Analytical dynamics

In this study, differential transform method (DTM) is employed to investigate free vibration of
uniform shear beams with constant shear distortion and constant stiffness resting on Winkler
foundation DTM is an efficient technique for the solution of problems defined by linear or
non-linear differential equations. This research shows that DTM is an effective method for free
vibration analysis of uniform shear beam with constant shear distortion and constant stiffness
resting on Winkler foundation. The model equation is obtained and solved numerically using
DTM, tabular and graphical results are presented for the natural frequencies and mode shapes.

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.

The formulation of the dynamic equations of motion proposed by Udwadia-Kalaba is discussed from the point of view of numerical efficiency. Since this formulation requires the computation of a pseudoinverse matrix, it is investigated the influence of the method of pseudoinverse computation on the dynamic simulation of an overconstrained linkage. Finally, it has been proposed a new 1 dynamic equation which explicitly computes accelerations also in the case of mathematical models with rank deficient Jacobian and mass matrices.

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.

This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov con-straint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.

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.

The explicit equations of motion for a general n-body planar pendulum are derived in a simple and concise manner. A new and novel approach for obtaining these equations using mathematical induction on the number bodies in the pendulum system is used. Assuming that the parameters of the system are precisely known, a simple method for its control that is inspired by analytical dynamics is developed. The control methodology provides closed-form nonlinear control and makes no approximations/linearizations of the nonlinear system. No a priori structure is imposed on the controller. Globally, asymptotic Lyapunov stability is achieved along with the minimization of a userprovided control cost at each instant of time. This control methodology is then extended to include uncertainties in the parameters of the system through the use of an additional continuous controller. Simulations showing the simplicity and efficacy of the approach are provided for a 10-body pendulum system whose model is only known imprecisely. The ease with which the uncertain system can be controlled to move from any initial

The formulation of the dynamic equations of motion proposed by Udwadia-Kalaba is discussed from the point of view of teaching effectiveness and numerical efficiency. Since this formulation requires the computation of a pseudoinverse matrix, it is investigated the influence of the method of pseudoinverse computation on the dynamic simulation of an overconstrained linkage.

In this paper, an analytical procedure for the determination of the dynamic parameters of a remainder body after mass separation is developed. The method is based on the general principles of momentum and angular momentum of a body and system of bodies. The kinetic energy of motion of the whole body and also of the separated and remainder body is considered. The derivatives of kinetic energies with respect to the generalized velocity determine the velocity and angular velocity of the remainder body. To confirm the proposed procedure, the results are compared with those obtained using the method of momenta and angular momenta. In the paper, the theorem about increase of kinetic energies of the separated and remainder bodies for perfectly plastic separation is proved. The increase of the kinetic energies correspond to the relative velocities and angular velocities of the separated and remainder bodies. As an example, the mass separation from a pendulum is considered. The kinematic properties of the remainder pendulum are obtained using the analytic procedure. The results are in agreement with those obtained by applying the basic principles of Newton's mechanics.

A two-step formation-keeping control methodology is proposed that includes both attitude and orbital control requirements in the presence of uncertainties. Based on a nominal system model that provides the best assessment of the real-life uncertain environment, a nonlinear controller that satisfies the required attitude and orbital requirements is first developed. This controller allows the nonlinear nominal system to exactly track the desired attitude and orbital requirements without making any linearizations/approximations. In the second step, a new additional set of closed-form additive continuous controllers is developed. These continuous controllers compensate for uncertainties in the physical model of the satellite and in the forces to which it may be subjected. They obviate the problem of chattering. The desired trajectory of the nominal system is used as the tracking signal, and these controllers are based on a generalization of the concept of sliding surfaces. Error bounds on tracking due to the presence of uncertainties are analytically obtained. The resulting closed-form methodology permits the desired attitude and orbital requirements of the nominal system to be met within user-specified bounds in the presence of unknown, but bounded, uncertainties. Numerical results are provided, showing the simplicity and efficacy of the control methodology, and the reliability of the analytically obtained error bounds.

This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov con-straint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.

This paper presents simple and exact formation-keeping guidance schemes that use a new method that is rooted in some recent advances in analytical dynamics. As a result of this new approach, explicit control inputs to exactly maintain a given formation
configuration are easily determined using continuous thrust propulsion systems. The complete nonlinear problem is addressed, and no linearizations and/or approximations are made. The approach provides a marked improvement over existing results in that the control forces, which cause geometric formation-keeping constraints to be exactly satisfied for arbitrary reference orbits, are found in closed form. For Keplerian reference orbits, a much simpler and explicit expression for the control needed to exactly satisfy formation-keeping constraints than hereto available is obtained. The paper also includes explicit control results when the follower is inserted into orbit with incorrect initial conditions, as usually happens in practice. The Hill reference frame, which is
often more intuitive for formation-keeping, is used in the analysis. While this paper takes an example of a projected circular formation, the methodology that is developed can be applied to any desired configuration or orbital requirements. Extensive  computational simulations are performed to demonstrate the ease of implementation, and the numerical accuracy provided by the approach developed herein.

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