-to-one internal resonances. 5.1.1 One-to-one internal resonance Fig. 19 Frequency response of the two-beam system at its three-to-one internal resonance [17]; ηDC = 1.3363, ηAC = 0.0013, Q = 500, μNL = 1.2; solution corresponds to out-of-phase mode, solid line: stable, dashed line: unstable, triangles: numerical verification

term. The change of the system’s response from hardening to softening occurs for the two-element array at a DC-voltage parameter of ηDC = 0.9617. We note that without nonlinear damping the response curves are unbounded. Although this (and also following) section(s) look(s) exclusively at the dynamical analysis of coupled resonators theoretically, we refer to the work done by Karabalin et al. [26] which shows similar linear and nonlinear phenomena validated against experimental observations. Although a direct and thus quantitative comparison of our results to their experiments cannot readily be made here, we would like to point out that their scientific analysis about the dependency of the first-mode frequency shift on the vibration amplitude of the second mode relates to the coupling parameters and resonance regions (which correspond to the DCand AC-voltage inputs) of our work. Future work of fabricated two- and three-beam arrays (see Fig. 37) will reveal statements on the quantitative accuracy of the presented theoretical results.

5 Numerical analysis As we include a necessary portion of previously published results (regarding regions of internal resonances) in this section, more in-depth works can be found in Gutschmidt and Gottlieb [15, 16] (for one-to-

Numerical simulations at its one-to-one internal resonance stand in good agreement with asymptotical results for small values of the DC-voltage parameter (cf. Fig. 17). Typical simulations, including time series and phase-plane diagrams with overlayed Poincaré points are depicted for several initial conditions and regions. Figure 21 shows a period doubled response at Ω = 0.9922ω (region II in Fig. 17), where x1 and x2 vibrate with a phase shift of π + 2kπ (out-ofphase). The two Poincaré points in Fig. 21b correspond to principal parametric resonance. Figure 22 depicts period doubled response at Ω = 0.9973ω (region IV in Fig. 17) and the two beams happen to vibrate with the same amplitude and precisely in-phase. Figure 23 shows the response at Ω = 1.007ω (region IX in Fig. 17) and the two beams vibrate in-phase with unequal amplitudes. These three selected simulations represent typical periodic solutions of the twobeam system at its one-to-one internal resonance. 5.1.2 Parametric resonance Numerical validation points of the solution branches of the single- as well as of the two-beam system are depicted along with the frequency response graphs of Fig. 13 and Fig. 14, where triangles and diamonds denote out-of-phase and in-phase vibration modes, respectively. Numerically obtained amplitudes for the two-beam systems are in good agreement with the asymptotically obtained results. For the single-beam system better agreement would be achieved by decreasing the AC-voltage parameter along with increasing the quality factor. However, in this work we present all results for a selected Q-factor of Q = 500. Analytically (and for the case of parametric resonance), only


Fig. 20 Numerically obtained frequency response of the two-beam system in parametric resonance; ηDC = 0.5230, ηAC = 0.0785, Q = 500, μNL = 1.2; grey lines indicate analytical solutions (solid: out-of-phase, dashed: in-phase); triangles: out-of-phase, diamonds: in-phase, stars: in-phase; hollow markers: periodic response, solid markers: aperiodic response

Fig. 21 Periodic out-of-phase solution at Ω = 0.9922ω (region II in Fig. 17); (a) time series, (b) phase plane with Poincaré points Xn ; solid lines: x1 , dashed lines: x2

periodic solutions were found. However, for increasing AC-values the frequency responses of the out-ofand in-phase modes are closer together and show a region where the solutions overlap, see Fig. 14c. The larger the AC-parameter the less accurate numerical results match analytical ones. Furthermore, Fig. 20, plotted for ηDC = 0.5230, ηAC = 0.0785, Q = 500, μNL = 1.2 shows an additional solution branch which was not obtained by the asymptotic approach. This solution branch corresponds to an in-phase vibration

25

Fig. 22 Periodic in-phase solution at Ω = 0.9973ω (region IV in Fig. 17); (a) time series, (b) phase plane with Poincaré points Xn ; solid lines: x1 , dashed lines: x2

Fig. 23 Periodic in-phase solution at Ω = 1.007ω (region IX in Fig. 17); (a) time series, (b) phase plane with Poincaré points Xn ; solid lines: x1 , dashed lines: x2

mode (denoted by stars). This solution evolves from the in-phase solution in the previous region (marked with diamonds). In regions VIII and IX this in-phase mode occurs to be aperiodic. A selected simulation is shown and discussed in the next section. The behavior of the two-beam system in parametric resonance, capturing also the two phenomena, first, that there is an additional solution branch and second, that also aperiodic responses occur, can only be obtained assuming a higher order approach in the multiple-scales method.

26

5.1.3 Three-to-one internal resonance In the case of three-to-one internal resonance, asymptotic results are verified by numerical integrations which are shown in Fig. 19. Near the pull-in region the system becomes highly sensitive toward most of the parameters, but especially toward the AC-voltage parameters. As in case of the single-beam system, the oscillations are influenced by the linear and nonlinear damping parameters. The lower the AC-values the larger the Q-factors must be in order to preserve vibrations. Thus, a better agreement between analytical and numerical results in Fig. 19 are achieved by decreasing the AC-voltage along with increasing the quality factor. Due to longer integration times for higher Qfactors (reaching steady state), only the comparison for the selected Q-factor of Q = 500 is portrayed in this work. Figure 19 shows the frequency response of a set of parameters for which numerical and asymptotical results are qualitatively acceptable. The overall softening behavior as well as the matching of the frequencies are in good agreement. A selected simulation at Ω = 1.75ω1 (Fig. 27) shows a bias and asymmetric signals. The beams vibrate in the out-of-phase mode.


VIII in Fig. 17). The Poincaré maps Fig. 24b show complex double-looped tori. The double-looped tori evolve from the two Poincaré points (periodic solution, in-phase) corresponding to regions III and IV in Fig. 17 and evolve into the two Poincaré points (periodic solution, in-phase) corresponding to solutions in region IX. Figure 25 shows two additional Poincaré maps of the same solution branch, one at an excitation frequency near the first Hopf point H51 (at Ω = 1.000ω) and the other near the second Hopf point H52 (at Ω = 1.003ω). The simple figure eight-shaped Poincaré map in Fig. 25a becomes more complex in Fig. 25b.

5.2 Quasiperiodic response of the two-beam system 5.2.1 One-to-one internal resonance Figure 24 depicts a distinct quasi-periodic solution of the in-phase solution at Ω = 1.001ω (shaded region Fig. 25 Poincaré maps of the in-phase solution at (a) Ω = ω (near H51 ), (b) Ω = 1.003ω (near H52 ) (region VIII in Fig. 17)

Fig. 24 Quasi-periodic solution at Ω = 1.001ω (region VIII in Fig. 17); (a) time series, black line: x1 , grey line: x2 , (b1 ), (b2 ) Poincaré maps, black line: X1 , grey line: X2


27

Fig. 26 Poincaré maps of the in-phase solution at Ω = ω (region IX in Fig. 20)

5.2.2 Parametric resonance Figure 26 shows selected Poincaré maps of the inphase solution of the two-beam system in parametric resonance (shaded region in Fig. 20) at the excitation frequency Ω = 2.169ω1 . The tori of the two beams are similar to the tori of the Poincaré maps of the twobeam system at the one-to-one internal resonance (see Fig. 24). In Fig. 26 the double-loop tori for Xn have evolved from and evolve into the two Poincaré points of the in-phase solution in regions VII and X, respectively.

Fig. 27 Periodic out-of-phase solution at Ω = 1.75ω1 (region II in Fig. 19); (a) time series, (b) phase plane with Poincaré points Xn ; solid lines: x1 , dashed lines: x2

5.3 Periodic and aperiodic responses of the three-beam system For the remains of this paper we investigate in the dynamic response of a three-beam system by numerically integrating the equations of motion (15) for N = 3 (see also [18]). 5.3.1 One-to-one internal resonance The frequency response of the three-beam system for an AC-value of ηAC = 0.2354 is shown in Fig. 28. The different markers (e.g. triangles, diamonds, circles, squares, stars) denote various types of solutions which are depicted in Fig. 29. Hollow markers denote periodic and solid markers aperiodic responses, respectively. For the sake of clarity the emphasis in Fig. 28 is laid on the solutions of the middle beam (bolder black markers). (Outer beams are indicated by thin grey markers, not distinguishing between x1 and x3 at this stage.) The grey splines mark the trend of the solution (added eye guide). We find three out-of-phase (Fig. 29a–c) and one inphase (Fig. 29d) periodic solutions.

Fig. 28 Frequency response of the three-beam system at one-to-one-to-one internal resonance [16]; ηDC = 0.0785, ηAC = 0.2354, Q = 500, μNL = 0; bold black markers denote the response of the middle beam x2 , thin grey markers denote outer beams x1 and x3 ; hollow markers: periodic response, solid markers: aperiodic response; triangles, diamonds, squares, circles, and stars denote vibration modes (see Fig. 29)

(a) The solution branch denoted by triangles consists of a periodic out-of-phase response having similar amplitudes (see Fig. 29a). This solution is stable until the amplitudes reach a critical amplitude of |xn | < 0.5. We recall that for amplitudes of |xn | ≥ 0.5 the beams in an out-of-phase mode would penetrate one another and render the dynamic model invalid. (b) The solution branch depicted by circles is another periodic out-of-phase response having unequal amplitudes (see Fig. 29b). |x1 | and |x2 |

28


Fig. 29 Vibration modes of the periodic solution branches in Fig. 28 [16]; grey solid lines: x1 , black solid lines: x2 , grey dashed lines: x3 , solutions corresponding to (a) triangles, (b) circles, (c) squares, (d) diamonds in Fig. 28

vibrate out-of-phase and with equal amplitudes while the third beam is in-phase with the middle beam but vibrating with a smaller amplitude. (c) The third periodic out-of-phase steady state solution, denoted by squares and shown in Fig. 29c, is characterized by the out-of-phase motion of the two outer beams maintaining the same amplitudes while the middle beam is phase-shifted toward them by π/2, i.e. the middle beam undergoes no motion when the two outer beams undergo the crests. (d) Solution (d) in Fig. 29 denoted by diamonds portrays the periodic in-phase response. The two outer beams vibrate with similar amplitudes and with a small offset. The middle beam vibrates inphase with respect to the outer beams yet with a smaller amplitude. There exists an additional in-phase solution whose response is aperiodic. In the two- as well as in the three-beam system the aperiodic response occurs for the in-phase solution only. However, the three-to-one internal resonance study [18] reveals also aperiodic responses for the out-of-phase mode. Figure 30 shows a series of Poincaré maps of this aperiodic solution of Fig. 28, in which the middle beam is portrayed at three distinct excitation frequencies, (a) Ω = 1.002ω, (b) Ω = 1.015ω, and (c) Ω = 1.036ω. The complex-

Fig. 30 Poincaré maps of the middle beam X2 of the aperiodic in-phase solution of Fig. 28 (denoted by stars) at (a) Ω = 1.002ω, (b) Ω = 1.015ω, (c) Ω = 1.036ω

ity in the Poincaré shapes increases with an increase in the excitation frequency. While Fig. 30a shows singleloop tori, Fig. 30b shows three-loops tori. 5.3.2 Parametric resonance The frequency responses of the three-beam system for the case of parametric resonance (ηDC = 0.5230) and two different AC-voltage values (a) ηAC = 0.0262, (b) ηAC = 0.1308 are shown in Fig. 31. Three types of solutions are identified: two periodic out-of-phase solutions (according to the types shown in Fig. 29a and Fig. 29c) denoted by triangles and squares, respectively, and an aperiodic solution marked by solid stars. Recall that in the case of the two-beam system the first mode at Ω ≈ 2ω1 corresponds to the out-ofphase mode while the second at Ω ≈ 2ω2 is the in-


29

Fig. 32 Poincaré map of the middle beam X2 of the aperiodic solution shown in Fig. 31a at Ω = 2.1ω1

Fig. 31 Frequency response of the three-beam system at parametric resonance; ηDC = 0.5230, Q = 500, μNL = 1.2, (a) ηAC = 0.0262, (b) ηAC = 0.1308; bold black markers denote the response of the middle beam x2 , thin grey markers denote outer beams x1 and x3 ; hollow markers: periodic response, solid markers: aperiodic response

phase mode. In the three-beam system the first eigenfrequency at Ω ≈ 2ω1 corresponds to the out-of-phase vibration mode, followed by second at Ω ≈ 2ω2 ≈ 2.078ω1 which is another out-of-phase mode (according to the type of Fig. 29c). The expected in-phase response at actuation of the third eigenfrequency at Ω ≈ 2ω3 ≈ 2.15ω1 occurs to be unstable. The aperiodic solutions in Fig. 31 are not comparable to any of the preliminary aperiodic solutions. While in afore found aperiodic solutions mutual phase ratios of the beams in the array are maintained [15] the aperiodic solutions found for the three-beam system at parametric resonance experience variable phase ratios. Figure 32a portrays a Poincaré map of this aperiodic solution for the middle beam at an excitation frequency of Ω = 2.1ω1 , which is similar to the Poincaré map of the middle beam for the three-beam system at one-to-one internal resonance (see Fig. 30a). Figure 33b portrays Poincaré maps of the aperiodic solution of the outer beam x1 at two distinct excitation frequencies, (a) Ω = 1.988ω1 and (b) Ω = 2.02ω1 .

Fig. 33 Poincaré maps of the outer beam X1 of the aperiodic solution shown in Fig. 31b at (a) Ω = 1.988ω1 , (b) Ω = 2.02ω1 , [16]

5.3.3 Three-to-one internal resonance From Sect. 3.3 we recall that the three-to-one internal resonance for the three-beam system occurs for two frequency ratios, ω2 /ω1 ≈ 3 and ω3 /ω1 ≈ 3. The ω2 /ω1 ≈ 3 internal resonance occurs for ηDC = 1.2705 and the ω3 /ω1 ≈ 3 internal resonance for ηDC = 1.2403. The beams get pulled in for the DCvoltage parameter ηDC = 1.3030. We focus our numerical investigations on the specific parameter combination where the excitation near the principal parametric resonance Ω = 2ω1 is close to the ω3 ≈ 3ω1 internal resonance and the combination resonance of ω3 ≈ ω2 + ω1 . This is obtained by selecting ηDC =

30

Fig. 34 Frequency response of the three-beam system at three-to-one internal resonance (ω3 /ω1 ≈ 3) [16]; ηDC = 1.2396, ηAC = 0.0013, μNL = 0.06; bold black markers denote the response of the middle beam x2 , thin grey markers denote outer beams x1 and x3 , solid markers denote aperiodic behavior, triangles: Q = 500, squares: Q = 5000

1.2396 that yields ω1 = 0.5245, ω2 = 1.1669, and ω3 = 1.5647 and thus 2ω3 /(3ω1 ) ≈ 1.99 and 2(ω1 + ω2 )/(3ω1 ) ≈ 2.15. Figure 34 shows the frequency response of the three-beam system near the ω3 /ω1 internal resonance for Q = 500 and ηDC = 1.2396. The beams vibrate periodically out-of-phase and with similar amplitudes. The amplitude of the middle beam is slightly larger than the amplitudes of the two outer beams, which vibrate precisely in-phase and with the same amplitude. A typical phase plot of this out-of-phase mode for the middle beam is shown in Fig. 35a. A phase plane for Q = 500 at the shaded region (Ω = 1.991ω1 ) of Fig. 34 portrays in Fig. 35b two additional loops which reveal a more dense frequency spectrum than that of Fig. 35a. However, the number of Poincaré points are preserved. Figure 36 shows the Poincaré maps of the outer and middle beams for (a) Q = 3000 and (b) Q = 5000, respectively at Ω = 1.991ω1 (shaded region in Fig. 34). Figure 36a portrays a complex quasi-periodic response, whereas Fig. 36b depicts lengthy chaotic transients. We conjecture that the bifurcation governing the appearance of aperiodic response is associated with the loss of stability of the dominant out-of-phase vibration mode of two adjacent elements in the array. This behavior will be investigated further in the future.


Fig. 35 Phase diagrams with Poincaré points of the three-beam system at three-to-one internal resonance (ω3 /ω1 ≈ 3) [16]; (a) Ω = 2.01ω1 and (b) Ω = 1.991ω1 (see Fig. 34); bold black lines: middle beam x2 , thin grey lines: outer beams x1 and x3 , triangles: Poincaré points

6 Summary and closing remarks In this paper we have derived a nonlinear multielement dynamical model for a microbeam array subject to capacitive parametric actuation. The equilibrium analysis identifies three distinct regions (below the first pull-in instability of the system) of internal and combination, and parametric resonances based on the ratios of natural frequencies of individual members of the array, which in turn are a function of the DC-voltage. In Table 1 we summarize all findings classified according to resonance regions for the two-element array. In Table 2 we summarize all findings classified according to resonance regions for the three-element array. Although the asymptotic multiple-scales analysis for the two-element system identifies the solutions which are also found numerically for small and large DC-voltages, for an applied medium DC-voltage (region of parametric excitation) the asymptotic multiple-scales analysis fails to identify the aperiodic solution. A possible explanation for this is having infringed the assumption of the small parameter in the first place. A comprehensive stability analysis enabled derivation of the system bifurcation structure which incorporates multiple distinct


31

Fig. 36 Poincaré maps of the three-beam system at its three-to-one internal resonance (ω3 /ω1 ≈ 3) at Ω = 1.991ω1 (see Fig. 34) [16]; (a) Q = 3000, (b) Q = 5000; for figures to the left: grey—X1 , black—X3

regions with different behaviors. The governing parameters controlling the bifurcation structure are the alternating excitation and linear damping which determine the number of regions. The numerical analysis of the three-beam system reveals periodic, aperiodic and chaotic solutions. A recently performed in-depth study of the three-element system by Gutschmidt and Gottlieb [14, 18] near the three-to-one region has manifested two regions defined by four Neimark–Sacker (secondary Hopf) bifurcation points, within which the limit cycles lose their stability and turn into simple and complex attracting tori. The degree of complexity is governed by the transition from the internal resonance of ω2 ≈ 3ω1 to that of ω3 ≈ 3ω1 and a possible combination resonance of ω3 ≈ ω1 + ω2 . We note that while both lumped-mass and continuum-based models similarly yield the array natural frequencies and critical damping ratios (or quality factors) which govern the small amplitude array response, the continuum-based model ensures a minimal number of nonlinear system parameters, whereas lumped-mass models require a separate procedure for identification of nonlinear stiffness (and damping) parameters which, as shown, are not independent. Although a direct comparison to the scientific work of coupled oscillators by Danzl and Moehlis [7] cannot be readily made here, the analysis of the two- and three-beam systems reveals similar observations re-

garding symmetry classes, vibration modes and their stabilities. As in the Huygens’ clock and other coupled systems it occurs that the out-of-phase solution (e.g. in the two-beam array) is stable while the in-phase is not (unless members are uncoupled). The existence and stability of periodic solutions in the array significantly depend on the intensity of electrical coupling between members, which determines the regions of internal (one-to-one, three-to-one), combinational and parametric resonances, respectively. It would be an interesting future excursion to extend the array analysis in such ways to identify if all symmetry classes of Danzl and Moehlis’s work [7] exist as they may be dynamically unstable or have limited domains of attraction. Furthermore, future research will focus on the analysis of the system response in the region between the two three-to-one internal resonances as well as beyond the pull-in threshold which may include additional combination resonances. We explored the bifurcation structures of small-size arrays as function of the excitation frequency, coupling intensities and forcing amplitudes, which are meaningful to experimentalists in NEMS and MEMS research. Results and findings motivate future experimental work(a few smallsize arrays have been recently fabricated in the Buks’ laboratory at the Technion and first experimental investigations are documented in a work by Mintz [33]) and serve as a design and testing guideline to investigate the feasibility of new MEMS array applications.

32


Table 1 An overview of solutions and their conditions in the two-element MEMS array for small, medium and large DC-voltages

two-resonator array

1:1

parametric

3:1

Solutions/bifurcations

Conditions, behavior

Sections

Figures

periodic, co-existing stable and unstable solutions corresponding to out-of- and in-phase vibration modes

small DC, hardening

4, 5

16, 21, 22, 23

quasi-periodic, in-phase mode, Hopf bifurcation points

small DC, for a range of large AC, hardening

4, 5

17, 18, 24, 25

periodic, co-existing stable and unstable out-of-phase solutions, co-existing stable and unstable in-phase solutions

medium DC, hardening

4, 5

14, 15

quasi-periodic, in-phase mode

medium DC, large AC, hardening

5

20, 26

periodic, co-existing stable and unstable out-of-phase solutions

large DC, small AC, softening

4, 5

19, 27

Table 2 An overview of solutions and their conditions in the three-element MEMS array for small, medium and large DC-voltages

three-resonator array

1:1

parametric

3:1

Solutions/bifurcations

Conditions, behavior

Sections

Figures

periodic solutions corresponding to three out-of- and one in-phase vibration modes

small DC, hardening

5

28, 29

quasi-periodic, in-phase mode

small DC, large AC

periodic out-of-phase solutions

medium DC, hardening

quasi-periodic, in-phase mode (hardening)

medium DC, large AC, hardening

periodic, out-of-phase mode

large DC, small AC, softening

quasi-periodic, chaotic, out-of-phase mode

large DC, small AC, piecewise hardening [18]

Fig. 37 Photograph of the three-beam array [33]

28, 30

5

31 31, 32, 33

5

34, 35 34, 36

A significant practical importance is that actual energy transfer between in- and out-of-phase co-existing periodic modes can be predicted (in an asymptotic approximate form) and, as such, be suppressed or enhanced for either sensor, actuator or filter applications. Finally we remark that the ideal case of perfect symmetry, as was assumed throughout this work, is absent in real structures due to imperfections caused by fabrication processes and the like. Future work of coupled microstructural arrays should include the effects of broken symmetries as they can lead to the appearance of new solutions [34, 35].


33

Appendix B: Numerical values of the pull-in threshold per number of resonators in the array

Acknowledgements This work was supported by the Israeli Science Foundation, the Vatat (Council for Higher Education) and the Minerva for which we express our thanks. The first author would like to thank the people in the Mechanical Engineering Department at Technion—Israel Institute of Technology, Haifa (Israel) for their hospitality during the postdoctoral years. Soli Deo gloria!

Table 5 lists the values which are plotted in Fig. 10.

Appendix A: Parameters of an individual member of the array Appendix C: Coefficients and forcing terms of the multiple-scales analysis of the two-beam system

Table 3 Material and design parameters of the beam design parameter

symbol

value

unit

length

L

270 × 10−6

m

width

H

0.25 × 10−6

m

height

B

1 × 10−6

m

gap

g

4 × 10−6

m

material parameter

symbol

value

unit

Young’s modulus

ESi

130 × 109

N/m2

density

2330

kg/m3

DC-voltage

VDC

0, . . . , 18

V

AC-voltage

VAC

50 × 10−3 , . . . , 12

V

electric constant

ε0

8.854 × 10−12

F/m

C.1 Parametric resonance

Forcing terms of O( 2 ) and O( 3 ) (44) and (45) fn2 = −2D0 D1 xn1 − 2xl1 D02 xn1 + αxn1 2 ηDC n+1 n+1 1 2 + (−1) x11 x21 − (−1) x , 2 2 l1 3 fn3 = −D12 xn1 − 2D0 D2 xn1 − 2D0 D1 xn2 − βxn1 − μ¯ L D0 xn1 + (−1)n 2xl1 2D0 D1 xn1 + D02 xn2 + αxn2 2 2 + −xl1 + 2xn1 − 2x11 x21 + (−1)n 2xl2 × D02 xn1 + αxn1 2 − (−1)n μˆ NL −xl1 (D0 x11 − D0 x21 )

Table 4 Coefficients of the shape function for a cl-cl beam for κ1 = 0 parameter

value

z1

4.73

Φ1 ( 12 )

1.5881

J1

1

unit

+ 2x11 x21 (D0 x11 − D0 x21 )

2 − (−1)n xn1 (2D0 xn1 − D0 xl1 )

J2

−12.30

J3

−151.35

J4

500.56

ωs

2.9578 × 104

+ ηDC η¯ AC cos Ωt · (2xn1 − xl1 ) + (−1)n

rad/s

2 ηDC (xl1 xl2 − x11 x22 − x12 x21 ), 2

with l = −1n+1 + n. Table 5 Pull-in voltages of the microbeam array of one to 40 beams, see Fig. 10

N

2 ηˆ DC

α

1

2

3

4

5

1

2 3

2√ 2+ 2

4√ 5+ 5

2√ 2+ 3

1.0000

0.6667¯

0.5858

0.5528

0.5359

10

20

30

40

0.5103

0.5028

0.5013

0.5007

34


Coefficients of second order solution (49):

+

C11 = −C21 = −c61 A21 , C12 = C22 = −c72 A1 A2 , C13 = C23 = −c83 A1 A¯ 2 ,

(95)

C14 = −C24 = c94 A22 ,

2 1 c105 α 3 ηˆ DC + . 2 ω2 16 ω2

Definition of parameters of the slowly varying dynamical systems (51)–(54) and amplitude–frequency relationship (55): 1 ηDC η¯ AC , 4 ω1

3 β , 4 ω1

C15 = −C25 = −c105 3|A1 |2 − |A2 |2 ,

δex1 =

with

ζL =

μ¯ L , 2

(96)

δ32 =

3 β , 4 ω2

,

(97)

Definition of parameters of the slowly varying dynamical systems (58)–(61) and amplitude–frequency relationship (62):

,

(98)

c61 = c72 = c83 = c94 =

2 ηDC 3 , 2 2 2α − 3ηDC 2 3ηDC 2 + 3ηDC

−2α

− 4ω1 ω2

2 3ηDC 2 + 4ω ω −2α + 3ηDC 1 2 2 3 ηDC , 2 2 6α − ηDC

c105 =

(99)

2 ηDC 3 . 2 2 −2α + 3ηDC

(100)

Coefficients for the slowly varying dynamical systems (51)–(54) and (58)–(61) having applied the detunings Ω = 2ω1 and Ω = 2ω2 , respectively: 27 δDC11 = − 8 −3

2 c61 ηˆ DC

ω1

δDC21 = −

ω1

2 2 3 ηˆ DC 9 c72 ηˆ DC 1 c72 α + − 4 ω1 8 ω1 2 ω1

2 3 c105 ηˆ DC c105 α + , 4 ω1 ω1

2 2 1 c72 α 3 c83 ηˆ DC 3 c72 ηˆ DC + + 4 ω2 16 ω2 16 ω2

+

2 2 9 c105 ηˆ DC 1 c83 α 3 c105 α 21 ηˆ DC − − − 8 ω2 4 ω2 2 ω2 8 ω2

1 1 + c83 ω1 − c72 ω1 , 2 2 δDC22 = −

3 c94 α 5 + 4 ω2 16

2 c94 ηˆ DC

ω2

3 β , 4 ω1

ψ1 = σ T2 − 2θ1 , ζNL =

ζL =

(101)

μNL , 8

μ¯ L , 2

ψ2 = θ2 .

ψ1 = θ1 ,

1 ηˆ DC η¯ AC 3 β , δ32 = , 4 ω2 4 ω2 μNL ζNL = , ψ2 = σ T2 − 2θ2 . 8 δex2 =

(102)

C.2 One-to-one internal resonance Forcing terms of O( 2 ) and O( 3 ) (67) and (68) fn2 = −2D0 D1 xn1 − 2xl1 D02 xn1 + αxn1 ,

(103)

3 fn3 = −D12 xn1 − 2D0 D2 xn1 − 2D0 D1 xn2 − βxn1

− μ¯ L D0 xn1

2 c105 α 27 ηˆ DC + , ω1 8 ω1

δDC12 = −c72 ω2 + +

3 c61 α 9 + − 2 ω1 4

2 c105 ηˆ DC

δ31 =

δ31 =

−

3 8

2 c105 ηˆ DC

ω2

+ (−1)n 2xl1 2D0 D1 x11 + D02 xn2 + αxn2 2 2 + −xl1 + 2xn1 − 2x11 x21 + (−1)n 2xl2 × D02 xn1 + αxn1 2 − (−1)n μNL −xl1 (D0 x11 − D0 x21 ) + 2x11 x21 (D0 x11 − D0 x21 )

2 (2D0 xn1 − D0 xl1 ) + (−1)n xn1 η¯ DC η¯ DC 2 1 2 + +2 cos Ωt + η¯ AC η¯ AC 2 η¯ AC 1 1 ∗ xn1 − xl1 + cos 2Ωt 2 2 with l = −1n+1 + n.

(104)


35

Definition of parameters of the slowly varying dynamical systems (72)–(75) and amplitude–frequency relationships (76) and (77):

Definition of parameters of the slowly varying dynamical (88)–(91) and amplitude–frequency relationships (92):

2 1 η¯ AC δAC = , 4 ω

δAC1 =

μ¯ L ζL = , 2

3β δ3 = , 4ω ψ1 = θ1 − θ2 ,

2 η¯ DC

1 δDC = , 4 ω

1 ζL = μ¯ L , 2 (105)

μNL ζNL = , 8

δDC1 =

δDC2 = Forcing terms of O( 2 ) and O( 3 ) equations (83) and (84): fn2 = −2D0 D1 xn1 − 2xl1 D02 xn1 + αxn1 2 ηDC 1 2 ◦ , (−1)n+1 x11 x21 − (−1)n+1 xl1 + 2 2 fn3 =

3 − 2D0 D2 xn1 − 2D0 D1 xn2 − βxn1

− μ¯ 12 D0 xn1

+ (−1)n 2xl1 2D0 D1 x11 + D02 xn2 + αxn2 2 2 + −xl1 + 2xn1 − 2x11 x21 + (−1)n 2xl2 × D02 xn1 + αxn1 2 − (−1)n μˆ 3 −x21 (D0 x11 − D0 x21 ) + 2x11 x21 (D0 x11 − D0 x21 ) 2 + xn1 (2D0 xn1 − D0 xl1 ) + ηDC◦ (η¯ DC + η¯ AC cos Ωt)(2xn1 − xl1 ) + (−1)n

2 ηDC ◦

2

(x21 xll − x11 x22 − x12 x21 )

with l = −1n+1 + n. Coefficients of second order solution (87): C1 = 6A21 ,

C2 = 12/7A1 A2 ,

C3 = 12/5A¯ 1 A2 ,

C4 = 6/35A22 ,

C5 = 48|A1 |2 − 16|A2 |2 .

(106)

438 α 3 β − , 13 ω1 4 ω1

ψ1 = σ2 T2 − 2θ1 ,

971 α 3 β + , 455 ω2 8 ω2

1 ζNL = μNL , 8

C.3 Three-to-one internal resonance

δ310 =

1 ηDC◦ η¯ DC , 4 ω1

δ301 = −

ψ2 = 2σ T2 − θ1 − θ2 .

−D12 xn1

1 ηDC◦ η¯ AC , 4 ω1

ψ2 = θ2 − 3θ1 ,

1 ηDC◦ η¯ DC , 4 ω2

δ312 = −

(107)

δ321 =

183 α 3 β + , 35 ω2 4 ω2

4618 α 3 β − . 455 ω1 2 ω1

References 1. Baller, M., Lang, H., Fritz, J., Gerber, C., Gimzewski, J., Drechsler, U., Rothuizen, H., Despont, M., Vettiger, P., Battiston, F., Ramseyer, J., Fornaro, P., Meyer, E., Guntherodt, H.-J.: Ultramicroscopy 82, 1–9 (2000) 2. Barber, D.J.: Ultramicroscopy 52, 101 (1997) 3. Bennett, M., Schatz, M.F., Rockwood, H., Wiesenfeld, K.: Proc. R. Soc. Lond. A 458, 563–579 (2002) 4. Bromberg, Y., Cross, M.C., Lifshitz, R.: Phys. Rev. E 73, 016214 (2006) 5. Brown, E., Holmes, P., Moehlis, J.: In: Kaplan, E., Marsden, J., Screenivasan, K. (eds.) Perspectives and Problems in Nonlinear Science: A Celebratory Volume in Honor of Larry Sirovich, pp. 183–215. Springer, New York (2003) 6. Buks, E., Roukes, M.L.: J. Microelectromech. Syst. 11(6), 802–807 (2002) 7. Danzl, P., Moehlis, J.: Nonlinear Dyn. 59, 661–680 (2010) 8. Despont, M., Drechsler, U., Yu, R., Pogge, H.B., Vettiger, P.: J. Microelectromech. Syst. 13(6), 895–901 (2004) 9. Dick, A.J., Balachandran, B., Mote, C.D. Jr.: In: Proc. of IMECE 2005, Orlando, FL, Nov. 5–11 (2005) 10. Dick, A.J., Balachandran, B., Mote, C.D. Jr.: Nonlinear Dyn. 54, 13–29 (2008) 11. Erbe, A., Blick, R.H., Tilke, A., Kriele, A., Kotthaus, J.P.: Appl. Phys. Lett. 73, 3751 (1998) 12. Gottlieb, O., Champneys, A.R.: In: IUTAM Chaotic Dynamics and Control of Systems and Processes in Mechanics, pp. 117–126. Springer, Berlin (2005) 13. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983) 14. Gutschmidt, S., Ammon, A.: In: Proc. of 21st ICAST, State College, PA, Oct. 4–6 (2010)

36 15. Gutschmidt, S., Gottlieb, O.: In: Proc. of IDETC/CIE 2007, Las Vegas, NV, Sept. 4–7 (2007) 16. Gutschmidt, S., Gottlieb, O.: J. Sound Vib. 329, 3835–3855 (2010) 17. Gutschmidt, S., Gottlieb, O.: In: Proc. of 6th EUROMECH Nonlinear Dynamics Conference, St. Petersburg, Russia, June 30–July 4 (2008) 18. Gutschmidt, S., Gottlieb, O.: Int. J. Bifurc. Chaos 20(3), 605–618 (2010) 19. Hagedorn, P., DasGupta, A.: Vibrations and Waves in Continuous Mechanical Systems. Wiley, Chichester (2007) 20. Hikihara, T., Okamoto, Y., Ueda, Y.: Chaos 7(4), 810–816 (1997) 21. Hikihara, T., Torii, K., Ueda, Y.: Phys. Lett. A 281, 155– 160 (2001) 22. Hull, R., Stevie, F.A., Bahnck, D.: Appl. Phys. Lett. 66, 341 (1995) 23. Huygens, C.: Philos. Trans. R. Soc. Lond. 4, 937–953 (1969) 24. Ilic, B., Yang, Y., Aubin, K., Reichenbach, R., Krylov, S., Craighead, H.G.: Nano Lett. 5(5), 925–929 (2005) 25. Kaplan, W.: Advanced Calculus. Addison-Wesley, Cambridge (1952) 26. Karabalin, R.B., Cross, M.C., Roukes, M.L.: Phys. Rev. B 79, 165309 (2009) 27. Kimura, H., Shimuzi, K.: In: Materials Research Society Symposium Proceedings, vol. 480, p. 341. MRS, Warrendale (1997) 28. Krylov, S.: Int. J. Nonlinear Mech. 42, 626–642 (2007) 29. Lenci, S., Rega, G.: J. Micromech. Microeng. 16, 390–401 (2006) 30. Lifshitz, R., Cross, M.C.: Phys. Rev. B 67, 134302 (2003) 31. Meirovitch, L.: Elements of Vibration Analysis. McCrawHill, New York (1975) 32. Minne, S.C., Manalis, S.R., Quate, C.F.: Bringing Scanning Probe Microscopy up to Speed. Kluwer Academic, Boston (1999)

S. Gutschmidt, O. Gottlieb 33. Mintz, T.: Master Thesis, Technion—Israel Institute of Technology (2009) 34. Moehlis, J., Knobloch, E.: Phys. Rev. Lett. 80, 5329–5332 (1998) 35. Moehlis, J., Knobloch, E.: Physica D 135, 263–304 (2000) 36. Napoli, M., Zhang, W., Turner, K., Bamieh, B.: J. Microelectromech. Syst. 14(2), 295–304 (2005) 37. Nathanson, H.C., Newell, W.E., Wickstrom, R.A., Davis, J.R. Jr.: IEEE Trans. Electron Devices 14, 117–133 (1967) 38. Nayfeh, A.H.: Nonlinear Interactions. Wiley-Interscience, New York (2000) 39. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. WileyInterscience, New York (1979) 40. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M.: Nonlinear Dyn. 48, 153–163 (2007) 41. Porfiri, M.: J. Sound Vib. 315, 1071–1085 (2008) 42. Rhoads, J.F., Shaw, S.W., Turner, K.L.: J. Micromech. Microeng. 16, 890–899 (2006) 43. Sasoglu, F.M., Bohl, A.J., Layton, B.E.: J. Micromech. Microeng. 17(3), 623–632 (2007) 44. Senturia, S.D.: Microsystem Design. Kluwer Academic, Boston (2001) 45. Shabana, A.A.: Theory of Vibration. Springer, New York (1991) 46. Vettiger, P., Brugger, J., Despont, M., Drechsler, U., Diirig, U., Hgberle, W., Lutwyche, M., Rothuizen, H., Stutz, R., Widmer, R., Binnig, G.: Appl. Phys. Lett. 46(1–4), 11–17 (1999) 47. Wang, P.K.C.: J. Sound Vib. 213(2), 537–550 (1998) 48. Wang, P.K.C.: Int. J. Bifurc. Chaos 13, 1019–1027 (2003) 49. Zalatdinov, M.K., Baldwin, J.W., Marcus, M.H., Reichenbach, R.B., Parpia, J.M., Houston, B.H.: Appl. Phys. Lett. 88, 143504 (2006) 50. Zhang, W., Turner, K.: Sens. Actuators A 134, 594–599 (2007)

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