Sharif Digital Repository

In this paper, a new approach is used to obtain the nonlinear mode for a flexible cantilever beam. The dependency of resonance of the beam on the exciting force amplitude is observed experimentally and theoretically. In this regard, the end response of a cantilever beam under the base excitation is measured for various force amplitudes. Then, by using the nonlinear normal modes, the reduced-order governing equation is obtained. The discrete model obtained according to identification of nonlinear terms can predict the frequency response of the beam and jump phenomenon properly. Comparison between experimental data and theoretical outputs show acceptable agreement. © 2020 Faculty of... 

The differential equation governing the motion of an electrically excited capacitive microcantilever beam is a nonlinear PDE [1]. Accurate analysis about its motion is of great importance in MEMS' dynamical response. In this paper first the nonlinear 4th order 2 point boundary value problem (ODE) governing the static deflection of the system is solved using three methods. 1. The nonlinear part is linearized and its exact solution is obtained. 2. For low applied DC voltages (not near pull-in) the solutin is found using the direct straight forward perturbation analysis. 3. Numerical computer solutions which are used for the previous solution's verifications. The next parts are devoted to the... 

This paper presents the modeling steps towards development of frequency equations for a cantilever beam with a tip mass under general base excitations. More specifically, the beam is considered to vibrate in all the three directions, while subjected to a base rotational motion around its longitudinal direction. This is a common configuration utilized in many vibrating beam gyroscopes and well drilling systems. The governing equations are derived using Extended Hamilton's Principle with general 6-DOF base motion. The natural frequency equations are then extracted in closed-form for the case where the base undergoes longitudinal rotation. For validation purposes, the resulting natural... 

Double cantilever beam (DCB) specimens were prepared according to standard surface mount technology (SMT). The samples were stored inside an environmental chamber at two different relative humidity values (i.e. 40 or 100%) for either 2 or 4 h. Then they were aged at 25 °C, 75 °C or 125℃ for the same time. Finally, after being cooled to room temperature, fracture tests were performed under mode-I loading conditions at a strain rate of 0.03 s−1 at room temperature. Storage time did not have a significant effect on the fracture behavior. The fracture load and energy of the solder joints decreased significantly when the temperature was increased from 25 to 75℃. By further increase in the... 

This study investigated the mode I fracture behavior of double cantilever beam (DCB) epoxy adhesive joints with similar adherends on the both sides (i.e., aluminum-aluminum or copper-copper) at different strain rates; i.e., quasi-static (∼10−3 s−1), low (∼7 s−1) and intermediate (∼14 s−1) rates. The fracture energy of the DCB joint in Al-adhesive-Al specimens decreased (i.e., by ∼62%, p = 0.0013) with an increase in the applied strain rate from quasi-static to low values, while it remained almost unchanged with further increase of stain rate to intermediate range (p > 0.05). For Cu-adhesive-Cu cases, however, the fracture energy was found to be almost insensitive to the applied strain rate... 

In this study the investigation of large deflections subject in compliant mechanisms is presented using homotopy perturbation method (HPM). The main purpose is to propose a convenient method of solution for the large deflection problem in compliant mechanisms in order to overcome the difficulty and complexity of conventional methods, as well as for the purpose of mathematical modeling and optimization. For simplicity, a cantilever beam of linear elastic material under horizontal, vertical and bending moment end point load is considered. The results show that the applied method is very accurate and capable for cantilever beams and can be used for a large category of practical problems for the... 

An analytical modified method is presented to investigate the clearance effects on the dynamic behavior of cantilever beams. In this work, apart from clearance, all other nonlinearities are avoided to be considered during the analytical solution. The beam is idealized using an equivalent single-degree-of-freedom structure. A tri-linear stiffness system is also adopted to simulate the beam together with the clearance. Subsequently the analytical solution is derived for a cantilever beam structure with and without considering structural damping. The experimental method is employed to verify the results from the presented analytical solutions. It is noted that the results obtained by the... 

In this paper, the two-point boundary value problem (BVP) of the nano-cantilever deflection subjected to Casimir and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the nano-beam. In the analytical treatment of the BVP, the nonlinear differential equation of the model is transformed into the integral form by using the Green's function of the cantilever beam. Then, closed-form solutions are obtained by assuming an appropriate shape function for the beam deflection to evaluate the integrals. The pull-in parameters of the beam are computed under the combined effects of electrostatic and Casimir forces. Electrostatic microactuators... 

A general modeling framework is presented for the development of the frequency equation of a microgyroscope, which is modeled as a suspended cantilever beam with a tip mass under general base excitation. Specifically, the beam is considered to vibrate in all the three directions, while subjected to a base rotational motion around its longitudinal direction. This is a common configuration utilized in many vibrating beam gyroscopes and well drilling systems. The governing equations are derived by using the Extended Hamilton's Principle with a general 6-dof base motion. The natural frequency equation is then extracted in a closed-form for the case where the beam support undergoes longitudinal... 

In this paper, the influence of the van der Waals force on two main parameters describing an instability point of cantilever type nanomechanical switches, which are the pull-in voltage and deflection are investigated by using a distributed parameter model. The fringing field effect is also taken into account. The nonlinear differential equation of the model is transformed into the integral form by using the Green's function of the cantilever beam. The integral equation is solved analytically by assuming an appropriate shape function for the beam deflection. The detachment length and the minimum initial gap of the cantilever type switches are given, which are the basic design parameters for... 

Micro-electromechanical systems (MEMS) have wide application in the development of sensors for the detection of magnitudes in almost any domain [1]. Resonant mode operation of micro and nano-scale oscillators have gained wide interest for applications including filters, amplifiers, non-linear mixers, atomic scale imaging, biological and chemical sensors. The device that we propose is an electrically actuated microcantilever beam. More precisely, in our design the microcantilever constitutes the movable plate of a micro-capacitor and its displacement is controlled by the voltage applied across the plates. Review of literature shows that the electric field pattern between the beam and the... 

In this paper, the influence of the Casimir force on two main parameters describing an instability point of cantilever type nanomechanical switches, which are the pull-in voltage and deflection are investigated by using a distributed parameter beam model. The nonlinear differential equation of the model is transformed into the integral form by using the Green's function of the cantilever beam. The integral equation is solved analytically by assuming an appropriate shape function for the beam deflection. The detachment length and the minimum initial gap of the cantilever type switches are given, which are the basic design parameters for NEMS switches. The pull-in parameters of micromechanical... 

Based on the non linear terms appearing in the strain tensor in classical continuum mechanics, two expressions for large strain in the Cosserat continuum are proposed. The generalized form of principal of virtual work together with the constitutive equations for an Isotropic elastic Cosserat continuum are used to derive the finite element formulations for elastic large deformation analysis based on the Cosserat theory. The finite element formulations are then applied to a four-node quadrilateral element with three degrees of freedom at each node including two translational and one rotational degrees of freedom. The tension of a semi-infinite plate with a circular whole in the center is... 

In this paper the combined optimal feedback-adaptive feedforward controller proposed to attain better performance of active vibration suppression of flexible structures subjected to different type of disturbances. The structure considered here is a cantilever beam actuated with a PZT patch actuator. The proposed controller consists of two individual parts, a filtered-x controller as a feedforward part and an optimal linear controller as a feedback part. Recursive Least Square algorithm (RLS) is used for the adaptive filtering scheme in Filtered-x adaptive feedforward controller. LQG optimal controller is also used in the feedback part of the controller. This research investigates the... 

A numerical procedure is proposed for obtaining the static deflection, pull-in (PI) deflection and PI voltage of electrostatically excited capacitive microcantilever beams. The method is not time and memory consuming as Finite Element Analysis (FEA). Nonlinear ordinary differential equation of the static deflection of the beam is derived, w/wo considering the fringing field effects. The nondimensional parameters upon which PI voltage is dependent are then found. Thereafter, using the parameters and the numerical method, three closed form equations for pull-in voltage are developed. The results are in good agreement with others in literature. Copyright © 2006 by ASME  

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural... 

This paper discusses the effects of substrate motions on the performance of microgyroscopes modeled as suspended beams with a tip mass. The substrate movements can be motions along as well as rotations around the three axes. Using Extended Hamiltonian Principle and Galerkin approximation, the equations of the motion of the beam are analytically derived. In these equations, the effects of beam distributed mass, tip mass, angular accelerations, centripetal and coriolis accelerations are clearly apparent. The effect of electrostatic forces inducing the excitation vibrations are considered as linear functions of beam displacement. The response of the system to different inputs is studied and the... 

The atomic force microscope (AFM) system has evolved into a useful tool for direct measurements of intermolecular forces with atomic-resolution characterization that can be employed in a broad spectrum of applications. In this paper, the nonlinear dynamical behavior of the AFM is studied. This is achieved by modeling the microcantilever as a single mode approximation (lumped-parameters model) and considering the interaction between the sample and cantilever in the form of van der Waals potential. The resultant nonlinear system is then analyzed using Melnikov method, which predicts the regions in which only periodic and quasi-periodic motions exist, and also predicts the regions that chaotic... 

The electroelastic response of functionally graded piezoelectric cantilever beams which includes the effect of body force is presented in this paper. The material properties such as elastic compliance, piezoelectric and dielectric impermeability are assumed to be graded with different indices in the thickness direction according to exponential distributions. Systems of fourth order inhomogeneous partial differential equations (PDEs) which are satisfied by the stress and induction functions and involve the body force terms are derived. Spectral forms for electrical and mechanical variables in the x-axis are employed to convert the partial differential governing equations and the associated...