## Introduction: Classification of Vibrations

A vibration can be thought of as an oscillatory motion that repeats itself after a period of time. An example would be a pendulum swinging back and forth. You can probably come up with many examples based on your own experience. While we will focus on *mechanical* vibrations in this course, other systems (electrical, thermal, control) will also exhibit vibrational or oscillatory “motions” in their own way and many of the results obtained in this course can be applied to such systems as well (with some careful thought).

We study vibrations for many reasons.

- Almost all structures and machines are subjected to some form of vibration:
- Buildings and Structures
- Earthquakes
- Wind loads
- Service loads (compressors, etc.)

- Machines
- Rotating imbalances
- Reciprocating imbalances
- Other imbalances due to imperfections

- Buildings and Structures

Vibrations in these systems can be associated with very large forces and can lead to rapid wear and/or failure.

- Many human activities involve some form of vibrations
- Speaking
- Hearing
- Sight is based on electromagnetic waves

- There are also ergonomic issues to consider as human beings are affected by vibrations (often negatively) in many ways.

In many of these cases, the goal of the engineer is to

- minimize or reduce the levels of vibration, or
- design the machine/structure to withstand the vibrations to which it will be subjected.

At other times, however, the goal will be to take advantage of (and perhaps even increase) the vibrations:

- sound systems
- accelerometers and other measurement instruments
- clocks
- vibratory conveyors
- electrical circuits (behave similarly in many ways)

To accomplish these goals we need to understand the mechanics of vibration and what effect physical parameters have on the system.

Vibrations are often classified in several different ways. Some of the more important classifications are discussed in the following.

### Free vs. Forced Vibrations

**Free Vibration:** After some initial disturbance, a system vibrating on its own is said to be undergoing free vibrations. No external force is acting on the system to cause the vibrations.**Forced Vibration:** The vibrations are caused by an external force which acts on the system and drives the resulting vibrations.

### Undamped vs. Damped Vibrations

If no energy is lost in the system during the oscillations, the vibration is said to be * undamped*. If energy is lost in the oscillations the vibration is said to be

*. Note that all real systems have some damping, usually through friction or viscous effects. However, in many practical physical systems the damping present is small enough that an undamped model, which is simpler and easier to deal with, provides satisfactory results for short enough time periods.*

*damped*### Linear vs. Nonlinear Vibrations

**Linear Vibrations**

- All components which comprise the system behave linearly.
- Governing equations are linear.

**Nonlinear Vibrations**

- At least one component in the system is nonlinear.
- Governing equations become nonlinear.

This is a very critical distinction. If the vibrations are linear, then a great deal of mathematical techniques have been developed and are available to carry out the analysis. Importantly, the principle of superposition can be applied. If the system is nonlinear, then the techniques for dealing with these systems are much less well developed. Superposition cannot be applied so each individual problem needs to be considered on its own. Nonlinear systems can exhibit very interesting behaviour which is not present in linear systems.

Any real system will be nonlinear, particularly as the amplitudes of the motion get large (eventually the system will break, a very nonlinear phenomenon). However, even a nonlinear system can be treated as linear if we limit the analysis to motions with a small enough amplitude. Linear vibration analysis is often meant to imply small motions, with what is meant exactly by “small” depending on the particular application.

We will only be concerned with linear vibrations in this course.

### Discrete vs. Continuous Vibrations

**Discrete Systems**

- Motion can be described by a finite number of parameters (one angle needed to specify the position of a pendulum for example).
- System has a finite number of degrees of freedom.
- Also known as
systems.*lumped-parameter* - Typically modeled using ordinary differential equations.

**Continuous Systems**

- Requires an infinite number of parameters to describe the motion of the system (specifying the position of every point on a deformed beam for example).
- System has an infinite number of degrees of freedom.
- Also known as
*distributed*systems. - Typically modeled using partial differential equations.

We will consider both discrete and continuous systems in this course.

### Deterministic vs. Random Vibrations

**Deterministic Vibrations**

If the excitation (force or motion) acting on a system is known at any given instant in time, the excitation is said to be deterministic and the resulting vibrations are said to be deterministic as well.

**Random Vibrations**

In some cases, the excitation is not a known function of time and is said to be non-deterministic or random. Some examples include wind loads, earthquakes, and road roughness. In these cases, while we cannot predict the excitation at any given time, it does tend to exhibit some statistical regularity so we can speak about averages, standard deviations, mean squared values, etc. In these cases the vibration response is also “random”, but can similarly be described in terms of statistical quantities.

We will only be concerned with deterministic vibrations in this course.