The Practice of Engineering Dynamics. Ronald J. Anderson
are often subjected to harmonic external disturbances at known forcing frequencies and their response to these forces provides critical design information. The linearized equations of motion are used.
The response of the system in the time domain – the fully nonlinear equations of motion are solved numerically to simulate the response of the system to known external forces. Large scale motions and the nonlinear characteristics of system elements are included. This is the numerical equivalent to conducting performance experiments with a prototype of the system. The design information gleaned from these simulations is, perhaps surprisingly to some, not very useful in the early stages of the design. Accurate nonlinear simulations require precise knowledge of system parameters that simply isn't available in the early design phases of a project. The time domain simulations are best left to the prototype testing stage when they can be validated through comparison of predicted and measured system response. Validated time domain simulations are valuable tools to use when considering design changes aimed at improving the measured performance of the system.
The presentation of material in this book divides the practice of engineering dynamics into three parts.
Part 1. Modeling: Deriving Equations of Motion
Dynamic analysis is based on the use of accurate nonlinear equations of motion for a system. Deriving these complicated equations is a task that is prone to error. Because of this, it is important to derive the governing equations twice, using two different methods of analysis, and then prove to yourself that the two sets of equations are the same. This is a time consuming activity but is vital because predictions made from the equations of motion are critical in the design process. Predictions made using equations with errors are not of any use. The first part of the book discusses the generation of nonlinear equations of motion using, firstly, Newton's laws and, secondly, Lagrange's equation. Only when the two methods give the same equations can the analyst proceed to part 2 with confidence.
Part 2. Simulation: Using the Equations of Motion
The second part presents a logical progression of analysis techniques and methods applied to the governing equations of motion for systems. The progression is from equilibrium solutions that find in what states the system would like to be, to analyzing the stability of these equilibrium states (stability is usually considered only in textbooks on control systems but it is vitally important to dynamic systems), to considering small motions about the stable equilibrium states (this topic is covered in textbooks on vibrations but is, again, vital to engineers doing dynamic analysis), to frequency domain analysis (vibrations again), and finally to time domain solutions (these are rarely covered in textbooks).
Part 3. Working with Experimental Data
While not usually considered a part of the design process, analysis of experimental data measured on dynamic systems is critical to creating a successful product. To assist engineers in developing capabilities in this area, part 3 covers the practical use of discrete fourier transforms in analyzing experimental data.
In order to emphasize the idea that any dynamic mechanical system can be analyzed using the sequence of steps presented here, all the exercises at the ends of the chapters are based on 23 mechanical systems defined in an appendix. Any one of these systems could be used as an example of all of the types of dynamic analysis.
This book is based on course notes that I have developed while teaching a one‐semester graduate course on dynamics over more than two decades. It could just as well be used in a senior undergraduate dynamics course.
November, 2019
Ronald J. Anderson Kingston, Canada
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1 Kinematics
Kinematics is defined as the study of motion without reference to the forces that cause the motion. A proper kinematic analysis is an essential first step in any dynamics problem. This is where the analyst defines the degrees of freedom and develops expressions for the absolute velocities and accelerations of the bodies in the system that satisfy all of the physical constraints. The ability to differentiate vectors with respect to time is a critical skill in kinematic analysis.
1.1 Derivatives of Vectors
Vectors have two distinct properties – magnitude and direction. Either or both of these properties may change with time and the time derivative of a vector must account for both.
The rate of change of a vector
1 The rate of change of magnitude .
2 The rate of change of direction .
Figure 1.1 A vector changing with time.
Figure 1.1 shows the vector
The difference between
(1.1)
or,
(1.2)
Then, using the definition of the time derivative,
