Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations

Abstract

The multi-frequency excitation of a microbeam, resting on a nonlinear foundation, is investigated and the governing equation of motion of the microbeam system is developed. The viscoelastic-type foundation is considered by assuming nonlinear parameters for both Pasternak and Winkler coefficients. The well-known Galerkin approach is utilized to discretize the governing equation of motion and to obtain its nonlinear ordinary differential equations. The multiple time-scales method is employed to study the multi-frequency excitation of the microbeam. Furthermore, the resonant conditions due to the external excitation as well as the combination resonances for the first two modes are investigated. The influences of different parameters, namely the Pasternak and Winkler coefficients, the position of the applied force and the geometrical factors on the frequency response of the system are examined.

Keywords

Nonlinear vibration microbeams multi-frequency excitation micro-scale effect Winkler and Pasternak foundation

1. Introduction

Micro- and nano-structures can be identified as the important types of Micro-Electro-Mechanical Systems and Nano-Electro-Mechanical Systems (MEMS/NEMS), which have wide applications in a variety of technological fields (Guo, 2013). Microbeams can be categorized as one of the most applicable micro-structures in various systems, namely, sensing devices, communications, bio-technology and quantum mechanics. Owing to the small size of microbeams, they are quite applicable when we are designing small instruments similar to sensors and actuators (Subramanian et al., 2002; Li et al., 2007; Jamshidifar et al., 2017).

Recently, many researchers have devoted their time and efforts to investigate the mechanical and vibrational behavior of both micro- and nano- structures (Younis and Nayfeh, 2003; Farokhi et al., 2013; Askari, 2014; Askari et al., 2014; Farokhi and Ghayesh, 2015a, 2015b; Ghayesh and Farokhi, 2015a, 2015c; Han et al., 2015; Ghorbanpour Arani et al., 2016). A few more recent research works in this area include the study of the functionally graded microbeams carrying microparticles in a thermal environment by Shenas et al. (2016), the size-dependent behavior of functionally graded microbeams using various shear deformation theories by Trinh et al. (2016), the size-dependent vibration of a nickel cantilever microbeam by Lei et al. (2016), the free flexural vibration of geometrically imperfect functionally graded beams by Dehrouyeh-Semnani et al. (2016) and the size-dependent behavior of functionally graded microbeams using various shear deformation theories by Shafiei et al. (2016).

As an earlier effort on the study of vibration properties of micro-beams, one can refer to the characterization of the mechanical behavior of electrically actuated microbeams. They presented a nonlinear model of electrically actuated microbeams considering the electrostatic force of the air gap capacitor, restoring force of the microbeam and the axial load applied to the microbeam. It has been demonstrated that the ratio of the width of the air gap to the microbeam thickness can be tuned to extend the domain of the linear relationship between the direct current (DC) polarization voltage and the fundamental natural frequency(Abdel-Rahman et al., 2002).

Ross et al. (2005) investigated the temperature-dependent vibration of bilayer microbeams both experimentally and theoretically and found that the vibration frequency of the beam is a nonmonotone function of the beam’s temperature. Actually, the frequency decreases with an increase of the temperature at its lower values, but increases with an increase in the temperature at the higher values.

The effects of rotary inertia and shear deformation on the nonlinear free vibration of microbeams were studied by Ramezani et al. (2006) and Hamilton’s principle was used to derive the partial differential equation of the motion of the microbeam. The partial differential equation, employing the Galerkin method, is converted to an ordinary nonlinear differential equation. The method of multiple time-scales was used to determine a second-order perturbation solution for the ordinary differential equation.

Free vibration of a micro-scale Timoshenko beam was studied by Abbasion et al. (2009) and a comprehensive model was proposed to investigate the influence of the surface elasticity and the residual surface tension on the natural frequency of the flexural vibrations of microbeams in the presence of the rotary inertia and the shear deformation effects. They demonstrated that the nondimensional natural frequency of vibration of micro- and nano-beams is size dependent and, for a limiting case in which the beam length increases, the results tend to move toward the results obtained by the classical beam models (Abbasion et al., 2009).

The in-plane and out-of-plane motions of microbeams with modal interactions were investigated by Ghayesh et al. (2014). They showed that, because of the presence of one-to-one internal resonances between the in-plane and out-of-plane transverse modes, an in-plane excitation can give rise to an out-of-plane displacement. Moreover, they confirmed that the internal resonances would develop the occurrence of extra solution branches with presence of new bifurcation points. Ghayesh and Farokhi (2016) studied the parametric instability of microbeams in a supercritical regime. They examined the supercritical parametric instability of a microbeam subject to a time-dependent axial load. In another study, Ghayesh et al. (2015) analyzed the size-dependent nonlinear parametric dynamics of microbeams with a time-dependent longitudinal excitation load as a parametric excitation problem. Their study illustrates the effect of the length-scale parameter on the parametric response of the system.

Vibrations of a fixed–fixed narrow microbeam electrostatically actuated by applying a voltage difference to it and a parallel rigid conductor were analyzed by Batra et al., (2008). A distributed electromechanical model, which accounts for electrostatic fringing fields, finite deflections and residual stresses, was broached by them. It was shown that for relatively large gaps between the microbeam and the base rigid conductor, both the membrane stretching and the fringing field effects are crucial and omitting one of them may result in improper estimations of both the pull-in parameters and the resonance frequencies (Batra et al., 2008).

Ghayesh et al. (2013) investigated the nonlinear resonant behavior of a longitudinal and transverse motion of the microbeam with a distributed harmonic excitation force. They determined different types of bifurcations as well as the stability of solution branches. Also, they analyzed the coupled longitudinal–transverse nonlinear behavior of an imperfect microbeam numerically (Ghayesh and Amabili, 2014). Ghayesh and Farokhi (2015b) explored the coupled longitudinal–transverse-rotational behavior of shear deformable microbeams considering the small-scale effect. They showed that the frequency–response curves of the system have a hardening-type nonlinear behavior with two limit points of bifurcation.

An electromechanical fluidic coupled dynamic model of the microbeam was studied by Xu and Jia (2008), where they derived the governing equations of the electric field and air pressure fluctuations. They also analyzed the forced responses of the electromechanical–fluidic coupled system to the voltage excitation. Changes in the electromechanical–fluidic coupled vibrations along with the mechanical, electric and fluidic parameters were investigated (Xu and Jia, 2008).

Precise predictions of the DC dynamic pull-in voltages of a clamped–clamped microbeam, based on a continuous model, were developed by Chao et al. (2008). They established a dynamic model in partial differential equations based on the equilibrium among the beam flexibility, inertia, residual stress, squeeze film, distributed electrostatic forces and its electrical field fringing effects. It was found that the static pull-in phenomenon follows the dynamic instabilities, and the DC dynamic pull-in voltage is around 91% of its static counterpart (Chao et al., 2008).

Free vibrations of geometrically nonlinear micro-switches, under electrostatic and Casimir forces, have been investigated by Jia et al. (2010). They obtained the natural frequencies and mode shapes of micro-switches for four different types of boundary conditions (i.e. clamped–clamped, clamped–simply, simply supported and clamped-free). It has been demonstrated that the Casimir force, axial residual stress and material composition have most significant effects on the natural frequency of their considered system.

Resonance frequencies of asymmetric micro-bridge resonators were experimentally investigated by Hassanpour et al. (2011). They showed that the ratios of the higher resonant frequencies to the fundamental resonant frequency of asymmetric resonators can be adjustable and, hence, the asymmetric resonators are an appropriate choice for those applications in which the excitation of multiple resonant frequencies are desirable (Hassanpour et al., 2007).

The exact solution of the free vibration of a microbeam subjected to an axial force, when carrying a concentrated rotary mass along its length, were obtained by Hassanpour et al. (2007). They demonstrated that, depending on the location of the concentrated mass, the inclusion of its rotary inertia may either decrease or increase the natural frequencies of the resonator when compared with the case of no rotary inertia being included. They have also shown that higher natural frequencies can be achieved in the case with a lighter mass by a proper placement of the lumped mass.

The large-amplitude clamped–clamped microbeam resonator, excited near the higher order modes of vibration of a large bandwidth, employing partial electrodes and a multi-frequency electrical source, was investigated by Jaber et al. (2016).

The nonlinear forced vibrations of the strain gradient microbeams were investigated by Vatankhah et al. (2013). The nonlinearities of their considered structure are caused by the mid-plane stretching and nonlinear external forces, such as van der Waals force. The multiple time-scales method (MSM) was utilized to solve the nonlinear governing equations of the microbeams and the primary, super-harmonic and the sub-harmonic resonances of a microbeam were fully studied. It has been depicted that the nonlinear forced vibration behavior of microbeams is size-dependent and the ratio of the microbeam thickness to the material length scale parameter, being an additional material property appearing in the strain gradient theory, plays an important role.

Free vibrations of a cantilever microbeam submerged in a bounded incompressible and frictionless fluid cavity were investigated by Shabani et al. (2013). In accordance with the Fourier–Bessel series expansion, and by using the linear potential theory, an analytical method was proposed to analyze the eigenvalue problem, where the fluid effect emerges as an added mass. Influences of the geometrical configuration and the fluid density on the natural frequencies of the coupled system were evaluated and it was shown that the presence of fluid flow is more effective on the higher modes than on the lower modes.

The linear and nonlinear free vibration analyses of a microbeam, subjected to a symmetrical electrostatic field, were investigated by Fu and Zhang (2013) using the differential quadrature method (DQM). They also considered the effects of the fringing field and the mid-plane stretching in their study and showed that the fringing field has a significant effect on the linear free vibration frequency only when a large voltage has been applied. Moreover, the nonlinear free vibration frequency of the microbeam increases more rapidly with an increase of the vibration amplitude when either increasing the applied voltage or decreasing the beam thickness.

Transverse vibrations of micro-scale visco-thermo-elastic beam resonators were studied by Grover (2013). They obtained closed-form expressions for the transverse vibration of a homogenous isotropic, thermally conducting and Kelvin–Voigt type visco-thermo-elastic thin beam based on the Euler–Bernoulli beam theory. The influences of relaxation time, thermo-mechanical coupling, surface conditions and the beam dimensions on the energy dissipation induced by thermo-elastic damping in MEMS resonators were fully investigated for beams with the clamped and simply supported end conditions.

Static bending, post buckling and nonlinear free vibrations of nonclassical micro-scale beams were initially investigated by Xia et al. (2010). The nonlinear equation of motion, in which the nonlinear term is associated with the mean axial extension of the beam, was developed by using a combination of the modified couple stress theory and the Hamilton principle. They showed that the size effect is significant when the ratio of the characteristic thickness to the internal material length scale parameter is approximately equal to one, but diminishes with an increase of the ratio.

Closed-form expressions were obtained for the dynamic response of a micro-scale elastic beam by Saadatnia et al. (2013). The governing nonlinear equation of motion of the microbeam was derived and solved using the Variational Iteration Method (VIM) and the Energy Balance Method (EBM). The effects of the mechanical and geometrical properties of microbeams, such as the Poisson’s ratio, thickness and the initial amplitudes, were depicted in frequency responses.

The MSM is one of the most powerful approaches for probing the dynamical behavior of structures. It was implemented for the vibration analysis of many types of structures. For example, Diba et al. (2014) implemented MSM to investigate four modes of vibrations of an isotropic plate with an inclined crack. It was shown that according to the aspect ratio of the plate, the vibration modes can have either a nonlinear hardening effect or a nonlinear softening behavior.

Harmonically excited vibrations of a cracked beam were scrutinized using the MSM. Accordingly, the frequency response of the cracked beam was obtained, and the effect of different parameters, namely the geometry and location of the crack, loading position and the linear and nonlinear foundation parameters on the frequency response solution were studied (Younesian et al., 2013). Three-to-one internal resonances in a hinged–clamped beam were studied by Chin and Nayfeh (1997). The MSM was used to directly solve the governing equation of the vibrations of a hinged–clamped beam and to derive two sets of four first-order nonlinear ordinary differential equations of the objective system.

In addition, this method was employed for the primary and secondary resonant analysis of a microbeam. It was shown that the damping coefficient has a significant effect on the primary resonance of the microbeam. Furthermore, it was delineated there is a minimum point for the steady-state responses of all the sub-harmonic resonance of the microbeam (Younesian et al., 2014). Furthermore, they obtained the super-harmonic and sub-harmonic resonances for their model. In their work, a special kind of electrostatic symmetric actuator with two symmetrical potential walls for a fully clamped microbeam with a uniform thickness was considered. The present research aims to study the nonlinear vibration of microbeams considering Winkler and Pasternak foundations with multi-frequency excitation. The governing vibration equation of microbeams, based on the nonclassical theory of vibration, is developed and the discretized nonlinear differential equation for two modes of vibration was found using the Galerkin approach.

The MSM (Askari et al., 2014b) is utilized to investigate the vibration of a microbeam. Two modes of vibration are considered and the resonant cases are fully investigated. The resonant cases, owing to the external excitation as well as the combination resonances, are studied in depth. Frequency responses of the microbeams are depicted to investigate the influences of different parameters, namely, Pasternak and Winkler coefficients, applied force position and the geometrical factors. In pursuance of the stability investigation of the system and also to confirm the validity of the results obtained, using the MSM, the state-plane trajectories are also constructed.

2. Solution procedure

The system shown in Figure 1 is considered as a micro-scale beam of length L, mass density ρ, cross-sectional height h and the cross-sectional width b, which is placed between two immovable supports. The cross-section of the beam is assumed to be symmetrical (either being of a rectangular shape or a circular one).

The nonlinear equation of motion of the beam in terms of $w (x, t)$ is given by (Saadatnia et al., 2013)

(EI + GA c l 2) \frac{\partial 4 w (x, t)}{\partial x 4} + \bar{μ} \frac{\partial w (x, t)}{\partial t} + k 1 w (x, t) - k 2 \frac{\partial 2 w (x, t)}{\partial x 2} + k 3 w (x, t) 3 + [T 0 - \frac{EA c}{2 L} \int_{0}^{L} (\frac{\partial w (x, t)}{\partial x}) 2 d x] \times \frac{\partial 2 w (x, t)}{\partial x 2} + m c \frac{\partial 2 w (x, t)}{\partial t 2} = q (x, t)

(1)

The parameters $m c, E$ and G are the mass, modulus of elasticity and the shear modulus of the beam, respectively. Furthermore, A_c is the cross-sectional area, I denotes the corresponding moment of inertia, l is a material length scale parameter and $\bar{μ}$ represents the coefficient of foundation damping. The linear and nonlinear Winkler stiffness parameters are k₁ and k₃, respectively, while k₂ represents the Pasternak foundation parameter. Also, T₀ corresponds to the axially applied load, and $q (x, t)$ is the external transverse force, which are considered to be (Askari et al., 2014b)

q (x, t) = F 1 \cos (Ω 1 t) δ (x - x 1) + F 2 \cos (Ω 2 t) δ (x - x 2)

(2)

where parameters δ, F_i, x_i and

Ω i

(i = 1, 2) are the Dirac function, excitation amplitudes, their corresponding locations and the oscillation frequencies, respectively. Moreover, the expressions for the boundary conditions of the microbeam are given as

\frac{\partial 3 w}{\partial x 3} = 0 or w = 0 | x = 0 x = L

(3)

\frac{\partial 2 w}{\partial x 2} = 0 or \frac{\partial w}{\partial x} = 0 | x = 0 x = L

(4)

The Galerkin method is then utilized to obtain the response of the system. One could assume that

w (x, t) = \sum_{n = 1}^{N} ϕ n (x) . y n (t)

(5)

where

ϕ n (x)

represents the characteristic mode of a beam, with the pinned–pinned boundary condition, and

ϕ n (x) = \sin (\frac{n π}{L} x), n = 1, 2, 3, \dots

(6)

2.1. Single mode vibration resonance

Substituting the above equation into the main governing equation and using the principle of the orthogonality of the mode shapes for the fundamental mode, where n = 1 in equation (6), one could arrive at

\overset{··}{y} + ω_{1}^{2} y + \tilde{μ} \overset{\cdot}{y} + \tilde{α} y 3 = f 1 \cos (Ω 1 t) + f 2 \cos (Ω 2 t)

(7)

where

{ω_{1}^{2} = \frac{(EI + GA c l 2) (\frac{π}{L}) 4 + k 1 + (\frac{π}{L}) 2 (k 2 - \tilde{T} 2)}{m c}, \tilde{α} = \frac{(\frac{π}{L}) 4 EA c + 3 k 3}{4 m c}, \tilde{μ} = \frac{\bar{μ}}{m c}

(8)

and

f i = \frac{2 F i}{m c L} \int_{0}^{L} \sin (\frac{π x}{L}) δ (x - x i) d x, i = 1, 2

(9)

In order to analyze the resulting nonlinear equation with the multi-frequency excitation, the MSM has been utilized. Here, one should set $\tilde{μ} = ɛ μ and \tilde{α} = ɛ α$ , where ɛ refers to a very small perturbation parameter. According to the MSM, the solution of equation (7) can be written as

y (t) = y 1 (T 0, T 1) + ɛ y 2 (T 0, T 1) + \dots

(10)

where

T n = ɛ n t

and the derivatives with respect to time are

\frac{d}{dt} = \frac{d T 0}{dt} \frac{\partial}{\partial T 0} + \frac{d T 1}{dt} \frac{\partial}{\partial T 1} + \dots = D 0 + ɛ D 1 + ɛ 2 D 2 + \dots

(11)

\frac{d 2}{dt 2} = D_{0}^{2} + 2 ɛ D 0 D 1 + ɛ 2 (D_{1}^{2} + 2 D 0 D 1) + \dots

(12)

Substituting the above equations in the main equation and equating the same power of the coefficients of ɛ, one could obtain

ɛ 0 : D_{0}^{2} y 1 + ω_{1}^{2} y 1 = f 1 \cos (Ω 1 T 0) + f 2 \cos (Ω 2 T 0)

(13a)

ɛ 1 : D_{0}^{2} y 2 + ω_{1}^{2} y 2 = - 2 D 0 D 1 y 1 - 2 μ D 1 y 1 - α y_{1}^{3}

(13b)

The solution of equation (13a) can be expressed as

y 1 = A (T 1) \exp (i ω 1 T 0) + \sum_{k = 1}^{2} β k \exp (i Ω k T 0) + C . C .

(14)

In the above equation one will have

β k, k = 1, 2 = \frac{f k}{2 (ω_{1}^{2} - Ω_{k}^{2})}

(15)

where C.C. represents the complex conjugate of the first term. Substituting the solution found into equation (13b) would lead to

D_{0}^{2} y 1 + ω_{1}^{2} y 1 = - (2 i ω 1 (A' + μ A) + 3 α A (A \bar{A} + 2 β_{1}^{2} + 2 β_{2}^{2})) \exp (i ω 1 T 0) - α (A 3 \exp (3 i ω 1 T 0) + β_{1}^{3} \exp (3 i Ω 1 T 0) + β_{2}^{3} \exp (3 i Ω 2 T 0)) - (2 i μ Ω 1 + 3 α (2 A \bar{A} + β_{1}^{2} + 2 β_{2}^{2})) \exp (i Ω 1 T 0) - (2 i μ Ω 2 + 3 α (2 A \bar{A} + β_{2}^{2} + 2 β_{1}^{2})) \exp (i Ω 2 T 0) - 3 α A 2 (β 1 \exp (i (2 ω 1 + Ω 1) T 0) + β 2 \exp (i (2 ω 1 + Ω 2) T 0) + β 1 \exp (i (2 ω 1 - Ω 1) T 0) + β 2 \exp (i (2 ω 1 - Ω 2) T 0)) - 3 α A (β_{1}^{2} \exp (i (ω 1 + 2 Ω 1) T 0) + β_{2}^{2} \exp (i (ω 1 + 2 Ω 2) T 0) + β_{1}^{2} \exp (i (ω 1 - 2 Ω 1) T 0) + β_{2}^{2} \exp (i (ω 1 - 2 Ω 2) T 0)) - 6 α A β 1 β 2 (\exp (i (ω 1 + Ω 1 + Ω 2) T 0) + \exp (i (ω 1 + Ω 1 - Ω 2) T 0) + \exp (i (ω 1 - Ω 1 + Ω 2) T 0) + \exp (i (ω 1 - Ω 1 - Ω 2) T 0)) - 2 α β_{1}^{2} β 2 (\exp (i (2 Ω 1 + Ω 2) T 0) + \exp (i (2 Ω 1 - Ω 2) T 0)) - 2 α β 1 β_{2}^{2} (\exp (i (2 Ω 2 + Ω 1) T 0) + \exp (i (2 Ω 2 - Ω 1) T 0)) + C . C

(16)

Different types of resonance could occur from the above equation during the vibration of the system. Due to the multi-frequency load, let us consider the cases relevant to the presence of both frequencies, known as the combinational resonances. By assuming $Ω 2 > Ω 1$ , the resonances will occur when either $ω 1 \approx Ω 2 \pm 2 Ω 1$ , $ω 1 \approx 2 Ω 2 \pm Ω 1$ or $ω 1 \approx (1 / 2) (Ω 2 \pm Ω 1)$ .

In this work, the last case of the considered resonance has been analyzed. Accordingly, the relationship between the natural frequency and the external frequencies is formed as

{ω 1 + ɛ σ = (1 / 2) (Ω 2 \pm Ω 1) \Rightarrow Ω 2 \pm Ω 1 = 2 ω 1 + 2 ɛ σ (Ω 2 \pm Ω 1 - ω 1) T 0 = (2 ω 1 + 2 ɛ σ - ω 1) T 0 = 2 σ T 1

(17)

where σ represents the detuning parameter.

Applying the above definition and by eliminating the secular terms of equation (16), one obtains

2 i ω 1 (A' + μ A) + 3 α A 2 \bar{A} + 6 α A (β_{1}^{2} + β_{2}^{2}) + 6 α \bar{A} \exp (2 i σ T 1) = 0

(18)

The “prime” in the above equation represents the derivative with respect to T₁. Considering the polar notation given as $A = \frac{1}{2} a \exp (i β)$ , and by separating the imaginary part and the real part this would yield

a' = - μ a - \frac{3 α β 1 β 2}{ω 1} a \sin γ \frac{γ' a}{2} = a σ - \frac{3 α}{8 ω 1} a 3 - \frac{3 α}{ω 1} (β_{1}^{2} + β_{2}^{2}) a - \frac{3 α β 1 β 2}{ω 1} a \cos γ

(19)

where

γ = 2 (σ T 1 - β)

(20)

and, consequently, the first approximate solution for the system would be

y 1 (t) = a \cos ((1 / 2) (Ω 1 \pm Ω 2) t - γ) + β 1 \cos (Ω 1 t) + β 2 \cos (Ω 2 t) + O (ɛ)

(21)

In order to find the steady-state solution of equation (19), one can set $γ' = a' = 0$ . Thus, the frequency response equation after some mathematical computations will be obtained as

(\frac{3 α β 1 β 2}{ω 1}) 2 = μ 2 + (σ - \frac{3 α}{8 ω 1} a 2 - \frac{3 α}{ω 1} (β_{1}^{2} + β_{2}^{2})) 2

(22)

The above equation represents a quadratic expression in terms of a². Simplifying the equation reveals that the nontrivial amplitudes of oscillation will occur when the following conditions will be satisfied

{σ \geq \frac{3 α}{ω 1} (β_{1}^{2} + β_{2}^{2}) μ 2 \leq (\frac{3 α β 1 β 2}{ω 1}) 2

(23)

2.2. Two-mode vibration resonances

To find an exact analytical view and better understand the nonlinear vibration of the proposed system, one could consider two modes of vibration for the Galerkin method, that is, $n = 2$ in equation (6). Once again by applying the principle of the orthogonality of the mode shapes, one could arrive at the following coupled nonlinear system for the two vibration modes of microbeam

{\overset{··}{y} 1 + ω_{1}^{2} y 1 + μ \overset{\cdot}{y} 1 + α 1 y_{1}^{3} + α 2 y 1 y_{2}^{2} = f 11 \cos Ω 1 t + f 12 \cos Ω 2 t \overset{··}{y} 2 + ω_{2}^{2} y 2 + μ uy 2 + α 3 y_{2}^{3} + α 4 y 2 y_{1}^{2} = f 21 \cos Ω 1 t + f 22 \cos Ω 2 t

(24)

where the constant coefficients are given as

ω_{1}^{2} = \frac{1}{m c} [(EI + GA c l 2) (\frac{π}{L}) 4 + k 1 + (\frac{π}{L}) 2 (k 2 - \tilde{T} 2)]

(25)

ω_{2}^{2} = \frac{1}{m c} [(EI + GA c l 2) (\frac{2 π}{L}) 4 + k 1 + (\frac{2 π}{L}) 2 (k 2 - \tilde{T} 2)]

(26)

α 1 = \frac{1}{m c} [\frac{3}{4} k 3 + \frac{EA c}{4} (\frac{π}{L}) 4], α 3 = \frac{1}{m c} [\frac{3}{4} k 3 + 4 EA c (\frac{π}{L}) 4]

(27)

α 2 = α 4 = \frac{1}{m c} [\frac{3}{2} k 3 + EA c (\frac{π}{L}) 4], μ = \frac{\bar{μ}}{m c}

(28)

and

f ij = \frac{2 F j}{m c L} \int_{0}^{L} \sin (\frac{i π x}{L}) δ (x - x j) dx, i, j = 1, 2

(29)

To find the solutions of the proposed nonlinear system, having a multi-frequency condition, the MSM has been employed once again. Based on this method, and because the nonlinearity of the system is in the cubic form, the following asymptotic expansions have been considered (Nayfeh and Mook, 1995)

{y 1 = ɛ y 11 (T 0, T 2) + ɛ 3 y 13 (T 0, T 2) + \dots y 2 = ɛ y 21 (T 0, T 2) + ɛ 3 y 23 (T 0, T 2) + \dots

(30)

In order to consider the effects of the system damping and the external excitation, one could set $\tilde{μ} = ɛ 2 μ$ and $\tilde{f} ij = ɛ 3 f ij, with i, j = 1, 2$ .

Substituting equation (29) into (23) and by equating the same power coefficients of ɛ would lead to

ɛ : D_{0}^{2} y 11 + ω_{1}^{2} y 11 = 0 D_{0}^{2} y 21 + ω_{2}^{2} y 21 = 0

(31)

ɛ 3 : D_{0}^{2} y 13 + ω_{1}^{2} y 13 = - 2 D 0 (D 1 y 11 + μ y 11) - α 1 y_{11}^{3} - α 2 y 11 y_{21}^{2} + \tilde{f} 11 \cos Ω 1 T 0 + \tilde{f} 12 \cos Ω 2 T 0 D_{0}^{2} y 23 + ω_{2}^{2} y 23 = - 2 D 0 (D 1 y 21 + μ y 21) - α 3 y_{21}^{3} - α 4 y 21 y_{11}^{2} + \tilde{f} 21 \cos Ω 1 T 0 + \tilde{f} 22 \cos Ω 2 T 0

(32)

The solution of equation (31) can be written as

{y 11 = A 1 (T 2) \exp (i ω 1 T 0) + C . C y 21 = A 2 (T 2) \exp (i ω 2 T 0) + C . C

(33)

Substituting the above equations into equation (32) yields

D_{0}^{2} y 13 + ω_{1}^{2} y 13 = [- 2 i ω 1 (A' 1 + μ A 1) - 3 α 1 A_{1}^{2} \bar{A} 1 - 2 α 2 A 2 \bar{A} 2 A 1] . \exp (i ω 1 T 0) - α 1 A_{1}^{3} \exp (3 i ω 1 T 0) - α 3 A 1 A_{2}^{2} \exp [i (ω 1 + 2 ω 2) T 0] - α 3 A 1 {\bar{A}}_{2}^{2} \exp [i (ω 1 - 2 ω 2) T 0] + \frac{1}{2} \tilde{f} 11 \exp (i Ω 1 T 0) + \frac{1}{2} \tilde{f} 12 \exp (i Ω 2 T 0) + C . C

(34)

and

D_{0}^{2} y 23 + ω_{2}^{2} y 23 = [- 2 i ω 2 (A' 2 + μ A 2) - 3 α 3 A_{2}^{2} \bar{A} 2 - 2 α 4 A 1 \bar{A} 1 A 2] . \exp (i ω 2 T 0) - α 3 A_{2}^{3} \exp (3 i ω 2 T 0) - α 4 A 2 A_{1}^{2} \exp [i (ω 2 + 2 ω 1) T 0] - α 4 A 2 {\bar{A}}_{1}^{2} \times \exp [i (ω 2 - 2 ω 1) T 0] + \frac{1}{2} \tilde{f} 21 \exp (i Ω 1 T 0) + \frac{1}{2} \tilde{f} 22 \exp (i Ω 2 T 0) + C . C

(35)

where the “prime” denotes the derivative with respect to T₂. The external resonances due to the external excitation of the microbeam vibrations will be studied for three different main cases in the following sections.

2.2.1. The first Case when $Ω 1 \approx ω 1$

The deviations of Ω from $ω 1$ are shown as the following definition

Ω 1 = ω 1 + ɛ 2 σ 1

(36)

where

σ 1

is the detuning parameter. After considering the above definition and by eliminating the secular terms from equations (34) and (35), one will arrive at

{- 2 i ω 1 (A' 1 + μ A 1) - 3 α 1 A_{1}^{2} \bar{A} 1 - 2 α 2 A 2 \bar{A} 2 A 1 + \frac{1}{2} \tilde{f} 12 \exp (i σ 1 T 2) = 0 - 2 i ω 2 (A' 2 + μ A 2) - 3 α 3 A_{2}^{2} \bar{A} 2 - 2 α 4 A 1 \bar{A} 1 A 2

(37)

Using the polar notation of $A i, i = 1, 2 = \frac{1}{2} a i \exp (i θ i)$ in equation (37) and by separating the real and the imaginary parts, this yields

- ω 1 a' 1 - μ ω 1 a 1 + \frac{1}{2} \tilde{f} 1 \sin γ 1 = 0 - ω 2 a' 2 - μ ω 2 a 2 = 0 σ 1 ω 1 a 1 - γ' 1 ω 1 a 1 - \frac{3}{8} α 1 a_{1}^{3} - \frac{1}{4} α 2 a 1 a_{2}^{2} + \frac{1}{2} \tilde{f} 1 \cos γ 1 = 0 θ' 2 ω 2 a 2 - \frac{3}{8} α 3 a_{2}^{3} - \frac{1}{4} α 4 a 2 a_{1}^{2} = 0

(38)

in which

γ 1 = σ 1 T 2 - θ 1

. For the sake of the steady-state response, all the parameters

γ' 1, θ' 2, a' 1

and

a' 2

must be set to zero and, after some mathematical simplification of equation (38), the following equation is obtained

(μ ω 1 a 1) 2 + (σ 1 ω 1 a 1 - \frac{3}{8} α 1 a_{1}^{3} - \frac{1}{4} α 2 a 1 a_{2}^{2}) 2 = \frac{1}{4} {\tilde{f}}_{12}^{2}, a 2 = 0

(39)

2.2.2. The second case when $Ω 1 \approx ω 2$

Here, the detuning parameter is defined as $Ω 1 = ω 2 + ɛ 2 σ 2$ . Therefore, the secular terms would be

{- 2 i ω 1 (A' 1 + μ A 1) - 3 α 1 A_{1}^{2} \bar{A} 1 - 2 α 2 A 2 \bar{A} 2 A 1 - 2 i ω 2 (A' 2 + μ A 2) - 3 α 3 A_{2}^{2} \bar{A} 2 - 2 α 4 A 1 \bar{A} 1 A 2 + \frac{1}{2} \tilde{f} 21 \exp (i σ 2 T 2) = 0

(40)

Similarly, by applying the polar forms and separating the imaginary and the real parts, and after some mathematical simplification, the steady-state response will be obtained

(μ ω 2 a 2) 2 + (σ 2 ω 2 a 2 - \frac{3}{8} α 3 a_{2}^{3} - \frac{1}{4} α 4 a 2 a_{1}^{2}) 2 = \frac{1}{4} {\tilde{f}}_{21}^{2}, a 1 = 0

(41)

2.2.3. The third case when $Ω 1 \approx ω 1$ and $Ω 2 \approx ω 2$

This case could be found to be of more general form and, consequently, one needs to define two different detuning parameters at the same time

{Ω 1 = ω 1 + ɛ 2 σ 1 Ω 2 = ω 2 + ɛ 2 σ 2

(42)

Thus, the secular terms from equations (34) and (35) are represented as

{- 2 i ω 1 (A' 1 + μ A 1) - 3 α 1 A_{1}^{2} \bar{A} 1 - 2 α 2 A 2 \bar{A} 2 A 1 + + \frac{1}{2} \tilde{f} 11 \exp (i σ 1 T 2) = 0 - 2 i ω 2 (A' 2 + μ A 2) - 3 α 3 A_{2}^{2} \bar{A} 2 - 2 α 4 A 1 \bar{A} 1 A 2 + \frac{1}{2} \tilde{f} 22 \exp (i σ 2 T 2) = 0

(43)

Applying the polar forms given for $A 1, A 2$ , as defined before, and by separating the real and the imaginary parts, once again the following equations will be found

- ω 1 a' 1 - μ ω 1 a 1 + \frac{1}{2} \tilde{f} 11 \sin γ 1 = 0 σ 1 ω 1 a 1 - γ' 1 ω 1 a 1 - \frac{3}{8} α 1 a_{1}^{3} - \frac{1}{4} α 2 a 1 a_{2}^{2} + \frac{1}{2} \tilde{f} 11 \cos γ 1 = 0 - ω 2 a' 2 - μ ω 2 a 2 + \frac{1}{2} \tilde{f} 22 \sin γ 2 = 0 σ 2 ω 2 a 2 - γ' 2 ω 2 a 2 - \frac{3}{8} α 3 a_{2}^{3} - \frac{1}{4} α 4 a 2 a_{1}^{2} + \frac{1}{2} \tilde{f} 22 \cos γ 2 = 0

(44)

where

γ 1 = σ 1 T 2 - θ 1

and

γ 2 = σ 2 T 2 - θ 2

Simplifying the above equation, one could obtain the steady-state frequency response of the system for this type of resonance condition as

{(σ 1 ω 1 a 1 - \frac{3}{8} α 1 a_{1}^{3} - \frac{1}{4} α 2 a 1 a_{2}^{2}) 2 + (μ ω 1 a 1) 2 = \frac{1}{4} {\tilde{f}}_{11}^{2} (σ 2 ω 2 a 2 - \frac{3}{8} α 3 a_{2}^{3} - \frac{1}{4} α 4 a 2 a_{1}^{2}) 2 + (μ ω 2 a 2) 2 = \frac{1}{4} {\tilde{f}}_{22}^{2}

(45)

3. Numerical results

The developed solution procedures and the obtained natural frequencies are fully examined in this section. The physical and geometrical parameters used in the evolutions are initially selected as

E = 1.44 Gpa, G = E / (1 - υ), υ = 0.38, ρ = 1000 kg / m 3, l = 17.6 \times 10 - 6 m, b = 24 \times 10 - 6 m, h = 2 l, L = 20 \times l, A c = h \times b, m = ρ \times A c, T 0 = 0, \bar{μ} = 10 - 5, k 1 = 106, k 2 = 10 - 1, k 3 = 1014

The effect of the nonlocal parameter on the frequency response of the microbeam vibration considering multi-frequency excitation is shown in Figure 2.

The expression $ω 1 \approx (1 / 2) (Ω 2 \pm Ω 1)$ is initially assumed for the analysis of the resonance case of the vibration of the system. As depicted in Figure 2, increasing the nonlocal parameter results in an increase of the oscillation amplitude of the microbeam.

The effect of the nonlinear Winkler foundation coefficient on the considered resonance case is depicted in Figure 3. It is shown that the oscillation amplitude will decrease by increasing the Winkler foundation coefficient.

Furthermore, it can be seen from Figure 4 that the oscillation amplitude would decrease by increasing the coefficient of the Pasternak foundation. The influence of the nonlocal parameter on the frequency response of the system, for the first mode of vibration, assuming $Ω 1 \approx ω 1$ , is illustrated in Figure 5.

As can be seen, increasing the nonlocal parameter will result in decreasing the amplitude of oscillations. Moreover, the nonlinearity of the considered system will increase by increasing the nonlocal parameter. Hence, the nonlocal parameter can be well implemented for adjusting the oscillation amplitude and the nonlinearity of the microbeam.

The influence of the Winkler foundation coefficient on the frequency response for the first mode of vibration of the system, assuming $Ω 1 \approx ω 1$ , is shown in Figure 6. It can be seen from Figure 6 that increasing the coefficient of the Winkler foundation will result in an increase of the hardening behavior of the considered system.

The effect of the Pasternak foundation coefficient on the frequency response of the system for the first mode of vibration postulating $Ω 1 \approx ω 1$ is shown in Figure 7. It can be seen that increasing the Pasternak foundation coefficient would result in decreasing the oscillation amplitude and the nonlinearity of the system. Therefore, the Pasternak foundation coefficient can be implemented to simultaneously adjust the vibration amplitude and the frequency behavior of the microbeam.

The effect of the position of the applied force on the frequency behavior of the considered system is illustrated in Figure 8. Accordingly, the highest value of the amplitude of the oscillation will occur when the applied force has been imposed at the middle span of the microbeam.

The influence of the nonlocal parameter on the frequency response of the considered system for the first mode of vibration, assuming $Ω 1 \approx ω 2$ , is illustrated in Figure 9. As can be seen from this figure, increasing the nonlocal parameter results in decreasing the amplitude of oscillation. Furthermore, the nonlinearity of the considered system would increase by increasing the nonlocal parameter. Therefore, the nonlocal parameter can be utilized to adjust the oscillation amplitude and the nonlinearity of the microbeam.

Figure 10 shows the influence of the Winkler foundation coefficient on the frequency response of the considered system for the first mode of vibration, assuming $Ω 1 \approx ω 2$ . It can be seen from Figure 10 that increasing the coefficient of the Winkler foundation would result in increasing the hardening behavior of the system.

The effect of the Pasternak foundation parameter on the frequency response for the first mode of vibration of the considered system postulating $Ω 1 \approx ω 2$ is shown in Figure 11. It can be seen from this figure that increasing the values of the Pasternak foundation parameter results in decreasing the oscillation amplitude and the nonlinearity of the system.

Figure 12 is presented to show the effect of the position of the applied force on the frequency behavior of the system. Accordingly, the highest amplitude of oscillation will occur when the applied force is imposed at the mid-span of the microbeam.

The effect of the damping coefficient on the frequency response for the first mode of vibration of the system, assuming $Ω 1 \approx ω 1$ , is shown in Figure 13. Figure 14 also depicts the effect of the damping coefficient on the frequency response for the second mode when $Ω 1 \approx ω 2$ . From both figures, it can be observed that increasing the damping coefficient for the two modes of vibration will considerably reduce the amplitudes of vibration, while it does not change the system nonlinearity, as expected.

Figures 15 –18 all present the two cases of $Ω 1 \approx ω 1$ and $Ω 2 \approx ω 2$ . Figure 15 represents the influence of the nonlocal parameter for the aforementioned case. As can be seen from this figure, increasing the nonlocal parameter results in decreasing the oscillation amplitude and, in turn, the nonlinearity of the system for both modes of vibration.

The influence of the values of the Winkler foundation parameter is depicted in Figure 16. Accordingly, the nonlinearity of the system would increase by increasing the Winkler foundation parameter for the case of the combinational resonances.

Figure 17 illustrates the influence of the coefficient of the Pasternak foundation for the case of $Ω 1 \approx ω 1$ and $Ω 2 \approx ω 2$ . As can be seen from Figure 17, increasing the parameter of the Pasternak foundation will result in decreasing the oscillation amplitude and the nonlinearity of the system.

The effects of the position of the applied force for the aforementioned case are presented in Figure 18. As can be seen, the highest value of the amplitude of the oscillation and the nonlinearity for the first mode of oscillation would occur when the force is imposed at the middle of the microbeam. The highest value of the amplitude of oscillation for the second mode of vibration occurs when the force is applied either at $1 / 4$ of the length of the considered system or at $3 / 4$ of its length.

In order to confirm the validity of the obtained results and also to check the stability of the steady-state solutions, a state-plane graph has been plotted for the two different resonant cases of the primary and secondary. The attraction zone, shown in Figure 19, is quite apparent and is adjacent to the selected point; the stability is confirmed for the primary resonance case.

The stability of the steady-state response for the secondary resonance is presented in Figure 20. It is evident from this plot that the validity of the obtained results using the MSM is guaranteed. The attraction zone, shown in Figure 20, is quite apparent and is adjacent to the selected point; the stability is confirmed for the secondary resonance case.

4. Conclusion

Nonlinear oscillations of a microbeam considering both the Pasternak and Winkler foundations and considering multi-frequency excitation were investigated. The nonlinear ordinary differential equation of vibration of the microbeam was obtained using the Galerkin method. The forced damped vibrations considering two modes of oscillation were fully investigated by using the MSM and including the multi-frequency excitation. The effects of different parameters, namely, the Pasternak and Winkler foundation coefficients, the position of the applied force and the nonlocal parameters, on different types of resonant cases of the system were fully investigated. Results of the analysis show that increasing the value of the nonlocal parameter would decrease the amplitude of oscillation. Comparison with the published papers has confirmed the originality of the results obtained for the multi-frequency excitations of the microbeams considering three different resonance cases, including $Ω 1 \approx ω 1$ , $Ω 1 \approx ω 2$ and $Ω 1 \approx ω 1$ $Ω 2 \approx ω 2$ . The influence of the position of the applied force on the primary and secondary types of resonance of the microbeam vibration has been illustrated. It has been demonstrated that the highest value of the amplitude of oscillation, for the first mode of vibration, would occur when the external force is applied at the middle of the microbeam. It has been shown that highest value of the amplitude of oscillation will be achieved when the external force is applied at either ¼ or ¾ of the length of microbeam. Moreover, the effects of the Pasternak foundation parameter for three different resonant cases of vibration of the microbeam have been studied. It has been found that the amplitude of oscillation decreases by increasing the Pasternak coefficients.

State-plane graphs have been presented for the first and second modes of vibration of the system to check the stability of the steady-state solutions and to confirm the validity of the obtained results.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada.

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Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations

Abstract

Keywords

1. Introduction

2. Solution procedure

2.1. Single mode vibration resonance

2.2. Two-mode vibration resonances

2.2.1. The first Case when Ω 1 ≈ ω 1

2.2.2. The second case when Ω 1 ≈ ω 2

2.2.3. The third case when Ω 1 ≈ ω 1 and Ω 2 ≈ ω 2

3. Numerical results

4. Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

2.2.1. The first Case when $Ω 1 \approx ω 1$

2.2.2. The second case when $Ω 1 \approx ω 2$

2.2.3. The third case when $Ω 1 \approx ω 1$ and $Ω 2 \approx ω 2$