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This book presents a new approach to learning the dynamics of particles and rigid bodies at an intermediate to advanced level. There are three distinguishing features of this approach. First, the primary emphasis is to obtain the equations of motion of dynamical systems and to solve them numerically. As a consequence, most of the analytical exercises and homework found in traditional dynamics texts written at this level are replaced by MATLAB®-based simulations. Second, extensive use is made of matrices. Matrices are essential to define the important role that constraints have on the behavior of dynamical systems. Matrices are also key elements in many of the software tools that engineers use to solve more complex and practical dynamics problems, such as in the multi-body codes used for analyzing mechanical, aerospace, and biomechanics systems. The third and feature is the use of a combination of Newton-Euler and Lagrangian (analytical mechanics) treatments for solving dynamics problems. Rather than discussing these two treatments separately, Engineering Dynamics 2.0 uses a geometrical approach that ties these two treatments together, leading to a more transparent description of difficult concepts such as "virtual" displacements.
Some important highlights of the book include: Extensive discussion of the role of constraints in formulating and solving dynamics problems.
Implementation of a highly unified approach to dynamics in a simple context suitable for a second-level course.
Descriptions of non-linear phenomena such as parametric resonances and chaotic behavior.
A treatment of both dynamic and static stability.
Overviews of the numerical methods (ordinary differential equation solvers, Newton-Raphson method) needed to solve dynamics problems.
An introduction to the dynamics of deformable bodies and the use of finite difference and finite element methods.Engineering Dynamics 2.0 provides a unique, modern treatment of dynamics problems that is directly useful in advanced engineering applications. It is a valuable resource for undergraduate and graduate students and for practicing engineers.

### Chapter 1. Basic Elements of Dynamics

This chapter discusses some of the basic elements of dynamics, including the Newton-Euler laws, units, description of motion in various coordinate systems, and vector-matrix notation that is used in the book. A short summary is also given of the objectives of this book as well as an outline of the topics covered.
Lester W. Schmerr

### Chapter 2. Dynamics of a Particle

This chapter contains a detailed treatment of the dynamics of a particle. The main emphasis is on obtaining and solving the equations of motion when the particle is subject to constraints. In most cases the solution will be obtained numerically using MATLAB®. Both Newton-Euler and Lagrangian methods are used to obtain the equations of motion. Constraints are handled by either embedding them into the equations of motion (implicitly or explicitly) or solving an augmented system of equations that also yields an explicit expression for the constraint forces. Both ideal and nonideal constraints are considered. Issues that must be addressed when solving the equations of motion numerically are discussed, including the problem of constraint drift.
Lester W. Schmerr

### Chapter 3. Dynamics of a System of Particles

In Chap. 2 we used the motion of a single particle to discuss a wide range of topics including force versus moment equations, potential energy, work-energy concepts, constraint forces, generalized coordinates, Lagrange’s equations, and others. In this chapter we consider those topics and others for the case where a system of multiple, interacting particles is moving under the action of a set of forces.
Lester W. Schmerr

### Chapter 4. Kinematics and Relative Motion

This chapter describes the kinematics of point masses and rigid bodies when non-inertial coordinate systems (frames) are used to describe their motion. We obtain relative velocity and acceleration expressions for moving frames and then apply those expressions to find the velocities and accelerations of constrained systems of rigid bodies at specific instances of time, similar to what is done in many elementary dynamics texts. However, we also show that it is possible to determine the kinematics of a constrained system of rigid bodies more completely as a function of time. In some cases this can be done analytically but in general the solution must be done numerically since the positional constraints are normally nonlinear. To solve the positional constraints we use the Newton-Raphson method. Kinematics is treated in this chapter both by the traditional vector approach and by an equivalent matrix-vector method that is more readily suited to dealing with complex systems. The matrix-vector approach for planar problems is covered in Sects. 4.4 and 4.5 while more general three-dimensional problems are treated in Sect. 4.6 and those that follow. Three-dimensional rotations are described in terms of both Euler angles and Euler parameters as these are the most commonly used generalized rotational coordinates. Some classical examples of the dynamics effects seen in rotating coordinate systems, such as the Foucault pendulum, are also given.
Lester W. Schmerr

### Chapter 5. Planar Dynamics of Rigid Bodies

This chapter examines the dynamics of rigid bodies in planar motion and the forces/moments that act upon them. It will be shown how the equations of motion can be generated using a classical Newton-Euler approach as well as with Lagrange’s equations. The augmented approach considered for single particles and systems of particles is also discussed. The matrix-vector kinematics developed in Chap. 4 is used throughout this chapter and the emphasis is on obtaining complete numerical solutions.
Lester W. Schmerr

### Chapter 6. Dynamic and Static Stability

The motion of dynamical systems is strongly dependent on whether their behavior is stable or unstable. This chapter obtains the conditions which determine stability for certain types of systems that we have analyzed in previous chapters, using a direct method that involves the properties of the potential energy or the dynamic potential energy. We also discuss an indirect method that first requires the linearization of the equations of motion, but which is applicable to a wider class of systems. Since the direct dynamic stability method relies on the behavior of the potential energy, a quantity that in most dynamical systems is independent of time, this stability criterion is also applicable to static systems, i.e., systems that are designed to inherently be in equilibrium. Stability of static systems is rarely treated in any depth in traditional statics or dynamics courses so we will also analyze some of the important ways in which such static systems can lose stability.
Lester W. Schmerr

### Chapter 7. Vibrations of Dynamical Systems

Engineers must often analyze and design systems that vibrate. In many cases the vibrations are of very small amplitude so that the equations governing them are linear. This allows one to use a variety of analytical tools to solve for the motion and forces. This chapter examines vibrating systems with multiple degrees of freedom where matrix methods can be used to great advantage. The vibration of single degree of freedom systems is covered in Appendix D.
Lester W. Schmerr

### Chapter 8. General Spatial Dynamics of Rigid Bodies

The planar problems examined in Chap. 5 do not describe a number of the dynamical behaviors found in 3-D problems such as gyroscopic effects. In this chapter we describe the spatial dynamics of rigid bodies and the numerical solutions of the equations of motion using both Euler angles and Euler parameters. We again examine both a Newton-Euler approach and Lagrange’s equations. There are many other methods and issues that could be considered for 3-D problems that we will leave to more advanced treatments of dynamics.
Lester W. Schmerr

### Chapter 9. Dynamics of Deformable Bodies

Dynamics texts typically follow a sequence where the motion of a particle is considered first, then the motion of systems of particles, and finally the motion of continuous, rigid bodies. We have followed that same traditional path in this book. Treatments of the motion of continuous deformable bodies, however, are often left to more specialized texts at a higher undergraduate and graduate level. In this chapter we give an introduction to the dynamics of deformable bodies that can serve as a bridge to more detailed studies.
Lester W. Schmerr