Nonlinear Equations Of Motion Research Articles

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

Formulation and free vibration analysis of spinning Timoshenko composite beams using nonlinear geometrically exact theory

AbstractIn this study, a set of generalized nonlinear equations of motion for Timoshenko composite shafts is derived using the geometrically exact approach. These partial differential equations of motion are discretized by Galerkin method and the free vibration of an orthotropic spinning shaft with hinged‐hinged boundary conditions is investigated using the multiple‐scale method. We find that in the case of two mode discretization, composite couplings can have significant linear and nonlinear effects on the behavior of the shaft and that these effects were not predictable in the classical model. Also, in the investigating of composite layering we find that the effects of the fibers angle as well as the composite couplings can be noticeable, and these effects are greater in the case of the asymmetrical layering.

Transient Response of Sandwich Plates with Corrugated Core Under Mechanical-Thermal Loads

The nonlinear dynamic responses of the corrugated sandwich plate under mechanical-thermal loads are studied and analyzed. Based on hyperbolic parabola shear deformation plate theory (HPSDT), the partial differential equation of the sandwich plate with the corrugated core is established. Using the Galerkin truncation method, the nonlinear motion equation is derived. By the solution, the response of the corrugation plate with simply supported four edges at different temperatures is obtained and validated. Then the transient responses of the corrugated sandwich plate under different impact loads are analyzed, and the effects of the base materials, the corrugation types and the structural parameters of the corrugated plate on the transient responses are discussed in detail. The research provides a reference for improving the impact resistance of the sandwich plate in practical application.

Solving the Non-Linear Dynamic Equations of Motion of a Variable Curvature Continuum Robot Using Runge-Kutta Method and MATLAB Software

Solving the Non-Linear Dynamic Equations of Motion of a Variable Curvature Continuum Robot Using Runge-Kutta Method and MATLAB Software

Nonlinear Dynamic Analysis of FG Fluid Conveying Micropipes with Initial Imperfections

This paper investigates the nonlinear dynamics of FG-FCMPs with initial imperfections. Based on the MCST, the nonlinear equations of motion and the corresponding boundary conditions are established by applying Hamilton’s principle, the Euler–Bernoulli beam theory, and von-Kármán geometric nonlinearity. To describe the initial geometric imperfection of the FG-FCMP, the first-order vibrational mode is employed. Subsequently, in cases of primary parametric resonance, 1:2 subharmonic resonance for the first-order mode, as well as primary resonance for the second-order mode, Galerkin’s method, and the multiple scale method are utilized to analyze the amplitude–frequency responses of the imperfect FG-FCMP. The numerical simulations test the influence of flow velocity, micro-scale parameter, geometrical imperfections, and external loads on the nonlinear characteristics of a coupled system with two DOFs. It is found that, as the increase of flow velocity, micro-scale effect, and external load, the amplitude of the first two modes can be increased. The hardening characteristics are converted into the softening characteristics due to the imperfect effect. Furthermore, numerical results provide a more comprehensive understanding of the nonlinear dynamics of FCMPs for both periodic and chaotic motions.

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.

A dynamic study of a bead sliding on a wire in fractal space with the non-perturbative technique

Drawing on the principles of fractal properties and nonlinear vibration analysis, this paper delves into the investigation of a moving bead on a vertically rotated parabola. The dynamical nonlinear equation of motion, incorporating fractal derivatives, transforms traditional derivatives within continuous space. Consequently, the equation of motion takes the form of the Duffing-Van der Pol oscillator. Utilizing a non-perturbative approach, the nonlinear oscillator is systematically transformed into a linear one, boasting an exact solution. The analytical solution yields two valid formulas governing the frequency-amplitude relationships. Numerical solutions affirm that these proposed formulas offer highly satisfactory approximations to the analytical solution. Leveraging fractal properties through Galerkin’s method, the paper successfully determines the fractalness parameter of the medium, shedding light on the intricate dynamics of the system.

Non-linear dynamics of a test particle near the Lagrange points L4 and L5 (Earth-Moon and Sun-Earth case)

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits

The dependent coordinates in the linearization of constrained multibody systems: Handling and elimination

In this paper, the handling of the dependent coordinates in the linearization of constrained multibody systems is clarified. To this end, a brief but comprehensive overview of linearization approaches used in the field of Multibody System Dynamics is presented. These procedures are illustrated by using a common notation and classified into four groups, depending on the initial form of the nonlinear equations of motion and the selection of the generalized coordinates (a redundant or a minimal set). The handling of the dependent coordinates in each of the procedures is discussed. The linearized equations of motion of constrained multibody systems are of great importance in several applications, such as linear stability and modal analyses, the design of linear state feedback controllers or the building of state and input estimators, like Kalman filters. When modeling a constrained multibody system, a number of coordinates greater than the number of degrees of freedom of the system is usually used. While some linearization approaches make use of the complete set of coordinates, several procedures are based on a coordinate partition in terms of independent and dependent coordinates to reduce the number of linearized equations of motion. In this latter scenario, the role of the dependent coordinates in the linearization is important and, in a general case, these dependent coordinates cannot be ignored to obtain the correct linearized equations of motion. This aspect, which may seem obvious, is not straightforward and is addressed in this work. The importance of considering the set of dependent coordinates in the linearization is demonstrated with a well-acknowledged bicycle benchmark multibody model and an electric kickscooter multibody model with rear and front suspensions. To this end, the Jacobian matrix and the linear stability results of these case studies are computed by considering different choices of independent and dependent coordinates.

Non-linear AUV Controller Design Using Logic-Based Switching PID Control

The inspection of pipes without stopping pipeline operations is of industrial interest due to the inherent economic benefits. A pipe inspection robot design is proposed together with a suitable motion controller. The proposed robot is an Autonomous Underwater Vehicle (AUV) architecture that does not require shunting, draining or unearthing pipework for inspection purposes. This posed the challenge of controlling the AUV through a narrow pipe where manouvers are restricted. A set of general non-linear equations of motion were identified and refined using existing research and strip theory. The non-linear analytical model was implemented in Simulink to enable real-time monitoring and controller tuning. A non-linear controller based on a combination of classical PID theory and switching logic was developed to control the platform. The controller was transplanted into a Hardware-in-Loop (HIL) testing model developed with the Arduino and Simulink software suites. Pool tests measuring the parameters pitch and yaw angles showed that when operated independently with a 5° input, the maximum overshoot was 0.5° or 10% of the command value with a maximum angular velocity of 1.25°/s. When operated simultaneously, the overshoot rose to 30% with a constant error in the region of 1° over the target. Distance readings conducted with ultrasonic sensors to the pipe wall showed constant Sway-Heave bias errors from the centerline as −5 mm, and 4 mm, respectively, with an error range of 4 mm / −7 mm for Sway and 6 mm/ −4 mm for Heave. Pitch and heave motions were up to 18% faster than yaw and sway motions due to actuator orientation, with speeds of 6.25°/s and 40.57 mm/s, respectively. Despite the turbulence present during tests, the controller successfully drove the AUV to target positions through active flow and presented a reasonable basis for further refinement.

Nonlinear Vibration and Dynamic Buckling of Complex Curved Functionally Graded Graphene Panels Reinforced with Inclined Stiffeners

This paper presents a new semi-analytical approach for the nonlinear vibration and dynamic responses of functionally graded graphene platelet-reinforced composite (FG-GPLRC) panels with complex curvature reinforced by orthogonal and/or inclined stiffeners in the thermal environment resting on the nonlinear viscoelastic foundation with the nonlinearities of von Kármán in the framework of the higher-order shear deformation theory (HSDT). The three curvature types of complexly curved panels are considered to be cylindrical, sinusoid, and parabola panels. Orthogonal and/or inclined stiffeners are modeled utilizing an improved smeared stiffener technique. The stress function form is estimated by applying the like-Galerkin method for complexly curved panels. By applying the Lagrange function and Euler–Lagrange equation, the nonlinear equations of motion of the panels are obtained. The viscous damping effects of the viscoelastic foundation are considered by utilizing the Rayleigh dissipation function. Numerical results are examined utilizing the Runge–Kutta method to acquire the time-deflection curves, and by utilizing the Budiansky–Roth criterion, the critical dynamic buckling loads are determined. From the investigated results, it is possible to evaluate the nonlinear vibration and buckling dynamic responses of three forms of panels with stiffeners.

Analytical and numerical stability analysis of a toroidal wheel with nonholonomic constraints

In this paper, a detailed and comprehensive linear stability analysis of a rolling toroidal wheel is performed. The wheel is modeled as a rigid toroid-shaped body rolling without slipping on a horizontal surface. The nonlinear equations of motion constitute a Differential-Algebraic Equations system, given by the dynamic equilibrium equations augmented with the nonholonomic constraints, which arise from the no-slip condition. The circular steady motion and the linearized equations of motion along this relative equilibrium are obtained, for both the solid and hollow tori. The expressions of the linearized equations and the corresponding eigenvalues are derived analytically as a function of the torus aspect ratio. The variation of the stability boundary with the torus aspect ratio is shown. A comparison of the results obtained in the solid and hollow scenarios is included, and all the results are validated with the rolling hoop, which corresponds to a degenerate torus with zero aspect ratio. In the particular case of the steady straight-line rolling and spinning about a vertical diameter, which constitute limit motions of the circular steady motion, the critical rotational and angular speeds required for stabilization are obtained.

Nonlinear vibration and dynamic buckling responses of stiffened functionally graded graphene‐reinforced cylindrical, parabolic, and sinusoid panels using the higher‐order shear deformation theory

AbstractThe nonlinear vibration and dynamic responses of functionally graded graphene‐reinforced composite (FG‐GRC) laminated cylindrical, parabolic, and sinusoid panels stiffened by FG‐GRC stiffeners in the uniformly distributed temperature variation are presented in this paper. An improved smeared stiffener technique is used to model the added stiffnesses of stiffeners to the total stiffnesses of panels. The higher‐order shear deformation shell theory (HSDT) with the geometrical nonlinearities of von Kármán is applied to establish the governing formulations. The stress function form is estimated using the approximated technique for complex curvature panels. Lagrange function and Euler‐Lagrange equations are applied, and the Rayleigh dissipation function is taken into account to obtain the nonlinear equation of motion. Numerical examples are investigated using the Runge‐Kutta method to obtain the dynamic responses of panels, and the critical dynamic buckling loads of panels are considered using the Budiansky‐Roth criterion. Some significant remarks on the nonlinear vibration and dynamic buckling responses of three types of stiffened panels can be recognized from the numerical examples.

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.

Analytical study on nonlinear frequency response and vibration isolation of multistage clutch damper in automotive powertrain

This paper studies the nonlinear frequency response and vibration isolation performance of the multistage clutch damper at resonance. First, a 2-DOF drivetrain model is established considering the piecewise-linear stiffness, piecewise-linear damping and asymmetric transition angles of the multistage clutch damper. Then, the averaging method is employed to deduce the analytical solution of the nonlinear motion equation, and the stability of the periodic solution is studied using Routh-Hurwitz criterion. The nonlinear behaviors of the oscillator are discussed based on the theoretical results, and the numerical simulation is carried out to verify the credibility of the proposed method. Finally, the sensitivity analysis is conducted to investigate the influence of the system parameters on the amplitude-frequency response of the multistage clutch damper. The results indicate that the isolation performance of the multistage clutch damper is sensitive to the first and second stages of damping and the first-stage transition angles. By increasing the first and second stages of viscous damping by 50% and simultaneously reducing the first-stage angle range by 3° with 1° asymmetry, the isolation performance of the clutch damper is effectively improved, reducing the vibration amplitude at resonance by 28.8%.

Nonlinear forced vibration analysis of PFG-GPLRC conical shells under parametric excitation considering internal and external resonances

Comprehending the vibration dynamics of porous functionally graded-graphene platelet reinforced composite (PFG-GPLRC) structures is vital for accurate predictions and reliability in practical applications. This study addresses gaps in nonlinear dynamics, instability, and frequency response research within truncated PFG-GPLRC conical shells under parametric loading and ½ subharmonic and 1:1 internal resonance. To achieve this, three porosity distributions in metal foam (uniform, non-uniform symmetric, and non-uniform asymmetric) are considered, along with various graphene platelet dispersion patterns (GPL-O, GPL-V, GPL-U, GPL-A, and GPL-X) within the matrix. These considerations lead to a comprehensive conical shell model. Utilizing the first-order shear deformation theory and von-Karman's assumptions, stress-strain relations are extracted, yielding nonlinear motion equations for the truncated conical shell. Employing Galerkin's method and considering simply supported boundaries, two-degree-of-freedom equations of motion are derived. The research culminates in steady state frequency responses obtained through perturbation theory and the multiple scales method, encompassing ½-subharmonic excitation resonance and 1:1 internal resonance. Bifurcation points are analysed to highlight the influence of parametric excitation and system instabilities. A parametric study underscores the significance of porosity and graphene platelets within the metal foam in relation to system instability, revealing their intricate impact on PFG-GPLRC structure behavior.

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.

Numerical roll motion control by using fins based on the linear quadratic regulator and dynamic mode decomposition

Reducing the roll motion and acceleration of a ship is a critical concern for the safety and comfort requirement in a seaway. In this paper, the uncontrolled roll motion time series is obtained under the harsh beam sea simulation for a naval vessel using a single degree of freedom nonlinear equation of motion. Then the linear quadratic controller (LQR) controller is applied by utilizing one pair of fins (port and starboard) that were positioned amidship. The amplitude and rate saturations of the fins are taken into consideration in the simulations. The LQR controller is designed based on an analytical mathematical model, with the selection of appropriate controller parameters. Then, an estimation is made based on the dynamic mode decomposition method for the ship model by taking measurements from online states as if those are derived from the experimental measurements. Finally, the ‘estimated mathematical model’ is used for the LQR controller design. The results demonstrate the successfulness of the estimation process, with only minimal degradation in performance observed.

Nonlinear dynamic analysis of a geometrically imperfect sandwich beam with functionally graded material facets and auxetic honeycomb core in thermal environment

In this study, a model for the nonlinear dynamic analysis of the geometrically imperfect sandwich beam with functionally graded material (FGM) facets and auxetic honeycomb core is proposed. The auxetic feature of the honeycomb material is realized by a homogenization method, which gives the expressions of the effective material coefficients in terms of material property and cellular geometric parameters. The nonlinear motion equation of the geometrically imperfect beam is derived based on the Reddy's higher shear deformation theory, von Kármán nonlinear theory, as well as general geometric imperfection function. The thermal effect on the material properties and structural rigidity is also taken into account in this model. The Newmark method and Newton–Raphson iterative method are employed to solve the nonlinear motion equation of the sandwich beam. In numerical examples, the dynamic response of the sandwich beam under two types of load, i.e. moving load and impulse load, are discussed in details. The influences of various factors (such as material parameter, geometric imperfection, and thermal condition) on the nonlinear dynamic behaviors of the sandwich beam are elaborately investigated. The numerical results show those factors can play significant roles in the bearing ability, structural stiffness, or nonlinear dynamic behaviors of the sandwich beam in certain conditions. Some key conclusions on the influences are drawn from the numerical results, which can be a useful reference for future investigations.

On the detection of nonlinear normal mode-related isolated branches of periodic solutions for high-dimensional nonlinear mechanical systems with frictionless contact interfaces

This contribution introduces a methodology for the detection of isolated branches of periodic solutions to the nonlinear mechanical equation of motion for systems featuring frictionless contact interfaces. This methodology relies on a harmonic balance method-based solving procedure combined with the application of the Melnikov energy principle. It is able to predict the location of isolated branches of solutions in the vicinity of families of autonomous periodic solutions, namely nonlinear normal modes. The methodology is first applied on a two-degree-of-freedom phenomenological system in order to illustrate its relevance and accuracy. In particular, for this academic application, the proposed methodology yields an understanding of the discontinuous evolution of the first nonlinear resonance frequency as the forcing amplitude increases. Isolated branches of solutions featuring high amplitudes of vibration are detected far beyond nonlinear resonance frequencies obtained with a typical application of the harmonic balance method coupled with an arc-length continuation algorithm. Demonstration of the applicability of the proposed methodology to high-dimensional industrial finite element models is made with the nonlinear vibration analysis of a transsonic compressor blade, NASA rotor 37, subjected to an harmonic loading and blade-tip/casing structural contacts. The proposed methodology yields numerous isolated branches of solutions, which relevance is assessed by means of time integration simulations. In the end, the presented results underline that typical continuation algorithms may yield a significant underestimation of nonlinear resonance frequencies.