Pseudo-arclength Continuation Research Articles

Nonlinear Resonant Responses of a Novel FGM Sandwich Cylindrical Shell Structure for Supersonic Flight Vehicles based on 1:1 Internal Resonance between Conjugate Modes

A new design scheme for the functionally graded material (FGM) sandwich cylindrical shell structure made of ceramic-FGM-carbon fiber composites is proposed, which is geared towards supersonic flight vehicles and has high specific stiffness and high-temperature resistance. We focus on analyzing the nonlinear resonant responses of the novel FGM sandwich cylindrical shell subjected to external excitation and aerodynamic force. Firstly, the mechanical properties of the novel FGM sandwich material are calculated, and then the nonlinear dynamic model of the FGM sandwich cylindrical shell is established based on the first order shear deformation theory (FSDT) and Hamilton's principle. Due to the axisymmetric property of the cylindrical shell, a 1:1 internal resonance between the driven and companion modes always exists in the perfect cylindrical shell. Taking this into consideration, the nonlinear resonant response problems of the FGM sandwich cylindrical shell are numerically simulated by combining the Galerkin method and the pseudo-arc length continuation method. Finally, the effects of complex parameters such as external excitation, aerodynamic force, aspect ratio, gradient index, and skin-core-skin ratio on the nonlinear resonant responses of the FGM sandwich cylindrical shell are investigated.

Nonlinear resonant responses and chaotic dynamics of three-dimensional braided composite cylindrical shell

The advanced three-dimensional braided composite structure has excellent integrity, which can effectively avoid the problems of interlayer stress jump and interface debonding that may occur in traditional composite laminated structures. However, most of the existing references on the three-dimensional braided composite structure are fragmented, and there is little research on their vibrations. Taking the cylindrical shell of a rocket as the object and considering the complex flight environment, a general framework for the vibration problems of the three-dimensional braided composite shell is proposed for the first time. The equivalent mechanical parameters of the three-dimensional braided composite material are calculated based on the volume averaging method. Then, the nonlinear dynamic model of the three-dimensional braided composite cylindrical shell is established, and the natural vibration characteristics are analyzed. Under the case of 1:1 internal resonance, the pseudo-arc length continuation method is used to solve the nonlinear resonant responses of this nonlinear system. Finally, we apply the fourth-order Runge-Kutta algorithm to study the chaotic dynamics of the three-dimensional braided composite cylindrical shell model. This framework presented in this paper can be extended to the study of nonlinear vibrations and chaotic dynamics of other three-dimensional braided composite structures.

Design and optimization of quasi-zero-stiffness dual harvester-absorber system

The single quasi-zero-stiffness harvester-absorber system has shown its effectiveness in harvesting vibration energy and suppressing low-frequency vibrations simultaneously. Nevertheless, drawbacks of narrow frequency range of vibration suppression and low output power are still challenges. In this paper, a novel quasi-zero-stiffness dual harvester-absorber system (QZS DHAS) is proposed for broadband synergistic vibration suppression and energy harvesting at low frequency. The QZS feature is realized by mounting an equal-strength piezoelectric beam and a pair of magnet rings in parallel. Firstly, the restoring force of both the piezoelectric beam and the magnet rings are theoretically derived, respectively, which are verified by finite element simulations. Then, the governing equations of a host oscillator with QZS DHAS are derived based on Newton's law and Kirchhoff's law, which are solved by the harmonic balanced method combined with the pseudo arc-length continuation method. Moreover, the effects of mass ratios, frequency ratios and damping ratios on the performance of the QZS DHAS are investigated in details. Finally, the parameters of the QZS DHAS are optimized to improve the performance of vibration suppression and energy harvesting by using the multi-objectives genetic algorithm in combination with the technique of establishing order preference by similarity to the ideal solution. The results show that the peak response of the host oscillator is reduced by 33.48 % and the peak power is increased by 531.38 % after optimization, which indicates that the dual objectives of broadband vibration absorption and high-efficiency energy harvesting are fulfilled.

Nonlinear dynamic modelling and analysis of a rotating composite thin-walled beam considering ice coating

For rotating structures operating in severe weather conditions, ice coating phenomenon is an established issue that impairs the reliability and effectiveness of the structures by changing the dynamic behaviors. To explore the influence mechanisms of ice coating on the dynamic behaviors of rotating structures, this work presents an analytical dynamic model for nonlinear forced vibration of a composite thin-walled beam under both rotating and ice coating conditions. The ice coating is considered as C-shaped configuration which can affect the system parameters of the beam, especially aerodynamic performance. With effect of out-of-plane warping taken into account, the nonlinear dynamic equation of the beam is established using Hamilton's principle. The Galerkin discrete method together with the incremental harmonic balance method (IHB) as well as pseudo arclength continuation technique are adopted to obtain the periodic response of the nonlinear system. Then, a detailed parametric study on natural frequency, frequency-response and bifurcation behaviors is numerically conducted. Special attention is paid to the effects of ice coating, rotating angular velocity, inflow velocity and design parameters on the linear and nonlinear dynamic behaviors of the beam. Results reveal a prominent change in natural frequency, vibration amplitude and cusp bifurcation of multivalued region under the combined effects of ice coating and rotating motion. Such ice-induced changes exhibit the requirement of vibration control and optimization design consideration for rotating structures operating in icing coating conditions.

Multimodal Resonance Response of Incompressible Hyperelastic Moderately Thick Cylindrical Shells

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.

Nonlinear vibrations of porous-hyperelastic cylindrical shell under harmonic force using harmonic balance and pseudo-arc length continuation methods

The resonant responses are investigated for the porous-hyperelastic Mooney–Rivlin cylindrical shell subjected to a radial harmonic excitation. Considering the higher-order shear deformation theory (HSDT), fourth-order strain-displacement relations are derived, which include the radial geometric imperfections varying along the thickness direction. Using the porous-hyperelastic Mooney–Rivlin constitution relation and Lagrange equation, the differential governing equations of motion are obtained for the porous-hyperelastic cylindrical shells. The resonant conditions are presented and the accuracy of the mathematical models is verified. The harmonic balance and pseudo-arc length continuation methods are used to obtain the amplitude-frequency and forced-amplitude curves. The stability of the solutions is analyzed by Floquet theory. The effects of the external excitation amplitudes, structure parameters and porosity infill parameters on the linear frequencies, amplitude-frequency responses and force-amplitude responses are discussed for the imperfect porous-hyperelastic cylindrical shells. The results show that the linear frequencies of the porous-hyperelastic cylindrical shells increase obviously as the uniform and sigmoid function porosity parameters reach the certain values. The increase of the structure parameters enhances the response amplitudes of the first-order mode and minified response amplitudes of the second-order mode. The decreasing porosity ratios weaken the softening nonlinear behaviors of the porous-hyperelastic cylindrical shells. With the changes of the external excitation amplitudes and structure parameters, the motion of the porous-hyperelastic cylindrical shell indicates that the synchronous vibrations occur with the period and chaotic vibrations alternately.

Numerical continuation and stability of nonlinear systems with distributed delays: Application to fluid-induced impacts of tubes in cross-flow

Fluid–elastic instability is the main source of concern during the design of heat-exchanger tube bundles. The associated research has focused on the understanding of the physical mechanisms behind this phenomenon and the establishing of models capable of predicting its onset; also some concomitant work has been done to establish post-instability behaviour and to prevent from excessive wear in possible sliding contacts. Moreover, the existing studies make use of time-integration methods alone for this purpose, through which it is difficult to get a comprehensive insight of global dynamics. Continuation methods, which give access to unstable branches and precise bifurcation information, are a precious tool to unfold the attainable dynamic regimes. In this paper, the parametric behaviour of two representative systems under cross-flow excitation is explored through pseudo arc-length continuation with mean flow velocity as a main driving parameter, wherein the nonlinear modal equations of motion are solved by harmonic balance at each step. As the quasi-unsteady model used for fluid–elastic coupling introduces convolution integrals, this approach is quite natural and we show that, despite some difficulties regarding the treatment of stiff intermittent contacts, it allows for a thorough exploration of the system’s response. For the first case -a benchmark model-, increasingly complex dynamics arise as more modes are kept in the truncated modal basis, which is due to a series of modal interactions as the impacts distribute mechanical energy from the linearly-unstable first mode to the higher ones. This can be anticipated by studying the nonlinear normal modes of the system, as they expose the allowed internal resonances. In the second case, consisting of a realistic heat-exchanger tube configuration, a similar pattern is observed.

Nonlinear forced vibrations of functionally graded three-phase composite cylindrical shell subjected to aerodynamic forces, external excitations and hygrothermal environment

In this paper, the new functionally graded three-phase composite cylindrical shell is assumed as a common structure in the carrier rocket in the future, and we creatively study the nonlinear forced vibration of this cylindrical shell considering the interaction of different factors in the complex operating environment, including the aerodynamic forces, external excitations, and hygrothermal environment. Based on the first-order shear deformation theory, Von-Karman geometric nonlinear theory, and Hamilton's principle, we derive the nonlinear partial differential equations of motion of the functionally graded three-phase composite cylindrical shell. Considering the axisymmetry of the perfect circular shell, there is a 1:1 internal resonance between the conjugate modes of this cylindrical shell. On this basis, the nonlinear forced vibration of the cylindrical shell is investigated by a combination of Galerkin's method and the pseudo-arc length continuation method. Matcont toolbox can directly solve the ordinary differential equations to obtain the nonlinear frequency response curves. The method can effectively obtain both stable and unstable solutions, avoiding the mathematical difficulties encountered in the formulation process, and facilitating the study of the effects of parametric variables on the resonance response in complex environments. The results show that the variation of material parameters and the complex environment have important effects on the nonlinear resonance response of functionally graded three-phase composite cylindrical shell.

Nonlinear normal modes of highly flexible beam structures modelled under the SE(3) Lie group framework

This work presents a shooting algorithm to compute the periodic responses of geometrically nonlinear structures modelled under the special Euclidean (SE) Lie group formulation. The formulation is combined with a pseudo-arclength continuation method, while special adaptations are made to ensure compatibility with the SE framework. Nonlinear normal modes (NNMs) of various two-dimensional structures including a doubly clamped beam, a shallow arch, and a cantilever beam are computed. Results are compared with a reference displacement-based FE model with von Kármán strains. Significant difference is observed in the dynamic response of the two models in test cases involving large degrees of beam displacements and rotation. Differences in the contribution of higher-order modes substantially affect the frequency-energy dependence and the nonlinear modal interactions observed between the models. It is shown that the SE model, owing to its exact representation of the beam kinematics, is better suited at adequately capturing complex nonlinear dynamics compared to the von Kármán model.

Practical implementation of pseudo-arclength continuation to ensure consistent path direction

Pseudo-arclength continuation is a scheme used to generate a set of solutions for nonlinear equations and has been applied in numerous fields. In continuation schemes, solutions are obtained by tracking a parameter associated with the system as it varies, thereby creating a solution path. Pseudo-arclength continuation employs the nullspace vector, which is tangent to the path, as an approximation of the arclength in order to determine the direction of variation at each point on the path. However, the nullspace vector has both positive and negative solutions that are equally valid, leading to uncertainty in the direction of travel along the path at each location. Selecting the wrong nullspace vector can result in returning to a previously located solution instead of continuing the mapping of the solution path. Techniques to ensure consistent path direction have been developed; however, their efficacy does not extend to all scenarios. This paper addresses this gap by introducing a “flipping condition” to track previously determined path directions and ensure continuous following of the same path direction. A clear description of the process used to incorporate the flipping condition into the pseudo-arclength continuation scheme, practical implementation of the scheme, and examples demonstrating its application to the nonlinear circular restricted three-body problem (CR3BP), is provided. The inclusion of the flipping condition ensures a consistent direction of travel along the solution path and allows for families of solutions to be developed.

A Design Method to Eliminate Sub Synchronous Response in Automotive Turbochargers utilizing Nonlinear Dynamics of Time Periodic Systems

The present work introduces a methodology for the dynamic design evaluation of high-speed rotors, applying nonlinear dynamics of time periodic systems. The paper focuses on the definition of design variables so as the system to avoid sub synchronous components in the response. The response of an unbalanced turbocharger rotor mounted on oil bearings with wire mesh dampers (WMDs) is evaluated with an efficient collocation method coupled to the pseudo arc length continuation to evaluate response limit cycles in the entire range of the rotating speed, which acts as bifurcation parameter. The specific system is used to demonstrate the method without loss of generality. The analytical model of the WMD component is implemented directly from the literature, while rigid rotor model and short bearing approximations are used for the rest components. Floquet theory is applied in the non-autonomous dynamic system and the stability characteristics of the response limit cycles are evaluated through Floquet multipliers, the magnitude of which act in the algorithm as stability index. A sensitivity analysis is performed at first to evaluate the correlation between WMD key design parameters and the time response of the system.A multi objective optimization method is embedded to set Floquet multipliers at specific values; in this way the response limit cycles retain stability and periodicity, avoiding any type of bifurcation which would render sub-synchronous component. Acceptable sets for design variables of journal bearings and WMDs are defined by multi-objective optimization and these are verified in efficiency by evaluating the response of the system by time integration, as a virtual experiment, for several operating conditions. The method requires severely lower computational time and capacity to render an acceptable design compared to the conventional design of experiment DoE procedure, and it is not limited on the analytical models of rotor, bearing, or other component.

Three-dimensional nonlinear vibrations of slightly curved cantilevered pipes conveying fluid

In this paper, nonlinear static and dynamic behaviors of slightly curved cantilevered pipes conveying fluid under transverse base excitations are investigated. Incorporating geometrical and inertial nonlinearities, a new three-dimensional dynamic model is developed for the inextensible cantilevered pipe based on the Euler–Bernoulli beam theory. The resultant partial differential equations are discretized using the Galerkin method and solved utilizing the pseudo-arclength continuation technique and a time-integration method. The static equilibrium configuration, linear stability, and nonlinear response of the pipe are calculated for various cases, which allow us to investigate the effects of flow velocity, gravity coefficient, initial curvature amplitude and shape, and base-excitation amplitude and frequency. An emphasis is put on the importance of modal interactions on frequency-response and force-response relationships, evidencing the possible in-plane and out-of-plane transverse vibrations and identifying the preferred forms of pre- and post-instability behaviors. This study offers an efficient theoretical model for exploring the nonlinear forced response of slightly curved cantilevered pipes conveying fluid and provides a better insight for further investigations on the modal interaction in cantilevered pipe devices.

On the quality of stability and bifurcation sets in rotors with permanent shaft bow on nonlinear supports

The paper investigates the dynamics of simple and complex geometry rotors of various configurations with permanent bow, mounted on nonlinear bearings, in regards to nonlinear stability, quality of motion, and bifurcation set. Permanent shaft bow is a fault existing in most slender rotors in thermal machines, e.g. gas turbines, steam turbines, and turbopumps, and other applications of rotating machinery like transmission shafts and geared shafts. In this work, finite element models of multi-segment rotors are implemented, including permanent bow, and a generic rotor system of a uniform rotor with three masses is studied first on its stability characteristics for various bow values, and for different slenderness. A realistic rotor of complex geometry is used as the second application. The rotors are mounted on two plain short bearings to include the nonlinearity of impedance forces with analytical formulas; this is the unique source of nonlinearity in the systems.The rotor-bearing systems are found to produce the well-known oil whirl instability through a Hopf, or a secondary Hopf bifurcation, depending the unbalance grade, in speeds higher than the first critical speed, when bow is not included. Supercritical and subcritical Neimark-Sacker and period doubling bifurcations are produced in speeds lower than critical speed when bow is included above some threshold values. Stable and unstable limit cycles are generated both for unbalanced or balanced rotors with bow, and their periodicity is investigated. A bifurcation set of the rotor-bearing systems is evaluated utilizing the pseudo-arc length continuation method with embedded collocation method, for the evaluation of periodic limit cycles (where these appear). Neimark-Sacker bifurcations are found to collide and vanish as the bow magnitude changes from high to low values. Bow phase angle with respect to unbalance force is found not to influence the respective bifurcation sets.

Nonlinear in-plane thermomechanical stability of shallow sandwich micro-arches including strain gradient tensors

The microstructural-dependent nonlinear in-plane stability characteristics of functionally graded (FG) sandwich shallow micro-arches subjected to a uniformly distributed lateral load in conjunction with a temperature rise are investigated in the current study. In this regard, the third-order shear flexible arch formulations are established within the framework of the modified strain gradient continuum mechanics. The considered sandwich micro-arches are made of nanocomposites reinforced with graphene nanofillers based upon various FG lamination patterns. The Chebyshev–Gauss–Lobatto type of gridding scheme together with the pseudo arc-length continuation technique are employed to employ a numerical solution strategy for tracing the size-dependent lateral load–deflection and lateral load–axial load nonlinear stability plots. It is observed that by adding a rise in the temperature to the applied uniform lateral load, the first bifurcation lateral load enhances, while the second one reduces. Also, it is dedicated that there is a unique value of the applied uniform lateral load corresponding to which the axial resultant load as well as the induced lateral deflection in the micro-arch are the same for all values of the temperature rise. Moreover, it is revealed that by increasing the value of the rise in the temperature, the size dependency in the value of the upper limit lateral load becomes less prominent, while it enhances in the value of the lower limit lateral load.

Study on the vibration suppression of beam structures with nonlinear piezoelectric shunt damping

The vibration suppression of the nonlinear beam structure under harmonic excitations by employing the nonlinear piezoelectric shunt damping circuit (PSDC) is researched. The energy approach is utilized to deduce the dynamic equations of the nonlinear beams with nonlinear PSDC. The complexification-averaging approach and pseudo arc-length continuation approach are applied to calculate the frequency domain response of the beam structure. Numerous numerical examples are conducted to verify the present formulations, thereby demonstrating the remarkable efficiency and accuracy. The impacts of several parameters of the proposed PSDC are discussed and the optimal study of the nonlinear PSDC is further performed to obtain the optimal negative and nonlinear capacitances. The comparisons concerning the vibration of the beam structures with and without PSDC are implemented to examine the vibration mitigation performance of the designed PSDC. In the end, the design criterion of the nonlinear PSDC is given.

Non-Smooth Dynamics of Tapping Mode Atomic Force Microscopy

Abstract This study examines the nonlinear dynamics in tapping-mode atomic force microscopy (AFM) with tip-surface interactions that include van der Waals and Derjaguin-Müller-Toporov contact forces. We investigate the periodic solutions of the hybrid system by performing numerical pseudo-arclength continuation. Through the use of bifurcation locus maps in the set of parameters of the discontinuous model, the overall dynamical response scenario is assessed. We demonstrate the influence of various dissipation mechanisms that are related with the AFM touching or lacking contact with the sample. Local and global analyses are used to investigate the stability of the stable solution in the repulsive regime. The impacting nonsmooth dynamics framed within a higher-mode Galerkin discretization is able to capture windows of irregular and complex motion.

Vibration Suppression on the Composite Laminated Plates Subjected to Aerodynamic and Harmonic Excitations Based on the Nonlinear Piezoelectric Shunt Damping

This paper proposes a nonlinear piezoelectric shunt damping circuit to mitigate the vibration of the plate structures subjected to the aerodynamic and harmonic excitations. The nonlinear piezoelectric shunt damping circuit consisting of the piezoelectric patch, inductor, resistor, negative capacitor and nonlinear capacitor is considered. Firstly, the governing equations of the electro-mechanical coupling nonlinear system are formulated based on the energy method and modified Ritz method. Furthermore, the complexification-averaging method combined with pseudo arc-length continuation method is adopted to analyze the steady-state response of the coupled system with aerodynamic and harmonic forces. A considerable number of numerical cases are performed to validate the proposed model and the time domain and frequency domain dynamic responses of the plate structures with and without control are further compared to validate the effectiveness of the proposed control strategy. Meanwhile, the energy transfer analysis between the plate structure and external piezoelectric shunt damping circuit is investigated. Finally, the effects of key parameters of the nonlinear piezoelectric shunt damping circuit including the negative capacitance and nonlinear capacitance on the vibration suppression performance of the piezoelectric shunt damping are discussed.

Size-dependent nonlinear thermomechanical in-plane stability of shallow arches at micro/nano-scale including nonlocal and couple stress tensors

In the present exploration, the unconventional tensors of nonlocal stress and couple stress are incorporated simultaneously in the in-plane nonlinear stability analysis of functionally graded (FG) multilayer shallow micro/nano-arches under thermomechanical loading conditions. For this purpose, the nonlocal couple stress (NCS) mechanics of continuum is implemented into the third-order shear flexible arch theory incorporating the von Karman kinematical nonlinearity. The nanocomposite material of FG multilayer shallow micro/nano-arches is reinforced with graphene nanofillers in accordance with different patterns of FG lamination. The generalized differential quadrature numerical strategy in conjunction with the pseudo arc-length continuation procedure are employed to deduce the roles of unconventional nonlocal and couple stress tensors in the NCS-based nonlinear stability paths of thermomechanical loaded FG multilayer shallow micro/nano-arches. It is found that for all patterns of the lamination, the contributions associated with the nonlocality and couple stress small scale effects on the value of the upper limit as well as the first bifurcation compressive loads are less than those on the lower limit and the second bifurcation ones. Also, it is demonstrated that by combining a temperature rise with the applied compressive lateral load, the upper limit lateral load increases, while the lower limit one decreases. Furthermore, it is seen that by applying a temperature rise, an initial lateral deflection is induced in the shallow micro/nano-arch before applying the compressive lateral load.

Symmetric and asymmetric vibrations of rotating GPLRC annular plate

The present paper discusses the symmetric and asymmetric free and forced vibrations of a rotating graphene-platelets-reinforced composite annular plate subjected to the complex external excitation in thermal environment, where the aerodynamic force is considered. Displacement components are able to vary along the radial and circumferential directions and von Kármán’s geometrical nonlinearity is taken into account. The free vibration analysis is conducted first. Then the forced vibration equations are discretized by utilizing the Galerkin method with the aid of vibration mode. The incremental harmonic balance method and pseudo-arclength continuation algorithm are employed in combination to solve these equations. The effects of plate dimensions, graphene-platelets-reinforced composite (GPLRC) parameters, temperature rise, rotational motion, and external excitation on the symmetric and asymmetric vibration are studied by the parameter analysis for annular plate. Results implicate that temperature rise would affect the order in which the symmetric and asymmetric vibrations appear whereas the other factors not. Asymmetrical material properties along the thickness direction leads to that the periodic motion is asymmetric with respect to mid-plane. Under the complex external excitation, the dominant role of deformation shifts from symmetric vibration to asymmetric vibration as the frequency increases.

Nonlocal couple stress-based quasi-3D nonlinear dynamics of agglomerated CNT-reinforced micro/nano-plates before and after bifurcation phenomenon

In present research exploration, the nonlinear dynamic stability characteristics of axially compressed nanocomposite plates at micro/nano-scale reinforced with randomly oriented carbon nanotubes (CNTs) are investigated within the both prebuckling and postbuckling regimes. To accomplish this examination, the nonlocal couple stress (NCS) continuum elasticity is incorporated to a quasi-3D plate theory which separates the plate deformation to the bending and shear parts considering simultaneously the transverse shear and normal displacements. In addition, a two-parameter homogenization scheme is utilized to obtain the effective characters of the randomly oriented CNT-reinforced nanocomposites. The NCS-based nonlinear differential equations of motion are discretized using the Kronecker tensor product together with the shifted Chebyshev-Gauss-Lobatto gridding pattern. Thereafter, the Galerkin technique together with the pseudo arc-length continuation method are employed to achieve the NCS-based fRequency-load and nonlinear frequency ratio-deflection curves before and after of the bifurcation point. It is deduced that for a randomly oriented CNT-reinforced heterogeneous micro/nano-plate in which the most CNTs are located inside clusters, increasing the value of cluster volume fraction leads to increase a bit the significance of the softening and stiffing characters related to the nonlocal and couple stress tensors before the bifurcation phenomenon, but it causes to decrease them after the critical bifurcation point. Opposite patterns before and after the bifurcation phenomenon are predicted for the agglomeration in which the most CNTs are located outside clusters.