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This chapter is devoted to vibration theory fundamentals concerning undamped and damped freely oscillating systems. Application of vibrationtheory to solving rotor dynamics problems is then discussed. Next， critical speed analysis and balancing techniques are examined. The latter part of thechapter discusses important design criteria for rotating machinery， specif-ically bearing drivertypes， and design and selection procedures.
Mathematical Analysis
The study of vibrations was confined to musicians until classical mechanics had advanced sufficiently to allow an analysis of this complexphenomenon. Newtonian mechanics provides an approach which， concep-tually， is easy to understand. Lagrangian mechanics provides a more sophis-ticatedapproach， but it is intuitively more difficult to conceive. Since thisbook uses some basic concepts， we will approach the subject using New-tonian mechanics.
Vibration systems fall into two major categories: forced and free. A free system vibrates under forces inherent in the system. This type of system willvibrate at one or more of its natural frequencies， which are properties of the elastic system. Forced vibration is vibration caused by external force being impressed on the system. This type of vibration takes place at the frequencyof the excitingforce， which is an arbitrary quantity independent of the natural frequencies of the system. When the frequency of the exciting force
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and the natural frequency coincide， a resonance condition is reached， and dangerously large amplitudes may result. All vibrating systems are subject to some form of damping due to energy dissipated by friction or other resist-ances.
The number of independent coordinates， which describe the system motion， are called the degrees of freedom of the system. A single degree of freedom system is one that requires a single independent coordinate to completely describe its vibration configuration. The classical spring mass system shown in Figure 5-1 is a single degree of freedom system.
Systems with two or more degrees of freedom vibrate in a complex manner where frequency and amplitude have no definite relationship. Among themultitudes of unorderly motion， there are some very special types of orderly motion called principal modes of vibration.
During these principalmodes of vibration， each point in the system follows a definite pattern of common frequency. A typical system with two or more degrees of vibration is shown in Figure 5-2. This system can be a string stretched between two points or a shaft between two supports. The dotted lines in Figure 5-2 show the various principal vibration modes.

Most types of motion due to vibration occur in periodic motion. Periodic motion repeats itself at equal time intervals. A typical periodic motion is shown in Figure 5-3. The simplest form of periodic motion is harmonicmotion， which can be represented by sine or cosine functions. It is importantto remember that harmonic motion is always periodic; however， periodic motion is not always harmonic. Harmonic motion of a system can be represented by the following relationship:
x二A sin wt (5-1)
Thus， one can determine the velocity and acceleration of that system by differentiating the equation with respect to t
Velocity二生x生t二Aw cos wt二Aw sin(wt +Z 2)(5-2)
Acceleration二生生t 2x2二 Aw2 sin wt二Aw2 sin(wt +Z)(5-3)
The previous equations indicate that the velocity and acceleration are alsoharmonic and can be represented by vectors， which are 90 0and 1800ahead of the displacement vectors. Figure 5-4 shows the various harmonic motions ofdisplacement，velocity， and acceleration. The angles between the vectors arecalled phase angles; therefore， one can say that the velocity leads displacement

Figure 5-.. Periodic motion with harmonic components Figure 5-4.Harmonic motion ofdisplacement，velocity， and acceleration

by 900， and that the acceleration acts in a direction opposite to displacement， or that it leads displacement by 1800 .
　
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