Nonlinear Frequency Response Curves Research Articles

Nonlinear free and forced vibration analysis of FG‐CNTRC annular sector plates

This article deals with the large amplitude free and forced vibration analysis of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) annular sector plates based on a numerical approach. The modified rule of mixture is used to estimate the material properties. The equations of motion are developed based on the first‐order shear deformation theory (FSDT) and the von Kármán geometric nonlinearity. First, the discretized form of energy functional of structure is given with the aid of variational differential quadrature method. Then, a time periodic discretization is performed and the frequency response of the nanocomposite plate is determined using the pseudo‐arc length continuation method. After verifying the correctness of the proposed approach, a comprehensive parametric study is presented to investigate the effects of important factors on the nonlinear vibration characteristics of the FG‐CNTRC annular sector plates. The results imply that the volume fraction and distribution type of nanotubes have considerable effects on the fundamental frequency as well as nonlinear frequency response curves. POLYM. COMPOS., 40:E1364–E1377, 2019. © 2018 Society of Plastics Engineers

Influence of aspect ratio on the linear and non-linear steady state forced vibration response of cylindrical open shells

The paper deals with the linear and non-linear steady state periodic response estimation of isotropic cylindrical open shells. The displacement field of first order shear deformation theory has been employed and the analysis is based on finite element method. CO continuous, eight-noded serendipity quadrilateral shear flexible element with five nodal degrees of freedom has been used for the analysis. The geometric non-linearity is included in the analysis using von Karman’s assumption for small strains and moderately large deflection. The governing equations of motion have been obtained in the time domain and the nonlinear periodic responses are obtained using shooting technique. The entire non-linear steady state frequency response curve consisting of stable and unstable regimes has been obtained employing shooting technique along with arc length and pseudo-arc length continuation schemes. The present method does not involve any apriori assumption on the participating modes. The comparison of linear and nonlinear frequency response curves reveals large differences between the two. Frequency response curve, response history and the phase plane plots have been obtained, for different aspect ratio (L/b), to explore the linear and non-linear forced vibration characteristics of cylindrical open shells. The asymmetric nature of the phase plane plots reveal significant higher harmonic contributions. With the increase in aspect ratio, thin open shells reveal increased peak amplitude in both linear and nonlinear analysis. The presence of secondary peak in the nonlinear frequency response for greater aspect ratio is due to modal interaction between first and higher modes.

Effect of crack location on vibration analysis of partially cracked isotropic and FGM micro-plate with non-uniform thickness: An analytical approach

An analytical model is presented for nonlinear vibration analysis of variable thickness, thin isotropic and functionally graded rectangular micro-plate containing a partial crack located within the centre line of the plate. Linear and parabolic thickness variation is considered in one or both the in-plane directions of the plate. The thickness variation is such that the volume of the plate is equal to that of the uniform thickness plate. The continuous line crack is parallel to one of the edges of the plate. The equations of motion are derived using the equilibrium principle based on Classical Plate Theory while the size effect is incorporated using the modified couple stress theory. The partial crack is represented by bending moment and in-plane force according to the simplified line spring model. The effect of in-plane forces is considered by employing the Berger's formulation. The derived governing equation is converted into a cubic nonlinear Duffing equation by employing Galerkin's method. The effect of nonlinearity is established by deriving the frequency response equation for the cracked variable thickness plate using the method of multiple scales. The nonlinear frequency response curves show the phenomenon of bending hardening or softening. The influence of crack length, crack location, internal material length scale parameter, boundary conditions, gradient index, unidirectional and bidirectional taper constant on the fundamental frequency of square plate is demonstrated. Similar to uniform thickness plate, it is found that the presence of crack affects the vibration characteristics of variable thickness plate. When compared to uniform thickness plate, the effect of crack can be reduced by varying the thickness of the plate. Thus, it is concluded that in a given volume, it is better to employ variable thickness plate as far as the vibration characteristics are considered.

Vibration analysis of partially cracked plate submerged in fluid

The present work proposes an analytical model for vibration analysis of partially cracked rectangular plates coupled with fluid medium. The governing equation of motion for the isotropic plate based on the classical plate theory is modified to accommodate a part through continuous line crack according to simplified line spring model. The influence of surrounding fluid medium is incorporated in the governing equation in the form of inertia effects based on velocity potential function and Bernoulli's equations. Both partially and totally submerged plate configurations are considered. The governing equation also considers the in-plane stretching due to lateral deflection in the form of in-plane forces which introduces geometric non-linearity into the system. The fundamental frequencies are evaluated by expressing the lateral deflection in terms of modal functions. The assessment of the present results is carried out for intact submerged plate as to the best of the author's knowledge the literature lacks in analytical results for submerged cracked plates. New results for fundamental frequencies are presented as affected by crack length, fluid level, fluid density and immersed depth of plate. By employing the method of multiple scales, the frequency response and peak amplitude of the cracked structure is analyzed. The non-linear frequency response curves show the phenomenon of bending hardening or softening and the effect of fluid dynamic pressure on the response of the cracked plate.

A coupled-field model of a rotating composite beam with an integrated nonlinear piezoelectric active element

The dynamics of a system consisting of a rotating rigid hub and a thin-walled composite beam with embedded active element is presented. The beam comprises of a generally orthotropic host made of an arbitrary laminate and an additional layer of a transversely isotropic piezoceramic material. The higher-order constitutive relations for piezoelectric are used to properly model its electromechanical structural behaviour when operated in near resonance conditions or subjected to strong electric fields. In the mathematical formulation of the problem, the full two-way coupling piezoelectric effect is considered by adopting the assumption of a spanwise electric field variation. To enhance the generality of the formulation, the model considers also the hub mass moment of inertia and a non-constant rotating speed case. A general nonlinear system of mutually coupled partial differential equations is derived using the Hamilton principle, and the Galerkin method is applied to reduce these governing equations to the ordinary differential ones. A specific case of CAS lamination scheme that exhibits flapwise bending and twist mode elastic coupling is discussed in detail. Numerical results for system free vibrations are obtained to investigate the natural mode shapes and electrical field spatial distribution depending on the system rotation speed and laminae fibre orientation angle. Next, a forced vibration case is studied by applying to the hub a periodic driving torque with zero mean value. Nonlinear frequency response curves for the combined beam–hub system together with time series plots are prepared for different driving torque scenarios.

Nonlinear Dynamics of Functionally Graded Material Plates Under Dynamic Liquid Load and with Longitudinal Speed

This paper investigates the dynamics of functionally graded material (FGM) plates under dynamic liquid load and with longitudinal speed. The liquid is assumed to be ideal so that it is incompressible, inviscid and irrotational. Based on the D’Alembert’s principle, the mathematical model of the system is developed by taking into account geometrical and material nonlinearities as well as velocity potential and Bernoulli’s equation. The Galerkin method is employed to discretize the partial differential governing equation to a series of ordinary differential ones, which are then analyzed via the use of the method of harmonic balance. Analytical results are compared with numerical ones to validate the present method. The stability of the steady-state response is examined by means of the perturbation technique. Linear analysis of the system shows the possible appearance of internal resonance, and nonlinear frequency-response curves demonstrate strong hardening-spring property of the system. A modal interaction behavior through 1:1 internal resonance is detected; the behavior can happen in a wide domain of constituent volume fraction, which is a unique phenomenon in moving FGM plates compared with their metallic counterparts. Furthermore, results show the modal interaction can be easily evoked in the moving FGM plate under dynamic liquid load, even while the plate is subjected to minimal exciting force or large damping. In addition, influence of the plate location on nonlinear dynamics of the system is examined; results show the dynamic response of the plate will change considerably when the plate is near the container wall.

Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity

Abstract This study investigates nonlinear dynamic thermoelastic response of functionally graded material (FGM) plates with longitudinal velocity for the first time. The large amplitude motion of FGM plates is considered so that the present model includes both geometry and material nonlinearities. Based on the D'Alembert's principle, the out-of-plane equation of motion of the system is obtained by considering the thermal effect and the longitudinal velocity. After that, the Galerkin method is employed to discretize the partial differential equation of motion to a set of ordinary differential equations. The method of harmonic balance is used to solve analytically the time-varying set of ordinary differential equations. The approximately analytical solutions are verified by numerical solutions utilizing an adaptive step-size fourth-order Runge-Kutta technique. Furthermore, the stability of steady-state response is analyzed for the approximately analytical solutions. The linear frequency characteristics and nonlinear frequency-response characteristics are both presented for the system. The nonlinear frequency-response relationships demonstrate strong hardening-type behavior of the system. Results are shown to examine the influences of different parameters including longitudinal velocity, temperature, constituent volume distribution, in-plane pretension, damping and force amplitude on the nonlinear dynamic thermoelastic response of FGM plates with longitudinal velocity.

Numerical methods and parallel algorithms for computation of periodic responses of plates

Numerical methods that compute the nonlinear frequency–response curves of plates are presented. The equation of motion of the plate is derived by assuming first order shear deformation theory and considering geometrical type of nonlinearity. It is discretized by the finite element method using bilinear quadrilateral elements. The frequency–response curve is computed by shooting and continuation methods. Parallel implementation of the shooting method is presented and its scalability is investigated.

Effect of fibre orientation on non-linear vibration of partially cracked thin rectangular orthotropic micro plate: An analytical approach

A nonlinear analytical model for vibration analysis of partially cracked thin orthotropic plate as affected by the fibre orientation and arbitrary position of a partial crack is presented. The approach is based on the classical plate theory and the effect of microstructure is incorporated using the modified couple stress theory. A single partial crack located at an arbitrary position along the line of symmetry is parallel to one of the edges of the plate. The crack considered is in the form of a continuous line and its effect is incorporated in the form of moment and in-plane forces based on a simplified line spring model. Non-linearity in the model is introduced by Berger׳s formulation for the in-plane forces while the Galerkin׳s method with modal functions is used to obtaining the solution of the governing differential equation. A nonlinear frequency response relation and peak amplitude is obtained by the application of the method of multiple scales. Extensive results for the fundamental frequencies and peak amplitude as affected by the fibre orientation, plate aspect ratio, crack length, crack position and internal material length scale parameter are presented for two boundary conditions. It is found that the fibre orientation and the microstructure of the plate affect the fundamental frequencies of the partially cracked orthotropic plates. The linear and nonlinear frequency response curves based on the classical plate theory and the modified couple stress theory are plotted which shows the effect of internal material length scale parameter and crack position on the bending hardening or softening phenomenon.

Experiments on nonlinear vibrations of a combined structure with segments of beams and a disc subjected to axial tensile force

This paper presents experimental results on nonlinear vibrations of a combined structure with segments of beams and a disc, which is a fundamental model of a micro scanner. Both ends of the structure are clamped for deflection and one of the ends of the structure is constrained with an axial spring. Under an axial tensile force, the structure is excited with lateral periodic acceleration. Sweeping the excitation frequency, nonlinear frequency response curves are obtained. In the typical frequency region, non-periodic responses are generated. The responses are inspected by the Fourier spectrum, the Poincare projection, the maximum Lyapunov exponents and the principal component analysis. The non-periodic responses are confirmed as chaotic responses coupled with torsional and the lowest flexural modes of the combined structure.

Confinement of Vibrations in Variable-Geometry Nonlinear Flexible Beam

In this paper, we propose a novel strategy for controlling a flexible nonlinear beam with the confinement of vibrations. We focus principally on design issues related to the passive control of the beam by proper selection of its geometrical and physical parameters. Due to large deflections within the regions where the vibrations are to be confined, we admit a nonlinear model that describes with precision the beam dynamics. In order to design a set of physical and geometrical parameters of the beam, we first formulate an inverse eigenvalue problem. To this end, we linearize the beam model and determine the linearly assumed modes that guarantee vibration confinement in selected spatial zones and satisfy the boundary conditions of the beam to be controlled. The approximation of the physical and geometrical parameters is based on the orthogonality of the assumed linear mode shapes. To validate the strategy, we input the resulting parameters into the nonlinear integral-partial differential equation that describes the beam dynamics. The nonlinear frequency response curves of the beam are approximated using the differential quadrature method and the finite difference method. We confirm that using the linear model, the strategy of vibration confinement remains valid for the nonlinear beam.

Nonlinear vibrations of laminated and sandwich rectangular plates with free edges. Part 2: Experiments & comparisons

Large-amplitude (geometrically nonlinear) forced vibrations of completely free sandwich rectangular plates are investigated experimentally. Harmonic excitation is applied by using an electro-dynamic exciter and the plate vibration is measured by using laser Doppler vibrometers. A scanning laser Doppler vibrometer is used for experimental modal analysis since it provides non-contact vibration measurements with very high spatial resolution. The large-amplitude vibration experiments are carried out by using a single point laser Doppler vibrometer and a stepped-sine testing procedure. The nonlinear frequency response curves are obtained by increasing and decreasing the excitation frequency in very small steps at specific force amplitudes controlled in a closed-loop. The experimental results are compared to numerical simulations obtained by reduced-order models and show very good agreement. The nonlinear damping is experimentally obtained as a function of the vibration amplitude.

Nonlinear responses of a symmetric cross-ply composite laminated cantilever rectangular plate under in-plane and moment excitations

The nonlinear dynamic responses of a composite laminated cantilever rectangular plate under the in-plane and moment excitations are studied. The Reddy’s higher-order shear deformation theory and the von Karman type equations for the geometric nonlinearity are used to establish the governing equations of motion. The nonlinear governing partial differential equations of motion for the composite laminated cantilever rectangular plate are derived by using the Hamilton’s principle, which are transformed into a two-degree-of-freedom nonlinear system by using the Galerkin approach. A new kind of expression of the displacement functions is given. The case of 1:2 internal resonance and primary parametric resonance is taken into account. The influence of the in-plane and moment excitations on the nonlinear vibrations of the composite laminated cantilever rectangular plate is discussed by using numerical simulation. The numerical results demonstrate that there exist the bifurcation and chaotic motions of the composite laminated cantilever rectangular plate. The nonlinear frequency–response curves of this system under different excitations are investigated to show the relationships between the excitations and the amplitudes of the first two modes.

Tunable, Dual-Gate, Silicon-on-Insulator (SOI) Nanoelectromechanical Resonators

Resonant nanoelectromechanical systems (NEMS) have the potential to have significant impact in mass sensing, signal processing, and field detection applications, if the challenges associated with processing, material, and geometric variability can be mitigated. The research presented here details a breakthrough in the design and development of resonant NEMS aimed at addressing these challenges. Specifically, this study details the fabrication, characterization, and tuning of dual-gate silicon nanoelectromechanical resonators, which are transduced electrostatically and realized with close to 100% yield. These devices are fabricated on a silicon-on-insulator (SOI) substrate using only top-down microfabrication techniques and can be easily integrated with SOI-CMOS transistors, enabling the development of fully integrated CMOS-NEMS with highly tunable nonlinear frequency response characteristics.

Free and forced nonlinear oscillations of a two-layer composite beam with interface slip

Free and forced flexural nonlinear vibrations of a two-layer beam are investigated. Each beam is assumed to have Euler–Bernoulli kinematics and free-free boundary conditions. The interface allows only nonlinear elastic slip between adjacent sides of the beams, so that the transversal displacement is unique. Free vibrations are considered first by the multiple time scale method, which allows to determine the amplitude dependent nonlinear natural frequencies of the system. It is shown that the nonlinear coefficient of the backbone curve is positive, so that hardening/softening behavior of the interface generates hardening/softening behavior of the whole structure. The modifications of the linear normal modes for moderate excitation amplitudes have been computed. Forced and damped nonlinear oscillations are then considered by the same mathematical method, and the nonlinear frequency response curves are obtained.

Efficient computation of higher order frequency response functions for nonlinear systems with, and without, a constant term

The behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation. These changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system's dynamics. In this article, a new and efficient algorithm is presented for computing the higher order Volterra frequency response functions from nonlinear time-domain models that may contain a constant term. A comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method. The algorithm is applicable to systems modelled by nonlinear differential, or difference, equations and is easily automated. Several examples are used to illustrate the method, and to highlight the importance of dc terms in nonlinear system analysis.

Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-of-freedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.

The effects of cubic stiffness on fatigue characterization resonator performance

Micromachined, kHz-frequency resonators are now routinely employed as testing structures to characterize the fatigue degradation properties of thin film materials such as polycrystalline silicon (polysilicon). In addition to stress-life (S–N) fatigue curves, important properties such as crack propagation rates may be inferred from proper resonant frequency measurements throughout a fatigue test. Consequently, any nonlinear dynamic behavior that would complicate the interpretation of resonant frequency changes should be avoided. In this paper, nonlinear frequency–response curves of a polysilicon fatigue structure are measured in a vacuum environment. Finite element models of the structure are used to identify the source of geometric nonlinearity leading to a Duffing-type cubic stiffness. Given the origin of the behavior, a parametric optimization strategy is performed to minimize the cubic stiffness. This study highlights the importance of considering the dynamic behavior when designing resonating structures, especially when they are used for mechanistic studies in various environments.

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.

Super-harmonics in a torsional system with dry friction path subject to harmonic excitation under a mean torque

The nonlinear frequency response characteristics of a two-degree-of-freedom torsional system with a significant dry friction controlled path are studied, when excited by sinusoidal torque under a mean load. An analytical solution is first developed for a simplified system subjected to continuous slipping motions. The nature of super-harmonic peaks as generated by the dry friction nonlinearity is efficiently found. The effect of a non-zero mean load is also determined and qualitatively understood. Further, a refined multi-term harmonic balance method (MHBM) is proposed that includes up to 12 terms. It is used to study an automotive drive train system that experiences significant stick–slip motions. Associated computational issues including the selection of initial conditions are addressed. Studies show that the mean load could induce asymmetric stick–slip motions and accordingly it has significant effect on time and frequency domain responses. Reasons for the occurrence of super-harmonic resonant peaks and transitional peaks are investigated. Finally, our MHBM is applied to the conventional single-degree-of-freedom system where the spring path exists in parallel with a dry friction damper (Den Hartog's problem). Our predictions match well with Den Hartog's analytical solution. Den Hartog's system differs, in terms of the dynamic behavior, from our torsional system (with a sole dry friction path).