Computer Oriented Analytical Dynamics of Machinery

Dynamics and Control of Machines

In this paper we show why the postulation scheme based on forces and couples is detrimental when designing novel metamaterials. Instead, the most effective postulation scheme for mechanics is that based on the Principle of Virtual Work formulated by, for example, d'Alembert, Lagrange, Piola, and Paul Germain. In fact, generalized continuum mechanics can be formulated in a coherent way only by basing its foundations on the Principle of Virtual Work while the problem of synthesis of novel metamaterials can be efficiently confronted only when introducing generalized continua as their model at macroscopic level. Once work functionals are postulated, the set of involved generalized forces are identified using the representation theorem for distributions by Laurent Schwartz, forces in particular being zeroth order distributions. Actually, only when limiting the class of considered internal work functionals to first order distributions is the Principle of Virtual Work equivalent to the balances of forces and couples. The predominant Role of the Principle of Virtual Work has been denied by those inductivist scholars who claim that balance laws can be induced by experimentation. The Principle of Virtual Work postulation more truly fits in a falsificationist approach to epistemology.

Kinematics and dynamics of a 4-PRUR Schönflies parallel manipulator by means of screw theory and the principle of virtual work

Differential equations serve as foundational tools in the modeling and analysis of dynamic mechanical systems. Their application spans various mechanical engineering domains, including vibrations, heat transfer, fluid mechanics, and structural analysis. This paper explores the role of ordinary and partial differential equations in representing physical laws governing mechanical systems. Analytical and numerical approaches to solving these equations are examined, and real-world applications are highlighted. The integration of modern computational techniques is also discussed, offering insights into the future of differential equation modeling in mechanical engineering.

Kinematics optimization and static analysis of a modular continuum robot used for minimally invasive surgery.

In this investigation, a new three-dimensional nonlinear train car coupler model that takes into account the geometric nonlinearity due to the coupler and car body displacements is developed. The proposed nonlinear coupler model allows for arbitrary three-dimensional motion of the car bodies and captures kinematic degrees of freedom that are not captured using existing simpler models. The coupler kinematic equations are expressed in terms of the car body coordinates, as well as the relative coordinates of the coupler with respect to the car body. The virtual work is used to obtain expressions for the generalized forces associated with the car body and coupler coordinates. By assuming the inertia of the coupler components negligible compared to the inertia of the car body, the system coordinates are partitioned into two distinct sets: inertial and noninertial coordinates. The inertial coordinates that describe the car motion have inertia forces associated with them. The noninertial coupler coordinates; on the other hand, describe the coupler kinematics and have no inertia forces associated with them. The use of the principle of virtual work leads to a coupled system of differential and algebraic equations expressed in terms of the inertial and noninertial coordinates. The differential equations, which depend on the coupler noninertial coordinates, govern the motion of the train cars; whereas the algebraic force equations are the result of the quasi-static equilibrium conditions of the massless coupler components. Given the inertial coordinates and velocities, the quasi-static coupler algebraic force equations are solved iteratively for the noninertial coordinates using a Newton–Raphson algorithm. This approach leads to significant reduction in the numbers of state equations, system inertial coordinates, and constraint equations; and allows avoiding a system of stiff differential equations that can arise because of the relatively small coupler mass. The use of the concept of the noninertial coordinates and the resulting differential/algebraic equations obtained in this study is demonstrated using the knuckle coupler, which is widely used in North America. Numerical results of simple train models are presented in order to demonstrate the use of the formulation developed in this paper.

The principle of virtual work is the starting point for the formulation of the finite element method for solids and structures and we dedicate this chapter to its study.We start by introducing the principle of virtual work for a 1-D model and by exploring the fundamental facts. Then, the principle is given for the 3-D elasticity model and for the remaining mathematical models discussed in Chapter 4.In the last section, we introduce some energy concepts in 3-D analysis which provide alternative ways to formulate the mathematical models of Chapters 3 and 4. These energy based formulations will be useful in Chapters 6 and 8.KeywordsVirtual WorkElasticity ProblemTotal Potential EnergyVirtual DisplacementDisplacement Boundary ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Analytical Mechanics. Foundations of Engineering Mechanics Series

In structural systems, impact-induced longitudinal elastic waves travel with finite speeds that depend on the material properties. Using Fourier method of analysis, the exact wave motion can be described as the sum of infinite number of harmonic waves which have the same phase velocity. In this case the medium is said to be nondispersive, since the phase velocities of the harmonic waves are equal and equal to the group velocity of the resulting wave motion. In mechanism systems with intermittent motion, on the other hand, elastic members undergo finite rotations. In this investigation, the effect of the finite rotation, coefficient of restitution, and impact conditions on the propagation of the impact-induced waves in costrained elastic systems is examined. The system equations of motion are developed using the principle of virtual work in dynamics. The jump discontinuities in the system variables as the result of impact are predicted using the generalized impulse momentum equations that involve the coefficient of restitution. It is shown that the phase velocities of different harmonic waves are no longer equal, that is, dispersion occurs in perfectly elastic mechanism members as the result of the finite rotation. The analysis presented in this paper shows that the finite rotation has more significant effect on the phase velocity of the low frequency harmonics as compared to the high frequency harmonics. A rotation-wave number that depends on the material properties and the wave length is defined for each harmonic wave. It is shown that if the angular velocity of the elastic member becomes large such that the rotation-wave number of a mode exceeds one, the associated modal displacement is no longer oscillatory.

Chapter 4 - Virtual work and energy methods

In this paper we present the kinematic and dynamic analysis of a parallel architecture manipulator with linear actuators, consisting of two 3-RRPS mechanisms through the screw theory and the principle of virtual work, respectively. The equations of input / output speed and acceleration are obtained by applying the Klein form of the Lie algebra. In addition, the generalized forces of the mechanism are determined by combining the above theory with the principle of virtual work. Finally, we present a case of study; the numerical results are compared with simulations using ADAMS software.

This paper presents an iterative automated method for optimizing structures with multiple deflection criteria and load cases. The method is based on the principle of virtual work. Discrete sections are selected for structures with fixed geometries. An optimal structure is one which meets all strength and deflection criteria using minimal material. Four case studies are considered in this paper. A simple portal frame is presented to show how the method works. A 60-story frame is optimized to demonstrate the effectiveness of the method for large structures. A warehouse designed by professional engineers is presented to show how the method can be used for structures subjected to complex loading conditions and deflection criteria. The automated method’s solution is 4.5% lighter than the engineers’. Finally, a stepped cantilever is optimized and compared to results in literature. Material savings of up to 14.4% are realized.

In this paper, the kinematics and inverse dynamics of a novel kind of mechanism called a general 3-PRS parallel mechanism is investigated. In the kinematics study, the inverse kinematics solution is derived in closed form, and the forward kinematics problem is resolved by the Newton iterative method seeking for an on-line solution to this issue. The inverse dynamics analysis is approached with two methods: Lagrangian formulations and principle of virtual work. After deriving the dynamic model by a Lagrangian formulation approach, the simulation results of two introduced examples quantitatively and qualitatively verify the accuracy of the derived dynamic equations. By introducing a simplifying hypothesis, a simplified dynamic model is set up using principle of virtual work, also a computer simulation is performed on this reduced model. The simulation results demonstrate that the simplified dynamic model is reasonable under such kind of assumptions through comparison with the precise model derived from the Lagrangian formulation. The inverse dynamics analysis provides a sound basis to develop controllers for controlling over a general 3-PRS parallel robot.

<div class="htmlview paragraph">Effective computer-aided design of engineering systems requires comprehensive computer-application software that conveniently adapts to the particular engineering design considered. This paper discusses the theoretical background and initial experience with a program intended for computer-aided design of machine-like mechanical systems, and outlines the use of a time-shared graphic terminal for schematic display of program output. The program is applicable to three types of mechanical systems: linear and nonlinear unconstrained dynamic systems, kinematic systems (that is, constrained systems), and multifreedom, constrained systems.</div>

This paper aims to present an analytical model to predict the static analysis of laminated reinforced composite plates subjected to sinusoidal and uniform loads by using a simple first-order shear deformation theory (SFSDT). The most important aspect of the present theory is that unlike the conventional FSDT, the proposed model contains only four unknown variables. This is due to the fact that the inplane displacement field is selected according to an undetermined integral component in order to reduce the number of unknowns. The governing differential equations are derived by employing the static version of principle of virtual work and solved by applying Navier\'s solution procedure. The non-dimensional displacements and stresses of simply supported antisymmetric cross-ply and angle-ply laminated plates are presented and compared with the exact 3D solutions and those computed using other plate theories to demonstrate the accuracy and efficiency of the present theory. It is found from these comparisons that the numerical results provided by the present model are in close agreement with those obtained by using the conventional FSDT.