Showing papers on "Multiple-scale analysis published in 2022"

Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

Abstract: In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.

Steady-state response of an axially moving circular cylindrical panel with internal resonance

Abstract: With the 3:1 internal resonance , the primary and secondary resonances of an axially moving thin circular cylindrical panel are investigated in the present work. The governing equation and the compatibility equation are established based on the Donnell's nonlinear shell theory and solved to obtain the nonlinear steady-state responses by combining the Galerkin method and the method of multiple scales. The analytical solutions are verified by numerical solutions based on the Runge-Kutta Method. The governing equation includes both the quadratic nonlinearity and the cubic nonlinearity, so the perturbation solutions need to consider three time scales. The quadratic nonlinearity causes the softening behavior of the system. Natural frequencies and the 3:1 internal resonance condition are obtained by the linear analysis. Under the primary resonance , the internal resonance causes the coupling of the first two modes to complicate the nonlinear dynamic response. The response for the second mode possesses an extra bulge or peak due to the internal resonance. The quadratic nonlinearity results in the zero frequency drift and the second-order harmonic. Under the secondary resonance, the exciting force only arouses the second mode. Results are shown to examine the effects of the internal resonance, the exciting force and viscous damping coefficients on the nonlinear dynamic response of an axially moving thin circular cylindrical panel.

Steady-state response of an axially moving circular cylindrical panel with internal resonance

Abstract: With the 3:1 internal resonance, the primary and secondary resonances of an axially moving thin circular cylindrical panel are investigated in the present work. The governing equation and the compatibility equation are established based on the Donnell's nonlinear shell theory and solved to obtain the nonlinear steady-state responses by combining the Galerkin method and the method of multiple scales. The analytical solutions are verified by numerical solutions based on the Runge-Kutta Method. The governing equation includes both the quadratic nonlinearity and the cubic nonlinearity, so the perturbation solutions need to consider three time scales. The quadratic nonlinearity causes the softening behavior of the system. Natural frequencies and the 3:1 internal resonance condition are obtained by the linear analysis. Under the primary resonance, the internal resonance causes the coupling of the first two modes to complicate the nonlinear dynamic response. The response for the second mode possesses an extra bulge or peak due to the internal resonance. The quadratic nonlinearity results in the zero frequency drift and the second-order harmonic. Under the secondary resonance, the exciting force only arouses the second mode. Results are shown to examine the effects of the internal resonance, the exciting force and viscous damping coefficients on the nonlinear dynamic response of an axially moving thin circular cylindrical panel.

Nonlinear Dynamics and Bifurcation Analysis of a Hinged-Clamped Beam Under Parametric and Internal Resonances

Abstract: The present study examines the stability and bifurcation of axially moving hinged-clamped beams subjected to parametric excitation originating due to speed alterations. Attention is paid to the principal parametric resonance amid 3:1 internal resonance in the subcritical speed regime. The direct method of multiple scales is applied to predict the nonlinear behavior of the beam modeled in the form of the nonlinear integro-partial differential equation. A continuation algorithm is used to solve numerical examples to illustrate the stability and bifurcations of periodic solutions for the given set of system parameters. The fixed point solution is different for different modes. Three kinds of equilibrium solution curves like trivial, two-mode, and single-mode solutions are obtained from numerical computations. The first two kinds are available for both first and second modes while the third one is available for the second mode only. The introduction of material damping converts the isolated two-mode solution into an isolated closed-loop form. The system displays saddle-node and Hopf bifurcation points in both modes while super and subcritical pitchfork bifurcation points in the second mode only. Decreasing viscous damping strengthens the effect of internal resonance. The numerically simulated results are unique, interesting, and are not available in the existing literature.

Multiple internal resonances of rotating composite cylindrical shells under varying temperature fields

Abstract: Abstract Composite cylindrical shells, as key components, are widely employed in large rotating machines. However, due to the frequency bifurcations and dense frequency spectra caused by rotation, the nonlinear vibration usually has the behavior of complex multiple internal resonances. In addition, the varying temperature fields make the responses of the system further difficult to obtain. Therefore, the multiple internal resonances of composite cylindrical shells with porosities induced by rotation with varying temperature fields are studied in this paper. Three different types of the temperature fields, the Coriolis forces, and the centrifugal force are considered here. The Hamilton principle and the modified Donnell nonlinear shell theory are used to obtain the equilibrium equations of the system, which are transformed into the ordinary differential equations (ODEs) by the multi-mode Galerkin technique. Thereafter, the pseudo-arclength continuation method, which can identify the regions of instability, is introduced to obtain the numerical results. The detailed parametric analysis of the rotating composite shells is performed. Multiple internal resonances caused by the interaction between backward and forward wave modes and the energy transfer phenomenon are detected. Besides, the nonlinear amplitude-frequency response curves are different under different temperature fields.

Forced-self-excited system of iced transmission lines under planar harmonic excitations

Abstract: This work involved analyzing the self-excited and forced vibrations of iced transmission lines. By introducing an external excitation load, the effect of dynamic wind on nonlinear vibration equations was reflected by the vertical aerodynamic force. The approximate analytic solution of the non-resonance of the forced-self-excited system was obtained using the multiple scale method. With an increase in excitation amplitude, the nonlinearity of the system was enhanced, and the forced-self-excited system experienced three vibration stages—namely, self-excited vibration, the superposition forms of self-excited and forced vibrations, and forced vibration controlled by nonlinear damping. Among these, the accuracy of the approximate analytic solution decreased with increase in nonlinear strength variations. When the excitation amplitude was greater than the critical value, the quenching phenomenon appeared in the forced-self-excited system, and the discriminant formula was derived in this work. In addition, the third-order Galerkin method, which considered the small sag effect, was used to discretize the nonlinear galloping governing equation. The response (principal resonance, harmonic resonance) of the forced-self-excited system was analyzed by time history displacement curves and phase diagrams. The conclusions of this work may contribute to the practical engineering of iced transmission lines. More importantly, as a combination of the Duffing equation and Rayleigh equation, the forced-self-excited system may have high theoretical research value.

Subharmonic Resonance of Duffing Oscillator with Dry Friction Under Foundation Excitation

Abstract: The 1/3 subharmonic resonance response of Duffing oscillator with Coulomb dry friction under foundation excitation is investigated, and the approximate analytical solution of the subharmonic resonance of the system is obtained by using the incremental averaging method. Based on the approximate analytical solution of the primary resonance obtained by the averaging method, the approximate analytical solution of subharmonic resonance is solved by the averaging method according to the incremental equation, and the amplitude-frequency response equation of subharmonic resonance is obtained. It is found that the Coulomb friction affects the amplitude-frequency response of both the primary resonance and subharmonic resonance of the nonlinear dry friction system in the form of equivalent damping. The comparison between the approximate analytical solution and the numerical solution shows that, the approximate analytical solutions of the primary resonance and subharmonic resonance are both in very good agreement with the numerical solution. The existence condition of the 1/3 subharmonic resonance for the nonlinear dry friction system is presented, and the stability of the steady-state solution of subharmonic resonance is also judged. Based on the approximate analytical solution, the effects of the nonlinear stiffness and the Coulomb friction on the amplitude-frequency response of resonance and critical frequency of 1/3 subharmonic resonance of the nonlinear dry friction system are analyzed in detail. The analysis results show that the incremental averaging method can effectively obtain the approximate analytical solution in unified form for the subharmonic resonance of nonlinear system with Coulomb friction.

High order analysis of a nonlinear piezoelectric energy harvesting of a piezo patched cantilever beam under parametric and direct excitations

Abstract: In this paper, the nonlinear dynamics of a piezoelectric beam under simultaneous parametric and external excitation is studied. The energy sensor considered is composed of a substrate and a piezoelectric layer used as a vibratory sensor located in the middle of the beam. The model thus constituted takes into account the electromechanical nonlinearities. Based on the Galerkin’s method, a strongly coupled nonlinear system is obtained. The multiple scales method is used to determine the analytical expressions of the deflection of the beam, the output voltage generated by the piezoelectric patch and the power harvested around the resonant frequency of the structure. The nonlinear characteristics of the energy harvester are explored under parametric and direct excitations. Analytical formulations of the deflection and the output voltage have been provided leading to investigate new insights into the effects of certain parameters such as the electromechanical coupling coefficient, the thickness and width of the elastic beam and the piezoelectric patch. In particular the position of the piezoelectric patch on the performance of the energy harvester is studied. The results show that for large values of the damping coefficient and thickness, the frequency-amplitude response curves of the energy harvester decrease. On the other hand, the frequency response curves of the displacement and voltage increase for large values of the excitation width and amplitude. We also note the fact that the length ratio influences not only the vibration amplitudes but also the output voltage. The deflection amplitudes and the output voltage obtained by the elaborated second-order multiple scale method are higher than those obtained by the first order, these results show the importance of nonlinearities in the dynamic study of vibrational energy harvesting systems by piezoelectric materials.

Free and forced nonlinear vibrations of Bi-Directional functionally graded Euler–Bernoulli porous beams

Abstract: In this study, the free and forced vibrations of a bi-directional functionally graded porous beam are investigated. The governing equations of motion are derived by Hamilton’s principle, and the reduced temporal equation of motion with cubic and quintic nonlinear terms is obtained using the Galerkin approach. Analytical solutions for the nonlinear natural frequencies in addition to the primary resonance response curves are established by the method of multiple scales. The effects of the axial and transverse functionally graded indexes, initial amplitude, porosity parameter, and the elastic and mass density ratios on the nonlinear frequencies and the forced responses are examined.

Vibration control of primary and subharmonic simultaneous resonance of nonlinear system with fractional-order Bingham model

Abstract: The passive vibration control of primary and subharmonic simultaneous resonance for the Duffing-type nonlinear system under the base excitation and external excitation by using magnetorheological (MR) fluid damper is studied, where the fractional-order derivative Bingham model of MR fluid damper is considered. The approximate analytical solution of the system is obtained by using the incremental averaging method. On the basis of obtaining the primary resonance of the system under base excitation by the averaging method, the subharmonic resonance solution of the system is obtained by taking the subharmonic resonance of the system under base excitation and external excitation as an increment, so as to obtain the approximate analytical solution of the simultaneous resonance of primary and subharmonic resonance. And the amplitude–frequency equation and phase–frequency equation of the steady-state solutions for the primary and subharmonic resonance of the system are derived respectively. According to the approximate analytical solutions, the stability conditions of the steady-state solution of the primary resonance and subharmonic simultaneous resonance are obtained by Lyapunov method. Compared with the numerical solution, the correctness and accuracy of the analytical solution of the primary resonance and subharmonic simultaneous resonance are demonstrated. The influence of system parameters on the resonant response of the system is analyzed in detail, with emphasis on the resonance bifurcation behavior of the system. The analysis results show that the damping passive vibration reduction control has obvious vibration suppression effect on the primary and subharmonic simultaneous resonance of the Duffing-type nonlinear system. Under certain conditions, there are 9 branches in the steady-state amplitude–frequency response of the primary and subharmonic simultaneous resonance of the Duffing-type system, of which 5 branches are stable solutions.

Parametric Analysis of Nonlinear Bi-Stable Piezoelectric Energy Harvester Based on Multi-Scale Method

Abstract: Given the geometric nonlinearity of the piezoelectric cantilever beam, this study establishes a distributed parameter model of the nonlinear bi-stable cantilever piezoelectric energy harvester, following the generalized Hamilton variational principle. The analytical expressions of the dynamic response were obtained for the energy harvesting system using Galerkin modal decomposition and the multi-scale method. The investigation focuses on how the performance of the energy harvesting system is influenced by the gap distance between magnets, external excited amplitude, mechanical damping ratio and external load resistance. The calculation results were compared with those obtained neglecting the geometric nonlinearity of the beam. The results show that the system responses contain jump and multiple solutions. The consideration of the geometrical nonlinearity significantly amplified the peak displacement and peak output power of the intra-well and inter-well motions. There is an evident hardening effect of the inter-well motion frequency response curve. By reasonable adjusting the parameters, it is possible to improve the output power of the piezoelectric energy harvesting system and broaden the operating frequency of the system.

Three-to-One Internal Resonance of L-Shaped Multi-Beam Structure with Nonlinear Joints

Abstract: In this paper, a reduced-order analytical model for an L-shaped multi-beam structure with nonlinear joints is presented to investigate the nonlinear responses of the system with three-to-one internal resonances conditions. Firstly, the global mode shapes are used to obtain an explicit set of nonlinear ordinary differential equations of motion for the system. Then, the first two natural frequencies of the system are calculated to determine the specific tip mass that results in three-to-one internal resonance. Subsequently, an approximation of the analytical solution of the dynamic model with two-degree-of-freedom is derived by using the multi-scale method. The accuracy of the approximation solution is verified by comparing it with the numerical solution obtained from the original motion equations. Based on the nonlinear dynamical model obtained by this paper, the frequency response curves are given to investigate the nonlinear dynamic characteristic of the L-shaped multi-beam structure with nonlinear joints. The results show that the nonlinear stiffness of the joints has a great influence on the nonlinear response of the system with three-to-one internal resonance conditions.