Mechanical Vibrations

J. P. Den Hartog

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486131856
• Topic: Tecnologia e ingegneria
• Subtopic: Ingegneria civile

This classic textbook by J. P. Den Hartog, retired professor of mechanical engineering at MIT, reflects the author's unique ability to combine the scholarly insight of a distinguished scientist with the practical, problem-solving orientation of an experienced industrial engineer. Although mathematics plays a role in the subject, Den Hartog employs the simplest possible mathematical approaches. His lucid explanations of complex problems are presented in a direct style and supported by illustrative models. Numerous figures in the text enhance its value as a basic foundation in a field which Den Hartog calls `a necessary tool for almost every mechanical engineer.` The author examines such topics as the kinematics of vibration (including harmonic motions and non-harmonic periodic motions), degrees of freedom, gyroscopic effects, relaxation oscillations, Rayleigh's method, natural frequencies of torsional vibration, Karman vortices, and systems with variable elasticity. Drawing on his experience as an engineer in private industry and in the U.S. Navy's Bureau of Ships, Den Hartog applies theory to practice, discussing the effects of vibrations on turbines, electrical machines, helicopter rotors and airplane wings, diesel engines and electrical transmission lines.
As a special aid to classroom work or self-study, this practical text includes an extensive selection of 233 problems and answers that test the student's mastery of every section of the book. In addition, a highly useful Appendix contains `A Collection of Formulas` for determining the load per inch deflection of linear springs, the load per radian rotation of rotational springs, the natural frequencies of simple systems, the longitudinal and torsional vibration of uniform beams, the transverse or bending vibrations of uniform beams, and the vibrations of rings, membranes, and plates.
When Mechanical Vibrations was first published in 1934, it was a pioneering work in a field which had just been introduced in America's technical schools. In fact, the author wrote it to assist him in teaching the subject at Harvard. `During the life of the book,` he says, `from 1934 on, the art and science of engineering has grown at an astonishing rate and the subject of vibration has expanded with it.` Professor Den Hartog's constant revisions have kept his book at the forefront of this vital subject, as useful today as its earlier versions were to students of the past.

CHAPTER 1

KINEMATICS OF VIBRATION

1.1 Definitions. A vibration in its general sense is a periodic motion, i.e., a motion which repeats itself in all its particulars after a certain interval of time, called the period of the vibration and usually designated by the symbol T. A plot of the displacement x against the time t may be a curve of considerable complication. As an example, Fig. 1.1a shows the motion curve observed on the bearing pedestal of a steam turbine.

The simplest kind of periodic motion is a harmonic motion; in it the relation between x and t may be expressed by

as shown in Fig. 1.1b, representing the small oscillations of a simple pendulum. The maximum value of the displacement is x₀, called the amplitude of the vibration.

FIG. 1.1. A periodic and a harmonic function, showing the period T and the amplitude x₀.

The period T usually is measured in seconds; its reciprocal f = 1/T is the frequency of the vibration, measured in cycles per second. In some publications this is abbreviated as cyps and pronounced as it is written. In the German literature cycles per second are generally called Hertz in honor of the first experimenter with radio waves (which are electric vibrations).

In Eq. (1.1) there appears the symbol ω, which is known as the circular frequency and is measured in radians per second. This rather unfortunate name has become familiar on account of the properties of the vector representation, which will be discussed in the next section. The relations between ω, f, and T are as follows. From Eq. (1.1) and Fig. 1.1b it is clear that a full cycle of the vibration takes place when ωt has passed through 360 deg. or 2π radians. Then the sine function resumes its previous values. Thus, when ωt = 2π, the time interval t is equal to the period T or

.

Since f is the reciprocal of T,

For rotating machinery the frequency is often expressed in vibrations per minute, denoted as v.p.m. = 30ω/π.

FIG. 1.2. Two harmonic motions including the phase angle φ.

In a harmonic motion for which the displacement is given by x = x₀ sin ωt, the velocity is found by differentiating the displacement with respect to time,

so that the velocity is also harmonic and has a maximum value ωx₀.

The acceleration is

also harmonic and with the maximum value ω²x₀.

Consider two vibrations given by the expressions x₁ = a sin ωt and x₂ = b sin (ωt + φ) which are shown in Fig. 1.2, plotted against ωt as abscissa. Owing to the presence of the quantity φ, the two vibrations do not attain their maximum displacements at the same time, but the one is φ/ω sec. behind the other. The quantity φ is known as the phase angle or phase difference between the two vibrations. It is seen that the two motions have the same ω and consequently the same frequency f. A phase angle has meaning only for two motions of the same frequency: if the frequencies are different, phase angle is meaningless.

Example: A body, suspended from a spring, vibrates vertically up and down between two positions 1 and 1½ in. above the ground. During each second it reaches the top position (1½ in. above ground) twenty times. What are T, f, ω, and x₀?

Solution: x₀ = 1/4 in., T = 1⁄20 sec, f = 20 cycles per second, and ω = 2πf = 126 radians per second.

1.2. The Vector Method of Representing Vibrations. The motion of a vibrating particle can be conveniently represented by means of a rotating vector. Let the vector

(Fig. 1.3) rotate with uniform angular velocity ω in a counterclockwise direction. When time is reckoned from the horizontal position of the vector as a starting point, the horizontal projection of the vector can be written as

a cos ωt

and the vertical projection as

a sin ωt

FIG. 1.3. A harmonic vibration represented by the horizontal projection of a rotating vector.

Either projection can be taken to represent a reciprocating motion; in the following discussion, however, we shall consider only the horizontal projection.

This representation has given rise to the name circular frequency for ω. The quantity ω, being the angular speed of the vector, is measured in radians per second; the frequency f in this case is measured in revolutio...