The Dynamics of MEMS Arches of Non-Ideal Boundary Conditions

2011 ◽  
Author(s):  
Sami A. Alkharabsheh ◽  
Mohammad I. Younis
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Electrostatic Force ◽  
Dynamical Behavior ◽  
Experimental Investigations ◽  
Harmonic Load ◽  
Ideal Boundary ◽  
Shooting Technique ◽  
Softening Behavior ◽  
The Stability

We present an investigation into the dynamics of MEMS arches when actuated electrically including the effect of their flexible (non-ideal) supports. First, the eigenvalue problem of a nonlinear Euler-Bernoulli shallow arch with torsional and transversal springs at the boundaries is solved analytically. Several results are shown to demonstrate the possibility of tuning the theoretically obtained natural frequencies of an arch to match the experimentally measured. Then, simulation results are shown for the forced vibration response of an arch when excited by a DC electrostatic force superimposed to an AC harmonic load. Shooting technique is utilized to find periodic motions. The stability of the captured periodic motion is examined using the Floquet theory. The results show several jumps in the response during snap-through motion and pull-in. Theoretical and experimental investigations are conducted on a microfabricated curved beam actuated electrically. Results show softening behavior and superharmonic resonances. It is demonstrated that non-ideal boundary conditions can have significant effect on the qualitative dynamical behavior of the MEMS arch, including its natural frequencies, snap-through behavior, and dynamic pull-in.

Nonlinear Dynamics of MEMS Arches

2010 ◽  
Author(s):  
Sami Alkharabsheh ◽  
Mohammad Younis
Keyword(s):  
Electrostatic Force ◽  
Equations Of Motion ◽  
Amplitude Dependence ◽  
Harmonic Load ◽  
Order Model ◽  
Shooting Technique ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Dynamic Phenomena ◽  
Unstable States

In this paper, the dynamic response of electrostatically actuated clamped-clamped arch microbeam is investigated when excited by a DC load superimposed to an AC harmonic load. The dynamic analysis is carried out using a Galerkin-based reduced order model along with a shooting technique to find periodic motions and analyzing its stability using a Floquet theory. Results are presented for the cases of primary and super harmonic resonances. We found several nonlinear dynamic phenomena due to the inherent nonlinear electrostatic force and geometric nonlinearity of the arch. These include frequency-amplitude dependence, jumps, tangent bifurcations, coexistence of solutions, and softening and hardening behaviors. The shooting technique showed high robustness in capturing both the stable and unstable states of the system. Hence, it helped clarify vague behaviors that were previously reported using longtime integration of the equations of motion.

An Analytical Model for a Clamped Isotropic Beam Under Thermal Effects

2002 ◽  
Author(s):  
Rodrigo F. A. Marques ◽  
Daniel J. Inman
Keyword(s):  
Boundary Conditions ◽  
Analytical Model ◽  
Thermal Effects ◽  
Natural Frequencies ◽  
Dynamical Behavior ◽  
Beam Theory ◽  
Optimization Procedure ◽  
Temperature Changes ◽  
Thermally Expand ◽  
Spring Constants

Structures and industrial equipment often operate in environments where temperature variations take place. Although thermal effects may be negligible in some cases, they have caused the unexpected failure of mechanical systems many times. Whether or not temperature has significant effects on the dynamical behavior of such machines and structures depends upon several aspects, amongst which are geometry, material properties and boundary conditions. In this paper we investigate the dynamical behavior of a clamped beam under the influence of a uniform, quasi-statically varying temperature field. An analytical model was used, based on Euler-Bernoulli’s beam theory with the introduction of the proper boundary conditions. Temperature effects are included in terms of an axial force that shows up when the beam tends to thermally expand, but this expansion is restrained by the clamping. Preliminary results do not agree with experimental data, since perfect clamping is difficult to achieve in practice. Finally the model is updated with the inclusion of axial and torsional springs connecting the beam to the support. The spring constants were calculated through optimization procedure to minimize the differences between the natural frequencies obtained from the analytical model and the corresponding experimental ones. Agreement with experimental results is reasonable up to the 4th mode of the beam. In the future, this analytical model is to be used for design and simulation of an active controller that accounts for temperature changes in the structure.

Dynamic Responses of Functionally Graded Sandwich Beams Resting on Elastic Foundation Under Harmonic Moving Loads

2018 ◽  
Vol 18 (09) ◽  
pp. 1850112 ◽  
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Moving Load ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Harmonic Load ◽  
Sandwich Beams

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

scholarly journals On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams

Materials ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 1707 ◽  
Author(s):  
Ali Shariati ◽  
Dong won Jung ◽  
Hamid Mohammad-Sedighi ◽  
Krzysztof Kamil Żur ◽  
Mostafa Habibi ◽  
...  
Keyword(s):  
Critical Velocity ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Dynamical Behavior ◽  
Functionally Graded ◽  
Fine Tuning ◽  
Dynamical Response ◽  
Power Law Distribution ◽  
Axially Moving ◽  
The Stability

In this article, size-dependent vibrations and the stability of moving viscoelastic axially functionally graded (AFG) nanobeams were investigated numerically and analytically, aiming at the stability enhancement of translating nanosystems. Additionally, a parametric investigation is presented to elucidate the influence of various key factors such as axial gradation of the material, viscosity coefficient, and nonlocal parameter on the stability boundaries of the system. Material characteristics of the system vary smoothly along the axial direction based on a power-law distribution function. Laplace transformation in conjunction with the Galerkin discretization scheme was implemented to obtain the natural frequencies, dynamical configuration, divergence, and flutter instability thresholds of the system. Furthermore, the critical velocity of the system was evaluated analytically. Stability maps of the system were examined, and it can be concluded that the nonlocal effect in the system can be significantly dampened by fine-tuning of axial material distribution. It was demonstrated that AFG materials can profoundly enhance the stability and dynamical response of axially moving nanosystems in comparison to homogeneous materials. The results indicate that for low and high values of the nonlocal parameter, the power index plays an opposite role in the dynamical behavior of the system. Meanwhile, it was shown that the qualitative stability of axially moving nanobeams depends on the effect of viscoelastic properties in the system, while axial grading of material has a significant role in determining the critical velocity and natural frequencies of the system.

scholarly journals On the Nonlinear Dynamics of a Doubly Clamped Microbeam Near Primary Resonance

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Nizar R. Jaber ◽  
Karim M. Masri ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Primary Resonance ◽  
Single Frequency ◽  
Response Curves ◽  
Shooting Technique ◽  
Softening Behavior ◽  
Dc Bias ◽  
The Stability ◽  
Good Agreement ◽  
Structural Layer

This work aims to investigate theoretically and experimentally various nonlinear dynamic behaviors of a doubly clamped microbeam near its primary resonance. Mainly, we investigate the transition behavior from hardening, mixed, and then softening behavior. We show in a single frequency–response curve, under a constant voltage load, the transition from hardening to softening behavior demonstrating the dominance of the quadratic electrostatic nonlinearity over the cubic geometric nonlinearity of the beam as the motion amplitudes becomes large, which may lead eventually to dynamic pull-in. The microbeam is fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from the bottom. Frequency sweep tests are conducted for different values of direct current (DC) bias revealing hardening, mixed, and softening behavior of the microbeam. A multimode Galerkin model combined with a shooting technique are implemented to generate the frequency–response curves and to analyze the stability of the periodic motions using the Floquet theory. The simulated curves show a good agreement with the experimental data.

Stability of a Rotor Whose Cavity Has an Arbitrary Meridian and Is Partially Filled With Fluid

1985 ◽  
Vol 107 (4) ◽  
pp. 440-445 ◽  
Author(s):  
R. Wohlbru¨ck
Keyword(s):  
Boundary Conditions ◽  
Close Correlation ◽  
Experimental Investigations ◽  
The Finite Element Method ◽  
Field Equations ◽  
Equilibrium Conditions ◽  
Conical Cavity ◽  
Rotor Spinning ◽  
The Stability ◽  
Elastically Supported

The stability of an elastically supported rotor spinning with constant angular velocity is studied. The rotor has a cavity of arbitrary meridian and is partially filled with an ideal fluid. The motions of the system are governed by linearized equilibrium conditions for the rotor and field equations as well as boundary conditions for the fluid. Due to the arbitrary shape of the meridian, it is not possible to solve the boundary value problem in closed form. Therefore a variational expression is developed which satisfies the boundary conditions naturally. The variational problem is solved approximately by the finite element method. The results, incorporated in the equilibrium conditions for the rotor, lead to stability statements. For numerical and experimental investigations, two rotors, one with an elliptical and one with a conical cavity are used. The fill medium is water. There is a close correlation between numerical and experimental results.

Vibration and Stability of Cylindrical Shells Containing Flowing Fluid

2002 ◽  
Author(s):  
Igor Zolotarev
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Finite Length ◽  
Natural Frequencies ◽  
Critical Flow ◽  
Thin Shells ◽  
Fluid Viscosity ◽  
Critical Flow Velocity ◽  
Flowing Fluid ◽  
The Stability

Natural frequencies and the thresholds for loosing the stability of thin-walled cylindrical shell conveying by flowing fluid are theoretically studied. Potential flow theory for fluid and 3D theory for thin shells are used. The shells of finite length are considered for the different case of boundary conditions at the edges of the shell, and their influence on the critical flow velocities for flutter are demonstrated. The fundamental importance of boundary conditions considered for fixing the edges of the cylindrical shell of finite length is shown. When the clamped - simply supported boundary conditions are assumed, the critical flow velocity for flutter is very low, even if the energy dissipation due to the fluid viscosity was taken into account.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

Buckling of Circular Cylindrical Shells Using a Displacement Function

1974 ◽  
Vol 96 (4) ◽  
pp. 1322-1327
Author(s):  
Shun Cheng ◽  
C. K. Chang
Keyword(s):  
Boundary Conditions ◽  
Axial Compression ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Displacement Function ◽  
Combined Loading ◽  
Trial Solution ◽  
Governing Differential Equation ◽  
Circular Cylindrical Shells ◽  
The Stability

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

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