Showing papers on "Multiple-scale analysis published in 2019"

Nonlinear dynamic characteristics of functionally graded sandwich thin nanoshells conveying fluid incorporating surface stress influence

Abstract: In the present work, nonlinear dynamics of fluid-conveying functionally graded material (FGM) sandwich nanoshells is investigated. In order to describe the large-amplitude motion, the von Karman nonlinear geometrical relations are taken into account. Compressibility and viscidity of the fluid are neglected, and the velocity potential and Bernoulli's equation are used to describe the fluid pressure acting on the nanoshells. Based on the classical shell theory and incorporating the surface stress effect, the governing equations are derived by using Hamilton's principle. After that, the Galerkin method is used to discretize the equation of motion, resulting in a set of ordinary differential equations with respect to time. The ordinary differential equations are solved analytically by utilizing the method of multiple scales. Results show that the surface stress plays important roles on the nonlinear vibration characteristics of fluid-conveying FGM sandwich thin-walled nanoshells. Furthermore, the fluid speed, the power-law index, the fluid mass density, the core thickness and the initial surface tension can also influence the vibration characteristics of fluid-conveying FGM sandwich nanoshells.

Nonlinear analysis of a bio-inspired vertically asymmetric isolation system under different structural constraints

Abstract: Inspired by the limb configuration of animals in their jumping and landing motions, a systematic investigation on the properties of a class of bio-inspired vertically asymmetric X-shaped (vaX) structures is carried out to explore the advantage of nonlinear characteristics in practical engineering. The nonlinear properties of two different vaX structures are studied under different constraint conditions. Formulations of the nonlinear vibration frequency and absolute displacement transmissibility of the structures are derived by the method of multiple scales. Considering practical conditions, three different constraints (i.e., (a) the same isolations height and assembling angle; (b) the same total rod length and assembling angle; (c) the same total rod length and isolation height) are summarized in this manuscript. Under these conditions, nonlinear properties including nonlinear vibration frequency, isolation performance and static stiffness are systematically discussed. Furthermore, the influences of the assembling pattern (i.e., normal and reverse assembling) on the isolation performance are investigated in detail. The results reveal that there exists rod-length ratio $$s_{1}$$ such that the nonlinear frequency ratio of the vaX-I vibration system is lowest; the natural frequency of the vaX-I structure is independent of the assembling pattern; however, compared with the normally assembled vaX-I structure, a lower resonant peak of the transmissibility can be obtained for the reverse-assembled structure, which suggests that the nonlinear damping of the vaX-I structure is affected by the assembling pattern. Experiments are carried out to verify the influence of the assembling pattern on the natural frequency and isolation performance of the vaX structures.

Vibration energy harvesting under concurrent base and flow excitations with internal resonance

Abstract: In this study, internal resonance is investigated to further explore the potential of energy harvesting under concurrent base and flow excitations. The effects of system parameters on the performance of energy harvester with three-to-one internal resonance are analyzed analytically. At first, a lumped-parameter model for the energy harvester, which consists of a two-degree-of-freedom airfoil with the piezoelectric coupling introduced to the plunging motion, is established by using a nonlinear quasi-steady aerodynamic model. Subsequently, the method of multiple scales is implemented to derive the approximate analytic solution of the energy harvesting system under three-to-one internal resonance. Then, the bifurcation characteristics of the energy harvester with respect to various system parameters are analyzed. Finally, the numerical solutions are presented to validate the accuracy of the approximate analytic solutions. The study shows that the harvested voltage and power of the energy harvester can be significantly improved in the presence of internal resonance. In addition, the analytic solutions of internal resonance and the bifurcation analysis can provide an essential reference for design of such a kind of energy harvester.

Nonlinear dynamics of parametrically excited piezoelectric energy harvester with 1:3 internal resonance

Abstract: In this work, the nonlinear dynamic analysis of a parametrically base excited cantilever beam based piezoelectric energy harvester is carried. The system consists of a cantilever beam with piezoelectric patches in bimorph configuration and attached mass at an arbitrary position. The attached mass is placed in such a way that the system exhibits 3:1 internal resonance. The governing spatio-temporal equation of motion is discretized to its temporal form by using generalized Galerkin’s method. To obtain the steady state voltage response and stability of the system, Method of multiple scales is used to reduce the resulting equation of motion into a set of first-order differential equations. The response and stability of the system under principal parametric resonance conditions has been studied. The parametric instability regions are shown for variation in different system parameters such as excitation amplitude and frequency, damping and load resistance. Bifurcations such as turning point, pitch-fork and Hopf are observed in the multi-branched non-trivial response. By tuning the attached mass an attempt has been made to harvest the electrical energy for a wider range of frequency. Such kind of smart self-sufficient systems may find application in powering low power wireless sensor nodes or micro electromechanical systems.

Vibration control for the nonlinear resonant response of a piezoelectric elastic beam via time-delayed feedback

Abstract: The paper presents time-delayed feedback control to reduce the nonlinear resonant vibration of a piezoelectric elastic beam.a#13; Specially, we examine three single-input linear time-delayed feedback control methodologies: displacement, velocity anda#13; acceleration time-delayed feedback. Moreover, the multi-input time-delayed feedback control methodologies are discussed. Utilizinga#13; the method of multiple scales, the modulation equation and the first order approximations of the primary resonances are derived and the effect of time delay on the resonances is analyzed. Then the effect of time delays and control gains on the stability, amplitude,a#13; frequency-response behavior, peak amplitude, critical excitation amplitude are investigated. Optimal values of the controllers gainsa#13; and delay are obtained, simulated, and compared. The time-delayed feedback control acts as a vibration absorber at specific values of time delay. On the other hand, using using mixed delay feedback controllers demonstrates an excellent improvement in mitigating the first-mode vibration.

Analytical approximation solutions of a dielectric elastomer balloon using the multiple scales method

Abstract: Dielectric elastomers (DEs) are soft electromechanical devices, which operate under a high voltage. The majority of methods for calculating the nonlinear vibration of DEs are the numerical ones. However, the analytical methods may also be capable to achieve the reliable general and specific solutions for DEs. This paper investigates the vibration of a dielectric elastomer balloon (DEB) using the method of multiple scales (MMS). The equations of motion are derived by the method of Euler-Lagrange. Using the Taylor expansion , the governing equation of motion is transformed into a general form, then the MMS is applied to solve the problem. Two cases of voltage are considered; in the first one, the balloon is under a static voltage while in the second one the balloon is under a sinusoidal voltage . When the voltage is static, the time-history responses and the phase diagrams are depicted using the MMS and the Runge-Kutta numerical integration to verify the accuracy of the proposed method. For the sinusoidal voltage, the effect of jump phenomenon and variations of pressure and electrical potential difference (Voltage) on the frequency-response curves are studied. The results show that the MMS is in a good agreement with the Runge-Kutta numerical method. Moreover, with the presentation of various values of the pressure and the electrical potential difference, the softening behavior and the jump phenomenon are observed in the frequency response curves .

Performance analysis of parametrically and directly excited nonlinear piezoelectric energy harvester

Abstract: The performance of bimorph cantilever energy harvester subjected to horizontal and vertical excitations is investigated. The energy harvester is simulated as an inextensible piezoelectric beam with the Euler–Bernoulli assumptions. A horizontal base excitation along the axis of the beam is converted into the parametric excitation. The governing equations include geometric, inertia and electromechanical coupling nonlinearities. Using the Galerkin method, the electromechanical coupling Mathieu–Duffing equation is developed. Analytical solutions of the frequency response curves are presented by using the method of multiple scales. Some analytical results are obtained, which reveal the influence of different parameters such as the damping, load resistance and excitation amplitude on the output power of the energy harvester. In the case of parametric excitation, the effect of mechanical damping and load resistance on the initiation excitation threshold is studied. In the case of combination of parametric and direct excitations, the dynamic characteristics and performance of the nonlinear piezoelectric energy harvesters are studied. Our studies revealed that the bending deformation generated by direct excitation pushes the system out of axial deformation and overcomes the limitation of initial threshold of parametric excitation system. The combination of parametric and direct excitations, which compensates and complements each other, can be served as a better solution which enhances performance of energy harvesters.

Free vibration analysis of a spinning piezoelectric beam with geometric nonlinearities

Abstract: The linear and non-linear free vibrations of a spinning piezoelectric beam are studied by considering geometric nonlinearities and electromechanical coupling effect. The non-linear differential equations of the spinning piezoelectric beam governing two transverse vibrations are derived by using transformation of two Euler angles and the extended Hamilton principle, wherein an additional piezoelectric coupling term and different linear terms are present in contrast to the traditional shaft model. Linear frequencies are obtained by solving the standard eigenvalues of the linearized system directly, and the non-linear frequencies and non-linear complex modes are achieved by using the method of multiple scales. For free vibrations analysis of a spinning piezoelectric beam, the non-linear modal motions are investigated as forward and backward precession with different spinning speeds. The responses to the initial conditions for this gyroscopic system are studied and a beat phenomenon is found, which are then validated by numerical simulation. The influences of some parameters such as electrical resistance, rotary inertia and spinning speeds to the non-linear dynamics of a spinning piezoelectric beam are investigated.

Super-harmonic resonances of a rotating pre-deformed blade subjected to gas pressure

Abstract: The present work investigates the super-harmonic resonances combined with 2:1 internal resonance of a rotating blade subjected to strong gas pressure. The dynamic model includes the effect of the initial curved axis of the blade induced by the thermal gradient. The dimensionless gas excitation amplitude is assumed to be the same magnitude of the dimensionless vibration displacement. The vibration of the blade in the plane of rotation and the plane perpendicular to it are described by a set of coupled ordinary differential equations with quadratic and cubic nonlinearities. The steady-state response of the rotating blade is calculated via the method of multiple scales. The stabilities of the steady-state responses are determined via Lyapunov theory. Parametric studies are performed to clarify the influences of system parameters on dynamic response and unstable regions. The various typical phenomena including jump, hysteresis and saturation are observed in the dynamic model. The stable and unstable regions of the solution are analyzed in the plane of external detuning parameter and excitation amplitude. The evaluation of the blade dynamic response is revealed in the unstable region. The theoretical results obtained via the method of multiple scales coincide with the numerical solutions.

Nonlinear performance analysis of forced carbon nanotube-based bio-mass sensors

Abstract: In this effort, an analytical solution is proposed for the large amplitude nonlinear vibrations of doubly clamped carbon nanotube (CNT)-based nano-scale bio-mass sensors. The single walled CNT is modeled as an elastic Euler–Bernoulli nano-scale beam and the size effects are introduced into the mathematical model of the system through Eringen’s nonlocal elastic field theory. The nonlinearity arises due to mid-plane stretching of the bridged CNT, and is accounted for as the von Karman nonlinearity. The impacts of deposited nano-scale bio-object, its geometrical properties, and its landing position along the longitudinal axis of the CNT-based resonator are considered. The nonlinear equations of motion are derived based on Hamilton’s principle and then the Method of Multiple Scales is employed to derive an analytical approximate solution for the system’s response. To verify the analytical solution and show its limits of applicability, the equations of motion are discretized by multi-mode Galerkin’s method and then the obtained set of equations are numerically solved by Runge–Kutta method and compared with those obtained by analytical solution. The potential applications of the CNT-based resonators for both of nonlinear frequency-/amplitude-based mass sensing are investigated and discussed. The obtained results show that the amplitude-based mass sensing has higher performance than frequency-based one in high quality-factor environments, such as in vitro biological mass sensing in the air or vacuum and inversely the frequency-based mass sensing method has higher mass sensibility in low quality factor environments, such as in vivo biological mass sensing in the liquid solution samples.

Dynamic analysis of pneumatic artificial muscle (PAM) actuator for rehabilitation with principal parametric resonance condition

Abstract: In this present work, the dynamic behavior of a pneumatic artificial muscle (PAM) has been studied by varying different muscle parameters and air pressure into it. The governing equation of motion has been derived using Newton’s law of motion to study the various responses in the system at principal parametric resonance condition. The temporal equation of motion contains various nonlinear parameters with forced and nonlinear parametric excitation. Then, the second-order method of multiple scales is used to find the approximate solutions and to study the dynamic stability and bifurcations of the system. The results are found to be in good agreement with the solutions obtained by solving the temporal equation of motion numerically. The instability regions by varying different system parameters have been plotted. The time responses and phase portraits have been plotted to study the system behavior with the nonlinearity. The influences of the different system parameters in the amplitude for the muscle have also been studied with the help frequency responses. In order to verify the solution, the basin of attraction has also been plotted. The obtained results will be very useful for designing the desired PAM use in different rehabilitation robotics and exoskeleton system.

On Parametrically Excited Vibration and Stability of Beams with Varying Rotating Speed

Abstract: In this paper, principal parametric resonance of rotating Euler–Bernoulli beams with varying rotating speed is investigated. The model contains the geometric nonlinearity due to the von Karman strain–displacement relationship and the centrifugal forces due to the rotation. The rotating speed of the beam is considered as a mean value which is perturbed by a small harmonic variation. In this case, when the frequency of the periodically perturbed value is twice the one of the axial mode frequencies, the principal parametric resonance occurs. The direct method of multiple scales is implemented to study on the dynamic instability produced by the principal parametric resonance phenomenon. A closed-form relation which determines the stability region boundary under the condition of the principal parametric resonance is derived. Numerical simulation based on the fourth-order Runge–Kutta method is established to validate the results obtained by the method of multiple scales. The numerical analysis is applied on the discretized equations of motion obtained by the Galerkin approach. After validation of the results, a comprehensive study is adjusted for demonstrating the damping coefficient and the mode number influences on the critical parametric excitation amplitude and the parametric stability region boundary. A discussion is also provided to illustrate the advantages and disadvantages of the fourth-order Runge–Kutta method in comparison with the method of multiple scales.

Magnetoelastic axisymmetric multi-modal resonance and Hopf bifurcation of a rotating circular plate under aerodynamic load

Abstract: In this article, an investigation of magnetoelastic axisymmetric multi-mode interaction and Hopf bifurcations of a circular plate rotating in air and uniform transverse magnetic fields is presented. The expressions of electromagnetic forces and an empirical aerodynamic model are applied in the derivation of the dynamical equations, through which a set of nonlinear differential equations for axisymmetric forced oscillation of the clamped circular plate are deduced. The method of multiple scales combined with the polar coordinate transformation is employed to solve the differential equations and achieve the phase–amplitude modulation equations for the interaction among the first three modes under primary resonance. Then, the frequency response equation for the single-mode vibration, the steady-state response equations for three-mode resonance and the corresponding Jacobian matrix are obtained by means of the modulation equations. Numerical examples are presented to show the dependence of amplitude solutions as a function of different parameters in the cases of single mode and three-mode response. Furthermore, a Hopf bifurcation can be found in three-mode equilibrium by choosing appropriate parameters, where a limit cycle occurs and then evolves into chaos after undergoing a series of period-doubling bifurcations.

Vibration Analysis of Geometrically Nonlinear and Fractional Viscoelastic Cantilever Beams.

Abstract: We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize the extended Hamilton principle to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model. By spectral decomposition in space, the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs. We use direct numerical integration in the decoupled system, in which we observe the anomalous power-law decay rate of amplitude in the linearized model. We further develop a semi-analytical scheme to solve the nonlinear equations, using method of multiple scales as a perturbation technique. We replace the expensive numerical time integration with a cubic algebraic equation to solve for frequency response of the system. We observe the super sensitivity of response amplitude to the fractional model parameters at free vibration, and bifurcation in steady-state amplitude at primary resonance.

On a non-ideal magnetic levitation system: nonlinear dynamical behavior and energy harvesting analyses

Abstract: Nowadays, a novelty of devices that use magnetic restoring forces to generate oscillations has increased substantially. These kinds of devices have been commonly used to energy harvesting area. Therefore, in this paper, numerical and analytical analyses of a non-ideal magnetic levitation system are carried out. The mathematical modeling of the magnetic levitation device is developed and examined considering an electrodynamical shaker to base-excite the main system, which is a non-ideal excitation. The magnetic levitation system has the form of a Duffing oscillator; thus, the nonlinear analysis is required to investigate the energy harvesting potential of this nonlinear system. The novelty here is the use of the shaker to the excitation which is non-ideal. The method of multiple scales is applied to investigate the modes of vibration of the coupled system, which will remark the non-ideality and nonlinear phenomena of the system. The average harvested power is described by through expressions related to the coupling between the mechanical and electrical domains. Moreover, it was developed an expression for the excitation frequency where the maximum harvested power is obtained. The results were obtained based on the numerical method of Runge–Kutta of fourth order with fixed step whose results are shown through phase planes, Poincare maps and parametrical variation. Such results showed multiple existence of behaviors (periodic, quasiperiodic and chaos), coexistence of attractors in a high sensibility of the initial conditions and interesting results of the maximum average power, obtaining high and continuous amount of energy in periodic and chaotic regions.

Vibration and deflection analysis of thin cracked and submerged orthotropic plate under thermal environment using strain gradient theory

Abstract: Based on a non-classical plate theory, a nonlinear analytical model is proposed to analyze transverse vibration of thin partially cracked and submerged orthotropic plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory in conjunction with the strain gradient theory of elasticity. The effect of centrally located surface crack is deduced using appropriate crack compliance coefficients based on the simplified line spring model, whereas the effect of thermal environment is introduced using moments and in-plane forces. The influence of fluidic medium is incorporated in the governing equation in the form of fluid forces associated with its inertial effects. The equation has been solved by transforming the lateral deflection in terms of modal functions. The shift in primary resonance due to crack, length scale parameter and temperature has also been derived with central deflection. To demonstrate the accuracy of the present model, a few comparison studies are carried out with the published literature. The variation in fundamental frequency of the cracked plate is studied considering various parameters such as crack length, plate thickness, level of submergence, temperature and length scale parameter. It has been concluded that the frequency is affected by crack length, temperature and level of submergence. A comparison has also been made for the results obtained from the classical plate theory and Strain gradient theory. Furthermore, the variation in frequency response and peak amplitude of the cracked plate is studied using method of multiple scales to show the phenomenon of bending hardening or softening as affected by level of submergence, temperature, crack length and length scale parameter .

Parameter sensitivity of cantilever beam with tip mass to parametric excitation

Abstract: The sensitivity of the response of a parametrically excited cantilever beam with a tip mass to small variations in elasticity (stiffness) and the tip mass is performed. The governing equation of the first mode is derived, and method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications.

Complexity of resonances exhibited by a nonlinear micromechanical gyroscope: an analytical study

Abstract: Dynamics behavior of the micromechanical gyroscope designed for measuring one component of the angular velocity is studied in the paper. The Cardan suspension is applied to connect the sensing plate with the substrate whose angular velocity is measured. The gimbal and the plate with sensors are connected via torsional joints. Vibrating motion of the sensing plate is excited mainly by a torque resulting from the Coriolis effect. The mathematical model equations have been derived using the Lagrange equation of the second kind. Both nonlinear effects of the geometrical nature and the nonlinear characteristics of the torsional joints are taken into account. The governing equations are solved with help of the method of multiple scales in time domain that belongs to the broad class of asymptotic methods. The approximate solution of analytical form has been obtained for non-resonant vibration as well as for the case of the main and internal resonances that occur simultaneously. Analytical form of solution allows for extensive analysis of the behavior of the system. The desirable state of the gyroscope work is steady-state vibration in resonance that is discussed in detail.