Introduction to Mechanical Vibrations. Ronald J. Anderson

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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

      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,