TERMINOLOGY OF VIBRATION

8/7/2019 TERMINOLOGY OF VIBRATION

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(b) What Causes Noise?A common cause is the impact or vibration of a solid material, which sets air In

motion; for example, a hammer striking a nail, or a vibrating equipment panel. Inmachinery, in particular, there are many commonly found sources of noise. These

are usually associated with the operating frequency of machine motions, e.g. theRPM of an electric motor or of gears, the rate of tooth engagement in gear teeth,

the frequencies associated with reciprocating machinery, etc. It is possible alsothat vibrations may be generated in one part of the equipment, but may set upnoise and vibration in another part of the equipment, such as doors, panels,

chassis, flexible lines, printed-circuit boards etc.

(c) The Adverse Effects of Uncontrolled NoiseThere are several. First, noise may be an indication of faulty equipment operation,

e.g. cracked parts, faulty bearings, excessive rotor unbalance, improperlubrication, loose parts, etc. It is possible also, however, for a machine to function

satisfactorily mechanically, but to be rejected by the Customer, if it is too noisy,

e.g. noisy household appliances, air conditioners, etc. Second, human efficiencyand fatigue may be adversely affected, for example in production lines in a noisy

factory, or in the office. Short of expensive and difficult investigations, a goodvibration mount can often be an effective way to reduce noise levels to within

acceptable limits.

(d) How Can Noise be Reduced?There are many ways. One of the most practical and effective may be the use of

vibration mounts. As a general rule, a well designed vibration isolator will also helpreduce noise. In the case of panel flutter, for example, a well designed vibration

mount could reduce or eliminate the noise. This can be achieved by eliminating the

flutter of the panel itself, or by preventing its transmission to ground, or by acombination of the two. The range of audible frequencies is so high that the

natural frequencies of a vibration mount can usually be designed to be well belowthe noise-producing frequency.

In order to reduce noise, try to identify the source of the noise, e.g. transformer

hum, panel flutter, gear tooth engagement, rotor unbalance, etc. Next identify thenoise frequency. A vibration mount designed in accordance with the guidelines for

vibration and shock control can then act as a barrier either in not conducting thesound, or in attenuating the vibration, which is the source of the noise.

2.0 BASIC DEFINITIONS AND CONCEPTS IN VIBRATIONANALYSIS


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2.1 Kinematic Characteristics

COORDINATE - A quantity, such as a length or an angle, which helps define the

position of a moving part. In Figure 1, x is a coordinate, which defines the position

of the weight, W.

DISPLACEMENT - A change in position. It is a vector measured relative to aspecified position, or frame of reference. The change in x (Figure 1) measured

upward, say, from the bottom position, is a displacement. A displacement can bepositive or negative, translational or rotational. For example, an upward

displacement may be positive; and a downward displacement negative. Similarly,

a clockwise rotation may be positive and a counterclockwise rotation negative.Units: inches, feet, or, in the case of rotations: degrees, radians, etc.

VELOCITY - The rate of change of displacement. Units: in/sec, M.P.H., etc.Velocity has a direction. It is a vector. Its magnitude is the speed. Angular velocity

might be measured in radians/sec or deg/sec, clockwise or counterclockwise.ACCELERATION - The rate of change of velocity. Units: in/sec etc. It is a vector

and has magnitude and direction, e.g. 5 in/sec North. Angular acceleration might

be measured in radians/sec or deg/sec, clockwise or counterclockwise.

VIBRATORY MOTION - An oscillating motion; for example, that of the weight W,

in Figure 1.

SIMPLE HARMONIC MOTION - A form of vibratory motion. The motion as a

function of the time is of the form X = a sin t, where a, are constants. Themaximum displacement, a, from the mean position (X = 0) is the amplitude; the

frequency (rate at which motion repeats itself) is /2 cycles/sec, where has thedimensions of reciprocal time, e.g. reciprocal seconds. The motion is also called

harmonic or sinusoidal motion.

AMPLITUDE - Figure 2 shows a vibrating motion, which repeats itself every Tseconds. The maximum values of the displacement, x, from the reference position(x = 0) are called amplitudes. These are (a1, a2. . .). The largest of these is called

the peak amplitude.


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Figure 2

FREQUENCY - Rate at which motion repeats itself per unit time. If the motion

repeats itself every T seconds, the frequency is 1/T cycles per second.

PERIOD, CYCLE The interval of time within which the motion repeats itself. InFigure 2, this is T seconds. The term cycle tends to refer also to the sequence of

events within one period.

STEADY-STATE MOTION - A periodic motion of a mechanical system, e.g. a

continuously vibrating pendulum of constant amplitude.TRANSIENT MOTION - A motion which changes with time in a non-periodic

manner; often the motion declines (attenuates) to a negligible value after a finite

period of time (e.g. impact effects which decay with time, etc.).

PERIODIC AND NON-PERIODIC MOTIONS - A motion, which repeats itself isperiodic; a motion, which does not repeat itself, is non-periodic.

HARMONICS - Any motion can be considered as made up of a series of simple

harmonic motions of different frequencies and amplitudes. The lowest-frequencycomponent is usually called the fundamental frequency; higher frequency

components are called harmonics or super-harmonics. Their frequencies are exactmultiples of the fundamental frequency. Sometimes, components of the

frequencies of which are a fraction of driving frequency are significant (e.g. the"half-frequency" whirl of rotating shafts, etc.). Such components are called

subharmonics.

PULSE - Usually a displacement-time or force-time function describing an inputinto a dynamical system.

PULSE SHAPE - The shape of the time-displacement or force-displacement curve

of a pulse. Typically, this might be a square wave, a rectangular pulse, or a half

sine-wave pulse. In general, however, the shape can be an arbitrary function of

the time.


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SHOCK MOTION - A motion in which there is a sharp, nearly sudden change invelocity, e.g. a hammer blow on a nail, a package falling to ground from a height,

etc. its mathematical idealization is that of a motion in which the velocity changessuddenly. The mathematical idealization of a sudden velocity change often

represents a close approximation to the real dynamic behavior of the system.

2.2 Rigid-Body Characteristics

MASS - Weight in lbs. divided by the gravitational constant (g = 32.2 ft/sec, or

386 in/sec).

CENTER OF GRAVITY - Point of support at which a body would be in balance.

MOMENT OF INERTIA - The moment of inertia of a rigid body about a given axis

in the body is the sum of the product of the mass of each volume element and the

square of its distance from the axis. Units are in-lb-sec, for example. Moments ofinertia of the standard shapes are tabulated in handbooks (see Par. 4.2,

Mechanical System Characteristics). If instead of the mass of the element, thevolume is used, the result is also called a moment of inertia. Depending on the

application, mass-, volume-, or area moments of inertia can be used.

PRODUCT OF INERTIA - The product of inertia of a rigid body about twointersecting, perpendicular axes in the body is the sum of the product of the mass

(volumes, areas) of a constituent element and the product of the distances of theelement from the two perpendicular axes. Units same as moment of inertia.

Tabulations are available in handbooks.

PRINCIPAL AXES OF INERTIA - At any point of a rigid body, mutually

perpendicular (orthogonal) axes can be chosen so that the products of inertiaabout these axes vanish. The orthogonal set of axes are called principal axes of

inertia. These can often be identified by the symmetry of the body, because theprincipal axes coincide with the axes of symmetry. (An axis of symmetry is a line

in the body, such that the body can be rotated a fraction of a turn about the line

without changing its outline in space).


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2.3 Spring and Compliance Characteristics

TENSION - When a body is stretched from its free configuration, its particles aresaid to be in tension (e.g. a stretched bar). The tensile force per unit area is called

the tensile stress (Units: lbs/in).

COMPRESSION - When a body is compressed from its free configuration, its

particles are said to be in compression (e.g. a column in axial loading). Thecompressive force per unit area is called the compressive stress (Units: lbs/in).

SHEAR- When a body is subject to equal and opposite forces, which are notcollinear, the forces tend to shear the body in two. The body is said to be in a

state of shear, e.g. a rubber pad under forces parallel to its upper and lower faces.

The shear force per unit area is called the shear stress (Units: lbs/in). Mostbodies in a state of stress are in tension, compression and shear simultaneously,

e.g. a beam in bending.

SPRING CONSTANT - When a helical spring is stretched or compressed (amount

x, say), the applied force, F, is proportional to the displacement, x (Hooke's law).The constant of proportionality (k) is called the spring constant or gradient: F =

kx. More generally, k is the ratio of a force increment to the correspondingdisplacement increment of the spring. In the usual helical spring this ratio is a

constant (independent of displacement) within the elastic limit of the spring

material. However, k may be variable, e.g. in a rubber pad, since the rubber isnearly incompressible. Units: lbs/in. If the spring deflect in torsion, the units of k

may be in-lb/radian or in-lb/degree, etc.

FORCE-DEFLECTION CHARACTERISTIC OF A SPRING - This refers to the shape of the force-deflection curve. The most familiar case is a straight line through the origin of coordinates (constant k).

It is possible, however, for the spring "constant" to vary. If it increaseswith displacement (e.g. asin a rubber pad), the spring is called a hard spring. If it decreases with

displacement (e.g. as in a Belleville spring), the spring is called a soft spring.ENERGY STORAGE - This is the area under the force-deflection curve of the

spring. It represents the strain energy stored in the spring (Units: in-lbs, or ft-lbs,etc.).

PRELOAD - A spring used in equipment may or may not have a rest (on the shelf)position in which it has its natural, free, or unstretched length. If its rest-position

length is not its free length, the spring is in tension or compression. The amount of

this tension or compression is called the preload. When measured in force units, it


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is a preload force; when measured in deflection from free position, it is a preloaddeflection.

ELASTIC MODULUS (E) AND SHEAR MODULUS (G) - These are material

properties, which characterize material compliance in tension or in compression

(E) and in shear (G). They are defined as the ratio of stress to strain, where strainrefers to the change in length (or deformation) per unit length. E involves tensile

or compressive stress and G involves shear stress. Units: lbs/in. In manypractical applications, E and G can be regarded as constants, within a limit of

material stress known as the proportional limit. Metals loaded below the

proportional limit are examples. Rubber and plastics, however, usually have nowell defined proportional limit.

2.4 Damping, Friction and Energy-Dissipation Characteristics

STATIC FRICTION, SLIDING FRICTION, COULOMB DAMPING - These are all

terms used for the frictional resistance encountered when one body slides relativeto another, e.g. a weight dragged on the ground. The frictional force is

approximately proportional to the contact force between the two bodies andopposed to the direction of relative motion. The constant of proportionality, m, is

known as the coefficient of friction. It a 10 lb weight is dragged along a horizontal

floor with a coefficient of friction, = 0.2, the frictional resistance is 0.2 x 10 = 2lbs. Sometimes a distinction is made between the value of the coefficient of

friction when motion is just impending (starting friction) and the value duringmotion (kinetic friction). The coefficient of friction in the latter case is generally

somewhat lower. Table 2 shows typical values of the coefficient of friction forvarious materials and operating conditions.

VISCOUS DAMPING - If a body moves relative to a second body, viscousdamping refers to a resisting force, which is proportional to the relative velocity

between the two bodies and opposes the direction of relative velocity between

them. The constant of proportionality is known as the coefficient of viscousdamping, c. Units: lbs per unit velocity, i.e. lbs/(in/sec). Such damping is

encountered, for example, in hydraulic dashpots and devices, which meter a liquid

through an orifice. The more viscous the fluid, the greater the damping. If c = 0.5lbs/(in/sec) and the body moves through a viscous fluid at 10 in/sec, the viscous

damping force is 0.5 x 10 = 5 lbs. Typical example: hydraulic door closers.

CRITICAL DAMPING - Value of damping constant just sufficiently high in a

mass-spring-damping system so as to prevent vibration.


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DAMPING RATIO - The ratio of the damping constant to the critical damping

constant for that system.

2.5 Vibration Characteristics of Mechanical Systems

MATHEMATICAL MODEL - An idealized representation of the real mechanical

system, simplified so that it can be analyzed. The representation often consists of

rigid masses and dashpots. Hopefully, the representation is sufficiently realistic sothat the results of the analysis correspond reasonably closely to the behavior of

the physical system from which it was derived.

LUMPED-AND DISTRIBUTED-PARAMETER SYSTEMS - In a lumped-parameter

system, the mass-, elastic- and damping properties are separated or lumped into

distinct components each of which has only mass or only elasticity or onlydamping, but not more than one of these per component. In a distributed-

parameter system, a component may have combined mass and elasticity anddamping, distributed continuously through the component. The latter models tend

to be more realistic, but more difficult to analyze.

DEGREE OF FREEDOM - This is the number of independent quantities

(dimensions), which must be known in order to be able to draw the mechanical

system in any one position, the fixed dimensions of the system being known. The

simple mass-spring system of Figure 1 has one degree of freedom; a mechanicaldifferential, for example, has two degrees of freedom; a rigid body moving freelyin space has 6 degrees of freedom.

FORCE AND MOTION EXCITATION - If a force is applied to a dynamical system,it usually is a source of vibration (e.g. centrifugal force due to an unbalanced

rotor). The vibrations are then said to be due to force excitation. If, on the otherhand, the foundation (or other part) of a machine is subject to a forced motion

(vibration or shock), the resulting machine vibration is said to be due to motion

excitation, e.g. an earthquake actuating a seismograph.

FREE VIBRATION - If the weight in Figure 1 is moved out of its equilibrium

position, and released, the system will vibrate without the action of any externalforces. Such an oscillation is called a free vibration.

FORCED VIBRATION - If an external force is applied to the weight in Figure 1,


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which causes it to vibrate (e.g. a force varying harmonically with time, say), theresulting motion of the spring-mass system is called a forced vibration. If the

base, which supports the spring, undergoes a forced motion, which in turn causesthe weight to vibrate, the vibration is also forced.

RANDOM VIBRATION - Equipment may be caused to vibrate by applied forces ormotions, the frequency (or frequency components) of which vary in a random

manner with time (e.g. wind gusts on a missile). The resulting vibration is calledrandom.

NATURAL FREQUENCY - When mechanical equipment vibrates freely, theresulting number of oscillations per unit time is called the frequency (cycles/sec).

According to whether the system is free without damping, or free with damping,

the frequency is called the free-undamped natural frequency or the free-dampednatural frequency. The natural frequency is a function of the mass distribution and

compliance of the system. For a simple mass-spring system (Figure 1), whichrepresents a reasonable approximation to many real mechanicalsystems, the

natural frequency is equal to radians per second, where k is thespring constant, lbs/in; W is the weight, lbs; g is the gravitational constant, 386

in/sec; and xst is the static deflection of the spring, in. Thus, flexible systems tend

to have low natural frequencies and rigid systems tend to have high naturalfrequencies. At the same time, the natural frequency can be changed by altering

the compliance and mass distribution of the system. The simple expressions fornatural frequency just given, yield the natural-frequency curve of the basic

vibration chart given in Par. 4, Case A. In the chart they are plotted on alogarithmic scale and the frequency is given in cycles per minute, rather than inradians per second. A system may have more than one natural frequency, in

which case the lowest of these is often the most significant. In general, thenumber of natural frequencies is equal to the degree of freedom of the system.

FORCING FREQUENCY - The number of oscillations per unit time of an externalforce or displacement, applied to a vibrating system.


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The springs are in parallel and combine accordingly, but because of the gearreductions all spring constants are multiplied by the square of the gear ratio

between the shaft involved and the reference shaft.In computing the equivalent moment of inertia, l, the moment of inertia at an

intermediate shaft is multiplied by the square of the gear ratio between that shaft

and the reference shaft.


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(i) Experimental Determination of System Compliance

It is often easier to measure the system compliance, instead of computing it. Thecompliance between any two members of a mechanical system can be measured

by holding one of them fixed and measuring the force (torque) required to deflect

the other member by a unit distance (angle). When the system is complex, this

may be the only way. For example, in calculating damage in freight cars due toshock loading, it is necessary to know the compliance of the lading. For a typicallading, such as canned dog food packed in fibre cartons and loaded in block

pattern, the compliance is more readily determined by experiment than by

computation.

4.3 Damping

In the case of viscous damping, it is usually required to make an estimate of thedamping ratio (the ratio of damping to critical damping). In most mechanical

systems, this ratio ranges from near zero to about 5%. A convenient way to

measure the ratio is by noting the decay in amplitude of a damped free vibration

of the system. The calculation is given in Case A, Par. 4.In the case of sliding friction (Coulomb damping or static friction), the coefficient

of friction is a measure of the resistance to the sliding motion. Sometimes, it is

convenient to work with equivalent viscous damping, rather than with static

friction, because viscous damping gives rise to mathematically linear equations,while static friction does not. In such cases, it is sometimes possible to compute

an equivalent coefficient of viscous damping. In the case of a single-degree-of-freedom system of Case A, Par.4, subject to a sinusoidally varying force acting

vertically on the weight W, and assuming for the time being that the dashpot is

replaced by sliding friction, It is possible to find the equivalent viscous damping

coefficient by equating the work done during a quarter cycle of vibration betweenthe actual system and the equivalent viscously damped system. However, caremust be taken that such equivalence is valid for the purposes of the vibration-

mount design. This is not always the case and requires separate investigation.

4.4 Vibration Analysis of Basic Vibration-Mount and Shock-MountConfigurations

The design of a vibration mount, or shock mount, involves a calculation of the


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force- or motion transmission from equipment to foundation or vice versa. In orderto facilitate this

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computation, it has been recognized that many mechanical systems can be

represented by reasonably simple mathematical models. Once these models havebeen analyzed, the results can be tabulated and used with minimum effort. Below,

we have selected 6 mathematical models of vibrating machinery and systems,which we believe will be applicable to a large number of vibration-isolation

problems encountered by the engineer. The basic results are summarized for each

case. Solved problems are given in the subsequent section and illustrate theapplication of these cases to vibration-mount design.

CASE A: UNBALANCED SINUSOIDAL FORCE ACTING ON EQUIPMENT

p = frequency of applied force or forcing frequency, rad/sec

= damping ratio = c/cc, where cc = critical damping coefficient

= (undamped natural frequency of system, rad/sec)g = gravitational constant, 386 in/sec

p/ = frequency ratio (ratio of forced to natural)x = equipment displacement, measured from equilibrium, in.

The performance of the vibration moment can be measured by factors oftransmissibility. The force transmissibility, TF, is defined as follows:

TF = Max. force transmitted to baseMax. unbalanced force acting on equipment

The quantity (1 - TF) is sometimes called the isolation or insulation and is

expressed as a percentage.The motion amplification of the equipment can be measured by comparing the
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max. equipment displacement to the static displacement under the unbalancedforce Fmax. This leads to the displacement transmissibility of the equipment, TDE:

TDE = Max. displacement of equipment from staticStatic displacement of equipment under force Fmax

(A-1)

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