Hamed Farokhi

Internal energy transfer in dynamical behaviour of Timoshenko microarches

The internal energy transfer and modal interactions in the motion characteristics of Timoshenko microarches are investigated numerically. The length-scale parameter is introduced to the strain energy of the system and the equations of motion are obtained via Hamiltons principle based on the modified couple stress theory; these equations are discretized into a set of nonlinear ordinary differential equations through use of the Galerkin scheme. The possibility of the occurrence of modal interactions and internal energy transfers is verified by obtaining the ratios of the linear natural frequencies of the system. The nonlinear response of the system is obtained for the cases with the modal interactions and internal energy transfer.

Resonance responses of geometrically imperfect functionally graded extensible microbeams

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori - Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

A higher-order mathematical modeling for dynamic behavior of protein microtubule shell structures including shear deformation and small-scale effects

Microtubules in mammalian cells are cylindrical protein polymers which structurally and dynamically organize functional activities in living cells. They are important for maintaining cell structures, providing platforms for intracellular transport, and forming the spindle during mitosis, as well as other cellular processes. Various in vitro studies have shown that microtubules react to applied mechanical loading and physical environment. To investigate the mechanisms underlying such phenomena, a mathematical model based on the orthotropic higher-order shear deformation shell formulation and Hamiltons principle is presented in this paper for dynamic behavior of microtubules. The numerical results obtained by the proposed shell model are verified by the experimental data from the literature, showing great consistency. The nonlocal elasticity theory is also utilized to describe the nano-scale effects of the microtubule structure. The wave propagation and vibration characteristics of the microtubule are examined in the presence and absence of the cytosol employing proposed formulations. The effects of different system parameters such as length, small scale parameter, and cytosol viscosity on vibrational behavior of a microtubule are elucidated. The definitions of critical length and critical viscosity are introduced and the results obtained using the higher order shell model are compared with those obtained employing a first-order shear deformation theory. This comparison shows that the small scale effects become important for higher values of the wave vector and the proposed model gives more accurate results for both small and large values of wave vectors. Moreover, it is shown that for higher circumferential wave number, the torsional wave velocity obtained by the higher-order shell model tend to be higher than the one predicted by the first-order shell model.

Global dynamics of an axially moving buckled beam

A parametric study for post-buckling analysis of an axially moving beam is conducted considering four different axial speeds in the supercritical regime. At critical speed, the trivial equilibrium configuration of this conservative system becomes unstable and the system diverges to a new non-trivial equilibrium configuration via a pitchfork bifurcation. Post-buckling analysis is conducted considering the system undergoing a transverse harmonic excitation. In order to obtain the equations of motion about the buckled state, first the equation of motion about the trivial equilibrium position is obtained and then transformed to the new coordinate, i.e. post-buckling configuration. The equations are then discretized using the Galerkin scheme, resulting in a set of nonlinear ordinary differential equations. Using direct time integration, the global dynamics of the system is obtained and shown by means of bifurcation diagrams of Poincaré maps. Other plots such as time traces, phase-plane diagrams, and Poincaré sections are also presented to analyze the dynamics more precisely.

Modal interactions and energy transfers in large-amplitude vibrations of functionally graded microcantilevers

Modal interactions and internal energy transfers are investigated in the large-amplitude oscillations of a functionally graded microcantilever with an intermediate spring-support. Based on the Mori–Tanaka homogenization technique and the modified couple stress theory, the energy terms of the functionally graded microsystem (kinetic and size-dependent potential energies) are developed and dynamically balanced. Large-amplitude deformations, due to having one end free, are modeled taking into account curvature-related nonlinearities and assuming an inextensibility condition. The continuous model of the functionally graded microsystem is reduced, by means of the Galerkin method, yielding an inertial- and stiffness-wise nonlinear model. Numerical simulations on this highly nonlinear reduced-order model of the functionally graded microcantilever are performed using a continuation method; a possible case of modal interactions is determined by obtaining the natural frequencies of the microsystem. The nonlinear oscillations of the microcantilever are examined, and it is shown how the energy fed to the functionally graded microsystem (from the base excitation) is transferred between different modes of oscillation.

Viscoelastic resonant responses of shear deformable imperfect microbeams

A viscoelastic model for the nonlinear analysis of the coupled transverse, longitudinal, and rotational oscillations of an imperfect shear deformable microbeam is developed, for the first time, based on the modified couple stress theory. An energy dissipation mechanism is developed via use of the Kelvin–Voigt internal energy dissipation mechanism. For the stress and deviatoric part of the symmetric couple stress tensors, the viscous components along with the corresponding work terms are obtained. The size-dependent elastic energy along with the kinetic energy of the viscoelastic microsystem is formulated in terms of the displacement field together with system geometric and physical parameters. The internal energy dissipation is developed via the work done by the viscous components of the stress and the deviatoric part of the symmetric couple stress tensors by means of the Kelvin–Voigt mechanism. These work and energy terms are inserted into Hamilton’s principle together with the work due to an external force in order to obtain three viscoelastically coupled equations governing the transverse, longitudinal, and rotational motions with cubic and quadratic nonlinear terms. A high-dimensional Galerkin approximation method is applied for all the three equations, yielding three sets of second-order coupled ordinary differential equations with cubic and quadratic nonlinearities. Upon application of a transformation, a continuation technique along with the backward differentiation formula (BDF) is employed in order to obtain the time-variant response of the system subject to a harmonic load. Special attention is paid to the effect of the Kelvin–Voigt type viscoelasticity on the system response in the presence of the length-scale parameter.

Stability and Bifurcations in Three-Dimensional Analysis of Axially Moving Beams

The geometrically nonlinear dynamics of a three-dimensional axially moving beam is investigated numerically for both sub and supercritical regimes. Hamilton’s principle is employed to derive the equations of motion for in-plane and out-of plane displacements. The Galerkin scheme is applied to the nonlinear partial differential equations of motion yielding a set of second-order nonlinear ordinary differential equations with coupled terms. The pseudo-arclength continuation technique is employed to solve the discretized equations numerically so as to obtain the nonlinear resonant responses; direct time integration is conducted to obtain the bifurcation diagrams of the system. The results are presented in the form of the frequency-response curves, bifurcation diagrams, time histories, phase-plane portraits, and fast Fourier transforms for different sets of system parameters.Copyright