Pseudo-arclength Continuation Method Research Articles

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.

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.

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.

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.

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.

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.

Nonlinear planar and non-planar vibrations of viscoelastic fluid-conveying pipes with external and internal resonances

In this study, nonlinear vibrations of a viscoelastic fluid-conveying pipe under in-plane and out-of-plane transverse excitations are investigated, with emphasis on external and internal resonances. For this purpose, a three-dimensional pipe model that couples the in-plane and out-of-plane vibrations is developed based on the Euler–Bernoulli beam theory together with the Kelvin–Voigt viscoelasticity. The Galerkin method is performed on the nonlinear governing equations, and the resultant discretized equations are solved via the pseudo-arclength continuation method and a direct integration method. The dynamical responses of the pipe are examined in both sub- and super-critical regions and presented by plotting the frequency–amplitude curves, force–amplitude curves, basins of the attraction, time histories, and phase portraits. A comparison between the simplified two-dimensional model and the present three-dimensional model is presented to showcase the possibility of the in-plane and out-of-plane vibrations in regions of primary and principal parametric resonances, highlighting the modal interaction between the in-plane and out-of-plane transverse modes. In the presence of a two-to-one internal resonance, in addition to typical jumping and hysteresis, the two-peak resonance behavior is observed. The numerical results not only show complex nonlinear dynamics of the fluid–structure interaction system, but also provide hints for safe design and novel application of such structures.

Instability analysis of fluid-filled angle-ply laminated circular cylindrical shells subjected to harmonic axial loading

This paper presents the dynamic instability and nonlinear vibration analysis of fluid-filled laminated composite circular cylindrical shells subjected to harmonic axial loading. The shells are assumed to be simply supported and filled with still fluid. The mathematical model is prepared by using higher-order shear deformation theory (HSDT) which considers skew modes for accurate results in the analysis of angle-ply laminated composite cylindrical shells. The contained fluid is assumed to be non-viscous and incompressible. The nonlinear governing partial differential equations (PDEs) are obtained using Hamilton's principle, and these equations are discretised into ordinary differential equations (ODEs) by employing Galerkin's method. The pseudo-arclength continuation method along with the incremental harmonic balance (IHB) method is used in order to compute the nonlinear responses, which are plotted in the frequency domain. The regions of instability are determined by adopting Bolotin's method. The time history responses and phase portraits are also studied for empty and fluid-filled laminated cylindrical shells using the Newmark-beta method. A complete set of results that examine the effect of fluid filling, static load factor, dynamic load factor and lamination scheme are obtained.

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

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.

Thermodynamic Bethe Ansatz past turning points: the (elliptic) sinh-Gordon model

We analyze the Thermodynamic Bethe Ansatz (TBA) for various integrable S-matrices in the context of generalized T overline{mathrm{T}} deformations. We focus on the sinh-Gordon model and its elliptic deformation in both its fermionic and bosonic realizations. We confirm that the determining factor for a turning point in the TBA, interpreted as a finite Hagedorn temperature, is the difference between the number of bound states and resonances in the theory. Implementing the numerical pseudo-arclength continuation method, we are able to follow the solutions to the TBA equations past the turning point all the way to the ultraviolet regime. We find that for any number k of resonances the pair of complex conjugate solutions below the turning point is such that the effective central charge is minimized. As k → ∞ the UV effective central charge goes to zero as in the elliptic sinh-Gordon model. Finally we uncover a new family of UV complete integrable theories defined by the bosonic counterparts of the S-matrices describing the Φ1,3 integrable deformation of non-unitary minimal models mathcal{M} 2,2n+3.

Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

Nonlinear vibration analysis of horizontal axis wind turbine blades using a modified pseudo arc-length continuation method

The horizontal axis wind turbine (HAWT) blades are exposed to parametric and direct excitations of harmonic nature due to the gravity and aerodynamic loads. Hence, the study of the periodic response of the HAWT blades under these operating loads is of great importance in structural design. In this paper, a new computationally efficient dynamic model for the blade is presented including the pre-twist effects. The Rayleigh-Ritz technique is applied to the derived dynamic model and a reduced order model (ROM) is devised in a systematic manner. An efficient method based on the pseudo arc-length continuation method is introduced to find out the edge-wise and flap-wise amplitude-frequency response (AFR) curves of the blade under harmonic wind inflow. Furthermore, the forced vibration of the blade under different operating loads including gravity, centrifugal and aerodynamic loads (due to the uniform and shear wind inflows) is studied separately. Applying the proposed method, the effects of structural damping, nonlinear terms, pre-twist ratio, and cyclic excitation due to the gravity force as well as the centrifugal force on the AFR curves are investigated. Eventually, the dynamic model and the presented method for the frequency analysis are validated with the available data.

Nonlinear forced vibrations of functionally graded piezoelectric cylindrical shells under electric-thermo-mechanical loads

This paper develops a new solution approach to solve nonlinear forced vibrations of functionally graded (FG) piezoelectric shells in multi-physics fields. The FG piezoelectric shells are subjected to electric-thermo-mechanical loads, and the effect of micro-voids is considered here. Motion equations are obtained by using Hamilton's principle, and combining with the Donnell nonlinear shallow shell theory. Afterwards, a new method combining multi-mode Galerkin scheme and Pseudo-arclength continuation method is used to solve the nonlinear multiple internal resonances and bifurcations of the multi-degree-of-freedom systems. The novel feature of this approach is that it can efficiently obtain the unstable solution and tackle the difficult problems in mathematics encountered during formulation. The results show that the external applied voltage, temperature change, external excitation, power-law exponent, and porosity volume fraction play important roles on nonlinear vibration response and bifurcation analysis of FG piezoelectric shells with micro-voids.

STUDY ON EQUILIBRIUM POINTS CHARACTERISTICS OF BINARY ASTEROID SYSTEMS BASED ON THE MASS DIPOLE MODEL

The local dynamics and periodic orbits in the vicinity of equilibrium points in binary asteroid systems are investigated. The dynamical model for synchronous binary asteroid system with one elongated asteroid and one small and regular asteroid is established. For the synchronous binary system, the influences of system parameters on location, stability, and variation trends of the equilibrium points are studied. The linear stability area for non-collinear equilibrium points is given. For the non-synchronous binary system, trajectories of equivalent equilibrium points are found. 1∶1 families of resonant orbits near the collinear equilibrium points of the synchronous binary asteroid system are obtained by the path searching correction method and the pseudo-arclength continuation method. This study can provide theoretical basis for the orbital design in binary asteroid explorations.

Nonlinear forced vibrations of FGM sandwich cylindrical shells with porosities on an elastic substrate

The nonlinear forced vibrations of functionally graded material (FGM) sandwich cylindrical shells with porosities on an elastic substrate are studied. A step function and a porosity volume fraction are introduced to describe the porosities in FGM layers of sandwich shells. Using the Donnell’s nonlinear shallow shell theory and Hamilton’s principle, an energy approach is employed to gain the nonlinear equations of motion. Afterwards, the multi-degree-of-freedom nonlinear ordinary differential equations are carried out by using Galerkin scheme, and subsequently the pseudo-arclength continuation method is utilized to perform the bifurcation analysis. Finally, the effects of the core-to-thickness ratio, porosity volume fraction, power-law exponent, and external excitation on nonlinear forced vibration characteristics of FGM sandwich shells with porosities are investigated in detail.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

A study on nonlinear, parametric aeroelastic energy harvesters under oscillatory airflow

In this study, based on parametric excitation originating from airflow oscillation, a novel nonlinear aeroelastic energy harvester is proposed. In this respect, first, the governing equation of the system is derived and studied thoroughly to understand the direct and indirect effects of airflow oscillation on the local and global responses of the system. Then, by using a pseudo-arclength continuation method based on the harmonic balance method, the stable and unstable periodic and quasi-periodic responses of the system are tracked and analyzed. It is demonstrated that the proposed self-parametric (combination parametric and self-excitation) energy harvester can extract more power than the respective nonparametric system for a wide range of amplitudes and frequencies. The gained knowledge of parametric, aeroelastic systems is applicable for both aero-harvesters and other aeroelastic systems undergoing flow oscillation.