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Taut cables such as those used in cable-stayed bridges are prone to exert a large amplitude response due to support motion. Support motion generally involve longitudinal and transverse components with respect to the cord. Such motions induce parametric and external excitation of the cable. For a certain excitation frequency, multimodal interaction due to simultaneous parametric and primary resonance can be expected. Previous studies have focused on changes in the boundary curves of the primary region of the parametric resonance affected by forced vibrations due to primary resonance. The hysteresis regions have not been determined; they have only been analyzed using the frequency-amplitude curves of the response. Moreover, the influence of the longitudinal and transverse displacement ratio has not been discussed thus far. This paper presents a complete formulation of the continuum equations of a cable model that is excited by support motion. A nonlinear discretized model that includes quadratic and cubic nonlinearities is obtained using the Galerkin method. Further, the analytical solution is determined using the method of multiple scales (MMS). By obtaining the expressions for the amplitudes and phases, the mathematical conditions are set for the amplitude of the parametric and primary resonance from which analytical expressions for the boundary curves of the interaction resonance region are derived. Local stability analysis is conducted for the steady-state response, and direct numerical integration is used for validating the frequency-amplitude curves obtained using the MMS. The solutions are verified using two independent numerical models. It is shown that the ratio of the transverse and longitudinal components of support motion significantly affects the resonance region, and several different response solutions can be obtained. It is also shown that different values of the mechanical cable parameters affect the resonance region and vibration amplitude.

The purpose of the present investigation is to reveal the primary resonance of a rotating pretwisted blade subject to a flapwise natural frequency gas excitation under thermal gradient in the presence of 2:1 internal resonance. The linear part of the dynamic model can be decoupled via the modal analysis. The method of multiple scales is applied to seek the steady-state responses. The stabilities of the responses are analyzed via the Lyapunov theory. Amplitude-frequency relationships are examined with the focus on the effects of thermal gradient, pre-deformation amplitude, rotating speed, gas pressure and damping coefficients. Plotting the response amplitudes as a function of the gas pressure amplitude exhibits the saturation and the jump phenomena in rotating blade. Colored distributions in resonance parameters plane demonstrate the occurrences of the saturation and the jump phenomena. The regions of stability and instability of the steady-state are presented in the plane of the gas pressure amplitude and the external excitation frequency tune. Detailed analysis is performed on an unstable region of the rotating blade. The analysis indicates the existence of quasi-periodic motion identified by time history, phase plane trajectories and Poincare sections. The numerical results show a good agreement with the available analytical solutions.

This paper is focused on the resonant responses and chaotic dynamics of a composite laminated circular cylindrical shell with radially pre-stretched membranes at both ends and clamped along a generatrix. Based on the two-degree-of-freedom non-autonomous nonlinear equations of this system, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equation. The resonant case considered here is the primary parametric resonance-1/2 subharmonic resonance and 1:1 internal resonance. Corresponding to several selected parameters, the frequency-response curves are obtained. From the numerical results, we find that the hardening-spring-type behaviors and jump phenomena are exhibited. The jump phenomena also occur in the amplitude curves of the temperature parameter excitation. Moreover, it is found that the temperature parameter excitation, the coupling degree of two order modes and the detuning parameters can effect the nonlinear oscillations of this system. The periodic and chaotic motions of the composite laminated circular cylindrical shell clamped along a generatrix are demonstrated by the bifurcation diagrams, the maximum Lyapunov exponents, the phase portraits, the waveforms, the power spectrums and the Poincare map. The temperature parameter excitation shows that the Pomeau-Manneville type intermittent chaos occur under the certain initial conditions. It is also found that there exist the twin phenomena between the Pomeau-Manneville type intermittent chaos and the period-doubling bifurcation.

In this paper, a nonlinear random model is built to describe the dynamics of planetary gear trains (PGTs), in which the time-varying mesh stiffness, tooth profile modification (TPM), tooth contact loss, and random tooth profile error are considered. A stochastic method based on the method of multiple scales (MMS) is extended to analyze the statistical property of the dynamic performance of PGTs. By the proposed multiple-scales based stochastic method, the distributions of the dynamic transmission errors (DTEs) are investigated, and the lower and upper bounds are determined based on the 3σ principle. Monte Carlo method is employed to verify the proposed method. Results indicate that the proposed method can be used to determine the distribution of the DTE of PGTs high efficiently and allow a link between the manufacturing precision and the dynamical response. In addition, the effects of tooth profile modification on the distributions of vibration amplitudes and the probability of tooth contact loss with different manufacturing tooth profile errors are studied. The results show that the manufacturing precision affects the distribution of dynamic transmission errors dramatically and appropriate TPMs are helpful to decrease the nominal value and the deviation of the vibration amplitudes.

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.

In this paper, the dynamic behavior of a rotor system with electromechanically coupled boundary condition under periodic axial load is studied, where the boundary condition is derived from a ring-shaped piezoelectric damper which is developed for vibration control of rotor system. This damper is based on the piezoelectric shunt damping technique, which can produce much or little damping performance depends on the selection of shunt circuit parameters. By using assumed mode method and Lagrange equation, the equations of motion are derived. Actually these equations can also be derived from a general forced parametrically excited gyroscopic system. Thus, to analyze such system, a method of multiple scales is developed firstly, where a general procedure is proposed to establish solvability conditions. Subsequently, this procedure is applied to obtain the analytical instability boundaries and forced vibration responses. The analytical results show that the additional combination instability regions are created due to the introduction of shunt circuit. When the parametrically excited eccentric rotor is rotating, the parametric vibrations are superimposed on the unbalanced responses. As the resistance value of shunt circuit is increasing, both parametric vibrations and unbalance vibrations can be significantly suppressed. For the unbalanced responses, both the resonance and anti-resonance phenomena can be observed; whereas for the parametric vibrations, only the resonance phenomena exist. These phenomena indicate that we may achieve great vibration control performance for the rotor parametric vibrations by introducing the piezoelectric shunt damping, as long as the circuit parameters are well adjusted. To validate the obtained analytical expressions, the numerical methods are applied. Specifically, the discrete state transition matrix method (DSTM) is applied to validate the analytical instability boundaries and the Runge-Kutta method is conducted to verify the analytical frequency response functions.

Nonlinear vibration absorbers have been extensively investigated for passive vibration control, motion isolation, and synchronous energy harvesting. This paper studies the exact nonlinear dynamics of a simply-supported beam carrying a nonlinear spring-inerter-damper energy absorber for primary resonance vibration reduction. The nonlinear governing equations of the system are derived from the energy method by considering the midplane stretching, structural discontinuity, and nonlinear boundary conditions at the spring-inerter-damper location of the beam and directly solved using the method of multiple scales. The nonlinear frequency correction factor, frequency response function, peak nonlinear frequency response, and bifurcation frequency are obtained and investigated for various system parameters. The influence of the location, spring stiffness, inertial mass, and damping of the nonlinear vibration absorber on the beam dynamics, including natural frequency, mode shape, and nonlinear frequency response, are studied. The stiffness and mass of the nonlinear vibration absorber are optimally tuned to minimize the peak nonlinear frequency response of the beam. The results show that ignoring the nonlinear boundary conditions at the spring-inerter-damper location could lead to serious underestimation of the nonlinear frequency responses. The nonlinear stiffness of the vibration absorber enhances the system nonlinearity but has no contribution to the peak nonlinear frequency response of the beam. Increasing the damping of the vibration absorber could effectively mitigate the beam vibration. When the nominal frequency of the absorber is tuned to be close to the natural frequency of the beam, the beam vibration is mostly reduced.

This study investigates the vibration of rotating elastic rings that are dynamically excited by an arbitrary number of space-fixed discrete stiffnesses with periodically fluctuating stiffnesses. The rotating, elastic ring is modeled using thin-ring theory with radial and tangential deformations. Primary and combination instability regions are determined in closed-form using the method of multiple scales. The ratio of peak-to-peak fluctuation to average discrete stiffness is used as the perturbation parameter, so the resulting perturbation analysis is not limited to small mean values of discrete stiffnesses. The natural frequencies and vibration modes are determined by discretizing the governing equations using Galerkin's method. Results are demonstrated for compliant gear applications. The perturbation results are validated by direct numerical integration of the equations of motion and Floquet theory. The bandwidths of the instability regions correlate with the fractional strain energy stored in the discrete stiffnesses. For rings with multiple discrete stiffnesses, the phase differences between them can eliminate large amplitude response under certain conditions.

A novel idea that applies the multiple scale analysis to a discretized decoupled system of gyroscopic continua is introduced and an axial moving string is treated as an example. First, the invariant manifold method is applied to the discretized ordinary differential equations of the axially moving string. Complex gyroscopic mode functions that agree well with true analytical results are obtained. The gyroscopic modes are subsequently used for the discretized ordinary differential equations with gyroscopic and nonlinear coupling terms that yield a gyroscopically decoupled system. Further the method of multiple scales is used to obtain the equations at a slow scale. This novel procedure is compared to solutions obtained by directly applying the classical multiple scale analysis to the gyroscopically coupled system without decoupling. The modal decoupled system analysis yields better frequency with comparing to the classic method. The proposed methodology provides a novel alternative for nonlinear dynamic analysis of gyroscopic continua.

The nonlinear characteristics of an autoparametric vibration system are investigated. This system consists of a base structure and a cantilever beam with a tip mass. The dynamic equations for the system are derived using the extended Hamilton's principle. The method of multiple scales (MMS) is used to determine an approximate analytical solution of the nonlinear governing equations and, hence, analyze the stability and bifurcation of the system. Compared with the numerical simulation, the first-order MMS is not sufficient. A Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems. The effects of the amplitude and frequency of the external force, damping and frequency of the attached cantilever beam, and the tip mass on the nonlinear responses of the autoparametric vibration system are determined. The results show that this system exhibits many interesting nonlinear phenomena including saturation, jumps, hysteresis and different kinds of bifurcations, such as saddle-node, supercritical pitchfork and subcritical pitchfork bifurcations. Power spectra, phase portraits and Poincare maps are employed to analyze the unstable behavior and the associated Hopf bifurcation and chaos. Depending on the application of such a system, its dynamical behaviors could be exploited or avoided.

Despite the shear amount of research studies on nonlinear flexural dynamics of cantilever beams, very few efforts address the practical geometry involving a constant thickness and linearly-varying width. This stems from the nature of the associated linear eigenvalue problem which cannot be easily solved in closed form. In this paper, we present a closed-form solution to this particular linear eigenvalue problem in the form of a general Meijer-G differential equation for which a solution is readily available in the shape of the Meijer-G functions. Using this approach, the exact linear modal frequencies and shapes are obtained and used in the discretization of the nonlinear partial-differential equation describing the dynamics of the system. The discretized system of ordinary-differential equations is then solved using the method of multiple scales to obtain an approximate analytical solution describing the primary resonance behavior of a given vibration mode. An analytical expression for the modal effective nonlinearity is obtained and used to analyze the influence of the beam's tapering on the nonlinear primary resonance behavior of the response (softening/hardening). Results are then compared to a finite element (FE) solution of the linear eigenvalue problem in which the modal shapes obtained using the FE method are fit into a set of orthogonal polynomial functions and used to discretize the nonlinear problem. It is shown that, while the modal frequencies obtained using the FE method approximate those obtained analytically with negligible error (less than 1%), there is a substantial error in the resulting estimates of the modal effective nonlinearity. This indicates that, even negligible errors in the approximate solution of the linear problem, can propagate to become significant when analyzing the nonlinear problem further reinforcing the importance of the exact solution.

Recently, it has been shown that nonlinear vibratory energy harvesters possessing a tri-stable potential function are capable of harvesting energy efficiently over a wider range of frequencies in comparison to harvesters with a double-well potential function. However, the effect of the design parameters of the harvester on the dynamic response and the effective bandwidth of such devices remains uninvestigated. To fill this void, this paper establishes an analytical approach to characterize the effective frequency bandwidth of harvesters that possess a hexic potential energy function. To achieve this goal, the method of multiple scales is utilized to construct analytical solutions describing the amplitude and stability of the intra- and inter-well dynamics of the harvester. Using these solutions, critical bifurcations in the parameter's space are identified and used to define an effective frequency bandwidth of the harvester. The influence of the electric parameters, namely, the time constant ratio (ratio between the period of the mechanical system and the time constant of the harvesting circuit) and the electromechanical coupling, on the effective frequency bandwidth is analyzed. Experimental studies performed on the harvester are presented to validate some of the theoretical findings.

The out-of-plane dynamic stability of an arch under a vertical periodical base excitation is investigated by using both analytical and experimental methods. When the excitation frequency is near twice the out-of-plane natural frequency of the arch, an out-of-plane divergent vibration is induced, which will eventually leads to an out-of-plane dynamic parametric instability of the arch. Firstly, Hamiltonian principle is employed to derive the in-plane dynamic equations, and analytical solutions for the dynamic axial compressive force and bending moment in pre-stability state are obtained and verified by numerical simulations. Then the mode shape functions for the out-of-plane dynamic instability are analytically determined and the out-of-plane dynamic instability regions are obtained by Bolotin's method. Factors affecting the critical excitation frequencies of the out-of-plane dynamic instability regions, such as the arch's included angle, the slenderness ratio, and the logarithmic decrement, are thoroughly analyzed and verified via the method of multiple scales. Lastly, laboratory tests are carried out for the out-of-plane parametric resonance instability of fix-ended circular arches with various included angles and slenderness ratios. The critical excitation frequencies of dynamic instability regions of the tested arches are determined by increasing and decreasing the excitation frequency, and the experimental results agree with the theoretical counterparts very well. The main contributions of this work include: (1) obtaining an analytical solution of out-of-plane dynamic instability regions for the arch under a vertical harmonic base excitation; (2) applying a newly-designed experimental system to conduct the out-of-plane dynamic instability tests; (3) proposing a method of time domain analytical solution to determine the critical excitation frequencies of the out-of-plane dynamic instability region, and (4) discussing the dynamic instability mechanism for base-excited arches.

A rich variety of resonance scenarios induced in thin elastic plates by periodic axial strains is investigated to describe the morphing capability of flexible, ultra-lightweight panels. The strain-induced excitation is provided by embedded piezoelectric strips actuated by time-varying voltages. The study deals with plates characterized by geometric and mechanical symmetry entailing cylindrical motions so that an ad hoc approximate nonlinear model of elastic beams is adopted to formulate the incremental equation of motion about the nonlinear equilibrium under dead loads. An asymptotic approach based on the method of multiple scales is employed to study the plate dynamics and, primarily, to investigate the effect of the plate slenderness on the nonlinearity of the lowest normal modes. A bifurcation analysis is carried out to investigate the wide array of resonances induced by the harmonic axial excitation, acting in nontrivial equilibrium states, which include primary, super-harmonic, sub-harmonic, and principal parametric resonances, respectively. For selected slendernesses of the plate, corresponding to nonlinear softening, hardening and quasi-linear behaviors, respectively, comprehensive parametric studies on the effect of the PZT cross-sectional size, the plate damping, and the piezoelectric coupling coefficient are carried out to identify the threshold voltages triggering the different resonances.

This paper studies the dynamic behaviors of a magnetically coupled oscillator with two-degrees-of-freedom. For the certain parameter combinations, the target component (TC), which is excited by a base movement, enters the saturated steady-state responses. Another component, which is called as the spatially constrained system (SCS), is considered as an energy absorber pumping out of the energy from the TC under 1:3 internal resonance. The governing equations of motion are derived using a magnetic dipole model and then are solved with the method of multiple scales. The energy distributions between the TC and SCS can vary with the base movement amplitude and excitation frequency. The saturation behavior is found and used for constraining the moving amplitude of an excited system. The bifurcation diagrams reveal the complex and interesting responses when the excitation frequency and amplitude change. The results include the comparisons among the analytical predictions, numerical simulations, and experimental tests. For the purpose of suppressing the primary resonance, the present system effectively transfer the energy from the excited component. The jumping and saturation phenomena are found in the experiments and the changes in the spectrum of two steady-state responses are also observed.

This work investigates the modal properties and parametric instabilities of high-speed spur epicyclic and planetary gears with an elastically deformable ring. Coriolis (i.e., gyroscopic) and centripetal acceleration effects are modeled. The gyroscopic effects lead to complex-valued (i.e., traveling-wave) modes. These modes fall into three categories: rotational, translational, and planet modes. Each category has highly structured modal deflections. The modal properties are proved analytically by discretizing the elastic ring such that the system falls into the class of general cyclically symmetric systems with proven modal properties. Such a proof readily extends to helical planetary gears with any or all of the sun, carrier, and ring modeled as elastic bodies. The time-varying sun-planet and ring-planet mesh stiffnesses are the sources of parametric excitation. The conditions where combinations of the mesh frequency and mesh stiffness amplitudes cause parametric instabilities are determined in closed form with the method of multiple scales. The analytical results agree with numerical results from Floquet theory. Many potential parametric instabilities are suppressed. A rule to determine whether a given potential parametric instability is suppressed or not is given. The rule is closely linked to the modal properties noted above and the planet mesh phasing parameters (sun and ring tooth numbers and number of planets).

This paper investigates the parametric resonance voltage response of nonlinear parametrically actuated Micro-Electro-Mechanical Systems (MEMS) cantilever resonators. A soft AC voltage of frequency near natural frequency is applied between the resonator and a parallel ground plate. This produces an electrostatic force that leads the structure into parametric resonance. The model consists of an Euler–Bernoulli thin cantilever under the actuation of electrostatic force to include fringe effect, and damping force. Two methods of investigation are used, namely the Method of Multiple Scales (MMS) and Reduced Order Model (ROM) method. ROM convergence of the voltage response and the limitation of MMS to small to moderate amplitudes with respect to the gap (gap-amplitudes) are reported. MMS predicts accurately both Hopf supercritical and supercritical bifurcation voltages. However, MMS overestimates the large gap-amplitudes of the resonator, and. misses completely or overestimates the saddle-node bifurcation occurring at large gap-amplitudes. ROM produces valid results for small and/or large gap-amplitudes for a sufficient number of terms (vibration modes). As the voltage is swept up at constant frequency, the resonator maintains zero amplitude until reaches the subcritical Hopf bifurcation voltage where it loses stability and jumps up to large gap-amplitudes, next the gap-amplitude decreases until it reaches the supercritical Hopf bifurcation point, and after that the gap-amplitude remains zero, for the voltage range considered in this work. As the voltage is swept down at constant frequency, the zero gap-amplitude of the resonator starts increasing continuously after reaching the supercritical Hopf bifurcation voltage until it reaches the saddle-node bifurcation voltage when a sudden jump to zero gap-amplitude occurs. Effects of frequency, damping and fringe parameters on the voltage response show that (1) the supercritical Hopf bifurcation is shifted to lower voltage values with the increase of any of the mentioned parameters, (2) the subcritical Hopf bifurcation is shifted to larger voltage values with the increase of damping, shifted to lower voltage values with the increase of the fringe parameter, and not significantly altered by the change in frequency, (3) the saddle-node bifurcation voltage decreases with the increase of frequency and damping, and decrease of fringe parameter, and (4) the saddle-node bifurcation gap-amplitude decreases with the increase of frequency and damping, and it is not significantly altered by the change of the fringe parameter.

As one of important characteristics of the nonlinear system containing the quadratic or cubic nonlinearity, the internal resonance between the system's modes may occur and affect the nonlinear properties of the system. In this paper, the internal resonance conditions of the spatial flexible beam suspended by two springs in a spatial on-orbit tethered system are obtained based on the method of multiple scales firstly. And then, the effects of the internal resonances on the attitude stability and the energy transfer tendency of the tethered system are investigated by the structure-preserving approach in detail. In the numerical simulation, the attitude angle always tends to zero in a finite interval when the internal resonance of the spatial flexible beam occurs even if the mass of the platform is close to that of the beam, which implies that, the occurrence of the internal resonance will improve the attitude stability of the tethered system. In addition, it can be found that, when the internal resonance occurs, the elastic potential energy stored in the springs tends to transfer to the beam even if the initial attitude angle is small, which will shorten the vibration control time of the spatial flexible beam if the damping of the beam is considered. The above findings permit us to optimize the system parameters or select the appropriate initial conditions to improve the attitude stability and to accelerate the total energy dissipation of the tethered system if the damping of the spatial flexible beam is considered.

Considerable attention has recently been given to the study of simultaneous energy harvesting and vibration attenuation in metastructures. However, only linear metastructures were investigated although nonlinear metastructures and nonlinear electromechanical devices offer superior interesting wave propagation phenomena (e.g., birth of solitary waves, tunable bandgap, acoustic nonreciprocity) and broadband energy harvesting. In this paper, we investigate the wave propagation in a weakly nonlinear metastructure with electromechanical resonators. Explicit expressions are derived for the nonlinear dispersion relations using the method of multiple scales. These expressions are validated via direct numerical integration. We carried out parametric studies to investigate the role of different parameters of the electromechanical resonators on the linear and nonlinear band structure. To obtain further detailed information on the nonlinear wave propagation, we employ spectro-spatial analysis on the numerical simulations. This spectro-spatial analysis can reveal the output voltage distortion due to different types of nonlinearities. The results indicate that nonlinear chain can enhance energy harvesting through the birth of solitary wave and without degrading the boundary of the bandgap. The results also suggest that such a system is suitable for designing electromechanical diodes and rectifiers.

The thermal dynamics in thermo-mechanical systems exhibits a much slower time scale compared to the structural dynamics. In this work, we use the method of multiple scales to reduce the thermo-mechanical structural models with a slowly-varying temperature distribution in a systematic manner. In the process, we construct a reduction basis that adapts according to the instantaneous temperature distribution of the structure, facilitating an efficient reduction in the number of unknowns. As a proof of concept, we demonstrate the method on a range of linear and nonlinear beam examples and obtain a consistently better accuracy and reduction in the number of unknowns than the standard Galerkin projection using a constant basis.

Forced vibration is investigated for two elastically connected cantilevers, under harmonic base excitation. One of the cantilevers is with a tip magnet repelled by a magnet fixed on the base. The cantilevers are uniform viscoelastic beams constituted by the Kelvin model. The system is formulated as a set of two linear partial differential equations with nonlinear boundary conditions. The method of multiple scales is developed to analyze the effects of internal resonances on the steady-state responses to external excitations in the nonlinear boundary problem of the partial differential equations. In the presence of 2:1 internal resonance, both the first and the second primary resonances are examined in detail. The analytical frequency–amplitude response relationships are derived from the solvability conditions. It is found that the frequency–amplitude response curves reveal typical nonlinear phenomena such as jumping and hysteresis in both primary resonances as well as saturation in the second primary resonance. The frequency–amplitude response curves may be converted from hardening-type single-jumping to double-jumpings, and further to softening-type single-jumping by adjusting the distance between two magnets. It is also found that the unstable parts of the frequency–amplitude response curves correspond to quasi-periodic motions. The finite difference scheme is proposed to discretize both the temporal and the spatial variables, and thus the numerical solutions can be calculated. The analytical results are supported by the numerical solutions.

The objective of this study was to optimize the damping effects of a lumped-mass vibration absorber (LMVA) attached to a hinged–hinged nonlinear beam held on a nonlinear elastic foundation. The LMVA was located at various points along the beam and supported by a spring from beneath. Analysis was performed on the internal resonance conditions with the aim of eliminating internal resonance and reducing the amplitude of vibrations in the beam by altering the location and mass of the LMVA as well as the spring constant. We employed the method of multiple scales (MOMS) for the analysis of frequency response in various modes. Fixed points plots were constructed and 3D maximum amplitude contour plots (3D MACPs) were compiled to identify the LMVA combinations with the optimal damping effects. The best damping results in the 3rd mode were achieved when the LMVA was placed between 1/4l and 1/2l in combinations of M ^ (mass of LMVA/mass of beam) and K ˜ (LMVA spring constant/elastic foundation spring constant) followed a linear relationship of K ˜ =10 M ^ .

In this paper, the transverse superharmonic resonances of a marine propulsion shafting are investigated under the first blade frequency excitation. A coupled longitudinal-transverse dynamic model due to geometrical nonlinearity is established by Hamilton’s principle and then is discretized by Galerkin method. The method of multiple scales is applied to these equations. The steady-state response and the stabilities are analyzed. The effect of the support stiffness, load, mass of propeller, damping ratio and slender ratio on the nonlinear effect is discussed. Research shows smaller values of slender ratio, bigger values of load and smaller values of damping ratio lead to stronger nonlinear effect. The nonlinear effect is reduced by increasing the back stern bearing stiffness and increased by increasing the front stern bearing and thrust bearing stiffness and the propeller mass. While the middle bearing makes small influence to it. It is also shown that these resonance curves are of the hardening type. Results of perturbation method are agreement with numerical simulations.

An analytical model is presented for vibration analysis of a thin orthotropic and general functionally graded rectangular plate containing an internal crack located at the center. The continuous line crack is parallel to one of the edges of the plate. The equation of motion of the orthotropic plate is derived using the equilibrium principle. The crack terms are formulated using Line Spring Model. The effect of location of crack along the thickness of the plate on natural frequencies is analyzed using appropriate crack compliance coefficients in the Line Spring Model. By using the Berger formulation for in-plane forces, the derived equation of motion of the cracked plate is transformed into a cubic nonlinear system. Applying the Galerkin׳s method, the equation is converted into well known Duffing equation. The peak amplitude is obtained by employing Multiple Scales perturbation method. The effect of nonlinearity is also established by deriving frequency response equation for the cracked plate using method of multiple scales. The influence of crack length, boundary conditions and crack location along the thickness, on the natural frequencies of a square and rectangular plate is demonstrated. It is found that the vibration characteristics are affected by the length and location of crack along the thickness of the plate. It is thus deduced that the natural frequencies are minimum when crack is internal and its depth is symmetric about the mid-plane of the plate for all the three boundary conditions considered. Further, it is concluded that the presence of crack across the fibers decreases the frequency more as compared to crack along the fibers. The Effect of varying elasticity ratio on the fundamental frequencies of the cracked plate is also established.

Flutter instability of a cantilever beam subjected to a follower force of constant magnitude is well understood: the beam oscillates with increasing amplitude when the force is larger than the critical load. Post-flutter analysis, similar to previous efforts, shows that the addition of damping results in steady-state oscillations. These oscillations are symmetric, but addition of a slope-dependent term to the magnitude of the follower force results in asymmetry. These asymmetric oscillations are investigated in this paper: the Ritz–Galerkin method is used to obtain a finite degree-of-freedom model of the cantilever and the method of multiple scales is used to analytically predict the amplitude and asymmetry of the oscillations. Numerical simulation results indicate a close match with analytically predicted results.

A method of multiple scales is developed for n-degree-of-freedom weakly nonlinear gyroscopic systems. A general procedure is proposed to establish solvability conditions. The conditions have n different versions whose equivalence cannot be mathematically demonstrated. The procedure is applied to a 4-degree-of-freedom nonlinear gyroscopic system that is the 4-term Galerkin truncation of the governing equation of a pipe conveying fluid flowing in the supercritical speed. The investigation focuses on the primary external resonance in the first frequency ω1 and the two-to-one internal resonance of the first two frequencies ω1 and ω2. The multi-scale analysis shows that the amplitude–frequency response curve in each of the first two modes has a peak bending to the left when ω2>2ω1, two peaks with the same height and the opposite bending directions when ω2=2ω1, and a peak bending to the right when ω2

This paper investigates the friction-induced instability and the resulting self-excited vibration of a propeller–shaft system supported by water-lubricated rubber bearing. The system under consideration is modeled with an analytical approach by involving the nonlinear interaction among torsional vibrations of the continuous shaft, tangential vibrations of the rubber bearing and the nonlinear friction acting on the bearing–shaft contact interface. A degenerative two-degree-of-freedom analytical model is also reasonably developed to characterize system dynamics. The stability and vibrational characteristics are then determined by the complex eigenvalues analysis together with the quantitative analysis based on the method of multiple scales. A parametric study is conducted to clarify the roles of friction parameters and different vibration modes on instabilities; both the graphic and analytical expressions of instability boundaries are obtained. To capture the nature of self-excited vibrations and validate the stability analysis, the nonlinear formulations are numerically solved to calculate the transient dynamics in time and frequency domains. Analytical and numerical results reveal that the nonlinear coupling significantly affects the system responses and the bearing vibration plays a dominant role in the dynamic behavior of the present system.

In this paper, vibration analysis of a rectangular thin isotropic plate with a part-through surface crack of arbitrary orientation and position is performed by using the Kirchhoff plate theory. Simply supported (SSSS), clamped (CCCC) and simply supported–clamped (SCSC) boundary conditions are considered for the analysis. First, the governing differential equation of a cracked plate is formulated. A modified line spring model is then used to formulate the crack terms in the governing equation. Next, by the application of Burger's formulation, the differential equation is transformed into the well-known Duffing equation with cubic and quadratic nonlinearities. The Duffing equation is then solved by the method of multiple scales (MMS) to extract the frequency response curve. Natural frequencies are evaluated for different values of length, angle and position of a part-through surface crack. Some results are compared with the published literature. Amplitude variation with different values of length, angle and position of a part-through surface crack are presented, for all three types of the plate boundary conditions.

In this paper, we study the nonlinear dynamics of a butterfly valve actuated by the induced electromotive force (emf) of a permanent magnet, with a focus on the on–off dynamics of the valve and its nonlinear response under ambient perturbation. The complex interplay between the electromagnetic, hydrodynamic and mechanical forces leads to a fundamentally multiphysical, nonlinear dynamical model for the problem. First, we analyze the stability of the on–off conditions in terms of three critical dynamical parameters – the actuating DC voltage, inlet velocity and the opening angle. Next, the response of the system to perturbations around the equilibrium points is studied in terms of the frequency response using the method of multiple scales. Finally, evidence of fractality is established using Melnikov analysis and a plot of the basins of attraction. The results reported in the article, in addition to being of fundamental theoretical interest, are expected to impact practical design considerations of electromechanical butterfly valves.

Nonlinear forced vibration of a viscoelastic pipe conveying fluid around the curved equilibrium configuration resulting from the supercritical flow speed is investigated with the emphasis on dynamics of external and internal resonance. The governing equation for the pipe system, a nonlinear integro-partial-differential model with variable coefficients, is truncated into a discrete perturbed gyroscopic system via the Galerkin method. A condition for the two-to-one internal resonance is established, and the condition implies that internal resonance is possible in the supercritical regime. The method of multiple scales is developed to present the solvability condition of approximate solutions. The modulation equations of the amplitude and the phase are derived from the condition. The first two primary resonances in the presence of the two-to-one internal resonance are examined. Steady-state solutions and their stabilities are determined. Some typical steady-state responses are demonstrated via the amplitude–frequency curves. In addition to jumping, hysteresis and saturation, other dynamical behaviors are observed to highlight the effects of the modal interaction in the internal resonance. The analytical results are confirmed by the numerical integrations.

Stability and bifurcation analyses were performed in this study on the turning process with a state-dependent and large time delay using the method of multiple scales (MMS). The turning system tool was modeled as an oscillator with two degrees of freedom, and both the cubic nonlinear stiffness and the nonlinear cutting force were considered. The nonlinear cutting force was appropriately expanded in a Taylor series considering the state-dependency of the time delay. The time delay and parameters were scaled through the proper ordering process to reflect the large delay effect on an asymptotic formulation of the MMS. Asymptotic solutions were then obtained by the MMS in the large delay regime and used to calculate the linear stability boundaries (i.e., Hopf bifurcation points) and coexisting one-period periodic solutions (i.e., limit cycles) of the turning system. To investigate the local and global behaviors of the tool chatter, bifurcation diagrams were obtained at various workpiece rotating speeds. The validity of the results was examined by comparison with those obtained through the method of harmonic balance and direct numerical integration. Additionally, using the bifurcation diagrams, the effects of the state-dependent time delay and nonlinear stiffness on the chatter vibration behaviors were examined.

We investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitation increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation.

This paper investigates the multi-pulse global bifurcations and chaotic dynamics for the nonlinear, non-planar oscillations of the parametrically excited viscoelastic moving belt using an extended Melnikov method in the resonant case. Using the Kelvin-type viscoelastic constitutive law and Hamilton's principle, the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation. Applying the method of multiple scales and Galerkin's approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:1 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics. The paper demonstrates how to employ the extended Melnikov method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear non-planar oscillations of the viscoelastic moving belt, the Shilnikov-type multi-pulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that the chaos for the Smale horseshoe sense in motion exists.

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.

s Nonlinear parametric vibration is investigated for axially accelerating viscoelastic beams subject to parametric excitations resulting from longitudinally varying tensions and axial speed fluctuations. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite axial support rigidity is also modeled. The governing equations of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle and the viscoelastic constitutive relation. The equation is simplified into a governing equation of transverse nonlinear vibration in small but finite stretching problems. The governing equation of transverse vibration is a nonlinear integro-partial-differential equation with time-dependent and space-dependent coefficients. The method of multiple scales is employed to analyze the combination and the principal parametric resonances with the focus on steady-state responses. In the difference resonance, there is only trivial zero response which is always stable. In the summation and the principal resonances, the trivial responses may become unstable and bifurcate into nontrivial responses for certain excitation frequencies. Some numerical examples indicate that the longitudinal tension variation makes the instability frequency intervals of trivial responses small and the nontrivial response amplitudes large (small) in the summation (principal) resonance. It is also found that the nontrivial responses are not sensitive to the axial support rigidity. Numerical solutions are calculated via the differential quadrature to support results via the method of multiple scales.

This paper investigates the flexibility afforded by the application of regular perturbation methods (in particular, the method of multiple scales) for the purpose of obtaining higher order approximations of the oscillatory response of a nonlinear dynamical system. It is shown that the non-uniqueness of these higher order approximations can be removed by enforcing additional conditions while the relationship between the frequency of oscillation and measurable quantities (the Hamiltonian, the time-averaged kinetic or stored energy) is unique and is thus not affected by these additional conditions.

In this paper, nonlinear free transverse vibrations of in-plane moving plates subjected to plane stresses are investigated. The Hamilton principle is applied to derive the governing equation and the associated boundary conditions. The method of multiple scales is employed to analyze the nonlinear partial differential equation. The solvability conditions are established in the cases without internal resonance and with 3:1 or 1:1 internal resonances. Some numerical examples are presented to demonstrate the effects of in-plane moving speeds on the frequencies. The nonlinear frequencies of the in-plane moving plate without internal resonances are numerically calculated. The relationship between the nonlinear frequencies and the initial amplitudes is showed at different in-plane moving speeds and the nonlinear coefficients, respectively. It is feasible to investigate resonances without the modes not involved in the resonances. The effects of the related parameters are demonstrated for the case of 3:1 and 1:1 internal resonances, respectively. The differential quadrature scheme is developed to solve numerically the governing equation and confirm results via the method of multiple scales.

A bifurcation analysis is performed on a nonlinear railway vehicle having dual-bogies to examine the coupling effect of the bogies on the vehicle's hunting behavior. Because of the coupled nature of these bogies, a pair of complex conjugate roots exists in the linearized system close to the origin in addition to the most dominant pair of roots at the hunting speed. Using these four principal modes and a scaling parameter introduced in a novel way, the original systems of equations are converted into new equations whose linear portions exhibit monofrequency oscillations. The solutions near the hunting speed are constructed with an asymptotic expansion of a small perturbation parameter using the method of multiple scales. Steady state solutions are sought near the hunting speed and the corresponding limit cycle behavior is investigated. The stability of these limit cycles is characterized using Lyapunov's indirect method for the steady state solutions, which is also a novel approach. To support the stability results validity, a series of numerical simulations are performed. Bifurcation diagrams for the lateral motion of the vehicle system are then obtained, and the effects of nonlinearity on the vehicle's hunting behavior are thoroughly examined.

A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler–Bernoulli beam, and (2) axially moving viscoelastic beam.

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass–spring–damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.

A detailed theoretical investigation into the first- and second-mode response of a parametrically excited slender cantilever beam, where, the narrow-band random excitation characters are taken into consideration, is presented. The method of multiple scales is used to determine a uniform first-order expansion of the solution of the nonlinear integro-differential equations. The first-order moment frequency- and force–response data (curves) of a specimen beam tested by other investigators are obtained. Further comparisons have been made and results show that whether the first-order moment frequency–response data (curves) or the first-order moment force–response data (curves) of the first two modes are all in agreement with other investigators’ experimental results. Furthermore, the stochastic jump and bifurcation have been investigated for the first modal parametric principal resonance by using the stationary joint probability of amplitude and phase. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency–response domain, if the bandwidth γ is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the higher branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the lower one when the bandwidth becomes higher; if the excitation central frequency f is a variable and others keep constant, the basic phenomena imply that the higher is f, the more probable is the jump from the higher branch to the lower one once f exceeds an certain value. For the force–response domain, there is a region of excitation acceleration a within which the joint probability density has two peaks: an upper peak and a lower peak. Results show that the upper peak decreases while the lower peak increases as the value of a decreases.

Nonlinear vibrations of a general model of continuous system is considered. The model consists of arbitrary linear and cubic operators. The equation of motion is solved by the method of multiple scales (a perturbation method). The primary resonances of external excitation is analysed. The amplitude and phase modulation equations are presented. Approximate analytical solution is derived. Steady-state solutions and their stability are discussed. Finally, the solution algorithm is applied to two different engineering problems. One of the application is the transverse vibration of an axially moving Euler–Bernoulli beam and the other is a viscoelastic beam.

The primary and parametric resonances of a directly and parametrically excited nonlinear cantilever beam of varying orientation with time-delay in the linear state feedback are investigated. The time-delay is presented in the proportional feedback and the derivative feedback, respectively. The method of multiple scales is used to obtain the first-order approximation of response. The effect of the feedback gains and time-delay on the steady state responses of two type resonances is investigated. It is found that a proper selection of the feedback gains and time-delay can enhance the control performance.

Dynamic behavior of smooth hysteretic systems subjected to harmonic parametric excitation is investigated. Wen's differential equation model for hysteresis, which can describe a large class of hysteretic systems, is used. Employing the piecewise power series expression for hysteresis proposed in the previous paper, the method of multiple scales is applied for the case of principal parametric resonance to obtain the second-order approximate solutions and their stability. It is shown that the hysteretic systems are unstable inside the principal resonance region but the responses are bounded having symmetric periodic solutions. It is also shown that the systems are stable outside the region even if viscous damping does not exist because of Hopf bifurcation. Numerical integrations are also performed to illustrate the nonlinear resonant characteristics of the systems and confirmed that trivial stationary solutions and stable symmetric periodic solutions in principal resonance region bifurcate to produce a pair of non-symmetric periodic solutions. Stability regions with regard to excitation parameters are also illustrated. The theoretical and numerical results are compared to examine the validity of the present analysis.

In the present study, the nonlinear response of a shallow suspended cable with multiple internal resonances to the primary resonance excitation is investigated. The method of multiple scales is applied directly to the nonlinear equations of motion and associated boundary conditions to obtain the modulation equations and approximate solutions of the cable. Frequency–response curves and force–response curves are used to study the equilibrium solution and its stability. The effects of the excitation amplitude on the frequency–response curves of the cable are also analyzed. Moreover, the chaotic dynamics of the shallow suspended cable is investigated by means of numerical simulations.

Based on Maxwell equations and corresponding electromagnetic constitutive relations, the electrodynamic equations and electromagnetic force expressions of a current-conducting thin plate in electromagnetic field are deduced. Nonlinear magneto-elastic vibration equations of the thin plate are given. In addition, nonlinear subharmonic resonances of the thin plate with two opposite sides simply supported which is under the mechanic live loads and in constant transverse magnetic field are studied. The corresponding vibration differential equation of Duffing type is deduced by the Galerkin method. The method of multiple scales is used to solve the equation, and the frequency-response equation of the system in steady motion under subharmonic responses is obtained, and the stability of solution is analyzed. According to the Liapunov stability theory, the critical conditions of stability are obtained. By the numerical calculation, the curves of resonance amplitude changing with the detuning parameter, the excitation amplitude and the magnetic intensity and corresponding state planes are obtained. The existing regions of nontrivial solutions and the changing law of stable and unstable solutions are analyzed. The time history response plots, the phase charts and the Poincare mapping charts are plotted. And the effect of the magnetic intensity on the system is discussed, and some complex dynamic performances as period-doubling motion and quasi-period motion are analyzed.

Anstract The parameter stability and global bifurcations of a strong nonlinear system with parametric excitation and external excitations are investigated in detail. Using the method of Multiple scales, the nonlinear system is transformed to the averaged equation. The parameter stability of solution in the case of principal parametric resonance is developed. Based on the averaged equation, the continuation algorithm is utilized to analyze the detailed bifurcation scenario as the parameter f0 is varied. The results indicate that there exist two limit points and neutral saddle points. Finally, a series of branching points were obtained by changing the parameters f0 and ρ.

The trivial equilibrium of a controlled van der Pol–Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship. With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions.

In this paper, transversal nonlinear vibration of an axially moving viscoelastic string supported by a partial viscoelastic guide is analytically investigated. The string is traveling under time-variant velocity, which includes a mean velocity along with small harmonic fluctuations. The model of the viscoelastic guide is also a parallel combination of springs and viscous dampers. The governing partial-differential equation is derived from Hamilton's principle and geometrical relations. The method of multiple scales is applied to the governing partial-differential equation to obtain solvability conditions for both non-resonance and principal parametric resonance cases. Additionally, in the case of principal parametric resonance, the stability and bifurcation of trivial and non-trivial steady-state responses are analyzed through the Routh–Hurwitz criterion. Eventually, numerical simulations are presented to highlight the effects of mean velocity, guide length, stiffness and damping coefficient of the guide and viscosity coefficient of the string on the natural frequencies, stability, frequency-response curves and bifurcation points of the system.

This paper studies the nonlinear vibration of a three-phase AC motor–linkage mechanism system with links fabricated from three-dimensional braided composite materials. Taking the drive motor and the linkage mechanism as an integrated system, the dynamic equations of the system are established by the finite element method. The relation between the nonlinear vibration of the system and the parameters of the system is obtained by the method of multiple scales. Results show that not only the structural parameters, but also the electromagnetic parameters and the material parameters have significant effects on the nonlinear vibration of the system. Finally, a numerical example is presented.