B04 - Dynamics of Rigid Bodies

Title: Dynamics of rigid bodies
Type: Book
Edition: First
Authors: Tsiatas G, Charalampakis A, Tsopelas P
Publisher: Tsotras publications
ISBN: 978-618-217-044-1
Format: Papercover
Pages: 741
Language: Greek
Date: September 2023


The primary aim of the book is to introduce the undergraduate student to basic concepts of Rigid Body Dynamics, while developing his/her ability to analyse, using logic, simple or more complex problems into sub-problems, which are easier to solve. More advanced mathematical tools, which are generally taught in the first years of university studies, are used in practice rather than in a sterile manner. A summary of these tools is provided in Chapter 1. The acquisition of the above-mentioned skills is an excellent weapon in every student's quiver, which will be useful in their future studies and professional careers.

In terms of subject covered, this textbook fully covers Newtonian Mechanics of Particles and Rigid Bodies, the study of which is divided into kinematics, which studies the geometrical properties of motion, and kinetics, which studies the relationship between motion and its causes, namely forces and moments.

Specifically, chapter 2 studies the kinematics of the particle, i.e. the general motion of a particle without considering the cause of the motion. The basic kinematic quantities (position vector, velocity, acceleration) are analysed and three-dimensional general motion in space, planar motion as well as rectilinear motion are studied. A variety of coordinate systems (Cartesian, physical, spherical, polar) are considered and various related topics are studied (pulleys, non-extending cables, relative motion with respect to a moving transport reference frame).

Chapters 4, 5, and 6 study the kinetics of the particle, i.e. the motion of the particle as a result of the forces acting on it. In particular, chapter 4 formulates Newton's equation of motion for particles and discusses the basic concepts of the free-body diagram, the kinetic diagram, inertial reference frames, and the dynamic equilibrium equation, which is essentially a transformation of Newton's equation of motion known as D'Alembert's principle. The motion of the particle in various coordinate systems (Cartesian, physical, cylindrical, spherical and polar) is studied, as well as the relative motion with respect to a moving (translational and rotational) reference frame. Due to the rotational motion of the reference system, additional inertial forces such as the centrifugal force, the Euler force and the curious Coriolis force make their appearance. At the end of the chapter, an attempt is made to understand these inertial forces with the help of simple examples from our everyday life.

In chapter 5, the kinetics of the particle is studied using the concepts of work and energy by formulating the work-energy theorem. The formulation of the work-energy theorem essentially follows from the integration of the equation of motion of the particle with respect to its position. For this reason there are no problems which can be solved by the work-energy theorem and cannot be solved by Newton's equation of motion. However, the work-energy theorem is ideally applicable to problems where the velocity of the particle must be correlated at two positions in its orbit and the forces acting on it are functions of its position. In addition, the related concepts of kinetic energy, generalized work-energy theorem, conservative and non-conservative forces, dynamic energy function, mechanical power and efficiency are presented and various cases of forces such as sliding friction, gravitational force, linear and non-linear spring force are considered.

In chapter 6, the kinetics of the particle is studied using the concepts of momentum and momentum by formulating the momentum-momentum theorem. For the rotational motion of particles with respect to some fixed point O, the corresponding angular thrust-momentum theorem is formulated. The formulation of these theorems essentially follows from the integration of Newton's equation of motion with respect to time. The two theorems are ideally applicable to problems where the velocity of the particle must be correlated at two positions in its orbit and the forces acting on it are functions of time. Also, the concepts of thrust and momentum find particular application in the study of the impact of two bodies where relatively large thrust forces are developed at their interface.

Chapter 7 studies the kinetics of a particle system and the relevant concepts that differentiate it from the case of a single particle, namely the centre of mass, kinetic energy, the work-energy theorem, the generalised work-energy theorem, the mechanical energy conservation theorem, the momentum-momentum theorem, the angular momentum-momentum theorem, and the conservation of momentum and angular momentum. Furthermore, the motion of a system of particles is a complex motion which is a combination of a translational and a rotational motion. For this reason, Euler's first law is formulated for translational motion and Euler's second law for the rotational motion of a particle system.

Next, Newtonian Mechanics of Rigid Bodies is presented, i.e. kinematics in chapter 3, and kinetics in chapters 8 and 9, which deal, respectively, with three-dimensional and planar kinetics of rigid bodies. Concepts such as translational and rotational degrees of freedom, mass and centre of mass, composite bodies, mass moment of inertia tensor, angular momentum, angular momentum - angular momentum theorem are introduced and analysed, conservation of angular momentum, kinetic energy and the generalised work-energy theorem, and the equations of rotational motion as a function of angular velocity components or as a function of Euler angles are given. At the end of Chapter 8, typical problems of rotational motion of axiosymmetric bodies or bodies in rotation are studied. One of these is gyroscopic motion which occurs when the axis about which a body rotates also rotates about another axis.

In addition, the book innovates over the existing literature by having been enriched with chapters on Dynamics-related topics, which, while particularly interesting, are rarely offered. Thus, Chapter 10 provides a detailed introduction to Analytic Dynamics, where the basic concepts of Lagrangian mechanics such as the calculus of variations are introduced and the related concepts of generalized coordinates, constrained motion, and constraint categorization are discussed. In addition, the principle of possible or virtual works, Hamilton's principle, Rayleigh damping function are presented and the famous Lagrange equations are formulated.

Chapter 11 deals with mechanical oscillations, which constitute a very important branch of the study of the dynamic behaviour of bodies and concern the determination of their response to dynamic loads in the presence of linear or non-linear restoring forces. By the term response of a body we mean the determination of the kinematic quantities of its motion, namely displacement, velocity and acceleration by solving the differential equation of motion. The study of the response of bodies is carried out in both the time and frequency domains. Initially, the motion of the single-stage oscillator in free oscillations with or without damping, in forced oscillations under periodic, impulsive and pulse loading is analyzed and the Duhamel integral, the Dirac delta function and the Fourier integral are presented. Then an extension is made to the motion of the two-stage and multi-stage oscillator and interesting related topics such as the resonant mass damper and frequency response functions are studied.

Chapter 12 deals with the stochastic oscillations of one- and two-stage linear and nonlinear oscillators. First, the basic principles of probability theory are introduced (concept of probability, bounded probability, random variable, mass and probability density function, distribution functions, statistical moments, Gaussian distribution, stochastic processes and fields, probability function of stochastic processes, nonstationary, stationary and ergodic stochastic processes, power function of spectral density, normal or Gaussian stochastic process, calculus of stochastic processes) and then stochastic oscillations are presented (relations between loading and response in the time and frequency domain, relations between statistical quantities of stochastic loading and response, stochastic response of linear and nonlinear oscillators).

Chapter 13 is devoted to the rocking of rigid bodies, a problem that has interested the engineer since ancient times, as it concerns one of the simplest human constructions. Corresponding to the simplicity of the construction, however, the dynamics of a rocking body is unexpectedly complex. Its behaviour is that of an inverted pendulum, where small changes in the initial conditions or in the characteristics of the external excitation result in large changes in the response. A rocking body may exhibit remarkable stability under strong seismic excitations, yet be overturned by some weaker ones. Its stability does not depend monotonically on its ductility, its size or the maximum acceleration of the earthquake. Even the most fundamental question, namely whether or not the body will ultimately overturn without any external excitation but only under given initial conditions, had no direct answer until very recently and required numerical integration of the equations of sway. The material in this chapter includes the formulation of the rocking equation of motion as well as the initial condition, and the exact nonlinear stability/reversal criterion for free rocking, with or without energy losses during base impacts, is given. Forced sway problems due to ground pulsations are studied, and rollover acceleration spectra are presented.  Finally, a methodology is proposed for the design of bodies against overturning due to seismic motion.

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