Nonlinear Dynamic Equations Of Motion Research Articles

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

Numerical simulation and behavior prediction of a space net system throughout the capture process: Spread, contact, and close

AbstractIn this paper, we develop an exhaustive numerical simulator for the dynamic visualization and behavior prediction of the tether‐net system during the whole space debris capture phases, including spread, contact, and close. First of all, to perform its geometrically nonlinear deformation, discrete different geometry theory is applied to model the mechanical response of a flexible net. Based on the discretization of the whole structure into multiple vertexes and lines, the internal force and associated Hession are derived in a closed form to solve a series of nonlinear dynamic equations of motion. The spread and deployment of a packaged net can be realized using this well‐established net solver. Next, a multidimensional incremental potential formulation is selected to achieve the intersection‐free boundary nonlinear contact and collision between the deformable net and rigid debris. Finally, for the closing mechanism analysis, a log‐like barrier functional is derived to achieve the nondeviation condition between the ring–rod linkage system. The continuous log barrier functionals constructed for both the contact model and the linkage system are smooth and differentiable, and, therefore, the nonlinear net capture dynamic system can be efficiently solved through a fully implicit time integrator. Overall, as a demonstration, the whole capture process of a defunct satellite using a hexagon net is simulated through our well‐established numerical framework. We believe that our comprehensive numerical methods could provide new insight into the optimal design of active debris removal systems and promote further development of the online control of tether tugging systems.

A discrete model for the geometrically nonlinear mechanics of hard-magnetic slender structures

Hard-magnetic soft (HMS) materials and structures, which can undergo rapid configurational transformation under non-contact magnetic stimuli, have attracted extensive attentions in a wide range of engineering applications, such as soft robotics, biomedical devices and stretchable electronics, etc. In order to realize the full potentials of HMS structures, it is crucial to be able to predict their mechanical (both static and dynamic) responses. In this work, we propose a discrete magneto-elastic rod model with the aim to analyse the mechanical behaviours of slender structures made of HMS materials under magnetic loading. The configuration of a slender object is described by multiply connected nodes and edges, from which the force vector and the associated Hessian matrix are derived by taking the variation of Kirchhoff-like magneto-elastic potentials. The nonlinear dynamical equations of motion are next evaluated through Newton-type optimization, and the static analysis can be obtained using dynamical relaxation method. For verification, we quantitatively compare our numerical results with either analytical solutions or experimental data for several representative cases. Good agreements in all these cases indicate the correctness and accuracy of our discrete model. We further extend the model to take the dipole–dipole interaction and viscous effect into consideration. Finally, as a demonstration, we perform simulations to reproduce the locomotion of a magnetic crawling robot. The developed discrete magneto-elastic model significantly improves the computational efficiency, enabling the practicability of simulating the mechanical, especially dynamic, behaviours of the hard-magnetic slender structures.

New modeling on vibrations and bifurcations of FGGP reinforced pretwisted composite rotating blade under axial aerodynamic force: Theoretical and numerical researches

A new modeling on the vibrations and bifurcations is proposed for the pretwisted composite rotating blade reinforced by the functionally graded graphene platelet (FGGP) subjected to the leakage flow at the tip clearance, aerodynamic force and high rotating speed for the first time. The blade is simplified to the dynamic model of a FGGP reinforced pretwisted composite rotating cantilever plate subjected to the in-plane and external excitations. The novel non-uniform axial excitation is applied to simulate the aerodynamic pressure of the airflow at the tip clearances subjecting to the FGGP reinforced pretwisted rotating composite blade. Three different distribution types of the graphene platelets (GPs) are studied. According to the first-order shear deformation theory and von-Karman nonlinear geometric relationship, the new differential governing equation of motion is established for the FGGP reinforced rotating pretwisted composite cantilever plate. The nonlinear dynamic equations of motion are derived by utilizing the energy functions and Lagrange equation. The validity is verified by comparing the obtained results with those of other references. The natural frequencies, bifurcations and stability are respectively studied by utilizing Rayleigh–Ritz and Newton Raphson methods. The instability regions are obtained for the FGGP reinforced pretwisted composite rotating cantilever plate. The influences of the rotating speeds, setting angles and weight fractions on the natural frequencies and vibrations are presented for the pretwisted rotating cantilever plate under three different distributions of the GPs. The obtained results illustrate that the static critical loads increase with the increase of the rotating speeds, setting angles and weight fractions under the X distribution plate for the FGGP reinforced pretwisted composite rotating cantilever plate subjected to the axial aerodynamic force and high rotating speed.

Dynamic snap-through of bi-stable laminates with simply supported at four corners under impact loads

In this paper, nonlinear dynamic responses of bi-stable laminate being simply supported on four corner points, which is subjected to various impact loads, are analyzed from theoretical viewpoints. Consider a four layers asymmetric cross-ply bi-stable laminate composited by the carbon fiber reinforced resin matrix. A novel configuration function that satisfies the boundary condition is proposed to predict the equilibrium configuration. Then, employing the first order shear deformation theory, the von Kàrmàn strains and Hamilton principle, nonlinear dynamic equations of motion expressed by two curvatures are derived. To obtain the two stable configurations of the laminate, the inertia terms are ignored. With the aids of static equilibrium equation, the side length and curvature of the system are solved by using NLPSolve functions in the Maple software. The stable configurations are verified by comparing present results with those reported in available literatures. Consider the incremental load loading, the decreasing loading, the sinusoidal loading, the step loading, the triangular loading and periodic loading, the nonlinear dynamic response of the bi-stable laminate are discussed. The dynamic snap-through behavior, the intra-well oscillations and inter-wall oscillations are studied by numerical simulation in detail.

The probabilistic dynamic stability analysis of fluid-filled porous cylindrical shells reinforced with graphene platelets

The aim of the current paper is the probabilistic dynamic stability analysis of graphene-reinforced functionally graded porous cylindrical shells under dynamic axial loads. For this purpose, various sources of uncertainty including the material and geometrical properties of the cylinder and loading are considered. By combining a Lagrangian solution method and a semi-analytical method, some formulations are developed for determining the dynamic buckling loads of the cylindrical shells. In order to model the fluid–reservoir interaction, a semi-analytical boundary solution is used. The functionally graded porous shells reinforced with graphene platelets (FGP-GPL) are modeled with three porosity distributions and three GPL dispersions in the shell thickness based on the Halpin–Tsai formulation. The kinetic equilibrium equations of the shells are derived based on Sanders’ nonlinear theory and the higher order shear theory. Moreover, in order to solve the nonlinear dynamic equations of motion, the Newton–Raphson and the Newmark methods are combined. Finally, the results are presented in terms of the probabilistic density functions (PDFs) and cumulative density functions (CDFs) of the buckling loads. In addition, a parametric study is conducted for the effects of the different material and physical conditions on the probabilistic dynamic buckling loads of the composite cylinders. As a result, in the probabilistic dynamic stability analysis of the composite shells, the distributions of the different buckling loads may have overlaps with each other. This phenomenon indicates that in probabilistic cases, the first buckling mode of a structure is not necessarily always the dominant failure mode.

Aerodynamic behaviour of double hinged articulated loading platforms

Articulated loading platforms (ALPs) belongs to a class of offshore structures known as compliant. ALP motions have time periods falling in the wind excitation frequency range due to their compliant behaviour. This paper deals with the dynamic behavior of a double hinged ALP subjected to low-frequency wind forces with random waves. Nonlinear effects due to variable submergence, fluctuating buoyancy, variable added mass, and hydrodynamic forces are considered in the analysis. The random sea state is characterized by the Pierson-Moskowitz (P-M) spectrum. The wave forces on the submerged elements of the platform's shaft are calculated using Morison's Equation with Airy's linear wave theory ignoring diffraction effects. The fluctuating wind load has been estimated using Ochi and Shin wind velocity spectrum for offshore structures. The nonlinear dynamic equation of motion is solved in the time domain by the Wilson-e method. The wind-structure interactions, along with the effect of various other parameters on the platform response, are investigated. The effect of offset of aerodynamic center (A.C.) with the center of gravity (C.G.) of platform superstructure has also been investigated. The outcome of the analyses indicates that low-frequency wind forces affect the response of ALP to a large extent, which otherwise is not enhanced in the presence of only waves. The mean wind modifies the mean position of the platform surge response to the positive side, causing an offset. Various power spectral densities (PSDs) under high and moderate sea states show that apart from the significant peak occurring at the two natural frequencies, other prominent peaks also appear at very low frequencies showing the influence of wind on the response.

Motion control of a caterpillar robot using optimized feedback linearization and sliding mode controllers

Caterpillar robots are widely used in various industrial applications, including pipe inspection, access to dangerous locations in factories, passing through narrow holes during natural disasters, and even in processes such as endoscopy and colonoscopy. In this paper, two control schemes, namely optimized feedback linearization controller and optimized sliding mode controller are designed and compared for forward-motion control of a bio-inspired caterpillar robot, which contains five bars connected via torque joints. At first, the governing nonlinear dynamic equations of motion are presented. Then, feedback linearization controller and sliding mode controller are designed to deliver angular positions of the robot to desired set points. To compare the performance of the mentioned controllers, the related parameters are optimized. In this regard, an appropriate objective function is defined to minimize the stabilization errors and control inputs, simultaneously. It is noted that the optimization process is performed using a genetic algorithm. Finally, the performance of the optimized controllers in the presence of parameter uncertainties, sensor noise, and actuator disturbances is compared quantitatively in the viewpoints of the stabilization error and control effort.

Moving Element Analysis of High-Speed Train-Slab Track System Considering Discrete Rail Pads

This paper presents a study of the dynamic behavior of a coupled train-slab track system considering discrete rail pads. The slab track is modeled as a three-layer Timoshenko beam. The study is carried out using the moving element method (MEM). By introducing a convected coordinate system moving at the same speed as the vehicle, the governing equations of motion of the slab track are formulated in a moving frame-of-reference. By adopting Galerkin’s method, the element stiffness, mass and damping matrices of a truncated slab track in the moving coordinate system are derived. The vehicle is modeled as a multi-body with 10 degrees of freedom. The nonlinear Hertz contact model is used to account for the wheel–rail interaction. The Newmark integration method, in conjunction with a global Newton–Raphson iteration algorithm, is employed to solve the nonlinear dynamic equations of motion of the vehicle–track coupled system. The proposed MEM model of the system is validated through comparison with available results in the literature. Further study is then made to investigate the vehicle–track system accounting for track irregularities modeled as short harmonic wave forms. Results showed that irregularities with short wavelengths have a significant effect on wheel–rail contact force and rail acceleration, and the dynamic response of the track structure does not increase monotonously with the increase of the vehicle speed.

Effect of current on the dynamic response of a bi-articulated offshore tower

Wind and wave loadings have a predominant role in the design of articulated offshore towers for its successful service and survival. Such towers are very sensitive to the dynamic effect of environmental loads. The compliant nature of these towers with environmental loads introduces geometric nonlinearity due to large displacements, which becomes an important consideration in the analysis of these towers. This article deals with the dynamic response of a bi-articulated offshore tower to wind, wave, and current forces. The exposed portion of the tower is subjected to the action of wind, while the submerged portion is acted upon by random wave and current forces. Wind load is modeled by Ochi and Shin spectrum, while the wave load is characterized by Pierson–Moskowitz spectrum. The nonlinear dynamic equations of motion are derived by Hamilton’s principle. Response of the tower is determined by a time domain iterative (Wilson-θ) method. Power spectral density of important parameters such as surge, tilting motion, hinge shear, and bending moment are presented under high and low sea states. It is observed that response of tower due to current modifies the peak energy of power spectral density functions.

Effect of disturbances and sensorimotor deficits on the postural robustness of an ankle–hip model of balance on a balance board

This paper investigates the effect of postural disturbances and sensorimotor deficits on the robustness of the upright posture (UP) for a human body model balancing on a balance board (BB). The robustness is investigated by computing the gradient field along the basin of attraction (BoA) of an asymptotically stable equilibrium point. The human model is modeled as a double-inverted pendulum (hip and ankle joints). The human-BB system is assumed actuated by torques at the hip, BB hinge, and ankle joints. Postural disturbances induce an initial joint angle velocity either at the ankle or at the hip joint. Moreover, either proprioceptive or visual and vestibular deficits are considered in the human-BB model. The nonlinear dynamic equation of motion of the human-BB system is numerically solved to obtain the BB, ankle, and hip joint angle position and velocity profiles. The BoA of the human-BB UP equilibrium is built as the set of initial conditions whose resulting time-series profiles converge to the equilibrium. It was shown that UP is more robust to disturbances that induce hip joint initial angle velocity. That is probably due to the fact that disturbances that induce ankle joint initial velocity affect the whole body, while disturbances that induce hip joint angle initial velocity only affect the trunk. Whenever visual and vestibular deficits are considered, the UP is more robust if proprioceptive gain and BB stiffness are small. Contrarily, whenever proprioceptive deficits are considered, the UP is more robust if visual and vestibular gain and BB stiffness are large. The method proposed here (the BoA and the gradients) can be used to systemically provide understanding about the robustness of the human-BB UP to external disturbances, which may help to identify people with a higher risk of fall.

Nonlinear planar secondary resonance analyses of suspended cables with thermal effects

This paper deals with a geometrically nonlinear model of suspended cables in thermal environments. In particular, the secondary resonant behaviors with thermal effects are investigated and discussed analytically and numerically. First, considering the influence of temperature changes, a dimensionless coefficient is generated. The nonlinear dynamic equations of motion in thermal environments are established, and the governing PDEs are reduced to a set of ODEs by using the Galerkin procedure. Four super and sub-harmonic resonant cases are studied, and frequency responses equations and steady-state solutions for each case are obtained. Extensive numerical results show that the hardening/softening behaviors, the response amplitudes, the number of nontrivial solutions, the range of resonant responses, the jump points, and the peak values are all closely connected to temperature. In some specific warming/cooling conditions, the nonlinear system just shows linear vibration behaviors or even just opposite spring characteristics. The time–displacement curves and phase portraits are extremely sensitive to temperature changes, but the number of cluster points in Poincaré sections seem to be independent of temperature changes.

Velocity control of a two-wheeled inverted pendulum mobile robot: a fuzzy model-based approach

This paper presents the design of a fuzzy tracking controller for balancing and velocity control of a Two-Wheeled Inverted Pendulum (TWIP) mobile robot based on its Takagi-Sugino (T-S) fuzzy model, fuzzy Lyapunov function and non-parallel distributed compensation (non-PDC) control law. The T-S fuzzy model of the TWIP mobile robot was developed from its nonlinear dynamical equations of motion. Stabilization conditions in a form of linear matrix inequalities (LMIs) were derived based on the T-S fuzzy model of the TWIP mobile robot, a fuzzy Lyapunov function and a non-PDC control law. Based on the derived stabilization conditions and the T-S fuzzy model of the TWIP mobile robot, a state feedback velocity tracking controller was then proposed for the TWIP mobile robot. The balancing and velocity tracking performance of the proposed controller was investigated via simulations. The simulation result shows the effectiveness of the proposed control scheme.

Nonlinear transverse vibrations of angle-ply laminated composite piezoelectric cantilever plate with four-modes subjected to in-plane and out-of-plane excitations

In this paper, the nonlinear vibrations of the laminated composite piezoelectric cantilever plate subjected to the transverse and in-plane excitations are investigated. Based on the Reddy’s third-order plate theory and von Karman nonlinear strain-displacement relation, the nonlinear partial differential governing equations of motion are established for the laminated composite piezoelectric cantilever plate under combined the transverse and in-plane excitations by applying Hamilton’s principle. Employing Galerkin’s approach, we discretize the continuous nonlinear dynamic equations of motion into the ordinary differential equations with four modes. Substituting the actual geometric and physical parameters of the laminated composite piezoelectric cantilever plate into the governing equation of motion, we use numerical method to investigate the effects of the lay-up parameters and the geometric parameters on the dimensionless natural frequencies of the cantilever plate. The amplitude-frequency response curves and the basin boundary diagram of two coexisting steady-state solutions are plotted for the laminated composite piezoelectric cantilever plate. The nonlinear dynamic responses of the laminated composite piezoelectric cantilever plate under combined the transverse and in-plane excitations are investigated by changing the lay-up parameters, the transverse excitation, the voltage excitation and the in-plane excitation. The periodic, the quasi-periodic, the chaotic motions and the nonlinear stiffness characteristics can be found. The nonlinear dynamic behaviors help us to optimize the structural and physical parameters of the laminated composite piezoelectric cantilever plate under combined the transverse and in-plane excitations.

Taming the Beast: Diffusion Method in Nonlocal Gravity

We present a method to solve the nonlinear dynamical equations of motion in gravitational theories with fundamental nonlocalities of a certain type. For these specific form factors, which appear in some renormalizable theories, the number of field degrees of freedom and of initial conditions is finite.

Continuum-Based Geometry/Analysis Approach for Flexible and Soft Robotic Systems

Control and stability of flexible and soft robotic systems (FSRS), which have complex geometry and experience desirable and undesirable deformations, are of major concern, particularly when lightweight soft materials are used. Nonetheless, there is no unified continuum-based geometry/analysis approach that can be used for the efficient FSRS virtual prototyping and design. The goal of this article is to propose a new FSRS geometric modeling and analysis methodology by addressing fundamental virtual prototyping challenges that include: (1) integration of the robot geometry and analysis; (2) implementation of general and unconventional material models and actuation forces; (3) use of new concepts for modeling FSRS joints; and (4) development of efficient and robust algorithms for the FSRS virtual prototyping. To address these challenges, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system computational algorithms are used. ANCF FEs allow for modeling arbitrarily large and coupled displacements, correctly capture complex geometries, allow for implementing general and nonconventional material models, provide accurate definitions of conventional and nonconventional actuation forces, lead to a constant inertia matrix that defines optimally sparse matrix structure of the dynamic equations, and allow for exploiting new geometry concepts to define linear and more general joint constraints instead of the less general and nonlinear joint constraints currently used for robot systems. Using the general structure of the FSRS nonlinear dynamic equations of motion, a nonmodal continuum-based approach can be developed and used to correctly capture the FSRS complex geometry and large deformations.

Dynamic analysis of open-loop mechanisms with multiple spatial revolute clearance joints

Dynamic model of a typical open-loop mechanism with multiple spatial revolute clearance joints were established based on the Newton–Euler equations and the Hertzian contact deformation theory. An augmented constraint violation correction method was presented to solve the nonlinear dynamic equations of motion, which improved the global convergence and stability effectively. The nonlinear dynamic behaviors of a serial robot manipulator with two spatial revolute clearance joints were studied to demonstrate the effects of the location and coupling relationship of the clearance joints. Numerical results show that the influence of spatial revolute clearance joint on the dynamic behaviors of the open-loop mechanism is relatively stronger than that of the planar ones. The location of the spatial revolute clearance joints is an important factor to dynamic behavior of the system. The closer the spatial revolute clearance joint is to the end-effector, the stronger influence it has on the system. The spatial revolute clearance joints interact significantly with each another, which exhibits vigorous vibration with a higher frequency, larger amplitude, and deeper penetration. This work provides new insights into investigating the nonlinear dynamic behaviors of the systems with spatial revolute clearance joints.

Near Fault Effect on the Response of Single Hinged Compliant Offshore Tower

The response of compliant offshore structure under simultaneous action of random waves and earthquake loading has been analyzed. Since earthquake forces play a significant role in affecting the response of these offshore structures, comparative studies have been carried out considering near fault and far fault seismic excitations in the presence of moderate random sea waves. The offshore tower is modeled as an inverted pendulum with a cylindrical shaft connected by an articulated joint at the base. Seismic forces are evaluated by dividing the tower shaft into finite elements with masses lumped at the nodes. The nonlinearities associated with the system owing to variable submergence, drag force, variable buoyancy along with the geometry are considered in the analysis. The nonlinear dynamic equation of motion is formulated considering Lagrangian approach, which is solved in time domain by the Newmark-beta integration scheme. The sea state conditions, more precisely the water particle kinematics are evaluated using Airy’s wave theory along with the stretching modifications, as suggested by Chakrabarti. To minimize the dynamic responses, emphasis has to be given to the variations in height and position of the buoyancy chamber in extreme wave condition. The results are expressed in the form of time histories of deck displacement, hinge rotation, hinge shear and the bending moment. Parameters like maximum, minimum, mean and standard deviation are also determined by statistical analysis of response time histories of the dynamic responses at articulated joint.

A novel time integration formulation for nonlinear dynamic analysis

To solve the linear and nonlinear dynamic equation of motion, a family of explicit higher-order time integration algorithms is presented. The differential transformation method in conjunction with the state–space concept is used. In this study, four members of this family are considered thoroughly. To validate the performance of the new algorithms, comparison studies are accomplished with the well-known time integration schemes. By solving some linear and nonlinear dynamic problems, the merits of the authors' formulations will be clearly demonstrated.

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.