Ordinary Differential Equations Of Motion Research Articles

Variational Gaussian approximation for the magnetic Schrödinger equation *

In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we derive ordinary differential equations of motion for the parameters of the variational solution. Moreover, we prove L 2-error bounds and observable error bounds for the approximating Gaussian wave packet.

A Novel Approach to Linear and Nonlinear Time-History Analysis of Structures: Gauss–Lobatto–Hermite 4-Point (GLH-4P) Method

An efficient time integration scheme has been devised for conducting time-history dynamic analysis of single-degree-of-freedom (SDOF) structural systems. The developed scheme belongs to Runge-Kuta family. The primary innovation of the new formulation lies in the constructive cooperation between the Hermite interpolation and the Gauss–Lobatto integration, linked to address the second-order ordinary differential equation of motion (DEOM). To put the idea into practice, several Hermite interpolators are generated to exploit the information of the internal points of Gauss-Lobatto integrator. The 4-point Gauss–Lobatto formula is employed to numerically integrate acceleration and velocity functions. This novel methodology has been designated the Gauss–Lobatto–Hermite 4-point (GLH-4P) method. GLH-4P furnishes a generally applicable implicit algorithm for conducting both linear and nonlinear analyses of structures subjected to seismic excitation. The advantage of the proposed algorithm over the conventional techniques is its higher accuracy level. Besides, it offers better stability compared with the conventional methods such as the Newmark-β and Wilson-θ methods. Unlike the other methods, GLH-4P can handle high-frequency systems without reducing the step size. Numerical examples reveal the efficacy of the GLH-4P method when juxtaposed against existing methodologies.

Large amplitude free vibration analysis of circular arches with variable thickness

This study aims to investigate the large amplitude nonlinear vibration of variable thickness arches with different cross-sectional shapes under different boundary conditions. The governing equations of the arches are derived using Hamilton's principle by considering their geometric nonlinearities. Then, the differential quadrature method (DQM) is employed for the discretization to obtain the ordinary differential equations of motion for the circular arches with variable thickness. The theoretical model is validated by comparing the linear and nonlinear natural frequencies of arches. The nonlinear vibration characteristics of arches are solved using the direct iteration scheme. The effects of the slenderness ratio, modified slenderness ratio of the arch, and variable thickness parameters under different cross-sectional shapes on the nonlinear to linear frequency ratios of the arches with clamped and pin-ended boundary conditions are investigated. The results indicate that appropriate variable thickness parameters can improve the linear frequency and the nonlinear to linear frequency ratio of arches, which offers some suggestions for the design of variable thickness arches.

Обоснование безопасных расстояний по отношению к окружающим объектам при проведении камуфлетных взрывов в горных породах

Introduction. Despite the simplicity and accessibility of camouflage blasting, the issues of ensuring the safe conduct of such work are important. A camouflage explosion eliminates the occurrence of an air shock wave and the harmful effects of the explosion products on the environment, but leads to strong seismic vibrations. The impact of seismic loads on nearby objects can lead to the loss of their carrying capacity, damage and destruction, therefore, when choosing a place for installing camouflage charges, the task arises of assessing the field of environmental velocities during an explosion and determining the distances at which the velocities do not exceed the maximum allowable values established for objects under consideration. The aim of the study is to solve the problem of determining the velocity field in a continuous elastic-plastic medium during a camouflage explosion. Research methodology. The solution of the central symmetric problem of the propagation of explosive disturbances in solid media is based on the assumption that a deep spherical charge of a certain radius is placed in an unbounded half-space, which instantly turns into a high-pressure gas without changing the volume. It is assumed that the pressure in the cavity decreases according to a power law, and the relationship of pressure with the radius, velocity and acceleration of the expanding cavity is determined by the camouflage equation , whose constants A, B and C are functions of the parameters of rocks and soils. The perturbed state of the medium caused by the expanding cavity is also characterized by densities ρ0 and ρ respectively, in the elastic and plastic regions of its deformation, and the transition from the elastic state to the plastic state is accompanied by an instantaneous change in the density of the medium from ρ0 to ρ, introduced to approximately take into account the actual compressibility. Results. The determination of the velocity field in the medium surrounding the charge during the explosion is reduced to solving the ordinary differential equation of motion of a spherical cavity expanding due to internal pressure. It is shown that the unknown constant included in the obtained relation can be determined from the condition of conservation of the energy released during the explosion in the entire elastoplastic region of the medium motion. Discussion. The novelty of the result obtained lies in the substantiation of the possibility of using the assumptions about the vibrationless nature of the movement of the camouflage cavity and the incompressibility of the medium in the plastic and elastic regions to determine the velocity field that forms in a continuous medium during the explosion of a camouflage charge. Conclusion. Under the assumption of the vibrationless nature of the motion and the incompressibility of the medium in the plastic and elastic regions, a solution is obtained for the centrally symmetric problem of determining the velocity field in a continuous elastoplastic medium during a camouflage explosion. The solution obtained makes it possible to estimate the sizes of expansion zones, plastic deformation of the medium, and the impact of explosive disturbances on various objects.

Analytical and differential reformulations of the Vaiana–Rosati model for complex rate-independent mechanical hysteresis phenomena

A recent analytical model of hysteresis, developed by the authors to simulate a great variety of complex uniaxial rate-independent mechanical hysteresis phenomena, is reformulated in a twofold manner. First, it is proposed an analytical reformulation that, compared to the current version of the model, does not require the evaluation of any internal variable. Second, the closed form expressions provided by the analytical reformulation are expressed in rate form to foster its use especially in the field of nonlinear dynamics. To verify their accuracy, both formulations are first employed to reproduce four different complex hysteresis phenomena and the related results are compared in terms of generalized force–displacement hysteresis loops, tangent stiffness functions, and work-displacement relations. Subsequently, nonlinear time history analyses are carried out on four single degree of freedom mechanical systems for three different types of external generalized forces. To this end, two solution strategies are adopted. The former combines the model analytical reformulation with an explicit structure-dependent time integration method used to directly integrate the second-order ordinary differential equation of motion. The latter employs the model differential reformulation in conjunction with the Runge–Kutta method implemented to solve the equivalent system of three first-order ordinary differential equations. The comparison of the numerical results obtained by the two solution procedures shows an excellent agreement.

Three-to-One Internal Resonance of L-Shaped Multi-Beam Structure with Nonlinear Joints

In this paper, a reduced-order analytical model for an L-shaped multi-beam structure with nonlinear joints is presented to investigate the nonlinear responses of the system with three-to-one internal resonances conditions. Firstly, the global mode shapes are used to obtain an explicit set of nonlinear ordinary differential equations of motion for the system. Then, the first two natural frequencies of the system are calculated to determine the specific tip mass that results in three-to-one internal resonance. Subsequently, an approximation of the analytical solution of the dynamic model with two-degree-of-freedom is derived by using the multi-scale method. The accuracy of the approximation solution is verified by comparing it with the numerical solution obtained from the original motion equations. Based on the nonlinear dynamical model obtained by this paper, the frequency response curves are given to investigate the nonlinear dynamic characteristic of the L-shaped multi-beam structure with nonlinear joints. The results show that the nonlinear stiffness of the joints has a great influence on the nonlinear response of the system with three-to-one internal resonance conditions.

Nonlinear and chaotic vibrations of rotating functionally graded GPL reinforced composite pre-twisted blade subjected to aerodynamic force

The nonlinear and chaotic vibrations of the rotating functionally graded graphene platelet reinforced composite (FG-GPLRC) pre-twisted blade subjected to the aerodynamic force are investigated by taking into account the cantilever boundary condition. The rotating FG-GPLRC pre-twisted blade is simplified to a rotating FG-GPLRC pre-twisted cylindrical panel. The graphene platelets (GPLs) are dispersed uniformly and oriented randomly in every layer while the GPL volume fractions vary from the layer to layer, which leads to four different GPL distribution patterns. Based on the strain–displacement relationship of the pre-twisted cylindrical panel derived by Green strain tensor, the natural frequencies of the rotating FG-GPLRC pre-twisted cylindrical panel are obtained by using the first-order shear deformation theory and Chebyshev–Ritz method. Lagrange’s formulation is used to derive the nonlinear ordinary differential equations of motion. The complex chaotic vibrations of the rotating FG-GPLRC pre-twisted cylindrical panel are investigated by performing numerical simulations on the basis of Runge–Kutta algorithm. The bifurcation diagrams, Lyapunov exponents, time histories, phase portraits and Poincare maps are depicted to detect the periodic and chaotic vibrations of the rotating FG-GPLRC pre-twisted cylindrical panel with different parameters, such as Coriolis force, GPL distribution pattern, steady-state rotating speed, perturbation rotating speed and aerodynamic force. The double-parameter maximum Lyapunov exponent is regarded as an effective method to detect the chaotic regions of the rotating FG-GPLRC pre-twisted cylindrical panel.

Nonlinear dynamic responses of functionally graded graphene platelet reinforced composite cantilever rotating warping plate

In this study, the nonlinear dynamic behaviors are investigated for the functionally graded graphene platelet (FGGP) reinforced composite rotating warping blade under combined the aerodynamic force and centrifugal force. The blade is simplified to a FGGP reinforced composite rotating warping cantilever plate with a rectangular cross-section. Four distribution patterns of the graphene platelets (GPLs) along the thickness of the plate are considered when establishing the dynamic model. The effective material parameters, Poisson's ratio, mass density and Young's modulus of the FGGP reinforced composite rotating warping cantilever plate are calculated based on the modified Halpin-Tsai model and mixture rule. Considering the warping effect, the dynamic governing equations of motion for the rotating FGGP reinforced composite cantilever plate subjected to the aerodynamic force and centrifugal force are established by using the first-order shear deformation theory and Hamilton's principle. Galerkin approach is exploited to transform the partial differential equations of motion to the ordinary differential equations of motion the FGGP reinforced composite rotating warping cantilever plate. Four-dimensional autonomous averaging equations of the system are obtained by using the asymptotic perturbation method. The parameter analyzes on the amplitude-frequency and force-amplitude responses of the FGGP reinforced composite rotating warping cantilever plate are carried out by considering the 1:2 internal resonance and principal parametric resonance. The stability of the steady solution of the autonomous system is discussed. The influences of the GPL distribution patterns, rotating speed, aerodynamic force and damping on the chaotic dynamics are investigated for the FGGP reinforced composite rotating warping cantilever plate.

Vibration Control of Coupled Wind Tower–Nacelle–Blade System

Wind turbines are slender and flexible structures which can undergo large amplitude vibrations when subjected to external loads and, an important way to reduce these excessive vibrations is to apply structural control. In this work, the vibration control of the coupled wind tower–blade system subjected to external lateral loads and rotating blade is studied. To model the tower and blade, the nonlinear Euler–Bernoulli beam theory was applied, and the structural control device considered is a nonlinear inverted tuned mass damper pendulum (ITMDP), located at the top of the tower. The Rayleigh–Ritz method, together with Hamilton’s principle, is applied to obtain a set of coupled nonlinear ordinary differential equations of motion which are in turn solved by the Runge–Kutta method. First, the dynamic instability of the system is studied by considering the variation of the natural frequencies of the tower, as a function of the speed of rotation of the blade and the veering phenomenon is observed. The optimum parameters for the ITMDP are obtained and, a nonlinear dynamic analysis is performed to evaluate the influence of the tuned mass damper (TMD) in the nonlinear regime, by obtaining the time responses, resonance curves, Poincaré maps and basins of attraction. The obtained results show the importance of considering the coupled system and the good performance of the structural control in the dynamic behavior of the system.

Moving load response of ring-stiffened sandwich truncated conical shells with GPLRC face sheets and porous core

As a first attempt, the dynamic responses of sandwich truncated conical shells with graphene platelets reinforced composite (GPLRC) face sheets, porous core and circumferential stiffeners under asymmetric internal ring-shaped moving load are analyzed. The face sheets and porous core are reinforced by uniformly distributed and randomly oriented graphene platelets (GPLs). The spatially discretized equations of motion subjected to different boundary conditions are derived based on the first-order shear deformation theory (FSDT) using the Chebyshev–Ritz method. The admissible functions of the field variables are constructed using the trigonometric functions in the circumferential direction and the one-dimensional Chebyshev polynomials in conjunction with the suitable boundary functions in the meridional direction. The Newmark’s time integration scheme is employed to solve the resulting ordinary differential equations of motion. The robustness and accuracy of the method are demonstrated through the different examples. After that, the influences of core porosity, face sheets, GPLs parameters, shell geometric parameters, boundary conditions, and the arrangement, number and size of stiffeners on the results are explored. The results show that porosity, face sheets, the location, size and number of stiffeners play important roles on the dynamical response of the shells under investigation. Also, it is found that the critical load velocity decreases by increasing the porosity coefficients and increases by increasing the number of ring-stiffeners.

Nonlocal anisotropic elastic shell model for vibrations of double-walled carbon nanotubes under nonlinear van der Waals interaction forces

In this paper, a novel nonlocal anisotropic elastic shell model is developed to investigate the nonlinear vibrations of double-walled carbon nanotubes (DWCNTs) in the framework of Sanders–Koiter shell theory. Van der Waals interaction forces between the two concentric single-walled carbon nanotubes (SWCNTs) composing a DWCNT are modelled via Lennard-Jones potential and He’s formulation. In the linear vibration analysis, the displacement field of each SWCNT is expanded by means of a double mixed series in terms of Chebyshev orthogonal polynomials along the longitudinal direction and harmonic functions along the circumferential direction, and Rayleigh–Ritz method is considered to get approximate natural frequencies and modal shapes. In the nonlinear vibration analysis, the three displacements of each SWCNT are re-expanded by means of the approximate eigenfunctions derived in the linear analysis, and an energy approach based on Lagrange equations is adopted to obtain a set of nonlinear ordinary differential equations of motion, which is then solved numerically. Molecular dynamics simulations are performed in order to calibrate the proper value of nonlocal parameter to be inserted in the constitutive equations of the proposed elastic continuum model. A simplified linear distribution of van der Waals interaction forces is initially adopted to analyse the nonlinear vibrations of DWCNTs, obtaining a hardening nonlinear behaviour. By considering a more realistic nonlinear distribution of van der Waals interaction forces, a stronger hardening nonlinear behaviour is found.

Snap-through and nonlinear oscillations of bistable asymmetric FCSS composite laminated square plate subjected to excitation: Theoretical, numerical and experimental researches

This paper proposes the theoretical, numerical and experimental researches on the dynamic snap-through phenomena and nonlinear oscillations of the bistable asymmetric composite laminated square thin plate under the foundation excitation and novel boundary support. The boundary conditions of four corner simply-support (FCSS) are designed. The displacement solutions satisfying the boundary conditions are assumed to describe various nonlinear oscillations of the bistable asymmetric composite laminated square thin plate under the foundation excitation. According to the third-order shear deformation theory (TSDT), von-Karman nonlinear strain displacement relations and virtual work principle, the nonlinear differential governing equation of motion are theoretically derived for the bistable asymmetric composite laminated square thin plate. Based on Hamilton principle, the nonlinear ordinary differential equations of motion for the curvatures in different directions are obtained. Numerical simulations are used to analyze the interesting snap-through phenomena and nonlinear oscillations of the bistable asymmetric FCSS composite laminated thin plate. The influences of the amplitude and frequency for the basic excitation on the snap-through are discussed. It is seen that under a certain external excitation, the bistable plate goes through the single stable, periodic snap-through and chaotic snap-through oscillations. The experimental platform of the bistable asymmetric composite laminated thin plate is established with four corner simply-support under the foundation excitation. Through the vibration experiments, the dynamic snap-through phenomenon and nonlinear oscillations are studied for the bistable asymmetric FCSS laminated composite square thin plates.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

Manipulator Kinematics and Dynamics on Differentiable Manifolds: Part II Dynamics

Abstract The manipulator differentiable manifold kinematics formulation presented in Part I of this paper is employed for forward and inverse kinematic analysis and formulation of ordinary differential equations of motion on disjoint, maximal, singularity free, path-connected components of regular configuration space, for a broad spectrum of manipulators. Existence of forward and inverse configuration mappings throughout maximal singularity free manifold components yields analytical forward and inverse velocity mappings. Efficient computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of three manipulator categories. Manifold parameterizations are used to transform variational equations of motion in manipulator coordinates to second-order ordinary differential equations of manipulator dynamics, with both input and output coordinates as state variables, eliminating the need for ad hoc derivation of equations of motion. Criteria that define manipulator differentiable manifolds are shown to guarantee that the equations of motion derived are well-posed on maximal singularity free components of manipulator configuration space. This process is illustrated by presenting terms required for the evaluation of equations of motion for three model manipulators. It is shown that computation involved in the evaluation of equations of manipulator kinematics and dynamics can be carried out in real-time on modern microprocessors, supporting the in-line implementation of modern methods of manipulator control.

Two problems of movement of multi-link wheeled vehicles

The paper presents the results of a study of the dynamics of multi-link wheeled vehicles with nonholonomic connections superimposed on this mechanical system. The article is of an overview nature. The mathematical formulation is a system of ordinary differential equations of motion in the form of Lagrange with undefined multipliers, solved together with derivatives of nonholonomic constraints. The results are presented in the case of controlled motion, when the law of motion of the first link is known, as well as in the case of uncontrolled motion, when the law of motion of the leading link is also the desired function of time. General equations are obtained for a mechanical system consisting of an arbitrary number of links. Numerical results are presented for the case of three coupled trolleys. The software package Maple was used to perform numerical calculations and plotting of the desired mechanical parameters.

On the free vibration problem of thin shallow and non-shallow shells using tensor formulation

Linear free vibration of thin shells with rectangular base such as parabolic conoids, elliptic paraboloids and hyperbolic paraboloids are investigated. These shells are described by non-orthogonal curvilinear coordinates hence, Koiter theory, which uses a tensor formulation, is applied to accurately describe the strain–displacement relations. Four different boundary conditions are considered: simply supported with movable edges, simply supported with immovable edges; fully clamped with movable edges and fully clamped with immovable edges. Using an energy approach, Rayleigh–Ritz method is applied to obtain a set of ordinary differential equations of motion and Fourier series and Legendre polynomials are used to describe the displacements fields of the shell. Both, shallow and non-shallow shells were studied and obtained results of natural frequencies and vibration modes were compared with those from Maguerre theory and FEM using Abaqus, as can be seen, a good agreement was observed with FEM showing the efficacy of the applied theory. When comparing natural frequencies to those from Marguerre theory, there are substantial differences, indicating that lack of use of the tensor formulation which is not appropriate to shells with non-orthogonal coordinates.

Dynamic stability of axially moving graphene reinforced laminated composite plate under constant and varied velocities

The dynamic stability of an axially moving graphene reinforced laminated composite plate is studied. The graphene nanoplatelets (GPLs) are distributed in each polyvinylidene fluoride (PVDF) layer uniformly and in the whole plate symmetrically. The volume fraction of the first layer is used as the control variable to determine the distribution pattern of the GPLs. Halpin–Tsai’s model is used to predict the effective Young’s modulus of the axially moving graphene reinforced laminated composite plate. The effective mass density and Poisson’s ratio are calculated by using the rule of mixture. Based on von Kármán plate theory and Hamilton’s principle, the governing equations of motion are derived for the axially moving graphene reinforced laminated composite plate. To evaluate the stability of the axially moving graphene reinforced laminated composite plate with the constant velocity, the ordinary differential equations of motion are obtained by using Galerkin method. The effect of the graphene reinforcement on the critical and flutter velocities of the axially moving laminated composite plate is analyzed based on the eigenvalue of the coefficient matrix. The influence of the graphene reinforcement on the instability region for the axially moving laminated composite plate with the varied velocity is also evaluated by using the direct multiscale method.

Nonlinear vibrations of rotating pretwisted composite blade reinforced by functionally graded graphene platelets under combined aerodynamic load and airflow in tip clearance

The primary resonance and nonlinear vibrations of the functionally graded graphene platelet (FGGP)-reinforced rotating pretwisted composite blade under combined the external and multiple parametric excitations are investigated with three different distribution patterns. The FGGP-reinforced rotating pretwisted composite blade is simplified to the rotating pretwisted composite cantilever plate reinforced by the functionally graded graphene platelet. It is novel to simplify the leakage of the airflow in the tip clearance to the non-uniform axial excitation. The rotating speed of the steady state adding a small periodic perturbation is considered. The aerodynamic load subjecting to the surface of the plate is simulated as the transverse excitation. Utilizing the first-order shear deformation theory, von Karman nonlinear geometric relationship, Lagrange equation and mode functions satisfying the boundary conditions, three-degree-of-freedom nonlinear ordinary differential equations of motion are derived for the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations. The primary resonance and nonlinear dynamic behaviors of the FGGP-reinforced rotating pretwisted composite cantilever plate are analyzed by Runge–Kutta method. The amplitude–frequency response curves, force–frequency response curves, bifurcation diagrams, maximum Lyapunov exponent, phase portraits, waveforms and Poincare map are obtained to investigate the nonlinear dynamic responses of the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations.

Effect of magnetic field on the thermo-elastic response of a rotating FGM Circular disk with non-uniform thickness

In this study, the effect of the magnetic field on the thermo-elastic response of a rotating non-uniform circular disk of functionally graded material (FGM) is investigated for both steady and transient states of temperature. A single second-order ordinary differential equation of motion was developed for an FGM disk and solved along with the boundary conditions using the finite-difference method (FDM). The steady-state and transient heat conduction equations were also solved using the finite-difference method. Numerical results were presented and discussed for an Al/Al2O3 FGM disk of exponentially varying material properties keeping Poisson’s ratio and magnetic permeability uniform. Displacement and stress components were analyzed by increasing the intensity of the magnetic field for different cases of steady and transient states of temperature. The analysis suggests that the magnetic field has a remarkable effect on the displacement and stress distributions. It is also found that, high intensity of the magnetic field changes the nature and location of maximum stress. The transient analysis of magneto-thermo-elastic field suggests that the increase in the intensity of magnetic field results in the increase in stress intensity near the outer region of the disk and maximum radial stress always exceeds maximum circumferential stress. The effects of inner and outer surface radius, thermal gradient between inner and outer surface, and the outer surface thickness were also analyzed in detail. It was found that, with the decrease in outer surface radius and thermal gradient between inner and outer surface, maximum circumferential stress becomes higher than the maximum radial stress. In addition, the soundness and accuracy of the solutions were verified with the results from the standard computational method and analytical solution.

Multibody Dynamics on Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.