Analytical dynamics of mechanisms—a computer oriented overview

Dynamics and Control of Machines

In this paper piezoelectric films will be used as control sensors and actuators to actively damp vibrations of a cantilever beam. The approach taken here is to use Hamilton's energy method to derive the differential equation of motion for a beam/piezoelectric structure. The equation of motion shows that when the piezoelectric has a constant spatial distribution, then the actuator and sensor voltage are functions of the beam position at one discrete location. The equation of motion is then solved in the Laplace domain and is given in two transfer function forms. One of the derived transfer functions is verified experimentally and is shown to be an accurate model of the system dynamics. Using the transfer functions the effects of compensation are analyzed using classical techniques. Various compensators are analyzed to see their effects on the dynamics of the structure. One control system design is implemented in an experiment to evaluate the accuracy of the analytical results.

A coupled spring-pendulum system in a conservative field was studied where the equation of motion of the system using Lagrangian and Hamiltonian equation were obtained. The equation of motion represented by a second-order differential equation from the three generalized coordinate were used. The potential energy equal to zero when the system is in its equilibrium position. The generalized coordinate that being used were the angle of the first pendulum θ_1, the angle of the second pendulum θ_2, and the increase in the length of the spring x. The resulting equation of motion can be used to determine the dynamics behavior of the system at any time. Students' understanding is expected to be more complete by providing a procedure to derive the Lagrangian equation of motion.

Chapter 4 - Virtual work and energy methods

The yaw problems of a three-bladed, downwind, horizontal axis wind turbine are examined in this report. A four-degree-of-freedom system was chosen to model the turbine. Linearized equations of rotor and nacelle motion were developed using the energy method and Lagrange's equations. Quasi-steady blade element and momentum theories were used in developing the axial induction factor and aerodynamic loads. A computer code was developed to obtain the numerical values of coefficients of the equations of motion, thus allowing the cause of yaw instability to be studied. The study indicated that yaw tracking error is primarily caused by tower shadow. However, the wind turbine studied - besides being unstable in yaw under normal operating conditions - has an additional problem in that the nacelle shape contributes to additional instability. Blade coning, in the present design, is inadequate to overcome this instability. The sensitivity of the system stiffness coefficients to the selected input parameters was studied, and results indicate that the system stiffness coefficient is highly sensitive to the coning angle. Increasing the coning angle could significantly increase the stiffness coefficient and lead to improvement of system stability.

Corresponding relationships between certain incremental equilibrium equations and compatibility equations can be derived for structures undergoing gross deformation. Investigation of these relationships leads to the development of a new principle of virtual work and a form of energy called stress energy. The concept of stress energy has been developed by considering the potential energy of the forces applied to the structure or mechanism and its relationship to Libove's from of complementary energy and Charlton's virtual work method has been discussed.

<div class="htmlview paragraph">This paper is to propose the equations of motion in time of steady-state cornering motion, which are derived from the Lagranges's equations. The proposed equation is the nonlinear algebraic equation analyzable the handling torque characteristic, and applicable also to a large bank angle conditions.</div> <div class="htmlview paragraph">The amount of this paper is explained under the following four points. 1. The equations of motion in time of steady-state motorcycle cornering motion are derived from applying the Lagranges's equations of motorcycles R.S. Sharp proposed. The experiment value was used in the process to derive from. This new proposed equations include the additional new items which are not used in existing equations of motion in time of steady-state cornering motion, based on the equations of the straight-running stability. 2. The proposed new equations are considered the influence of tire section radius. 3. The equations of motion in time of steady-state cornering motion are derived as the nonlinear algebraic equation. 4. The handling torque characteristic over the large bank angle was analyzable by deriving the nonlinear algebraic equations to the linear equations for arbitrary bank angle around there.</div>

The static stability of an asymmetric sandwich beam with viscoelastic core on viscoelastic supports at the ends and subjected to an axial pulsating load and a steady, one-dimensional temperature gradient is investigated by computational method. The equations of motion and associated boundary conditions are obtained using the Hamilton's energy principle. Then, these equations of motion and the associated boundary conditions are non- dimensionalised. A set of Hill's equations is obtained from the non-dimensional equations of motion by the application of the general Galerkin method. The static buckling loads are obtained from the Hill's equations. The effects of shear parameter, geometric parameters, core loss factors, and thermal gradient on the non-dimensional static buckling loads zones have been investigated.

This paper presents a new formulation for the equations of motion of interconnected rigid bodies. This formulation initially uses Cartesian coordinates to define the position of the system, the kinematic joints between bodies, and forcing functions on and between bodies. This makes initial system definition straightforward. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates relative coordinates to Cartesian coordinates. It is derived using kinematic relationships for each joint type and graph theory for identifying the system topology. By using relative coordinates, the equations of motion are efficiently integrated. Use of both Cartesian and relative coordinates produces an efficient set of equations without loss of generality. The algorithm just described is implemented in a general purpose computer program. Examples are used to demonstrate the generality and efficiency of the algorithms.

A spatial redundantly actuated parallel mechanism (RAPM) constrained by two point contact higher kinematic pairs (HKPs) has been designed, arising from the inspiration of mastication in human beings: the end effector is the lower jaw, the six kinematic chains are the primary chewing muscles, and the constraints at HKPs are the temporomandibular joints. In this paper, firstly, the constrained motions of the mechanism are described in detail; thereafter, five models are formulated by the well-known Newton-Euler's law, the Lagrangian equations, and the principle of virtual work, to explore its rigid-body inverse dynamics. The symbolic results show that the model structures based on these approaches are quite different: the model via the Newton-Euler law well reflects the nature of the mechanism in terms of the constraint forces from HKPs with six equations and eight unknowns, and the existence of reaction forces at the spherical joints is tightly dependent on the number of kinematic chains. In comparison, from the latter two methods, the constraint forces and the reaction forces at spherical joints do not appear in the four models in which there are only four equations and six unknowns. Further, by using the dynamics model of the non-redundantly actuated counterpart as the core in both the second models from the energy and virtual work-related methods, their computational cost is only about 16.7% and 36.63% of the two first models, respectively. Finally, the comparisons between the dynamics models of the RAPM and its counterpart clarify that the HKP constraints greatly alter the model structures and raise the technical difficulties.

Hamiltonian systems, information transmittal and special relativity

: Systematic investigation is made of effects of kinematic assumptions and finite element approximations in the context of nonlinear flexible multibody dynamics. Two nonlinear beam finite elements are consistently derived from virtual work principle using Bernoulli Euler and Timoshenko beam kinematics. Initial assessment is made by studying convergence properties of element formulations with eigenvalue problems: free vibration, static buckling, and dynamic buckling. Equations of motion are derived for rigid central body with flexible appendage using virtual work principle. Virtual work principle allows natural and consistent discretization of flexible appendage using finite element method. Nonlinearities in flexibility are explored through dynamics examples using beam finite elements. Application of dynamics formulation is made to a realistic scenario involving space shuttle remote manipulator arm with attached payload. Contribution of nonlinear theory, in both formulation and solution, is assessed.

Unified Hamiltonian theory of relativistic particle equations

Electrical activity and synchronization of memristor synapse-coupled HR network based on energy method

A cross section of the rod is taken as object of investigation. The freedom of the section in free or constraint case is analyzed and the definition of virtual displacement of the section is given, which can be expressed by a variational operation. Assuming the variational and partial differential operations has commutativity, based on the hypothesis about surface constraint subjected to the rod, the freedom of the section on constraint surface is discussed and the equations satisfied by virtual displacements of the section are given. Combining D'Alembert principle and the principle of virtual work, D'Alembert-Lagrange principle is established. When constitutive equation of material of the rod is linear, the principle can be transformed to Euler-Lagrange form. From the principle, a dynamical equation in various forms such as Kirchhoff, Lagrange, Nielsen and Appell equation can be derived. For the case when a rod is subjected to a surface or a nonholonomic constraint, Lagrange equation with undetermined multipliers is obtained. Integral variational principle of dynamics of a super-thin elastic rod is also established, from which Hamilton principle formulation is obtained when the material of the rod is linear. Finally, canonical variables to describe the state of the section and Hamilton function are defined, and Hamilton canonical equation is derived. The analytical methods of dynamical modeling of a super-thin elastic rod have been constructed, which can serve as a theoretical framework of analytical dynamics of a super-thin elastic rod with two independent variables.