Engineering Dynamics
J. Alan Adams, Professor Emeritus
United States Naval Academy

This collection of undergraduate Engineering Dynamics analyses and solutions constitutes an Engineering Lecture Complement (ELC). It is a tool made possible by Internet technology and can be used to increase classroom productivity, teaching flexibility and personalized learning.

This computer oriented, WEB based presentation is not an engineering text in the traditional sense. Important fundamental definitions, concepts, and derivations found in many excellent texts are not repeated. Here one begins with a problem, or application, and builds upon the fundamental foundation as needed, just in time. This simulates the manner in which most engineering problems are addressed in practice. If this material is used to support an introductory course in college, the instructor should assure that the student is aware of all assumptions, limitations, and abstractions inherent in the mathematical definitions and theory. If this material is used by practicing engineers or graduate students, they should be aware of suitable references that can be used as needed to justify the selected approach to a problem.

Traditional texts in dynamics begin with the derivation of kinematic relationships. Dynamics (kinetics) is then applied to particle analysis and finally to a rigid body dynamic analysis. Many mathematical concepts from vector analysis, matrix theory, relative kinematics and various coordinate systems are presented before the dynamic analysis of a practical system is performed. This approach was developed in the 1960's when national priorities dictated a more scientific and analytical approach to engineering courses.

This material is presented in an order more suitable to a new national priority, e.g., the computer supported and Internet based instruction of engineering courses. Building on an assumed background in calculus and physics, rigid body analysis occurs early to make clear the distinction between dynamic analysis and static analysis for systems in rectilinear or rotational motion. Consideration is given to the non-linear effects that are necessary to accurately simulate many real systems. Computer software for equation solving is then utilized to allow the solution of more realistic mathematical models.
Particle analysis, when appropriate, is considered a special case of rigid body analysis. When the basic laws of mechanics are applied to systems where integration of a known acceleration is used to obtain the velocity and displacement, complex ideas such as the kinematics of relative motion, moving coordinates, Coriolis acceleration, etc., can be delayed until needed in more complex applications. When the kinematic analysis begins with non-lineardisplacement loop equations, numerical techniques are often needed.Computer solutions supported by animation are then available to produce the continuous kinematic behavior of interest during a specified time interval, rather than an instantaneous solution for a particular configuration typical of a traditional vector analysis approach. The ability to animate vectors which represent continuous kinematic behavior is made possible by the displacement loop equations. When a complete kinematic analysis is combined with a full dynamic analysis of a complex system, productivity demands that modern day engineers make use of available simulation and/or solid modeling software such as ADAMS, IDEAS, Working Model, ProEngineer, Algor, etc. One objective of this collection of problem solutions, coupled with formal lecture, is to create the experience and insight necessary for the eventual use of such software for dynamic analysis.
A secondobjective in presenting this material is to use software which allows a "lecture friendly" presentation of the mathematics, solution techniques, and discussions which are suitable for either a distance learning environment or for a complement to a traditional lectureenvironmentThe collection of problems and solutions which make up this textual material are generated using the equation solving software Mathcad, and its symbolic processor which is a subset of MAPLE. This software is available from MathSoft ( Each chapter contains a small sample of problem solutions which hopefully will inspire similar effort generated by the user.
Chapter 1 gives a short review of vector and calculus concepts that are used throughout. Vectors are used graphically more to understand resulting kinematic and dynamic behavior rather than to formulate a solution to the problem. Symbolic mathematics can often be used to actually obtain the required differentiation and integration as illustrated throughout later chapters.

Chapter 2 first treats 2-D projectile motion as particle motion without drag in the classical sense, but then includes motion with drag which requires a numerical solution to a non-linear differential equation to obtain velocity and displacement.

Chapter 3 applies Newton's Second Law to motion of rigid bodies where inertial and/or friction effects are important. Particle motion is a special case.
Chapter 4 deals with linear and angular impluse and momemtum, as applied to both rigid body motion and particle motion.

Chapter 5 applies work-energy methods to both rigid bodies and particles. The use of symbolic mathematics is predominate in this chapter.

Chapter 6 considers 1-D and 2-D elastic, inelastic, real and oblique impact
Chapter 7 gives a full dynamic analysis supported by graphics and animation for two simple mechanisms. Animation of the complete solution including kinematic and force vectors is created.

Chapter 8 addresses planar kinematics from the vector loop equation point of view. Both analytical and numerical solutions are used. Animations show the relationship between resulting vectors during complete cycles of operation. Components of acceleration vectors include normal, tangential, centripetal and Coriolis. Velocity vectors include absolute and relative components. In addition, .mpeg files are available to show further realistic animation of actual mechanisms if a broad band WEB connection is available to the user.

Chapter 9 is normally not part of a first course in dynamics. It deals with the computation of geometric properties such as area, weight, centroids and mass moment of inertia for flat parts, which cannot be obtained from tables of simple geometric objects. Green's Theorem is used along with contour integration. These properties are important parameters in the dynamic equations of motion for planar mechanisms.

Chapter 10 uses numericaltechniques suitable for advanced analysis.