Nonlinear Equations Of Motion Research Articles

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.

Functionally graded doubly-curved shell with temperature dependent material properties and surface-mounted MEE layers

In the present work, the geometrically nonlinear behavior of a simply supported functionally graded doubly curved shell with a surface-mounted magneto-electro-elastic layer is investigated based on Donnell’s theory. Kirchoff’s assumption and von Kármán’s strain-displacement relationship are used to obtain the governing differential equations. The through-the-thickness shell temperature is computed using Fourier’s heat conduction equation. Elastic material properties are assumed to be temperature-dependent and follow the power-law distribution function of the constituent ceramic-metal volume fractions. The magneto-electro-elastic layer is polarized along the thickness direction, which is subjected to electric and magnetic potentials. Gauss’s laws for electrostatics and magnetostatics are satisfied for magneto-electric materials. Nonlinear partial differential equations of motion are reduced to ordinary differential equations by introducing a stress function and using a single-term time-dependent assumed displacement function. Nonlinear free vibration response is obtained by direct numerical integration. After comparing the derived model with published literature results, numerical studies on the effects of applied electric and magnetic coupled fields of surface-mounted magneto-electro-elastic layers on shell vibration are reported. The nonlinear frequency ratios with respect to specified displacement for both spherical and cylindrical shells with thin Piezoelectric BaTiO 3 and Piezomagnetic CoF e 2 O 4 panels across material compositions, the radius of curvature, and thermal conduction are studied.

On the selected problems of time-dependent dynamics of composite truncated conical shells-like aerospace structures

In this paper, the nonlinear dynamic responses of ring-stiffened truncated conical sandwich shells-like aerospace structures fabricated by perfectly bonded graphene platelets (GPLs) reinforced composite porous core and face sheets (GPLRC-PC-FS) under asymmetric internal ring-shaped moving load are investigated. The sandwich shells have edge point supports, which are simulated using the translational and rotational artificial springs. In addition to the open cell porosities, closed cells are also considered and analyzed. The spatially discretized nonlinear motion equations are derived based on the first-order shear deformation theory (FSDT) under the von Kármán geometric nonlinearity assumptions using the Ritz method with Chebyshev polynomials as its admissible basis functions. Then, Newmark's time integration scheme along with the Newton–Raphson method is employed to obtain the shell dynamic responses. After demonstrating the robustness and accuracy of the approach, the influence of rings number, their size and arrangement, core porosity distribution and structure, face sheets, GPLs parameters, moving load velocity and boundary conditions on the shell responses are studied. The results show that the shell dynamic behaviors are significantly affected by porosity, but almost are independent of the type of porosity distribution pattern and its structure (open and closed cell porosities). It is shown that the shell deflection can be minimized by suitably choosing the number of rings, their dimensions and arrangement. Moreover, the nonlinear behaviors of the shells have a significant correlation with the number of point supports and by adjusting the support stiffnesses, the nonlinear dynamic responses of the shells can be reduced.

Nonlinear vibroacoustic response and sound transmission loss analysis of functionally graded doubly curved shallow shells

This paper presents the semi-analytical nonlinear vibroacoustic and sound transmission loss behaviors of functionally graded doubly curved shallow. The model is developed by considering the simple power law scheme of material distribution in the thickness direction, under thermal load and incident oblique plane sound wave, as well as Donnell’s nonlinear shallow shell theory. By defining the stress function in terms of the force resultants and applying differential operators to these resultants, the coupled compatibility equation and nonlinear equation of motion are obtained only with regard to the stress function and displacement components. The methods of inverse differential operator and average form approach are then utilized to solve the particular and homogeneous solutions of the stress function associated with different boundary conditions. Based on this solution and the use of the Galerkin method with trigonometric mode shape functions, the nonlinear partial differential equation of motion is turned into the Duffing equation. Next, the method of multiple scales is implemented to capture the frequency of the shell corresponding to transverse motion. Finally, the nonlinear vibration and acoustic responses of shells are studied by considering the variation of important parameters such as aspect ratio, dimensionless amplitude, volume fraction power of functionally graded material, phase portrait, sound transmission loss, velocity, average mean square velocity of drive point, and the sound power level of the shells.

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.

The Josephson junction as a quantum engine

We treat the Cooper pairs in the superconducting electrodes of a Josephson junction (JJ) as an open system, coupled via Andreev scattering to external baths of electrons. The disequilibrium between the baths generates the direct-current bias applied to the JJ. In the weak-coupling limit we obtain a Markovian master equation that provides a simple dynamical description consistent with the main features of the JJ, including the form of the current–voltage characteristic, its hysteresis, and the appearance under periodic voltage driving of discrete Shapiro steps. For small dissipation, our model also exhibits a self-oscillation of the JJ’s electrical dipole with frequency around mean voltage V. This self-oscillation, associated with ‘hidden attractors’ of the nonlinear equations of motion, explains the observed production of monochromatic radiation with frequency Ω and its harmonics. We argue that this picture of the JJ as a quantum engine resolves open questions about the Josephson effect as an irreversible process and could open new perspectives in quantum thermodynamics and in the theory of dynamical systems.

Nonlinear modeling and analysis of spatial pipeline with arbitrary shapes supported by clamps considering displacement-dependent characteristic

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes.

Dynamic Analysis of an Articulated Offshore Tower Using Stokes Fifth Order Nonlinear Wave Theory

In offshore engineering, a compliant offshore structure accommodates dynamic forces by flexibility rather than rigidly withstanding load effects. As an alternative to conventional fixed platforms for loading/mooring in deep waters, articulated offshore towers are highly appealing because of their reduced structural weight. This study presents the results of a probabilistic random wave with and without current force on a 500 m high articulated tower with two hinges. By using the Lagrangian approach, governing nonlinear equations of motion are derived. The motion equation has been resolved in the time domain with the Houbolt approach. Based on Stokes fifth order nonlinear wave theory, wave forces have been assessed. Hydrodynamic forces are determined using a modified Morison equation with stretching modifications developed by Chakrabarti. The study examined two marine states, i.e., 10 m/10 s and 15 m/15 s. The current intensity of 2 m/sec has been taken into account. The salient statistical parameters such as maximum, minimum, mean, and standard deviation are presented under both the ocean environments. The proposed model is feasible for both the sea states. When comparing sea states, those with a greater wave height and period and those with a smaller wave height and period have more energy in their PSDs for all responses.

A shortening pendulum: Hoisting control for medical evacuation rescues

Strategies to vary the length of a pendular payload maneuvered by a helicopter in an aerial medical evacuation scenarios and similar situations are discussed. The aim is to rapidly stabilize the swinging payload to decrease effects of cable swing and decrease rescue time. A variable length procedure is proposed for small changes to the line length as the pendulum swings. The 2D nonlinear equation of motion of the pendulum is implemented in MATLAB, and for a proof of concept, manual intervention is used to identify values of the time interval and line parameters to achieve payload stabilization. Three time-varying line procedures are proposed. Conservation of energy is used to determine expressions for the initial angle and angular velocity that decrease the energy of the pendular payload when it swings through its highest and lowest points. Results show that the proposed stabilization strategies are able to control the swing angle due to variable length reel-in and reel-out at precise timing intervals. Further, validation of the variable length idea is demonstrated with energy calculations determined from Mathematica simulations in our prior work.

Nonlinear low-velocity impact of magneto-electro-elastic plates with initial geometric imperfection

Magneto-electro-elastic (MEE) materials are widely used in the aviation and spaceflight area, attributed to magneto-electric coupling effect of MEE materials which can achieve mutual energy conversion in complex physical fields. This article takes the imperfect MEE plate as the research object to explore the nonlinear low-velocity behaviors, and the influence of multiple physical fields such as electric, magnetic and temperature fields are taken into account. Based on Maxwell's equations and first-order shear deformation theory (FSDT), the constitutive equation of MEE plate is obtained, and the nonlinear motion equations are derived using the Euler-Lagrange principle. Subsequently, with the aid of the Galerkin method, the ordinary differential equations are obtained by discretizing the partial differential equations. Then the low-velocity impact problem is solved using the Runge-Kutta method. Finally, the impacts of unknown factors upon the dynamic behaviors of the MEE plate are illustrated particularly. It can be found that the magnetic and electric potentials have opposite effects on the impact displacement of the plate, and the MEE plate with the lowest volume fraction of BaTiO3 has the smallest center displacement and the shortest contact time, but has the maximum contact force.