aliasing; leakage and windowing. The approach I have taken here is very old‐fashioned in that it treats the Discrete Fourier Transform as something that is derived from a least squares curve fit. This harks back to methods used more than fifty years ago but it enhances students' understanding of what lies behind the transformation to the frequency domain. This is a long chapter that presents methods that the students can program themselves so they don't have to be tied down to packaged FFT programs. Once they have their software working, they can experiment with data sets that clearly demonstrate aliasing, leakage, and so on.
Chapter 12 discusses a variety of topics in vibrations such as how to handle the mass of a spring, flow‐induced vibrations, self‐excited vibrations in rail vehicles, rigid body modes, and things you can determine from the static deflection of a system. I find that I am usually able to get somewhere into this chapter before the semester is over. The material in this chapter is interesting but certainly doesn't need to be covered to have a complete introduction to vibrations.
It is my hope that this book strikes the right balance for professors teaching introductory vibrations and for their students. I wish them all well.
Ronald J. Anderson
September, 2019
Kingston, Canada
About the Companion Website
The companion website for this book is at
www.wiley.com/go/anderson/introduction-to-vibrations
The website includes:
Animated GIFs
PDFs and
Modes software files
Scan this QR code to visit the companion website.
1 The Transition from Dynamics to Vibrations
Introductory undergraduate courses on dynamics typically consider large scale motions of systems of particles and/or rigid bodies and instantaneous solutions to their nonlinear, governing equations. You may recall working on dynamics problems where a system of bodies starts from rest at a prescribed position and your task was to determine, for example, the angular acceleration of a body or the forces acting on some part of the system. Solutions like this, while having some utility, provide only part of the understanding of the system that is required for a successful design. In most cases, the derived governing equations are complete enough but the “snapshot” solutions don't help much with the design process.
There are, in fact, many things that can be done with the equations governing the dynamic motion of the system. Briefly, they can be used to
1 Find where the bodies in the system would be if the system were at rest. These are the Equilibrium States.
2 Determine whether the equilibrium states are stable or unstable.
3 Determine how the system behaves for small motions away from a stable equilibrium state.
4 Determine the response of the system in the time domain through the use of numerical simulations. This is the most complex type of analysis and, perhaps surprisingly, gives the least information to the designer until the design has reached the fine tuning phase. The simulations are the analog of “cut and try” experiments where an unsuccessful result gives little information on what to change in order to improve the design.
While going through the material presented in this book, you will be concentrating on very small motions of systems about stable equilibrium states. In doing so, you will see connections to topics you may have covered in courses on statics, on dynamics, and on control systems. You will become very familiar with the linearized, differential, equations of motion for dynamic systems moving around stable equilibrium states and methods for deriving and solving them. This is the essence of Vibrations.
To get started and as a review of sorts we begin with the dynamic analysis to a relatively simple system – a bead sliding on a rotating semicircular wire.
1.1 Bead on a Wire: The Nonlinear Equations of Motion
First courses on the subject of Dynamics, whether for particles or rigid bodies, are primarily concerned with teaching the basics of kinematics, free body diagrams, and applications of Newton's Laws of Motion. Applying these three concepts sequentially will lead to a set of simultaneous force and moment balance equations that take account of kinematic constraints.
There are different ways of approaching these problems. One can use a formal vector‐based approach and we will start with that here because it gives a complete set of governing equations including solutions for all constraint forces that are required to enforce kinematic constraints on the motion. A shorthand version of this approach which may be called an “informal vector approach” is often used in practice and that will be the second method addressed here. It typically works with two‐dimensional views and leads to the governing equations of motion without necessarily solving for all constraint forces. The third approach will see the equations of motion derived using Lagrange's Equations. This is a work/energy approach that leads to the nonlinear differential equation of motion with minimal effort on the part of the analyst. The kinematic constraint forces are automatically eliminated as the governing equations are derived, leaving a designer with no information about forces acting on elements of the system unless extra work is done to find them. Lagrange's Equations are not typically introduced to undergraduate engineers as often as Newton's Laws are, so extra effort is made in this chapter to introduce the procedures for applying Lagrange's Equations to mechanical systems.
As an example, consider Figure 1.1. The figure shows a small bead with mass,
Figure 1.1 A bead on a wire.
1.1.1 Formal Vector Approach using Newton's Laws
Using the formal vector approach, the first step in the kinematic
