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,
Imagine now that Figure 1.1 is compressed to show only an infinitesimally small time interval,
1 A component aligned with the vector . This is a component that is strictly due to the rate of change of magnitude of . The magnitude of is where is the rate of change of length (or magnitude) of the vector . The direction of is the same as the direction of . Let be designated1 as .
2 A component that is perpendicular to the vector . That is, a component due to the rate of change of direction of the vector. Terms of this type arise only when there is an angular velocity. The rate of change of direction term arises from the time rate of change of the angle in Figure 1.1 and is the magnitude of the angular velocity of the vector. The rate of change of direction therefore arises from the angular velocity of the vector. The magnitude of is where is the length of . By definition the rate of change of the angle (i.e. ) has the same positive sense as the angle itself. It is clear that is the “tip speed” one would expect from an object of length rotating with angular speed .
The angular velocity is itself a vector quantity since it must specify both the angular
