Showing papers on "Multiple-scale analysis published in 2023"

Resonance and bifurcation of fractional quintic Mathieu-Duffing system.

Abstract: In this paper, the main subharmonic resonance of the Mathieu-Duffing system with a quintic oscillator under simple harmonic excitation, the route to chaos, and the bifurcation of the system under the influence of different parameters is studied. The amplitude-frequency and phase-frequency response equations of the main resonance of the system are determined by the harmonic balance method. The amplitude-frequency and phase-frequency response equations of the steady solution to the system under the combined action of parametric excitation and forced excitation are obtained by using the average method, and the stability conditions of the steady solution are obtained based on Lyapunov's first method. The necessary conditions for heteroclinic orbit cross section intersection and chaos of the system are given by the Melnikov method. Based on the separation of fast and slow variables, the bifurcation phenomena of the system under different conditions are obtained. The amplitude-frequency characteristics of the total response of the system under different excitation frequencies are investigated by analytical and numerical methods, respectively, which shows that the two methods achieve consistency in the trend. The influence of fractional order and fractional derivative term coefficient on the amplitude-frequency response of the main resonance of the system is analyzed. The effects of nonlinear stiffness coefficient, parametric excitation term coefficient, and fractional order on the amplitude-frequency response of subharmonic resonance are discussed. Through analysis, it is found that the existence of parametric excitation will cause the subharmonic resonance of the Mathieu-Duffing oscillator to jump. Finally, the subcritical and supercritical fork bifurcations of the system caused by different parameter changes are studied. Through analysis, it is known that the parametric excitation coefficient causes subcritical fork bifurcations and fractional order causes supercritical fork bifurcations.

A Non-Linear Non-Planar Coupling Mechanism of Suspended Cables in Thermal Conditions

Abstract: Slight variations induced by thermal effects may bring unexpected discrepancies in both the system’s linear and non-linear responses. The present study investigates the temperature effects on the non-linear coupled motions of suspended cables subject to one-to-one internal resonances between the in-plane and out-of-plane modes. The classical non-linear flexible system is excited by a uniform distributed harmonic excitation with the primary resonance. Introducing a two-mode expansion and applying the multiple scale method, the polar and Cartesian forms of modulation equations are obtained. Several parametric investigations have highlighted the qualitative and quantitative discrepancies induced by temperature through the curves of force/frequency-response amplitude, time history diagrams, phase portraits, frequency spectrum, and Poincaré sections. Based on the bifurcation and stability analyses, temperature effects on the multiple steady-state solutions, as well as static and dynamic bifurcations, it is observed that the periodic motions may be bifurcated into the chaotic motions in thermal environments. The saddle-node, pitch-fork, and Hopf bifurcations are sensitive to temperature changes. Finally, our perturbation solutions are confirmed by directly integrating the governing differential equations, which yield excellent agreement with our results and validate our approach.

Parametric resonance and bifurcation analysis of thin-walled asymmetric gyroscopic composite shafts: An asymptotic study

Abstract: Nonlinear geometries and cross-sectional asymmetry can be among the most critical contributing factors affecting the performance and stability of composite shafts. The influence of these factors, which should be devoted to particular attention in the design of these systems and have not been investigated yet, are evaluated analytically in this study. The shaft is simply supported, made of orthotropic multi-layers, and spinning at a constant speed. To express the nonlinear system’s behavior, which is due to the large amplitude of the vibrations, it is assumed that the shaft is under the stretching assumption. Moreover, a rectangular cross-section is used to model the asymmetry that results in parametric excitation in the system. To accurately investigate the behavior of composite material, an optimal lay-up is employed. The gyroscopic coupling is included because of Rayleigh beam theory, and Euler’s angles are employed to achieve the angular velocities. The analytical study of the parametrically excited system, obtained by the method of multiple scales, is performed in two categories of resonant and nonresonant cases. In the nonresonant case, the analytical investigation suggests that the asymmetric shaft behaves like a symmetric one therefore, the parametric excitations do not have a significant impact. This claim is confirmed by numerical results. Also, the presence of the gyroscopic coupling and hollowness of the shaft causes the beating phenomenon in the system. However, in the resonant case, the presence of parametric excitation plays a pivotal role. The results also show that under certain conditions and despite the presence of damping, asymmetric balanced rotors can have a nontrivial stable amplitude. This response exists as long as the parametric excitation effects dominate the damping effects. Although damping reduces the vibrations’ amplitude, it can improve the stability of the system and eliminates unstable responses. Furthermore, the time response and frequency response curve of the system is carefully evaluated for various geometric design parameters and operation speed. Depending on the operating speed, the system can experience supercritical or subcritical pitchfork bifurcation. In addition, a detailed description of the system’s Campbell’s diagram, damping effect, and bifurcation is provided. Furthermore, it is proved that internal resonance cannot occur in the system. The accuracy of the analytical responses of the resonant case is compared with numerical ones. Stability of trivial and nontrivial responses is discussed in the time response and phase portrait of the system simultaneously. Finally, in the undamped system, multi-frequency responses are appeared as homoclinic and heteroclinic closed orbits.

Parametrically Excited Nonlinear Pneumatic Artificial Muscle Under Hard Excitation: A Theoretical and Experimental Investigation

Abstract: In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle subjected to time-varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales. The time and frequency response, phase portraits, and basin of attraction are plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters are evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters is illustrated.

The constrained parameter-splitting multiple-scales method for the primary/sub-harmonic resonance of a cantilever-type vibration energy harvester

Abstract: In this paper, the approximate analytical solutions obtained by using the constrained parameter-splitting-multiple-scales (C-PSMS) method to the primary and [Formula: see text] sub-harmonic resonances responses of a cantilever-type energy harvester are presented. The C-PSMS method combines the multiple-scales (MS) method with the harmonic balance (HB) method. Different from the erroneous stability results obtained by using the Floquet theory and the classical HB method, accurate stability results are obtained by using the C-PSMS method. It is found that the correction to the erroneous solution when the HB method and Floquet theory are adopted in the stability analysis of the primary and [Formula: see text] sub-harmonic resonances of a largely deflected cantilever-type energy harvester is necessary. On the contrary, the C-PSMS method gives much improved results compared to those obtained by using Floquet theory and HB method when the numbers of terms in each response expression are the same. The frequency response curves of the primary resonance and the [Formula: see text] sub-harmonic resonance of the harvester obtained by the C-PSMS method are compared to those obtained by the HB method and verified by those obtained by the fourth-order Runge–Kutta method. Moreover, the basin of attraction based on the fourth-order Runge–Kutta method is presented to confirm the inaccurate stability results obtained by using the HB method and Floquet theory. The convergence examinations on the stability analysis carried out by the HB method and Floquet theory show that enough terms in the response assumption are needed to achieve relatively accurate stability results when studying the stability of the primary and sub-harmonic resonances of a cantilever by using the HB method and the Floquet theory. However, the low-order C-PSMS method is able to give an accurate frequency-amplitude response and accurate stability results of the primary and sub-harmonic resonances of a largely deflected cantilever-type energy harvester.

Nonlinearity modulation in a mode-localized mass sensor based on electrostatically coupled resonators under primary and superharmonic resonances

Abstract: A general model of a mode-localized mass sensor incorporating two weakly coupled clamped-clamped microbeams under electrostatic excitation is presented, and a reduced-order model considering quadratic and cubic nonlinearities is established. The multiple time scales method is used to solve the dynamic characteristics of the coupled resonators under primary resonance, simultaneous superharmonic and primary excitations, and one-third superharmonic resonance respectively, and to analyze the contribution of each harmonic excitation term. It is shown that the sensor can display softening, hardening, and linear behaviors by tuning the overall nonlinear coefficient in three different excitation scenarios. Furthermore, the conditions for restoring linear behavior with the highest possible amplitude without any hysteresis under different excitations are obtained. Finally, the mass sensitivities represented by the relative shift of amplitude ratio are calculated for all the resulting dynamic behaviors. The results show that the sensitivity is highest, for the hardening behavior in the in-phase mode and for the softening behavior in the out-of-phase mode. Interestingly, the sensitivities of the linear behavior obtained by nonlinearity modulation are the same for the two vibration modes, which is improve the output stability. Consequently, the sensor resolution can be significantly enhanced below the pull-in instability, while avoiding noise mixing.

Nonlinear Vibrations of Carbon Nanotubes with Thermal-Electro-Mechanical Coupling

Abstract: Carbon nanotubes (CNTs) have wide-ranging applications due to their excellent mechanical and electrical properties. However, there is little research on the nonlinear mechanical properties of thermal-electro-mechanical coupling. In this paper, we study the nonlinear vibrations of CNTs by a thermal-electro-mechanical coupling beam theory. The Galerkin method is used to discretize the partial differential equation and obtain two nonlinear ordinary differential equations that describe the first- and second-order mode vibrations. Then, we obtain the approximate analytical solutions of the two equations for the primary resonance and the subharmonic resonance using the multi-scale method. The results indicate the following three points. Firstly, the temperature and electric fields have a significant influence on the first-mode vibration, while they have little influence on the second-mode vibration. Under the primary resonance, when the load amplitude of the second mode is 20 times that of the first mode, the maximal vibrational amplitude of the second is only one-fifth of the first. Under the subharmonic resonance, it is more difficult to excite the subharmonic vibration at the second-order mode than that of the first mode for the same parameters. Secondly, because the coefficient of electrical expansion (CEE) is much bigger than the coefficient of thermal expansion (CTE), CNTs are more sensitive to changes in the electric field than the temperature field. Finally, under the primary resonance, there are two bifurcation points in the frequency response curves and the load-amplitude curves. As a result, they will induce the jump phenomenon of vibrational amplitude. When the subharmonic vibration is excited, the free vibration term does not disappear, and the steady-state vibration includes two compositions.

The magneto-thermo-elastic primary resonance of rotating ferromagnetic functional gradient cylindrical shell in a transverse magnetic field

Abstract: The primary resonance behavior of a rotating ferromagnetic functional gradient cylindrical shell in a magnetic field, temperature field, and excitation force is investigated. Based on the physical neutral surface deformation theory and the Donnell theory, considering the effect of geometric nonlinearity, expressions of strain energy and kinetic energy of the shell and the work of forces are given, respectively. Applying the Hamilton principle, the magneto–thermo–elastic equation of a functional gradient cylindrical shell is derived by considering the magnetization effect of ferromagnetic metal. The question is discretized by Galerkin method and solved by the multi-scale method to obtain the amplitude–frequency response equation. The stability of the solution is discriminated by using the Lyapunov theory. Through numerical examples, the response curves of the system under different parameters are plotted, and the parameter ranges corresponding to multi-valued solution regions and single-valued solution regions are determined. The effects of parameter changes on the dynamic response and stability of system are analyzed. The results show that a coupling mechanism between temperature field, magnetic field, and excitation force affects the response and stability of the system, and the change of parameters have a significant effect on the vibration characteristics and stability. The dynamics model established in this paper is a theoretical reference for investigation on the multi-physics field coupling dynamic behaviors of structures.

Vibration Study of Rotating Rigid Hub and a Flexible Composite Beam under Simultaneous Primary-internal Resonance

Abstract: In this paper, the vibration of a composite system consisting of a rotating rigid hub and a flexible thin-walled beam is considered and studied. The equation of motion is derived in the previous work of Warminski and Latalski. The method of multiple scale technique has been applied to obtain frequency response equations near the simultaneous internal and primary resonance case in the absence of the acceleration of the hub. The vibration stability at this resonance case is investigated from the frequency response equations and studied using Liapunov’s methods. The effects of changes in selected structural parameters on the vibrating system behavior are investigated and studied numerically. Through the performed studies of the effects of changes in selected system a shift of the steady state amplitudes and the multi-valued of the bent curves are observed and the steady state amplitudes of the beam and hub have decreasing in the instability regions for natural frequencies. Finally, a comparison with the papers of previously published work is reported.

Nonlinear dynamics of parametrically excited cantilever beams with a tip mass considering nonlinear inertia and Duffing-type nonlinearity

Abstract: Abstract The response of a parametrically excited cantilever beam (PECB) with a tip mass is investigated in this paper. The paper is mainly focused on accurate prediction of the response of the system, in particular, its hardening and softening characteristics when linear damping is considered. First, the method of varying amplitudes (MVA) and the method of multiple scales (MMS) are employed. It is shown that both Duffing nonlinearity and nonlinear inertia terms govern the hardening or softening behaviour of a PECB. MVA results show that for frequencies around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations, and results are validated by numerical results obtained from direct integration (DI) of the equation of motion, while the MMS fails to predict this critical frequency. A criterion for determining the hardening and softening characteristics of PECBs is developed and presented using the MVA. To verify the results, experimental measurements for a PECB with a tip mass are presented, showing good agreement with analytical and numerical results. Furthermore, it is demonstrated that the mass added at the cantilever tip can change the system characteristics, enhancing the softening behaviour of the PECB. It is shown that, within the frequency range considered, increasing the value of the tip mass decreases the amplitude response of the system and broadens the frequency range in which a stable response can exist.

Nonlinear vibrations of four-degrees of freedom for piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular plate with PPF control strategy

Abstract: The nonlinear vibration suppression of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever (PFG-GRLCC) rectangular plate with positive position feedback (PPF) control strategy is investigated firstly. The material properties of the graphene-reinforced structure are calculated through the Halpin–Tsai micromechanical model. Considering the transverse external excitation and the converse effect of piezoelectricity, the governing equations of motion are formulated through von Karman large deformation theory, the classical laminated plate theory, and Hamilton principle. After adding the PPF controllers, a four-degrees of freedom model for the close-loop vibration control system is achieved via Galerkin truncation technique. The average equations in the case of the primary resonance and 1:1:3:3 internal resonance can be obtained using the multiple scale perturbation (MSP) method. The amplitude–frequency response curves are studied by the numerical continuation algorithm. The detailed parametric analyses show that the PPF controller can effectively reduce the nonlinear vibration response amplitudes of the PFG-GRLCC rectangular plate. In addition, the results reveal the energy transform between the host system and the PPF controller. This work is expected to provide theoretical guidance for nonlinear large amplitude vibration reduction of graphene-reinforced structure.

Multiple Scales Analysis of the Nonlinear Dynamics of Coupled Acoustic Modes in a Quasi 1-D Duct

Abstract: Abstract We present novel solutions based on the method of multiple scales (MMS) to the nonlinear equations governing the time-dependent amplitudes of coupled acoustic modes in a quasi one-dimensional duct with non-uniform cross-section and axially inhomogeneous mean velocity, temperature and pressure. The modal amplitude equations constitute a multi-degree-of-freedom system of linearly and nonlinearly coupled ordinary differential equations with quadratic and cubic nonlinearities. Due to the presence of both quadratic and cubic terms, the perturbation expansion for the MMS solution necessary includes terms of three orders, 𝑂(𝜖), 𝑂(𝜖2) and 𝑂(𝜖3). In addition to the MMS solutions, those obtained using the Krylov-Bogoliubov method of averaging (KBMA) are presented. The MMS and KBMA solutions are in good agreement with each other and with the numerical solutions to the governing equations. Two representative internal resonance cases that arise in a two-mode system are considered, i.e., 𝜔2 ≈ 2𝜔1 and 𝜔2 ≈ 3𝜔1, where 𝜔1 and 𝜔2 are the linear natural frequencies of the modes. For the 𝜔2 ≈ 2𝜔1 case, both the numerical and KBMA solutions contain low- frequency oscillations in the outer envelope of the limit-cycle like oscillations, but the method of multiple scales does not capture these oscillations. It was observed that when the amplitude equations are linearly uncoupled, the low-frequency oscillations in the outer envelope disappear. The criteria for the stability of the limit cycles are analyzed using the MMS and the KBMA solutions and the stability boundaries illustrated.