Nonlinear Dynamic Equations Of Motion Research Articles

On the use of discontinuous nonlinear bistable dynamics to increase the responsiveness of energy harvesting devices

There is promise in the use of bistable devices to transduce ambient vibrations into electrical power. However, it is critical to sustain the relatively large amplitude snap-through motion, or interwell motion, to significantly improve the responsiveness of bistable devices as compared to linear resonance-based approaches. This work posits that relatively stiff structural elements can be placed in the vicinity of the equilibria of bistable devices such that the discontinuous change in dynamics will tend to eject an otherwise small amplitude motion into the large amplitude interwell orbit that is to be preferred for energy harvesting applications. The discontinuous nonlinear dynamic equations of motion are derived and a proxy system parametrically studied. These numerical studies demonstrate that discontinuous nonlinear bistable devices have a significantly broadened frequency range that elicits the large amplitude snap through behavior. It is also seen that interwell motion is achievable at significantly reduced excitation amplitudes through these discontinuous structural elements.

Optimal vibration energy harvesting from nonprismatic piezolaminated beam

The present article encompasses a nonlinear finite element (FE) and genetic algorithm (GA) based optimal vibration energy harvesting from nonprismatic piezo-laminated cantilever beams. Three cases of cross section profiles (such as linear, parabolic and cubic) are modelled to analyse the geometric nonlinear effects on the output responses such as displacement, voltage, and power. The simultaneous effects of taper ratios (such as breadth and height taper) on the output power are also studied. The FE based nonlinear dynamic equation of motion has been solved by an implicit integration method (i.e., Newmark method in conjunction with the Newton-Raphson method). Besides this, a real coded GA based constrained optimization scheme has also been proposed to determine the best set of design variables for optimal harvesting of power within the safe limits of beam stress and PZT breakdown voltage.

Dynamic response of articulated towers under correlated wind and waves

This paper deals with the dynamic behavior of a double-hinged articulated tower to wave alone, and correlated wind and waves. The analysis includes the nonlinearities due to nonlinear drag force, fluctuating buoyancy, variable added mass and instantaneous tower orientation. The fluctuating wind load is modeled by Ochi and Shin spectrum, while the wave load is characterized by Pierson–Moskowitz (P–M) spectrum. The nonlinear dynamic equation of motion is derived by Hamilton's principle. The equations of motion are solved in time domain by using Wilson-θ method. Power spectral density function (PSDF) of surge, tilting motion, hinge shear and bending moment are presented under high, moderate and low sea states. Studies of correlated wind and waves are found to be imperative for double hinged articulated towers to serve and survive in the extreme ocean environment. The response PSDF highlights the wind induced dynamic responses of the tower.

Multibody Dynamics Formulation for Modeling and Simulation of Roller Chain Using Spatial Operator

This paper proposes a new approach for modeling and simulating roller-chain drives using multibody formulations. The presented approach models the chain links as spatial decoupled dynamic bodies. The chain links are modeled using two different dynamic representations for the pin-link and the bushing-link. The bushing-link is modeled with two descendent dynamic bodies to represent the rollers. The adjacent pin and bushing links are connected by compliant bushing force elements. An efficient search algorithm is used to detect the contact between the roller and the sprocket teeth while a nonlinear force module is used to predict the contact force. A generalized sprocket representation is used to model the sprocket. The spatial motion of the chain links allows the out-of-plane vibrational motion of the links as well as simulating sprockets misalignment. Using the compliant connection between links avoids using the iterative calculation needed to satisfy the joint constraint equations leading to more efficient calculation scheme. The nonlinear dynamic equations of motion are solved using recursive approach. Complex roller chains drives, bicycle chain and conveyance systems can be easily modeled and analyzed using the proposed approach.

Analysis of warping deformation modes using higher order ANCF beam element

Most classical beam theories assume that the beam cross section remains a rigid surface under an arbitrary loading condition. However, in the absolute nodal coordinate formulation (ANCF) continuum-based beams, this assumption can be relaxed allowing for capturing deformation modes that couple the cross-section deformation and beam bending, torsion, and/or elongation. The deformation modes captured by ANCF finite elements depend on the interpolating polynomials used. The most widely used spatial ANCF beam element employs linear approximation in the transverse direction, thereby restricting the cross section deformation and leading to locking problems. The objective of this investigation is to examine the behavior of a higher order ANCF beam element that includes quadratic interpolation in the transverse directions. This higher order element allows capturing warping and non-uniform stretching distribution. Furthermore, this higher order element allows for increasing the degree of continuity at the element interface. It is shown in this paper that the higher order ANCF beam element can be used effectively to capture warping and eliminate Poisson locking that characterizes lower order ANCF finite elements. It is also shown that increasing the degree of continuity requires a special attention in order to have acceptable results. Because higher order elements can be more computationally expensive than the lower order elements, the use of reduced integration for evaluating the stress forces and the use of explicit and implicit numerical integrations to solve the nonlinear dynamic equations of motion are investigated in this paper. It is shown that the use of some of these integration methods can be very effective in reducing the CPU time without adversely affecting the solution accuracy.

Inverse solution technique of steady-state responses for local nonlinear structures

An inverse solution technique with the ability of obtaining complete steady-state primary harmonic responses of local nonlinear structures in the frequency domain is proposed in the present paper. In this method, the nonlinear dynamic equations of motion is first condensed from many to only one algebraic amplitude–frequency equation of relative motion. Then this equation is transformed into a polynomial form, and with its frequency as the unknown variable, the polynomial equation is solved by tracing all the solutions of frequency with the increase of amplitude. With this solution technique, some complicated dynamic behaviors such as sharp tuning, anomalous jumps, breaks in responses and detached resonance curves could be obtained. The proposed method is demonstrated and validated through a finite element beam under force excitations and a lumped parameter model with a local nonlinear element under base excitations. The phenomenon of detached resonance curves in the frequency response and its coupling effects with multiple linear modes in the latter example are observed.

Chain Drive Simulation Using Spatial Multibody Dynamics

This paper presents an efficient approach for modeling chain derives using multibody dynamics formulation based on the spatial algebra. The recursive nonlinear dynamic equations of motion are formulated using spatial Cartesian coordinates and joint variables to form an augmented set of differential-algebraic equations. The spatial algebra is used to express the kinematic and dynamic equations leading to consistent and compact set of equations. The connectivity graph is used to derive the system connectivity matrix based on the system topological relations. The connectivity matrix is used to eliminate the Cartesian quantities and to project the forces and inertia into the joint subspace. This approach will result in a minimum set of equation and can avoid iteratively solving the system of differential and algebraic equations to satisfy the constraint equations. In order to accurately capture the full dynamics of the chain links, each link in the chain is modeled as rigid body with full 6 degrees of freedom. To avoid singularities in closed loop configurations, the chain drive is considered a kinematically decoupled subsystem and the interaction between the links and other system components is modeled using force elements. The out-of-plane misalignment between the sprockets can be easily modeled using a compliant force element to model the joints between each two adjacent links. The nonlinear three dimensional contact forces between the chain links and the sprockets are modeled using elastic spring-damper element and accounts for the sliding friction. The proposed approach can be used to model complex drive chain, bicycle chain as well as conveyance systems. Results show that realistic behavior of the chain as well as out-of-plane vibration can be easily captured using the presented approach. The proposed approach for chain drive subsystem could be easily appended to any other multibody simulation system.

Takagi-Sugeno fuzzy modeling of a two-wheeled inverted pendulum robot

A two wheeled inverted pendulum TWIP mobile robot is an under-actuated mechanical system, which is inherently open-loop unstable with highly nonlinear dynamics. This property attracts the interest of researchers worldwide in recent years. This paper presents the development of a linearized model of the TWIP mobile robot based on its nonlinear dynamical model using Takagi-Sugeno T-S fuzzy modeling technique, which can allow the application of linear control techniques in analyzing and efficiently controlling such under-actuated nonlinear system. The complete nonlinear dynamical equations of motion of the TWIP mobile robot were first derived using Kane's Method. Based on the nonlinear dynamical equations of motion, a T-S fuzzy model of the TWIP robot was then developed using local approximation technique. To validate the proposed model, simulations study was carried out and the results show that the proposed model approximates the original nonlinear TWIP mobile robot in the pre-specified domains to certain degree of accuracy.

Stochastic response of a double hinged articulated leg platform under wind and waves

The displacement response of an articulated leg platform (ALP) is mainly governed by its rigid body mode of vibration which has a very low frequency. Since the fluctuating component of the wind velocity has very low frequency energy content, the wind induced vibration of the platform may be significant. In this paper, the response of an ALP to wave alone, wind alone, and correlated wind and waves are investigated. The fluctuating component of the wind is modeled by Simiu's spectrum, while the sea state is characterized by Pierson–Mosokowitz (P–M) spectrum. Random wind and waves are simulated by Monte Carlo simulation technique. For comparative studies of the platform responses, wind spectra suggested by Kareem, Davenport and API-RP2A are employed. The nonlinear dynamic equations of motion are derived by Lagrangian approach considering nonlinearities due to variable submergence, instantaneous tower orientation and drag force. The response of ALP is determined by a time domain iterative method using Newmark's- β integration scheme. It is concluded that response estimates based on the Davenport spectrum are lower than those based on the Simiu, Kareem and API RP 2A spectra due to higher energy available in these spectra at low frequencies. Furthermore, average hinge rotation of the platform under all cases is within permissible limits for platform operations.

Speed Control of Pipeline Pig Using the QFT Method

To provide an efficient inspection of pipeline, pigging operations must be executed at a constant speed. This paper presents an efficient and simple method based on Quantitative Feedback Theory (QFT), which is a well-known robust controller scheme, for speed control of a pig with bypass flow in a liquid pipeline. This classical-type controller commands the valve installed in the body of the pig to open or close. Then, the pig is controlled using the amount of bypass flow across its body. For this purpose, the nonlinear dynamic equation of motion of the pig is converted to a family of linear uncertain equivalent plants using Sobhani-Rafeeyan’s method (SR method). Then, for this family of uncertain equivalent plants, a QFT-type controller is synthesized. The presented method is developed for two-dimensional pipelines in two cases. The designed controllers are simulated numerically using the Simulink toolbox of the MATLAB software. The simulation results show that the designed controller can be used for speed control of the pig with good performance when it runs in the liquid pipelines.

The numerical solution of dynamic response of SDOF systems using cubic B-spline polynomial functions

In this paper, we present a new explicit procedure using periodic cubic B-spline interpolation polynomials to solve linear and nonlinear dynamic equation of motion governing single degree of freedom (SDOF) systems. In the proposed approach, a straightforward formulation was derived from the approximation of displacement with B-spline basis in a fluent manner. In this way, there is no need to use a special pre-starting procedure to commence solving the problem. Actually, this method lies in the case of conditionally stable methods. A simple step-by-step algorithm is implemented and presented to calculate dynamic response of SDOF systems. The validity and effectiveness of the proposed method is demonstrated with four examples. The results were compared with those from the numerical methods such as Duhamel integration, Linear Acceleration and also Exact method. The comparison shows that the proposed method is a fast and simple procedure with trivial computational effort and acceptable accuracy exactly like the Linear Acceleration method. But its power point is that its time consumption is notably less than the Linear Acceleration method especially in the nonlinear analysis.

Multihinged Articulated Offshore Tower under Vertical Ground Excitation

In this study, the response of a multihinged articulated offshore tower subjected to different seismic excitations in the presence of random waves was investigated. The study includes near-fault as well as far-fault earthquake ground motions. The contribution of the vertical component to the overall seismic behavior of the articulated tower is examined. The nonlinearities associated with the system owing to variable submergence, drag force, variable buoyancy, and added mass, along with the geometry, are also considered. The nonlinear dynamic equation of motion is formulated considering the Lagrangian approach, which is solved in time domain by the Newmark-beta integration scheme. The results are expressed in the form of time histories and spectral densities of the dynamic responses. Dynamic response quantities such as rotational angle, hinge shear, axial force at the articulation, and bending moment at peak ground acceleration in different seismic sea environment are discussed. The spectral responses unde...

A rational finite element method based on the absolute nodal coordinate formulation

The NURBS method is widely used for representing the geometry of engineering and physics systems in computer aided design (CAD) software. However, no finite element formulations currently used in computer aided analysis (CAA) can produce complex geometry identical to NURBS representations. Most contemporary finite element methods, in fact, are at best only crudely capable of representing the complex shapes that can be represented using NURBS. However, finite elements based on the nonlinear absolute nodal coordinate formulation (ANCF) are known to utilize similar shape functions to those employed in Bezier and B-spline representations of geometry. In order to facilitate the integration of computer aided design and analysis (ICADA), this investigation proposes the creation of a new finite element formulation based on the ANCF that utilizes rational polynomials for shape functions. This will lead to the new rational absolute nodal coordinate formulation (RANCF). To demonstrate the feasibility of developing RANCF finite elements, it is shown that rational cubic Bezier curve elements can be converted through a simple coordinate transformation into a rational ANCF form, in which the coordinates are the nodal positions at the end points of the element and the gradient vectors at those nodal points. Furthermore, it is shown that because a cubic NURBS curve can be converted to a series of rational cubic Bezier elements that such a NURBS curve can be converted into a series of these RANCF elements. This investigation only proposes the kinematic description of this new RANCF element. The full development of the nonlinear dynamic equations of motion as well as the development of fully parameterized RANCF finite elements will be explored in future investigations.

Stability and Chaotic Vibrations of a Fluid-Conveying Pipe with Additional Combined Constraints

AbstractThe stability and possible chaotic vibrations of a fluid-conveying pipe with additional combined constraints are investigated. The pipe, restrained by motion constraints somewhere along the length of the pipe, is modeled by a beam clamped at the left end and supported by a special device (a rotational elastic constraint plus a Q-apparatus) at the right end. The motion constraints are modeled by both cubic and trilinear models. Based on the Differential Quadrature Method (DQM), the nonlinear dynamical equations of motion for the system are formulated, and then solved via a numerical iterative technique. Calculations of bifurcation diagrams, phase portraits, time responses and Poincare maps of the oscillations establish the existence of chaotic vibrations. The route to chaos is shown to be via period-doubling bifurcations. It is found that the effect of spring constant of the rotational elastic constraint on the dynamics is significant. Moreover, the critical fluid velocity at the Hopf bifurcation point for the cubic model is higher than that for the trilinear model.

Algorithms by Design: Part I—On the Hidden Point Collocation Within LMS Methods and Implications for Nonlinear Dynamics Applications

In computational mechanics to date, even after five decades of research dealing with integration of the linear and nonlinear dynamic equations of motion, there exists for the general cases of algorithm designs still a clear lack of a fundamental understanding regarding evaluation of these equations of motion, and how, why, and what precise time levels the integrations need to take place. This has placed major limitations on commercial code developers, and a lack of confidence in general, thereby restricting the implementation of LMS methods to only a select few that happen to be correct, but without any rigorous proofs or underlying reasons explaining the ramifications. This is extremely critical for the general cases of integration of the equations of motion, in particular for the class of Linear Multi-Step (LMS) that are implicit, involve a single solve, and are second-order accurate in time with unconditional stability (this pertains to linear dynamic situations only), since these serve as the backbone and drivers for most finite element commercial and research software. In this regard, the present Part I of the three-part exposition puts the matter to rest and provides closure, while in Part II (see Ref [1]) and Part III (see Ref [2]) we describe extensions to nonlinear dynamics applications of the basic general framework for the classes of nondissipative and controllable numerical dissipative methods, respectively. Within the confines of these LMS methods, particular attention is paid to designing algorithms that are symplectic-momentum preserving, and with enhancements to include controllable dissipative features and optimal algorithm designs. Simply for illustration of the basic ideas, readily understandable numerical examples are presented that validate the overall concepts and developments.

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrodinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.

Three-axis Attitude Control for Flexible Spacecraft by Lyapunov Approach under Gravity Potential

Attitude control law synthesis for the three-axis attitude maneuver of a flexible spacecraft model is presented in this study. The basic idea is motivated by previous works for the extension into a more general case. The new case includes gravitational gradient torque which has significant effect on a wide range of low earth orbit missions. As the first step, the fully nonlinear dynamic equations of motion are derived including gravitational gradient. The control law design based upon the Lyapunov approach is attempted. The Lyapunov function consists of a weighted combination of system kinetic and potential energy. Then, a set of stabilizing control law is derived from the basic Lyapunov stability theory. The new control law is therefore in a general form partially validating the previous work in some sense.

Classical and Nonclassical Dynamic Problems of Sandwich Shells with a Transversely Soft Core

Based on the hypothesis of similarity of transverse displacements in thin-walled sandwich shells with a transversely soft core under dynamic and static loads, refined geometrically nonlinear dynamic equations of motion are constructed in the case of large variations in the parameters of the stress-strain state (SSS) in the tangential directions. For shells structurally symmetric across the thickness and loaded with initial static loads, linearized dynamic equations are derived, which, upon introducing the synphasic and antiphasic functions of displacements and forces, can be used to describe the synphasic and antiphasic buckling forms in the transverse and tangential directions. For nonshallow cylindrical and shallow spherical shells, the nonclassical problems on all possible vibration forms realized at zero indices of variability of the SSS parameters in the tangential directions are formulated and solved. For shallow shells of symmetric structure, the resolving equations are obtained by introducing, instead of tangential displacements and transverse tangential stresses in the core, the corresponding potential and vortex functions.

Spatial Dynamics of Multibody Tracked Vehicles Part I: Spatial Equations of Motion

SUMMARY In this paper, the nonlinear dynamic equations of motion of the three dimensional multibody tracked vehicle systems are developed, taking into consideration the degrees of freedom of the track chains. To avoid the solution of a system of differential and algebraic equations, the recursive kinematic equations of the vehicle are expressed in terms of the independent joint coordinates. In order to take advantage of sparse matrix algorithms, the independent differential equations of the three dimensional tracked vehicles are obtained using the velocity transformation method. The Newton-Euler equations of the vehicle components are defined and used to obtain a sparse matrix structure for the system dynamic equations which are represented in terms of a set of redundant coordinates and the joint forces. The acceleration solution obtained by solving this system of equations is used to define the independent joint accelerations. The use of the recursive equations eliminates the need of using the iterative Newton-Raphson algorithm currently used in the augmented multibody formulations. The numerical difficulties that result from the use of such augmented formulations in the dynamic simulations of complex tracked vehicles are demonstrated. In this investigation, the tracked vehicle system is assumed to consist of three kinematically decoupled subsystems. The first subsystem consists of the chassis, the rollers, the sprockets, and the idlers, while the second and third subsystems consist of the tracks which are modeled as closed kinematic chains that consist of rigid links connected by revolute joints. The singular configurations of the closed kinematic chains of the tracks are also avoided by using a penalty function approach that defines the constraint forces at selected secondary joints of the tracks. The kinematic relationships of the rollers, idlers, and sprockets are expressed in terms of the coordinates of the chassis and the independent joint degrees of freedom, while the kinematic equations of the track links of a track chain are expressed in terms of the coordinates of a selected base link on the chain as well as the independent joint degrees of freedom. Singularities of the transformations of the base bodies are avoided by using Euler parameters. The nonlinear three dimensional contact forces that describe the interaction between the vehicle components as well as the results of the numerical simulations are presented in the second part of this paper.

Panel Flutter Limit-Cycle Suppression with Piezoelectric Actuation

An optimal control design is presented to suppress panel flutter limit-cycle motions using piezoelectric actuators. First, the nonlinear dynamic equations of motion based on the classical continuum method are derived for a simply supported isotropic panel with a pair of patched piezoelectric layers. After linearizing the dynamic modal equations, an optimal controller is developed to provide an optimal combination of inplane force and bending moments through piezoelectric actuators for flutter suppression. For the panel configuration studied, numerical simulations based on the nonlinear model show that the maximum suppressible dynamic pressure X,.,. can be increased about three times of the critical dynamic pressure Xc, by the piezoelectric actuation, and the bending moment is much more effective in flutter suppression than the inplane force. Within the maximum suppressible dynamic pressure, limit-cycle motions can be completely suppressed. For the actuator design, the two-set actuators perform better than the one-patched actuator, and the one-patched actuator, may have better performance than the completely covered actuator. The results demonstrate that piezoelectric materials are effective in panel flutter suppression.