Analytical dynamics

In this paper, we present the general structure for the explicit equations of motion for general mechanical systems subjected to holonomic and non-holonomic equality constraints. The constraints considered here need not satisfy D'Alembert's principle, and our derivation is not based on the principle of virtual work. Therefore, the equations obtained here have general applicability. They show that in the presence of such constraints, the constraint force acting on the system can always be viewed as made up of the sum of two components. The explicit form for each of the two components is provided. The ÿrst of these components is the constraint force that would have existed, were all the constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs speciÿcation by the mechanician and depends on the particular situation at hand, this component nonetheless has a speciÿc form. The paper also provides a generalized form of D'Alembert's principle which is then used to obtain the explicit equations of motion for constrained mechanical systems where the constraints may be non-ideal. We show an example where the new general, explicit equations of motion obtained in this paper are used to directly write the equations of motion for describing a non-holonomically constrained system with non-ideal constraints. Lastly, we provide a geometrical description of constrained motion and thereby exhibit the simplicity with which Nature seems to operate. ?

This research work presents a discussion and a plan towards the analytical solving of Partial Differential Equation (Heat Equation) using manual solving and symbolic computation, the algorithm is developed in order to make the task of solving easier in the process of calculating exact solutions for partial differential equation. Finally, the algorithm was written in Maple programming language which is deployed on Windows Operating System.

The derivation of fourth order Runge-Kutta method involves in computation of many unknowns and the detailed step
by step derivation and analysis can hardly be found in many literatures. Due to the vital role played by the method in
the field of computation and applied science/engineering, we simplify and further reduce the complexity of its derivation
and analysis by exploring some possibly well-known works and propose a step by step derivation of the method. In this
paper, the improved Runge-Kutta Method of Order 4 with four (4) stages for solving First Order Ordinary Differential
Equation is proposed. The method is based on classical Runge-Kutta (RK) method which is considered as special class
of two-step method. Here, the coefficients of the method are obtained using the minimization of the error norm up to
order five. The improved method with only 4-stages is more accurate than fourth order 4-stages RK Method. A number
of test problems are solved and the numerical results compared with the analytical solution.

We present an analytical analysis of a continuous rotor shaft subjected to universal temperature gradients. To this end, an analytical model is derived to investigate the generic thermal vibrations of rotor structures. The analytical solutions are obtained in a rotating frame and include parameters related with both the thermal environment and the rotor dynamic structures. This provides an insight into the mechanisms for the rotor thermal vibration. Furthermore, numerical results based on the analytical solutions are given. An index denoting the temperature gradients is proposed for the occasions with
nonlinear cross-sectional temperature distributions. Finally, the factors influencing the thermal vibrations are analyzed. The results show that the thermal vibration is affected by
many factors including the shaft size, rotational speeds, heating locations, critical speed, etc. Moreover, it is investigated how the convection coefficient and the heat conductivity influence the thermal vibrations in order to provide an insight into the management of thermal vibrations from the perspective of thermal aspects.

In this research, we will use adomian decomposition method to obtain the numerical solution for one-dimensional Bratu’s problem. We compared with other existing methods that have been used for this problem to show its accuracy and the error of the numerical solution. It will be demonstrated that adomian decomposition method can be used to solve Bratu’s problem associated with initial conditions.

This paper presents a new, simple, and exact solution to the formation keeping of satellites when the relative distance between the satellites is so large that the linearized relative equations of motion no longer hold. We employ a recently proposed approach, the Udwadia-Kalaba approach, which makes it possible to explicitly obtain the desired control function without making any approximations related to the nonlinearities in the underlying dynamics. We use an inertial frame of reference to describe the motion of a satellite and since no approximations are made, the results obtained apply to situations even when the distance between the satellites is arbitrarily large. The paper deals with a projected circular formation, but the methodology in this paper can be applied to any desired configuration or orbital requirements. Numerical simulations confirm the brevity and the accuracy of the analytical solution to the dynamical control problem developed herein.

This study deals with the numerical solutions of the first natural frequency of uniform shear beams with constant
shear distortion and constant stiffness reading on Winkler Foundation. Adomian Decomposition Method was used to
solve for the first natural frequency at free-ends. The results were tabulated and graphical result for the mode shapes
was presented.

In this paper, we present the general structure for the explicit equations of motion for general mechanical systems subjected to holonomic and non-holonomic equality constraints. The constraints considered here need not satisfy D’Alembert’s principle, and our derivation is not based on the principle of virtual work. Therefore, the equations obtained here have general applicability. They show that in the presence of such constraints, the constraint force acting on the system can always be viewed as made up of the sum of two components. The explicit form for each of the two components is provided. The 5rst of these components is the constraint force that would have existed, were all the constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs speci5cation by the mechanician and depends on the particular situation at hand, this component nonetheless has a speci5c form. The paper also provides a generalized form of D’Alembert’s principle wh...

Lagrangians for classically damped linear multi-degrees-of-freedom dynamical systems are obtained using simple and elementary methods. Such dynamical systems are very widely used to model and analyze small amplitude vibrations in numerous naturally occurring and engineered systems. An invariant of the motion is also obtained.

This paper deals with the initial development of a methodology for controlling real-life, multi-body dynamical systems in the presence of uncertainties in our knowledge of their exact physical nature as well as uncertainties in the 'given' forces acting on them. We assume that both these uncertainties can be time-varying, yet bounded. The methodology is developed in two steps. In the first step an exact control of the so-called nominal system—our best assessment of the real-life physical system and the forces acing on it—is developed. This control is inspired by results from analytical dynamics and involves no linearizations and/or approximations of the nonlinear, nonautonomous system. It also minimizes a suitable norm of the trajectory errors at each instant of time. In the second step, a continuous controller using a generalized sliding mode approach is developed that guarantees that the real-life system tracks, without any chatter and to within pre-specified error bounds, the control requirements that are imposed on the nominal system. Thus an explicit closed-form control for an uncertain, nonlinear, nonautonomous multi-body system that satisfies trajectory control requirements placed on the nominal system (to within pre-specified error bounds), is obtained. An example of a triple pendulum in which there are uncertainties both in the description of its physical parameters and in the description of the gravity field in which it moves is considered. The example demonstrates the simplicity and efficacy of the approach when there are uncertainties both in the description of a dynamical system and in the given forces to which it is subjected.