Nonlinear Differential Equations Of Motion Research Articles

Size-dependent nonlinear dynamics of two-directional functionally graded microbeams in thermal environment under a moving mass

The size-dependent nonlinear dynamics of two-directional functionally graded (2D-FG) microbeams in a thermal environment under a moving mass is studied for the first time in the framework of the refined higher-order shear deformation theory (HSDT) and modified couple stress theory (MCST). The variation of the temperature-dependent material properties in the axial and thickness directions is estimated by the Mori–Tanaka homogenization scheme. Considering the rotary inertia and shear deformation, the nonlinear differential equations of motion, obtained from Hamilton’s principle, are discretized using an enriched beam element. Numerical results determined by the Newmark method in combination with Newton–Raphson iterative procedure confirm the efficiency of the derived beam formulation in predicting the nonlinear dynamics. The enriched beam element, which is derived herein by using hierarchical functions to enrich the conventional shape functions, is capable of furnishing accurate nonlinear dynamic characteristics with fewer elements compared to the conventional one. It is shown that the difference between the dynamic response predicted by the linear analysis and the nonlinear one increases by increasing the temperature, but it decreases by increasing the size scale parameter. The effects of the material gradation, temperature rise and micro-scale parameter on the nonlinear dynamic behavior of the 2D-FG microbeams are studied in detail and highlighted.

Nonlinear Vibrations and Stability Analysis of GPL Reinforced Pipes Conveying Fluid Excited by Arbitrary Initial Conditions: An Optimized Analytical Solution

In this study, nonlinear vibrations and stability of multilayer functionally graded (FG) pipes conveying fluid laying on nonlinear elastic foundation and reinforced by graphene nanoplatelets (GPLs) are investigated with an emphasis on optimized analytical method. Various types of GPL distribution patterns are considered and to estimate the mechanical characteristics of the reinforced pipe the Halpin–Tsai micromechanics theory is applied. The nonlinear differential equation of motion is obtained by using Von–Kármán strain relations in conjunction with the Euler–Bernoulli beam theory and applying Hamilton’s principle. The Galerkin technique has been used for discretizing the partial differential equation. The optimized homotopy analysis method (HAM), as a strong analytical method, is utilized to solve the nonlinear differential equation by considering different initial excitations. The convergence-control parameter is taken into account to guarantee the convergence of the analytical solution. In this study, an exact solution based on HAM for the critical flow velocity and [Formula: see text]th nonlinear frequency is presented. Additionally, series solutions for the nonlinear time responses of the transverse and longitudinal vibrations of fluid-conveying pipe based on optimized HAM are obtained for the first time. In the numerical results, the effects of arbitrary initial conditions and different physical characteristics on the nonlinear frequencies and time responses are extensively examined. It is shown that the nonlinear frequencies and critical flow velocity of the pipe depend not only on the initial excitation amplitude, but also on the entire initial excitation function applied on the pipe.

Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies

In spite of extensive studies on the nonlinear vibrations of cylindrical shells, the significant influences of some effective modes and the in-plane nonlinearity on the nonlinear coupled multi-mode vibrations are not clear. In this work, which is based on Donnell’s nonlinear shell theory and Amabili–Reddy’s third-order shear deformation theory, the nonlinear differential equations of motion of both the thin-walled and moderately thick cylindrical shells under the simply-supported boundary condition are established via Lagrange equations. These high-dimensional differential equations with the quadratic and cubic nonlinearities are solved by using an iteration procedure that is a combination of incremental harmonic balance method, pseudo-arclength method and extrapolation techniques. Turning and bifurcation points of the system are determined with the help of the direct method, the stability of solution of the frequency–response is examined by using the multi-variable Floquet theory. The numerical responses obtained by adopting two present shell theories are compared to investigate the influence of ignoring the in-plane nonlinearity on nonlinear vibrations. In the coupled multi-mode vibrations with respect to the fundamental mode m,n, apart from the regular modes i×m,j×ni=1,3,5,…,andj=0,1,2,3,… that include the axisymmetric and asymmetric modes, some irregular modes are taken into account to study the frequency–responses. Results show that present iteration procedure is efficient and successful to get the frequency–responses of the coupled multi-mode vibrations, and the influence of the irregular mode on the coupled multi-mode vibration is dependent on the relationship between the natural frequency of the irregular mode and that of the fundamental mode.

GPL-Reinforced composite piezoelectric microcantilever dynamics in atomic force microscope

Currently, Atomic Force Microscope (AFM) is utilized as an efficient tool for nanoscale measurements and nanomanipulation of particles. The central part of AFM is considered to be the microcantilever (MC) used in the device. AFM MC is generally made from silicon. Since functionally graded graphene-nanoplatelet-reinforced composite (GPLRC) materials are highly popular in industry and academic research, the present study aims to be the first attempt to investigate the possibility of utilizing these materials in AFM MC. Also, it seeks to compare the nonlinear vibrational behavior of the MC made from this type of material with that of the silicon MC and also to investigate the vibrational behavior of the GPLRC MC in interaction with the sample surface. To do so, the GPLRC MC vibrated with the piezoelectric layer on its top is utilized. In order to increase the sensitivity of the MC to the interaction force between the surface and the probe tip, the tip of the MC is considered narrower. The differential equation of the vibrational motion of the MC near the surface of the sample is derived by considering the Lennord-Jones model according to the Euler-Bernoulli theory and applying the Lagrange method. In order to solve the nonlinear differential equation of motion with partial derivatives in terms of the time and the position of each point on the MC, Galerkin variable-separation method and multiple time scale (MTS) method are applied. The results of the dynamic modeling near the surface of the sample indicate that the MC made of GPLRC is more sensitive to the nonlinearity of the interaction force as compared to the common silicon MC.

Application of Parametric Forced Tuned Solid Ball Dampers for Vibration Control of Engineering Structures

In this paper, parametric forced tuned solid ball dampers (TSBD) are considered for vibration control of engineering structures in an untypical way. The special feature of the presented investigation is to evaluate the potential application of parametric forcing of the rolling cylindrical or spherical body in the runway for reducing the vertical vibrations of a vibration-prone main system. Typically, tuned solid ball dampers are applied to structures that are prone to horizontal vibrations only. The coupled nonlinear differential equations of motion are derived and the phenomenon of parametric resonance of the rolling body in the runway is analyzed. A criterion for avoiding parametric resonance is given to achieve the optimal damping effect of the TSBD. In the second part of the article, a method for the targeted use of parametric resonance to reduce the vertical vibrations of engineering structures is presented and verified, considering a biaxially harmonic excited pedestrian bridge. It is shown that, with a suitable choice of damper parameters, a stable vibration of the rolling body in the runway is formed over the course of the vibration despite the occurrence of parametric resonance and that the maximum vertical vibration amplitudes of the main system can be reduced up to 93%. Hence, the here presented untypical application of parametric forced TSBD for reducing the vertical forced vibrations of vibration-prone main systems could be successfully demonstrated.

Multibody-Dynamics Approach to Study the Deformation and Aerodynamics of an Insect Wing

In this study, a multibody-dynamics simulation approach was developed for a hawkmoth flexible wing. The wing structure is modeled as a chain of rigid bodies connected by elastic springs, and the aerodynamic force is measured by the extended unsteady vortex-lattice method. The multibody-dynamics and aerodynamic solvers are combined by an implicit coupling approach, and the quasi-Newtonian method is adopted to solve the system of nonlinear differential equations of motion. For validation, numerical results were compared with measurement data from a robotic wing and a living insect. A parametric analysis was conducted to study the effects of several kinematic parameters on the deformation and aerodynamic performance of the wing in hover. In most cases, using a flexible wing is far more efficient in terms of force production in comparison with its rigid counterpart. In general, wing deformation may cause considerable differences between the wing-tip and wing-base kinematic parameters. In particular, elevation motion as observed in living insects may be due to the passive oscillations of elastic elements, as opposed to a deliberate motion.

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.

Effect of crack characteristics on the vibration behavior of post-buckled functionally graded plates

This study investigates the free vibration analysis of a cracked post-buckled functionally graded plate under a uniaxial compressive load. The crack is assumed to be open-edged and modeled using a linear, rotational spring. The material is distributed through its thickness gradually. The nonlinear differential equations of motion are derived using the Mindlin plate theory for an initially imperfect plate. First, the nonlinear static differential equations are converted to algebraic equations using the differential quadrature method and solved by an arc-length strategy. The solution of the post-buckling equations gives the equilibrium state of the vibration. Next, the results were implemented into the motion equations to derive the vibrational differential equations. Then, the solution of the standard linearized eigenvalue problem, using the differential quadrature method, results in the plate's natural frequencies and mode shapes. The study compares the effects of various plate parameters on its vibrational behavior before and after buckling. It can be an excellent reference to consider the effect of crack characteristics on vibrations of the functionally graded plates before and after buckling. The presented results agree well with those available in the literature and those computed from the finite element method.

Non-Linear Dynamic Modeling and Analysis of Split-Torque Face-Gear Drive Systems

Abstract A non-linear dynamic model of a multi-mesh involute spur pinion-driven face-gear split-torque drive system is developed. The lumped mass model consists of five pinions and two face-gears with seven rotational degrees-of-freedom. Model includes two inputs, two outputs, and three idler gears meshing with two identical face-gears at the bottom and top of the assembly. Face-gear tooth form and corresponding spur gear teeth are established by differential geometry and the theory of gearing. Face-gear drive system mesh stiffnesses are established by utilizing finite strip method for the first time in the literature. The mesh stiffness and damping are time-variant. The model includes clearance-type non-linearity. The harmonic balance method is utilized to solve the non-linear differential equations of motion. The results are compared with the time simulation results. The stability is checked with Floquet theory and bifurcation diagrams from Poincare sections. The model is fully capable of generating the tooth geometries, mesh stiffnesses, and gear train dynamics without reliance on any package programs. Hence, the model provides flexibility and fast computation for parametric studies compared to existing literature. The effect of the pinions’ orientations on the face-gear is investigated to evaluate the mesh phasing effects among system dynamic response. The effect of sub-harmonic motion on the dynamic response of the split-torque face-gear system is also demonstrated for the first time in the literature. The case studies demonstrate that sub-harmonic resonance peaks are avoided.

Resonant vibrations of a non-ideal gyroscopic rotary system with nonlinear damping and nonlinear stiffness of the elastic support

When motor performance characteristic is unknown, non-linear differential equations of motion of nonideal gyroscopic rigid rotary system with nonlinear cubic damping and nonlinear stiffness of the elastic support turn out to be numerically unsolvable.•In this case, the method uses the motor performance characteristic expression found from the frequency equation of forced stationary oscillations based on assumption that the angular acceleration is many times less than the square of the angular speed of rotation, replacing the stationary rotation angular speed with the shaft rotation angle derivative.•The method correctness is evidenced by a good consistency between the rotor motion equation numerical solution results and the analytical solution results, and by the nonlinear cubic damping of the shaft angular coordinate oscillograms obtained by direct simulation, as well as by comparison with the results of numerical simulation for a straight-line DC motor performance characteristic.•The method limitations are that it is used for the first approximation and weak nonlinear oscillations in the resonance region, where the shaft rotation speed is of the order of the oscillating system natural frequency.

Transient Response Sensitivity Analysis of Localized Nonlinear Structure Using Direct Differentiation Method

Based on the direct differentiation method, sensitivity analysis of transient responses with respect to local nonlinearity is developed in this paper. Solutions of nonlinear equations and time-domain integration are combined to compute the response sensitivities, which consist of three steps: firstly, the nonlinear differential equations of motion are solved using Newton–Raphson iteration to obtain the transient response; secondly, the algebraic equations of the sensitivity are obtained by differentiating the incremental equation of motion with respect to nonlinear coefficients; thirdly, the nonlinear transient response sensitivities are determined using the Newmark-β integration in the interested time range. Three validation studies, including a Duffing oscillator, a nonlinear multiple-degrees-of-freedom (MDOF) system, and a cantilever beam with local nonlinearity, are adopted to illustrate the application of the proposed method. The comparisons among the finite difference method (FDM), the Poincaré method (PCM), the Lindstedt–Poincaré method (LPM), and the proposed method are conducted. The key factors, such as the parameter perturbation step size, the secular term, and the time step, are discussed to verify the accuracy and efficiency. Results show that parameter perturbation selection in the FDM sensitivity analysis is related to the nonlinear features depending on the initial condition; the consistency of the transient response sensitivity can be improved based on the accurate nonlinear response when a small time step is adopted in the proposed method.

Finite strain-based theory for the superharmonic and subharmonic resonance of beams resting on a nonlinear viscoelastic foundation in thermal conditions, and subjected to a moving mass loading

We address the nonlinear free vibration, superharmonic and subharmonic resonance response of homogeneous Euler–Bernoulli beams resting on nonlinear viscoelastic foundations, under a moving mass and an abrupt uniform temperature rise. The nonlinear differential equation of motion stemming from the Hamiltonian principle and Finite Strain Theory is discretized according to a Galerkin decomposition method, and is solved by means of a multiple time scale method. A comparison between the Finite Strain theory and the Von-Karman approach is discussed, accounting for the effect of temperature rise, linear and nonlinear coefficients of the elastic foundation on the nonlinear vibration history and phase trajectory. At the same time, we check for the sensitivity of the frequency response of the system in superharmonic and subharmonic resonance for different input parameters, namely, location, velocity, and magnitude of the moving load, temperature rise and elastic foundation.

NONLINEAR VIBRATIONS OF THE "ROTOR – JOURNAL BEARINGS" SYSTEM

The equations of motion of a rotor system mounted on journal bearings with a non-linear characteristic are solved by high-precision analytical methods. A new technique has been developed for solving nonlinear differential equations of motion of rotor systems mounted on journal bearings, taking into account nonlinearity of reaction forces of the lubricating layer. Algebraic systems of equations were obtained that allow us to determine amplitudes of nonlinear oscillations of the rotor and supports, and construct the amplitude-frequency characteristics of the system for varying parameters of the rotor, supports and fluid depending on the angular velocity of the rotor. The conditions and frequency intervals for the presence of self-oscillations of the rotor and supports were determined. The amplitude-frequency characteristics of the nonlinear oscillations of the rotor system are obtained, taking into account nonlinearity of characteristics of journal bearings. The optimal parameters depending on the size of the gap and the oil film, the mass of the supports, the fluids used as a lubricating layer in the journal bearing, with rigidity and damping coefficients, at which the magnitudes of the amplitudes of self-excited oscillations have optimal values, are obtained.

Nonlinear forced transient response of rotating ring on the elastic foundation by using adaptive ANCF curved beam element

Based on the Euler-Bernoulli beam theory, nonlinear axial Lagrangian strain and simplified curvature, the nonlinear differential equations of motion for rotating ring on the elastic foundation (RREF) are established by absolute nodal coordinate formulation curved beam element. In non-rotating state, the double modes at the same frequency are observed. In rotating state, the first-order critical angular speed, the frequency of forward traveling wave and backward traveling wave are obtained. Then, the nonlinear forced transient response (radial deformation rate) of the RRE under the action of concentrated load at bottom point is analyzed. In order to improve computational efficiency, the adaptive ANCF curved beam element is proposed, and reasonable range of two key parameters, i.e., element length and rotation angle of adjacent nodal tangential vector are determined. And it is used to analysis the influences of internal pressure, radial and tangential elastic support stiffness, and Young's modulus on nonlinear transient response characteristics of RREF. In order to study the standing wave phenomenon for a period, the RREF-hub contact model is established and used to analyze the nonlinear transient response of RREF at the first-order critical angular speed.

Size-dependent behavior of a MEMS microbeam under electrostatic actuation

The size-dependent behavior of a silicon microbeam with an axial force in MEMS is studied using a nonlinear finite element procedure. Based on a refined third-order shear deformation theory and the modified couple stress theory (MCST), nonlinear differential equations of motion for the beam are derived from Hamilton’s principle, and they are transferred to a discretized form using a two-node beam element. Newton-Raphson based iterative procedure is used in conjunction with Newmark method to obtain the pull-in voltages and deflections of a clamped-clamped microbeam under electrostatic actuation. The influence of the axial force, applied voltage and material length scale parameter on the behavior of the beam is studied in detail and highlighted.

A comprehensive study on the coupled multi-mode vibrations of cylindrical shells

As is discussed in the previous work (Dong et al. 2021), the functions of beam-like modes are not suitable for simulating the cylindrical shells with the clamped and free ends. Also, ignoring the influences of the in-plane inertias can lead to large relative differences of the solutions of vibration characteristics. This work aims to accurately calculate the vibration characteristics of the cylindrical shells under various classical boundary conditions comprised of movable simply-supported, immovable simply-supported, sliding, clamped and free ends. Meanwhile, this work deals with the nonlinear coupled multi-mode vibrations of the cylindrical shell under different boundary conditions with the in-plane inertias taken into consideration. Firstly, several modified Chebyshev polynomials in terms of the axial coordinate and new harmonic functions in terms of the circumferential coordinate are employed to establish the equations of natural frequency in the frame of Lagrange equations for all the single-modes, where the polynomials are independent of geometries and material properties of the shells. And then, the circumferential-wave-dependent mode functions of the cylindrical shells are proposed, based on these mode functions coming from the linear vibration, a unified nonlinear differential equation of motion of the cylindrical shells under different boundary conditions is established for the first endeavor, in which the coupled multi-mode vibrations containing various vibration waves are considered. Finally, the high dimensional differential equation with the square and cubic nonlinearities is investigated by employing the incremental harmonic balance method. Results shown that polynomials accurately satisfy various boundary conditions, and the contribution of the single-mode to the coupled multi-mode vibration is related to the loading patterns.

КОЛЕБАНИЯ ПРИ НЕЛИНЕЙНОМ ПАРАМЕТРИЧЕСКОМ И ОГРАНИЧЕННОМ ВОЗБУЖДЕНИИ

Parametric oscillations are considered for a non-linear elastic force and non-linear parametric excitation - the presence of a non-linear expression with a periodic coefficient in the oscillation equation. The model for analysis is a rod, which is driven by an energy source - an engine of limited power, connected to the rod by means of a crank and a spring. Nonlinear differential equations of motion of the system are solved using the method of direct linearization. On the basis of this method, the equations of non-stationary, stationary mo-tions and the conditions for the stability of oscillations are derived. To obtain information about the influence of a nonlinear parametric action on the dynamics of the system, calculations were carried out. A comparison of the results obtained for linear and nonlinear parametric excitations shows a significant difference. With linear elas-tic force and linear parametric excitation, the amplitude turns to infinity. And with nonlinear parametric excitation, this does not happen. The resonance regions, the shapes of curves of amplitude and load on the energy source differ (respectively, physical effects when the source speed changes).

Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction.
 Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

Passive fluid-induced vibration control of viscoelastic cylinder using nonlinear energy sink

This study focuses on the performance of the nonlinear energy sink (NES) in passive controlling the cantilever cylinder vibrations subjected to the external fluid flow. The nonlinear differential equations of motion are obtained by considering the large strain-displacement relation and viscoelastic behavior. Wake oscillation in fluid-structure interaction is modeled based on the Van der Pol wake oscillator model with is the classic acceleration coupling between the cross-flow motion and wake. Based on the Von Karman strain-displacement relation, and Euler-Bernoulli beam theory, the nonlinear vibration equations which are coupled with attached NES motion are obtained using Newton's second law, and discretized by applying the Galerkin method. The fluid flow velocity and nonlinear stiffness, damping, and mass of the NES are studied to determine their effects on the vibration response of the system. The present study comprehensively evaluates the effects of adding a NES on the lock-in phenomenon and maximum oscillating amplitudes of a cantilever cylinder, and guides to determine the best design of NES for significant fluid-induced vibration mitigation.

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.