Mergen H. Ghayesh

Vibrations and stability of axially traveling laminated beams

Abstract In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh–Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system.

On the Natural Frequencies, Complex Mode Functions, and Critical Speeds of Axially Traveling Laminated Beams: Parametric Study

The dynamic response of an axially traveling laminated composite beam is investigated analytically, with special consideration to natural frequencies, complex mode functions and critical speeds of the system. The equation of motion for a symmetrically laminated system, which is in the form of a continuous gyroscopic system, is considered; the equation of motion is not discretized — no spatial mode function is assumed. This leads to analytical expressions for the complex mode functions and critical speeds. A parametric study has been conducted in order to highlight the effects of system parameters on the above-mentioned vibration characteristics of the system.

Impedance Control of an Intrinsically Compliant Parallel Ankle Rehabilitation Robot

Robot-aided physical therapy should encourage subjects voluntary participation to achieve rapid motor function recovery. In order to enhance subjects cooperation during training sessions, the robot should allow deviation in the prescribed path depending on the subjects modified limb motions subsequent to the disability. In the present work, an interactive training paradigm based on the impedance control was developed for a lightweight intrinsically compliant parallel ankle rehabilitation robot. The parallel ankle robot is powered by pneumatic muscle actuators (PMAs). The proposed training paradigm allows the patients to modify the robot imposed motions according to their own level of disability. The parallel robot was operated in four training modes namely position control, zero-impedance control, nonzero-impedance control with high compliance, and nonzero-impedance control with low compliance to evaluate the performance of proposed control scheme. The impedance control scheme was evaluated on 10 neurologically intact subjects. The experimental results show that an increase in robotic compliance encouraged subjects to participate more actively in the training process. This work advances the current state of the art in the compliant actuation of parallel ankle rehabilitation robots in the context of interactive training.

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.

Dynamics of a Fluid-Conveying Cantilevered Pipe With Intermediate Spring Support

The aim of this study is to investigate the three-dimensional (3-D) nonlinear dynamics of a fluid-conveying cantilevered pipe, additionally supported by an array of four springs attached at a point along its length. In the theoretical analysis, the 3-D equations are discretized via Galerkin’s technique, yielding a set of coupled nonlinear differential equations. These equations are solved numerically using a finite difference technique along with the Newton-Raphson method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincare maps for two different spring locations and inter-spring configurations. Interesting dynamical phenomena, such as planar or circular flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were conducted for the cases studied theoretically, and good qualitative and quantitative agreement was observed.Copyright

Nonlinear size-dependent dynamics of microarches with modal interactions

This paper investigates the nonlinear dynamics of microarches with internal modal interactions; the nonlinear size-dependent motion characteristics are analyzed for the system with two-to-one and three-to-one internal resonances. The partial differential equation of motion is discretized into a set of second-order nonlinear ordinary differential equations via the application of the Galerkin scheme. The linear natural frequencies of the system are obtained by eliminating the nonlinearities; these are used to verify the occurrence of modal interactions. The nonlinear resonant dynamics are examined via the pseudo-arclength continuation technique for the systems with internal modal interactions.

Coupled nonlinear dynamics of geometrically imperfect shear deformable extensible microbeams

This paper aims at analyzing the coupled nonlinear dynamical behavior of geometrically imperfect shear deformable extensible microbeams based on the third-order shear deformation and modified couple stress theories. Using Hamiltons principle and taking into account extensibility, the three nonlinear coupled continuous expressions are obtained for an initially slightly curved (i.e., a geometrically imperfect) microbeam, describing the longitudinal, transverse, and rotational motions. A high-dimensional Galerkin scheme is employed, together with an assumed-mode technique, in order to truncate the continuous system with an infinite number of degrees of freedom into a discretized model with sufficient degrees of freedom. This high-dimensional discretized model is solved by means of the pseudo-arclength continuation technique for the system at the primary resonance, and also by direct time-integration to characterize the dynamic response at a fixed forcing amplitude and frequency; stability analysis is conducted via the Floquet theory. Apart from analyzing the nonlinear resonant response, the linear natural frequencies are obtained via an eigenvalue analysis. Results are shown through frequency-response curves, force-response curves, time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The effect of taking into account the length-scale parameter on the coupled nonlinear dynamic response of the system is also highlighted.

Three-Dimensional Nonlinear Global Dynamics of Axially Moving Viscoelastic Beams

The three-dimensional nonlinear global dynamics of an axially moving viscoelastic beam is investigated numerically, retaining longitudinal, transverse, and lateral displacements and inertia. The nonlinear continuous model governing the motion of the system is obtained by means of Hamiltons principle. The Galerkin scheme along with suitable eigenfunctions is employed for model reduction. Direct time-integration is conducted upon the reduced-order model yielding the time-varying generalized coordinates. From the time histories of the generalized coordinates, the bifurcation diagrams of Poincare sections are constructed by varying either the forcing amplitude or the axial speed as the bifurcation parameter. The results for the three-dimensional viscoelastic model are compared to those of a three-dimensional elastic model in order to better understand the effect of the internal energy dissipation mechanism on the dynamical behavior of the system. The results are also presented by means of time histories, phase-plane diagrams, and fast Fourier transforms (FFT).

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.