Showing papers on "Multiple-scale analysis published in 2012"

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

Abstract: A global nonlinear distributed-parameter model for a piezoelectric energy harvester under para- metric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geomet- ric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler- Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the per- formance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester's behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a sub- critical pitchfork bifurcation.

Nonlinear vibrations of blade with varying rotating speed

Abstract: This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton’s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin’s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

Abstract: A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.

Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches

Abstract: Introduction.- Perturbation method (Lindstedt-Poincare).- The method of harmonic balance.- The method of Krylov and Bogolyubov.- The method of multiple scales.- The optimal homotopy asymptotic method.- The optimal homotopy perturbation method.- The optimal variational iteration method.- Optimal parametric iteration method.

Forced Vibrations of Supercritically Transporting Viscoelastic Beams

Abstract: This study focuses on the steady-state periodic response of supercritically transportingviscoelastic beams. In the supercritical speed range, forced vibrations are investigatedfor traveling beams via the multiscale analysis with a numerical confirmation. The forcedvibration is excited by the spatially uniform and temporally harmonic vibration of thesupporting foundation. A nonlinear integro-partial-differential equation is used to deter-mine steady responses. The straight equilibrium configuration bifurcates in multiple equi-librium positions at supercritical translating speeds. The equation is cast in the standardform of continuous gyroscopic systems via introducing a coordinate transform for nontri-vial equilibrium configuration. The natural frequencies and modes of the supercriticallytraveling beams are analyzed via the Galerkin method for the linear standard form withspace-dependent coefficients under the simply supported boundary conditions. Based onthe natural frequencies and modes, the method of multiple scales is applied to the govern-ing equation to determine steady-state responses. To confirm results via the method ofmultiple scales, a finite difference scheme is developed to calculate steady-state responsenumerically. Quantitative comparisons demonstrate that the approximate analyticalresults have rather high precision. Numerical results are also presented to show the con-tributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity tothe response amplitude for the first and the second mode. [DOI: 10.1115/1.4006184]Keywords: supercritical, vibration, nonlinearity, transporting beam, Galerkin trunca-tion, multiple scales method

Internal resonance of pipes conveying fluid in the supercritical regime

Abstract: This paper treats nonlinear vibration of pipes conveying fluid in the supercritical regime. If the flow speed is larger than the critical value, the straight equilibrium configuration becomes unstable and bifurcates into two possible curved equilibrium configurations. The paper focuses on the nonlinear vibration around each bifurcated equilibrium. The disturbance equation is derived from the governing equation, a nonlinear integro-partial-differential equation, via a coordinate transform. The Galerkin method is applied to truncate the disturbance equation into a two-degree-of-freedom gyroscopic systems with weak nonlinear perturbations. The internal resonance may occur under the certain condition of the supercritical flow speed for the suitable ratio of mass per unit length of pipe and that of fluid. The method of multiple scales is applied to obtain the relationship between the amplitudes in the two resonant modes. The time histories predicted by the analytical method are compared with the numerical ones and the comparisons validate the analytical results when the nonlinear terms are small.

Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades

Abstract: A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation with cubic nonlinearity to model its dynamic behavior. This equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. Order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and they are similar to those of the Duffing equation.

Dynamic response of slacked single-walled carbon nanotube resonators

Abstract: This paper presents an investigation of the dynamics of electrically actuated single-walled carbon nanotube (CNT) resonators including the effect of their initial curvature due to fabrication (slack). A nonlinear shallow arch model is utilized. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response due to DC and small AC loads for various slacked CNTs of higher and lower aspect ratio. Results of the perturbation method are verified with those obtained by numerically integrating the equations of a multi-mode reduced-order model based on the Galerkin procedure. The effective nonlinearity of the CNT is calculated as a function of the slack level and the DC load. To handle computational problems associated with CNTs of small radiuses, results based on a nonlinear cable model are also demonstrated. The results have indicated that the quadratic nonlinearity due to slack has dominant effect on the dynamic behavior of the CNT.

Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance

Abstract: The nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to the in-plane and transversal excitations are investigated in this paper. Based on the Reddy’s third-order plate theory and the von Karman type equations for the geometric nonlinearity, the nonlinear partial differential governing equations of motion for the composite laminated cantilever rectangular plate are established by using the Hamilton’s principle. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two degree-of-freedom nonlinear system under combined parametric and forcing excitations. The case of foundational parametric resonance and 1:1 internal resonance is taken into account. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. The numerical method is used to find the periodic and chaotic motions of the composite laminated cantilever rectangular plate. It is found that the chaotic responses are sensitive to the changing of the forcing excitations and the damping coefficient. The influence of the forcing excitation and the damping coefficient on the bifurcations and chaotic behaviors of the composite laminated cantilever rectangular plate is investigated numerically. The frequency-response curves of the first-order and the second-order modes show that there exists the soft-spring type characteristic for the first-order and the second-order modes.

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

Abstract: In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales. [DOI: 10.1115/1.4004672]

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 2: flapping nonlinear normal modes

Abstract: The geometrically exact equations of motion about the prestressed state discussed in part 1 (i.e., the nonlinear equilibrium under centrifugal forces) are expanded in the Taylor series of the incremental displacements and rotations to obtain the third-order perturbed form. The expanded form is amenable to a perturbation treatment to unfold the nonlinear features of free undamped flapping dynamics. The method of multiple scales is thus applied directly to the partial-differential equations of motion to construct the backbone curves of the flapping modes and their nonlinear approximations when they are away from internal resonances with other modes. The effective nonlinearity coefficients of the lowest three flapping modes of elastic isotropic blades are investigated when the angular speed is changed from low- to high-speed regimes. The novelty of the current findings is in the fact that the nonlinearity of the flapping modes is shown to depend critically on the angular speed since it can switch from hardening to softening and vice versa at certain speeds. The asymptotic results are compared with previous literature results. Moreover, 2:1 internal resonances between flapping and axial modes are exhibited as singularities of the effective nonlinearity coefficients. These nonlinear interactions can entail fundamental changes in the blade local and global dynamics.

Primary resonance in forced vibrations of in-plane translating viscoelastic plates with 3:1 internal resonance

Abstract: Nonlinear forced vibrations of in-plane translating viscoelastic plates subjected to plane stresses are analytically and numerically investigated on the steady-state responses in external and internal resonances. A nonlinear partial-differential equation with the associated boundary conditions governing the transverse motion is derived from the generalized Hamilton principle and the Kelvin relation. The method of multiple scales is directly applied to establish the solvability conditions in the primary resonance and the 3:1 internal resonance. The steady-state responses are predicted in two patterns: single-mode and two-mode solutions. The Routh–Hurvitz criterion is used to determine the stabilities of the steady-state responses. The effects of the in-plane translating speed, the viscosity coefficient, and the excitation amplitude on the steady-state responses are examined. The differential quadrature scheme is developed to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the single-mode solutions of the steady-state responses.

Nonlinear, parametrically excited dynamics of two-stage spur gear trains with mesh stiffness fluctuation

Abstract: This work studies the nonlinear, parametrically excited dynamics of two spur gear pairs in a two-stage counter-shaft configuration. The dynamic model includes parametric excitations and contact loss of both tooth meshes with two different mesh frequencies. The time-varying mesh stiffnesses and nonlinear tooth separation functions are reformulated into forms suitable for perturbation analysis. The periodic steady-state solutions are obtained by the method of multiple scales and compared against a semi-analytical harmonic balance method as well as numerical integration for fundamental and subharmonic resonances for ranges of system parameters. The interaction of the two meshes is found to depend strongly on the relation of the two mesh periods. The dynamic influences of design parameters, such as shaft stiffness, mesh stiffness variations, contact ratios, and mesh phasing, are discussed. The closed-form solutions provide design guidelines in terms of the system parameters.

Parametric and internal resonances of in-plane accelerating viscoelastic plates

Abstract: This paper analytically and numerically investigates the nonlinear vibration in parametric and internal resonances of in-plane accelerating viscoelastic plates subjected to plane stresses. An approximate nonlinear plate theory was developed under the Kirchoff assumptions. The in-plane translating speed is characterized as a simple harmonic variation about the constant mean axial speed. The governing equation with the associated boundary conditions is derived from the generalized Hamilton principle and the Kelvin constitutive relation. The method of multiple scales is applied to establish the solvability conditions in principal parametric and internal resonances. The steady-state responses are predicted in three possible patterns: trivial, single-mode, and two-mode solutions. The stabilities of the steady-state responses are determined based on the Routh-Hurwitz criterion. The effects of the mean in-plane translating speed, the in-plane translating speed fluctuation amplitude, the viscosity coefficient, and the nonlinear coefficient on the steady-state responses are examined. The differential quadrature schemes are developed for the two-dimensional full plate model and the one-dimensional reduced plate model to solve the nonlinear governing equations numerically. The numerical calculations confirm the approximate analytical results regarding the trivial and single-mode solutions of the steady-state responses.

Analytical solution for nonlinear free vibrations of viscoelastic microcantilevers covered with a piezoelectric layer

Abstract: Nonlinear vibrations of viscoelastic microcantilevers with a piezoelectric actuator layer on the top surface are investigated. In this work, the microcantilever follows a classical linear viscoelastic model, i.e., Kelvin–Voigt. In addition, it is assumed that the microcantilever complies with Euler–Bernoulli beam theory. The Hamilton principle is used to obtain the equations of motion for the microcantilever oscillations. Then, the Galerkin approximation is utilized for separation of time and displacement variables, thus the time function is obtained as a second order nonlinear ordinary differential equation with quadratic and cubic nonlinear terms. Nonlinearities appear in stiffness, inertia and damping terms. Using the method of multiple scales, the analytical relations for nonlinear natural frequency and amplitude of the vibration are derived. Using the obtained analytical relations, the effects of geometric factors and material properties on the free nonlinear behavior of this beam are investigated. The results are also verified by numerical analysis of the equations.

Application of fractional calculus in the dynamics of beams

Abstract: This paper deals with a viscoelastic beam obeying a fractional differentiation constitutive law. The governing equation is derived from the viscoelastic material model. The equation of motion is solved by using the method of multiple scales. Additionally, principal parametric resonances are investigated in detail. The stability boundaries are also analytically determined from the solvability condition. It is concluded that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations.

Vibration control of a transversely excited cantilever beam with tip mass

Abstract: The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler–Bernoulli beam theory A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam An active control of the transverse vibration based on cubic velocity is studied Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response Then the stability and bifurcation of the system is investigated Frequency–response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3 The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively

Stability analysis of a rotor-AMB system with time varying stiffness

Abstract: A rotor-active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic nonlinear under tuned, and external excitation is studied. The method of multiple scales is applied to analyze the response of two modes of a rotor-AMB system near the simultaneous combined and sub-harmonic resonance. The stability of the steady-state solution for that resonance is determined and studied applying Rung–Kutta fourth order method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening and softening nonlinear and chaos in the second mode of the system. The effects of the different parameters on the steady-state solutions are investigated and discussed.

A Theoretical and Experimental Investigation on Free Vibration Vehavior of a Cantilever Beam with a Breathing Crack

Abstract: In this paper the free nonlinear vibration behavior of a cracked cantilever beam is investigated both theoretically and experimentally. For simplicity, the dynamic behavior of a cracked beam vibrating at its first mode is simulated using a simple single degree of freedom lumped parameter system. The time varying stiffness is modeled using a harmonic function. The governing equation of motion is solved by a perturbation method - the method of Multiple Scales. Results show that the correction term that is added to the main part of the response reflects the effect of breathing crack on the vibration response. Moreover, this part of response consists of the superharmonic components of the spectrum which is due to the system's intrinsic nonlinearity. Using this method an analytical relation is established between the system characteristics and the crack parameters in one hand and the nonlinear characteristics of system response on the other hand. Results have been validated by the experimental tests and a numerical method.

Nonlinear dynamic analysis of a Cartesian manipulator carrying an end effector placed at an intermediate position

Abstract: Nonlinear dynamic analysis of a Cartesian manipulator carrying an end effector which is placed at different intermediate positions on the span is theoretically investigated with a single mode approach. The governing equation of motion of this system is formulated by using the D’Alembert principle in addition to profuse application of Dirac delta function to indicate the location of the intermediate end effector. Then the governing equation is further reduced to a second-order temporal differential equation of motion by using Galerkin’s method. The method of multiple scales as one of the perturbation techniques is being used to determine the approximate solutions and the stability and bifurcations of the obtained approximate solutions are studied. Numerical results are demonstrated to study the effect of intermediate positions of the end effector placed at various locations on the link with other relevant system parameters for both the primary and secondary resonance conditions. It is worthy of note that the catastrophic failure of the system may take place due to the presence of jump phenomenon. The results are found to be in good agreement with the results determined by the method of multiple scales after solving the temporal equation of motion numerically. In order to determine physically realized solution by the system, basins of attraction are also plotted. The obtained results are very useful in the application of robotic manipulators where the end effector is placed at any arbitrary position on the robot arm.

Dynamics of an AMB-rotor with time varying stiffness and mixed excitations

Abstract: A rotor- active magnetic bearing (AMB) system with a periodically time-varying stiffness subjected to multi- external, -parametric and -tuned excitations is studied and solved. The method of multiple scales is applied to analyze the response of the two modes of the system near the simultaneous sub-harmonic, super-harmonic and combined resonance case. The stability of the steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. Also, the system exhibits many typical nonlinear behaviors including multi-valued solutions, jump phenomenon, softening nonlinearities. The effects of the different parameters on the steady state solutions are investigated and discussed. Simulation results are achieved using MATLAB 7.0 program.

Chaotic vibration and resonance phenomena in a parametrically excited string-beam coupled system

Abstract: The nonlinear behavior of a string-beam coupled system subjected to parametric excitation is investigated in this paper. Using the method of multiple scales, a set of first order nonlinear differential equations are obtained. A numerical simulation is carried out to verify analytic predictions and to study the steady-state response, stable solutions and chaotic motions. The numerical results show that the system behavior includes multiple solutions, and jump phenomenon in the resonant frequency response curves. It is also shown that chaotic motions occur and the system parameters have different effects on the nonlinear response of the string-beam coupled system. Results are compared to previously published work.