Fundamentals of Vibration Analysis

Index

CHAPTER 1

FUNDAMENTALS

SECTION 1. INTRODUCTION

The subject of vibration is essentially the study of oscillatory motion of machines and structures and the forces that create this motion. Since a vibration cannot arise under the action of constant forces only, the force creating and sustaining a vibration is always a fluctuating one. Fluctuating forces may vary in magnitude only and are then usually called reciprocating forces, or they may vary in direction only and are then usually called rotating forces. Sometimes they vary in both magnitude and direction. In general, a fluctuating force that causes vibration is called an exciting force, a disturbing force, an impressed force, a driving force, ora shaking force.

Such forces and motions are always present in moving machines, and in most cases they are undesirable. The undesirable effects that they can produce include increased stresses of machine parts; interference with the proper functioning of the machine itself and other machines and instruments in the neighborhood; physiological discomfort, particularly when the vibration is a noisy one; and loss of mechanical energy due to damping forces, which are always present.

Vibrating Systems. Degrees of Freedom. For a simple illustration of a vibrating system, consider the simple spring-mass system shown in Fig. 1-1, consisting of a mass of weight W suspended by means of a spring of stiffness k. The spring stiffness or spring constant k is given as the force necessary to stretch or compress the spring one unit of length. The mass is in equilibrium under the action of two equal and opposite forces, the weight W acting down and the spring force kδ acting up. The quantity δ is merely the static deflection of the spring due to the weight of the mass, and this position, in which all the forces are in equilibrium, is called the equilibrium position, neutral position, or central position. The displacement of the mass is usually measured from this position.

Suppose now that the mass were forced down an additional distance X and then suddenly released. At the time of release the spring force would be larger than the weight by an amount kX, and the mass would start moving up. As long as the mass is below the neutral position, the upward spring force is greater than the weight and the mass will increase its upward speed. When the neutral position is reached, the mass will keep right on going because of its momentum, but the downward force will now be greater than the upward force, and this will tend to slow the motion down. Finally, the upward velocity will become zero when the mass reaches the upper extreme position, and the motion will thereafter be downward. The downward velocity will increase until the neutral position is reached again ; it will then decrease and become zero when the mass reaches the lower extreme position. This is the position from which it started, and the mass has completed one cycle. The vibration just described is called a free vibration because it is performed under the action of forces inherent in the system itself without the benefit of fluctuating external forces of any kind. The force that tends to move the mass toward the neutral position is called the restoring force, and it is clear that any freely vibrating system must possess both mass and some kind of restoring force.

FIG. 1-1. A simple spring-mass system, consisting of a mass of weight W and a weightless spring of stiffness k.

A third element that is present to a larger or lesser degree in all practical vibrating systems is a friction force. This friction force, which may be of a complex form, is called damping, and it always resists the motion of the vibrating mass. The amount of damping present in most practical systems, however, is so slight that quite often it may be neglected altogether. The study of undamped vibrating systems is therefore of vital importance.

If no damping is considered to be present in the system, it will be found that the displacement of the mass at its upper extreme position is equal in magnitude to its displacement at the lower extreme position and this magnitude is called the amplitude of the motion. The length of time it takes the mass to complete one cycle of the motion is called the period, and the number of cycles completed in one unit of time is called the frequency.

It will be shown later that the frequency of the free vibration just described is independent of the amplitude but that it will increase with increasing spring stiffness and with decreasing weight of the vibrating mass. This frequency of free vibration of a system is called its natural frequency, and it increases as the square root of the spring stiffness k and inversely as the square root of the weight W.

If damping is considered to be present in the vibrating system, it will be found that the amplitude of free vibration gradually decreases and that the frequency is slightly lower than the natural frequency. The difference between these two frequencies, however, is almost always so small that it may be neglected.

Suppose now that the mass of Fig. 1-1 is being shaken up and down by a periodically fluctuating force. For some time after the application of the force the vibration is of an irregular type called a transient vibration, but because of damping the irregularities are soon damped out, and then only a steady-state vibration persists. This is called forced vibration. It is then obvious that both the amplitude and frequency of the vibration will depend on this force. It will be found that in this case the frequency of the forced vibration depends only on that of the force; the amplitude depends both on the magnitude of this fluctuating force and on the ratio of its frequency to the natural frequency of the system, and when this ratio becomes equal to unity, the amplitude of the vibration may build up to a dangerous value. This condition is called resonance, and the purpose of most vibration investigations is to avoid its occurrence. Hence the calculation of the natural frequency of a system is of the greatest importance. Since the natural frequency depends on the spring stiffness, it is clear that even for a forced vibration the restoring force is of the utmost importance, and it is present in all practically important vibrating systems. The damping will also affect the amplitude of the forced vibration so that for this case there are four elements of importance. These are (1) a mass, which executes the vibration; (2) a restoring force acting on the mass and tending to move it back to the neutral position; (3) a damping force, which always resists the motion of the mass; and (4) a driving force acting on the mass.

FIG. 1-2. Examples of single-degree-of-freedom systems.

The vibrating system mentioned above is of the simplest type in so far as only one coordinate is necessary to specify the motion of the mass. Systems of this type are called systems of one degree of freedom. Other examples of such systems are shown in Fig. 1-2. Figure 1-2a shows a compound pendulum. It differs from a simple pendulum in that the oscillating mass can no longer be considered as a particle, so that its mass moment of inertia about a gravity axis must also be considered. In this case the displacement is an angular one, measured by the angle θ, and the mass moment of inertia of the swinging body about the hinge axis takes the place of the mass of Fig. 1-1. Instead of an elastic restoring force, there is a restoring moment which is due to gravity. Figure 1-26 shows a U tube with an oscillating liquid inside, and in this case the restoring force is also due to gravity. Figure 1-2c shows an elastic shaft with a heavy disk at its middle, which is similar to the case of Fig. 1-1. Figure 1-2d shows a torsional pendulum, which consists of a shaft with an inertia at its end. An inertia is merely a body possessing a mass moment of inertia J about a particular axis, and the torsional stiffness K of the shaft takes the place of the spring constant k. The value of K is the moment necessary to twist the shaft a total angle of 1 rad.

If more than one coordinate is necessary to specify the configuration of a vibrating system, either because there are several masses or because one mass is capable of moving in more than one way, the system is said to have more than one degree of freedom. The number of degrees of freedom is equal to the minimum number of coordinates necessary to specify the configuration of the vibrating system at any time. A rigid body restrained to move in two directions or to rotate about two axes is said to have two degrees of freedom. A rigid body that is perfectly free in space (except for elastic restraints) has six degrees of freedom. Three coordinates are necessary to locate a point of the body (usually the center of gravity) and three more to indicate the amounts of rotation about three axes through this point. A continuous elastic body has at least three degrees of freedom for each of its molecules (if these are considered particles without dimensions), but for practical purposes the number may be considered to be infinite.

Figure 1-3 shows a few simple systems of two degrees of freedom. Figure 1-3a to c shows two masses with various combinations of elastic restraint. It will be noticed that the number of springs has no influence on the number of degrees of freedom, which is wholly determined by the freedom of motion of the masses. Figure 1-3d shows a simple torsional system of two degrees of freedom; Fig. 1-3e, a rigid body that is free to move up and down and also to rotate about a horizontal axis perpendicular to the paper; Fig. 1-3f, a mass capable of moving both horizontally and vertically.

In Fig. 1-4 are shown some simple systems of several degrees of freedom. Figure 1-4a shows one form of vibration of a tightly stretched elastic string with eight masses attached. The masses are considered to move vertically only, and the system has eight degrees of freedom. Figure 1-4b shows a torsional system of four degrees of freedom, and Fig. 1-4c a vibrating elastic cantilever beam which is considered to have an infinite number of degrees of freedom.

Systems of more than one degree of freedom are much more complicated to deal with than systems of only one degree of freedom. As will be shown later, such systems have more than one natural frequency. However, it is often only the lowest natural frequency that has practical importance, and this can in most cases be found approximately in the same manner as the natural frequency of a one-degree-of-freedom system. The method by which this is done is called the Rayleigh method and is described in Chap. 2.

FIG. 1-3. Examples of two-degrees-of-freedom systems.

FIG. 1-4. Examples of systems of several degrees of freedom.

SECTION 2. KINEMATICS OF SIMPLE VIBRATORY MOTION

Any irregular motion of a particle about some fixed position of equilibrium may be called a vibration. In most practical engineering problems of importance, however, the maximum displacements are limited by the constraints of the system, and the motion is considered to be periodic. A periodic motion is one that repeats itself over and over again at regular intervals. The length of this interval in units of time is called the period of the motion and will be designated by the letter τ.

For simplicity consider only the rectilinear motion of a single vibrating particle. The displacement from the neutral position is then called x, positive in one direction and negative in the other, and the time, measured from some zero point, is designated by the letter t. If a graph is plotted of the displacement x against the time t, the periodic curve will in general be of an irregular shape, such as that shown in Fig. 2-1. The numerical value of the maximum displacement to either side of the neutral position is called the amplitude in that direction, and the total displacement from one extreme position to the other, which is called the total excursion or the double amplitude, is thus the sum of the two amplitudes. In most practical vibration problems, however, the amplitude is about the same in either direction, and this will be considered to be the case throughout this book. Its value will be designated by the letter X, and the reader should bear in mind that this value is only one-half of the total excursion.

FIG. 2-1. A periodic function with period τ.

The simplest form of a periodic motion is a harmonic one, i.e., a sine or cosine function, and most vibrating systems have motions that are nearly harmonic. In the following, therefore, only harmonic motions will be considered. A harmonic motion may be written

x = X cos ωt

(2-1)

where ω is a constant and ωt an angle which is measured in radians. The angular period of this function is obviously 2π so that ωτ = 2π, where τ is the period in units of time. The physical dimension of ω is seen to be t–1, and it is given in radians per unit of time. In Fig. 2-2 are shown three Such harmonic curves, each having the same amplitude and period but a different origin of time. In Fig. 2-2a time is measured from the instant the particle is in one of the extreme positions, x = X. In Fig. 2-2b time is measured from the instant the particle is in the neutral position, x = 0. In Fig. 2-2c the origin is t0 units of time after the particle passes the neutral position, where ωt0 = α, or t0 = α/ω = (α/2π)τ.

FIG. 2-2. Harmonic motions with the same amplitude and frequency but different origins of time. (a) x = X cos ωt; (b) x = X sin ωt; (c) x = X sin (ωt + α).

The relationships between the period τ, the frequency f, and the constant ω are

τ is usually measured in seconds, f in cycles per second, and ω in radians per second. The constant ω is proportional to the frequency f, and it is called the angular frequency or circular frequency.

As pointed out previously, the three curves in Fig. 2-2 are exactly the same except that they are displaced relatively to each other along the t axis. This relative displacement is called the phase difference, and, when multiplied by ω, its magnitude in radians is called the phase angle between the curves. If

x = X cos ωt

is taken as the reference curve, the curve

is π/2 rad or 90° out of phase. Furthermore, it is said to be 90° behind the reference curve, while the curve

is α – π/2 rad or α – 90° ahead. In the same way the curve x = X sin ωt could be taken as the reference curve and all others compared with it.

In the above description it was assumed that τ was the same for all curves. If this were not true, one curve could not be obtained from another by merely sliding it along the t axis. The term phase angle is therefore only applicable to harmonic motions of the same frequency. In the future, whenever all the harmonic curves in question have the same frequency, the displacement x will be plotted against the angle ωt instead of against the time t. This is convenient as all the harmonic curves thus plotted will have the angular period 2π regardless of the value of ω. However, it must be remembered that the time t, and not the angle ωt, is the independent variable.

FIG. 2-3. Displacement, velocity, and acceleration of the harmonic motion x = X cos ωt.

If the displacement of a vibrating particle is given by the equation x = X cos ωt, the velocity of the particle will be

which is seen to be a harmonic function with the same frequency as the displacement and an amplitude ω times as large. If plotted against ωt, its phase angle will be 90° ahead of the displacement, as shown in Fig. 2-3. The derivative of a harmonic function with respect to time involves multiplying the amplitude by ω and increasing the phase angle by 90°. (Increasing the phase angle means moving the curve to the left in Fig. 2-3.)

A second differentiation with respect to time gives the acceleration

are universally used to designate the time derivatives dx/dt and d²x/dt², respectively. As these expressions are used in most books treating the subject of vibrations and in books on mechanics and physics, the author considers it advisable that all students of vibration become acquainted with them. They will therefore be used throughout these pages.

FIG. 2-4. Harmonic motion represented by a rotating vector.

As is shown in Fig. 2-3, the amplitude of the acceleration is ω² times that of the displacement, and its phase angle is 180°. Its frequency is of course the same as that of the displacement and velocity.

FIG. 2-5. The motions X cos ωt and X sin ωt represented by the horizontal projections of rotating vectors.

Both the velocity and the acceleration are harmonic functions of time, and their frequencies are the same as that of the displacement.

Vector Representation. The reciprocating harmonic motion of a vibrating particle can be represented by the projection of the end point of a rotating vector on a fixed line through the center of rotation. Figure 2-4 shows a vector of length X which is rotating counterclockwise about the point 0 with an angular velocity ω. It is assumed that t = 0 when the rotating vector coincides with the positive x axis. The horizontal projection of the vector X is then X cos ωt, and the vertical projection is X sin ωt. Any one of these two projections, therefore, will represent a harmonic motion, but for simplicity consider only the horizontal projection x = X cos ωt. The expression X sin ωt is then represented by the horizontal projection of a rotating vector with phase angle – π/2 with respect to the first one (X cos ωt), as the sine function lags behind the cosine function an amount τ/2. This is shown in Fig. 2-5. The frequency f = ω/2π is obviously represented by the number of revolutions of the rotating vector per unit of time, and the period τ is the time for one revolution of the vector. The name angular frequency for the constant ω comes from the fact that its value is the same as that of the angular velocity of the rotating vector.

Another rotating vector of the same length X and angular frequency ω, but always an angle α ahead of the first one, will form an angle ωt + α with the X axis and have a horizontal projection X cos (ωt + α). It describes a harmonic motion which is an angle α out of phase with the first one. From Eqs. (2-5) and (2-6) it follows that the velocity of the particle may be described by a rotating vector of length ωΧ, 90° ahead of the displacement vector, and the acceleration by a vector of length ω²Χ, 180° ahead of the vector X. This is illustrated in Fig. 2-6. Differentiation with respect to time is thus effected by advancing the rotating vector 90° and multiplying its length by ω.

FIG. 2-6. Vectors representing the displacement, velocity, and acceleration of a harmonic motion.

The vector method is particularly useful in adding and subtracting harmonic motions with the same frequency but with different phase angles. The vectors representing the motions are simply added or subtracted graphically, and the resultant vector will completely describe the resultant motion both as to amplitude and phase. Referring to Fig. 2-7, it is desired to add the two harmonic motions X1 cos ωt and X2 cos (ωt + α). The latter expression may be written as X2 cos α cos ωt – X2 sin α sin ωt. The combined motion is then given as

(X1 + X2 cos α) cos ωt – X2 sin α sin ωt

= X cos (ωt + β) = X cos β cos ωt – X sin β sin ωt

From this it follows that

X cos β = Χ1 + X2 cos α

X sin β = – X2 sin α

Squaring these two equations and adding will give the amplitude

By taking the ratio of these same equations the phase angle is found to be

As shown in Fig. 2-7, both the amplitude and phase angle