Force-response Curves Research Articles

Correlations between Axial and Oblique Loaded Column

In an impact, a structure is rarely subjected to pure axially loading. Load endured by an axially loaded column has been analytically derived by previous researchers, whereas it is very limited works on predicting the obliquely loaded column performance. In this study, the force response curve of the obliquely loaded empty aluminum column is analyzed and compared to its axially loaded column. Theoretical formula to calculate the axial mean force has been applied to obliquely loaded column, and it has yields a largely deviated value. Thus, a modified equation to cater for obliquely loaded column has been proposed.

Oscillations of functionally graded microbeams

The size-dependent oscillations of a third-order shear-deformable functionally graded microbeam are investigated taking into account all the longitudinal and transverse displacements and inertia as well as the rotation and rotary inertia. The modified couple stress theory along with the Mori–Tanaka homogenisation technique is employed to develop formulations for the elastic potential energy as well as the kinetic energy of the system. The energy of the system is balanced by the work of a harmonic excitation force via an energy method based on Hamilton's principle, yielding the size-dependent coupled nonlinear continuous models of the functionally graded system for the longitudinal and transverse displacements as well as the rotational motion. A model reduction procedure, on the basis of a weighted-residual method, is applied without any simplifications on the displacement/inertia/rotation. This operation yields three sets of second-order reduced-order coupled model of the functionally graded system for the longitudinal, transverse, and rotational motions. These reduced-order models are solved via use of a continuation method in order to construct the nonlinear frequency-response and force-response curves of the functionally graded system. A linear analysis is also performed by means of an eigenvalue extraction method in order to determine the linear natural frequencies of the system. It is shown that the material gradient index as well as the length-scale parameter of the functionally graded system affects the system dynamics substantially.

Stability analysis of a composite laminated piezoelectric plate subjected to combined excitations

The aim of this paper is to discuss the stability of a symmetric cross-ply composite laminated piezoelectric plate subject to combined excitations. Multiple timescale perturbation method is implemented to solve the nonlinear governing equations including the second-order approximation. The case of 1:1:3 internal resonance and primary resonance case is investigated. The stability of the system is discussed using frequency, force response curves. A bifurcation analysis was performed using the amplitude of parametric excitation force as the bifurcation parameter. It is found that there are two Hopf bifurcation points: the first one is at \(f_{11}=5.592\), and the other one is at \(f_{11}=13.96\). It is observed that the system is dominated by the periodic attractor in the ranges of \((5.592< f_{11 }< 7.21)\) and \((13.31< f_{11 }< 13.96)\). The system is enriched with period doublings which are the main way leading to chaotic behavior. Some recommendations regarding the parameters limit of the dynamic system are reported.

Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes

The aim of this paper is to numerically investigate the geometrically nonlinear primary resonance of third-order shear deformable functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates with various edge supports subjected to a harmonic excitation transverse force. The nanocomposite plates are composed of a mixture of matrix and single-walled carbon nanotubes (SWCNTs), and the effective mechanical properties are obtained by means of the rule of mixture. Employing the von Kármán hypotheses and Reddy’s third-order shear deformation plate theory, the nonlinear equations of motion for the in-plane and out-of-plane directions as well as the corresponding boundary conditions are derived using Hamilton’s principle. In the numerical solution procedure, the generalized differential quadrature (GDQ) method is used for the discretization, and a numerical Galerkin scheme is then employed to convert the discretized nonlinear partial differential equations (PDEs) into a Duffing-type nonlinear time-varying set of ordinary differential equations (ODEs). Afterwards, a time periodic discretization and the pseudo-arc length continuation method are utilized to determine the frequency- and force-responses of FG-CNTRC plates. Moreover, the curves corresponding to the nonlinear free vibration are provided. The influences of important parameters on the frequency-response and force-response curves of the FG-CNTRC plates are studied in the numerical results.

Nonlinear free and forced vibration analysis of embedded functionally graded sandwich micro beam with moving mass

In this study, nonlinear free and forced vibration analysis of an embedded functionally graded sandwich micro-beam with a moving mass is investigated. The velocity of moving mass is assumed constant. The structure is resting on nonlinear Pasternak foundation. The governing equation of motion is obtained using Hamilton's principle based on the Euler–Bernouli model with considering nonlinear terms in strain–displacement relation. Strain gradient elasticity theory is used to model the small scale effects. The micro-beam contains a homogenous core and two integrated functionally graded face-sheets. Mechanical properties except Poisson ratio are assumed to be variable based on the power-law distribution along the thickness direction. Galerkin's decomposition technique is implemented to convert nonlinear partial differential equation to a nonlinear ordinary differential equation. Multiple times scale method is applied to derive closed form approximate solution for free and forced vibration and nonlinear natural frequencies of the micro-beams. Accuracy of the obtained results using current issue may be justified by comparing with those obtained by existing results of the literature. The effect of some important parameters such as length scale parameter, power gradient index, nonlinear elastic foundation, aspect ratio, position, and velocity of moving mass and boundary conditions is studied on the various responses of the micro-beam such as nonlinear natural frequency, frequency response, and force–response curves.

Non-linear time delay saturation controller for reduction of a non-linear vibrating system via 1:4 internal resonance

In this paper, active control with time delay was implemented for eliminating the vibration of the non-linear system excited by external and parametric forcing amplitude in the presence of 1:4 internal resonances. Multiple time scale method is applied to determine first-order approximations of the controlled system. The stability of the system is investigated analytically using Lyapunov first method and numerically using frequency, force-response curves and phase-plane method for the considered resonance case. Effects of different parameters on the steady state response of the controlled system are investigated. Variation of some parameters leads to the bending of the frequency, force-response curves and hence to the jump phenomenon occurrence.

Characterization for Cassie-Wenzel wetting transition based on the force response in the process of squeezing liquid drops by two parallel superhydrophobic surfaces

Application of superhydrophobic surfaces is partly limited by the escaping of the entrapped air beneath the liquid sitting on the superhydrophobic surfaces, i.e., the so-called Cassie-Wenzel wetting transition. Here, to characterize this wetting transition, a linear force response relation with certain abnormal systematic deflection showing the wetting transition information is constructed for the process of squeezing the test liquid drop by two parallel structured (superhydrophobic) surfaces. The linear force response relation is validated by replotting the experimental data from the former work. And then the wetting transition information is investigated on a numerically generated force response curve with certain errors by taking into account the liquid pressure variation during the wetting transition. Results show that the wetting transition can cause an obvious bulge on the linear force response curve. We believe that this method has a potential application in characterizing the robustness of superhydrophobic surfaces.

Nonlinear Breathing Vibrations and Chaos of a Circular Truss Antenna with 1:2 Internal Resonance

This paper investigates the nonlinear breathing vibrations and chaos of a circular truss antenna under changing thermal environment with 1:2 internal resonance for the first time. A continuum circular cylindrical shell clamped by one beam along its axial direction on one side is proposed to replace the circular truss antenna composed of the repetitive beam-like lattice by the principle of equivalent effect. The effective stiffness coefficients of the equivalent circular cylindrical shell are obtained. Based on the first-order shear deformation shell theory and the Hamilton’s principle, the nonlinear governing equations of motion are derived for the equivalent circular cylindrical shell. The Galerkin approach is utilized to discretize the nonlinear partial governing differential equation of motion to the ordinary differential equation for the equivalent circular cylindrical shell. The case of the 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance is taken into account. The method of multiple scales is used to obtain the four-dimensional averaged equation. The frequency-response curves and force-response curves are obtained when considering the strongly coupled of two modes. The numerical results indicate that there are the hardening type and softening type nonlinearities for the circular truss antenna. Numerical simulation is used to investigate the influences of the thermal excitation on the nonlinear breathing vibrations of the circular truss antenna. It is demonstrated from the numerical results that there exist the bifurcation and chaotic motions of the circular truss antenna.

Modal interactions in primary and subharmonic resonant dynamics of imperfect microplates with geometric nonlinearities

This paper analyses the modal interactions in the nonlinear, size-dependent dynamics of geometrically imperfect microplates. Based on the modified couple stress theory, the equations of motion for the in-plane and out-of-plane motions are obtained employing the von Karman plate theory as well as Kirchhoff’s hypotheses by means of the Lagrange equations. The equations of motions are solved using the pseudo-arclength continuation technique and direct time-integration method. The system parameters are tuned to the values associated with modal interactions, and then nonlinear resonant responses and energy transfer are analysed. Nonlinear motion characteristics are shown in the form of frequency-response and force-response curves, time histories, phase-plane portraits, and fast Fourier transforms.

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 Hamilton's 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.

Coupled size-dependent behavior of shear deformable microplates

The aim of the present study is to investigate the nonlinear motion characteristics of a shear deformable microplate based on the modified couple stress theory. The microplate is modeled via the third-order shear deformation theory retaining in-plane displacements and inertia. Using the Lagrange equations together with an assumed-mode method, five sets of second-order nonlinear ordinary differential equations of motion with coupled terms are obtained. These five sets of equations (two for the in-plane motions, one for the out-of-plane motion, and two for rotations) are transformed into ten sets of first-order nonlinear ordinary differential equations. These resultant equations are then solved by means of a direct time integration technique and the pseudo-arclength continuation method in order to analyze the nonlinear response of the system. Apart from the nonlinear analysis, the linear natural frequencies of the system are obtained using an eigenvalue analysis. Results are shown through frequency–response and force–response curves. Points of interest in the parameter space in the form of time histories, phase-plane portraits, and fast Fourier transforms are also highlighted. Moreover, a comparison is made between the motion characteristics of the system based on the modified couple stress and classical continuum theories.

Nonlinear size-dependent dynamics of an imperfect shear deformable microplate

The nonlinear size-dependent dynamics of a geometrically imperfect shear deformable microplate is investigated based on the modified couple stress theory. An initial imperfection in the out-of-plane direction is taken into account. The third-order shear deformation theory is employed to model the system by retaining in-plane displacements and inertia. The governing equations of motion of the system are obtained employing an energy method through use of the Lagrange equations, which upon employing an assumed-mode technique, yields a set of second-order nonlinear ordinary differential equations with coupled terms. A high-dimensional discretised system is considered and made double-dimensional via a change of variables in order to obtain a set of first-order nonlinear ordinary differential equations. The resulting equations are solved using a direct time-integration technique, resulting in time-dependent generalised coordinates for the in-plane and out-of-plane displacements and two rotations. From these generalised coordinates, phase-plane portraits and fast Fourier transforms (FFTs) are also obtained. Moreover, the frequency–response and force–response curves of the system are obtained using a continuation technique; stability analysis is conducted via the Floquet theory. The effect of the initial imperfection as well as the length-scale parameter on the system response is also examined.

Parametric instability of microbeams in supercritical regime

The supercritical parametric instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear parametric instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincare sections.

Dynamical behaviour of electrically actuated microcantilevers

The current paper aims at investigating the nonlinear dynamical behaviour of an electrically actuated microcantilever. The microcantilever is excited by a combination of AC and DC voltages. The nonlinear equation of motion of the microcantilever is obtained by means of force and moment balances. A high-dimensional Galerkin scheme is then applied to reduce the equation of motion to a discrete model. A numerical technique, based on the pseudo-arclength continuation method, is used to solve the discretized model. The electrostatic deflection of the microcantilever and static pull-in instabilities, due to the DC voltage, are analyzed by plotting the so-called DC voltage-deflection curves. At the simultaneous presence of the DC and AC voltages, the nonlinear dynamical behaviour of the microcantilever is analyzed by plotting frequency-response and force-response curves.

Nonlinear resonant response of imperfect extensible Timoshenko microbeams

This paper investigates the nonlinear size-dependent dynamics of an imperfect Timoshenko microbeam, taking into account extensibility. Based on the modified couple stress theory, the nonlinear equations of motion for the longitudinal, transverse, and rotational motions are derived via Hamilton’s energy method. A high-dimensional finite degree-of-freedom system of ordinary differential equations is obtained by the application of the Galerkin scheme. This set of equations is solved through use of the pseudo-arclength continuation method. A stability analysis is conducted via use of the Floquet theory. The resonant motion characteristics of the microbeam are examined by plotting the frequency-response and force-response curves. The effect of system parameters on the resonant response of the system is highlighted.

Subcritical parametric dynamics of microbeams

The aim of this paper is to analyse the size-dependent nonlinear parametric dynamics of microbeams; the source of parametric excitation is a time-dependent longitudinal excitation load. Taking into account small-size effects, via the modified couple stress theory, the expressions for the potential and kinetic energies of the system are developed. A multi-degree-of-freedom discretised system is obtained by transforming the continuous model into a reduced-order one via the Galerkin scheme. For the system in the subcritical mean axial load regime, the parametric response is obtained via two different numerical techniques; first one is based on a continuation technique and the second one is via a direct time-integration method. A stability analysis is also conducted via the Floquet theory. Results for the nonlinear parametric response are illustrated in the form of parametric frequency–response diagrams, parametric force–response curves, time histories, phase-plane portraits, fast Fourier transforms (FFTs), and Poincaré maps. The effect of taking into account the length-scale parameter on the parametric response of the system is also highlighted.

Nonlinear motion characteristics of microarches under axial loads based on modified couple stress theory

The aim of the present study is to investigate the geometrically nonlinear size-dependent bending as well as resonant behaviour over the bent state of a microarch under an axial load. In particular, an axial load is applied on the system causing the initial curvature to increase by giving rise to a new bent configuration. A distributed harmonic transverse force is then exerted on the microarch and the nonlinear resonant response of the system over the new deflected configuration is investigated. The nonlinear partial differential equation of motion is obtained via Hamilton's principle based on the modified couple stress theory. The equation is discretized into a set of nonlinear ordinary differential equations through use of the Galerkin scheme. The pseudo-arclength continuation technique is then applied to the resultant set of ordinary differential equations. First, for the unforced system in the transverse direction, the axial load is increased and the new deflected configuration of the system is plotted versus the axial compression load; the nonlinear resonant response over the deflected configuration is then investigated through constructing frequency–response and force–response curves.

Bifurcations in Forced Response Curves

We consider small amplitude periodic forcing (with forcing frequency $\omega_F$) of a system of differential equations near a point of Hopf bifurcation (with Hopf frequency $\omega_H$). We follow Zhang and Golubitsky [SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1272--1306] and consider only those small amplitude periodic solutions to the forced system whose frequency is $\omega_F$; that is, the 1:1 phase-locked or entrained solutions. These authors assume sinusoidal forcing of a normal form Hopf system when $\omega_F$ is close to $\omega_H$, and they classify the existence and multiplicity of the entrained solutions. The forced response curve is a bifurcation diagram showing the amplitude of the entrained solutions as the forcing frequency is varied. Zhang and Golubitsky showed that there are six kinds of forced response curves with distinguished bifurcation parameter $\omega = \omega_H - \omega_F$. In this paper we show that there are 41 possible bifurcation diagrams when stability in addition to multiplicity of the periodic solutions in the forced response curves is included.

Post-buckling dynamics of Timoshenko microbeams under axial loads

The nonlinear buckling and post-buckling dynamics of a Timoshenko microbeam under an axial load are examined in this paper. The microbeam is assumed to be subject to an axial force along with a distributed harmonically-varying transverse force. As the axial load is increased, in the absence of the transverse load, the stability of the microbeam is lost by a super-critical pitchfork bifurcation at the critical axial load, leading to a static divergence (buckling). The post-buckling state, due to a sufficiently high axial load, is obtained numerically; the resonant response over the buckled state, due to the harmonic transverse loading, is also examined. More specifically, Hamilton’s principle is employed to derive the equations of motion taking into account the effect of the length-scale parameter via the modified couple stress theory. These partial differential equations are then discretized by means of the Galerkin scheme, yielding a set of ordinary differential equations. The resultant equations are then solved via a continuation technique as well as a direct time-integration method. Results are shown through bifurcation diagrams, frequency-responses, and force-response curves. Points of interest in the parameter space in the form of time histories, phase-plane portraitists, and fast Fourier transforms are also highlighted. Finally, the effect of taking into account the length-scale parameter on the buckling and post-buckling behaviour of the system is highlighted by comparing the results obtained through use of the classical and modified couple stress theories.

In-plane and out-of-plane nonlinear size-dependent dynamics of microplates

The aim of the current study is to examine the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries. Employing the von Karman plate theory as well as Kirchhoff’s hypotheses, the equations of motion for the in-plane and out-of-plane directions are derived by means of the Lagrange equations, based on the modified couple stress theory. The potential energies stored in a Winkler-type elastic foundation and the rotational springs at the edges of the microplate are taken into account. The set of second-order nonlinear ordinary differential equations, obtained via the Lagrange scheme, is recast into a double-dimensional set of first-order nonlinear ordinary differential equations with coupled terms by means of a change of variables. The linear natural frequencies of the system are obtained through use of an eigenvalue analysis upon the linear terms of the equations of motion. The nonlinear response, on the other hand, is obtained by means of the pseudo-arclength continuation method. The dynamical characteristics of the system are examined via plotting the frequency–response and force–response curves. The effect of the stiffness of the rotational and translational springs on the nonlinear size-dependent behaviour is also examined. Finally, the effect of employing the modified couple stress theory, rather than the classical theory, on the response is discussed.