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Chapter 19 MECHANICAL VIBRATIONS

Consider the free vibration of a particle , i.e., the motion of a particle P subjected to a restoring forceproportional to the displacement of theparticle - such as the force exerted bya spring. If the displacement x of theparticle P is measured from itsequilibrium position O, the resultant Fof the forces acting on P (including itsweight) has a magnitude kx and isdirected toward O . Applying Newtons

second law ( F = ma ) with a = x, thedifferential equation of motion is

O

+ xm

- xm

P Equilibrium

+ mx + kx = 0..

..

x

O

+ xm

- xm

P Equilibrium

+

mx + kx = 0setting n2 = k /m

x + n2 x = 0

..

..

The motion defined by this expressionis called simple harmonic motion .

x = xm sin ( nt + )

The solution of this equation, whichrepresents the displacement of theparticle P is expressed as

where xm = amplitude of the vibrationn = k /m = natural circular

frequency = phase angle

x

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O

+ xm

- xm

P Equilibrium

+

x + n2 x = 0..

x = xm sin ( nt + )The period of the vibration (i.e., thetime required for a full cycle) and itsfrequency (i.e., the number of cyclesper second) are expressed as

Period = n =2n

Frequency = f n = =n2

1

nThe velocity and acceleration of the particle are obtained bydifferentiating x, and their maximum values are

vm = xmn a m = xmn2

x

O

P

The oscillatory motion of the

particle P may be representedby the projection on the x axis of the motion of a point Qdescribing an auxiliary circle of radius xm with the constantangular velocity n. Theinstantaneous values of thevelocity and acceleration of P may then be obtained byprojecting on the x axis thevectors v m and a m representing,

respectively, the velocity andacceleration of Q .

xm

a m= xmn2

nt

v m= xmn

nt +

x QOQ

v

a

x

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While the motion of a simple pendulum is not truely a simple

harmonic motion, the formulas given above may be used withn 2 = g /l to calculate the period and frequency of the small oscillations of a simple pendulum.

The free vibrations of a rigid body may be analyzed bychoosing an appropriate variable, such as a distance x or anangle , to define the position of the body, drawing a diagramexpressing the equivalence of the external and effective forces,and writing an equation relating the selected variable and itssecond derivative. If the equation obtained is of the form

x + n2 x = 0 or + n2 = 0.. ..

the vibration considered is a simple harmonic motion and itsperiod and frequency may be obtained.

The principle of conservation of energy may be used as analternative method for the determination of the period andfrequency of the simple harmonic motion of a particle or rigidbody. Choosing an appropriate variable, such as , to definethe position of the system, we express that the total energyof the system is conserved, T 1 + V 1 = T 2 + V 2 , between theposition of maximum displacement ( 1 = m) and the positionof maximum velocity ( 2 = m). If the motion considered issimple harmonic, the two members of the equation obtainedconsist of homogeneous quadratic expressions in m and m ,respectively. Substituting m = m n in this equation, we may

factor out m and solve for the circular frequency n.2

..

. .

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xEquilibrium

P = P m sin f t

f t = 0

mm sin f t

f t

xEquilibrium

The forced vibration of a mechanical systemoccurs when the system is subjected to a

periodic force or when it is elasticallyconnected to a support which has analternating motion. The differentialequation describingeach system ispresentedbelow.

mx + kx = P m sin f t

mx + kx = k m sin f t

..

..

xEquilibrium

P = P m sin f t

f t = 0

mm sin f t

f t

xEquilibrium

mx + kx = P m sin f t

mx + kx = k m sin f t

..

..

The general solutionof these equations isobtained by adding aparticular solution of the form

x part = xm sin f t

to the general solution of the correspondinghomogeneous equation. The particular solution represents thesteady-state vibration of the system, while the solution of thehomogeneous equation represents a transient free vibrationwhich may generally be neglected.

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xEquilibrium

P = P m sin f t

f t = 0

mm sin f t

f t

xEquilibrium

mx + kx = P m sin f t ..

mx + kx = k m sin f t ..

x part

= xm

sin f

t

Dividing the amplitude xm of the steady-state vibration by P m/k in the case of aperiodic force, or by m in the case of anoscillating support, the magnification factor of the vibration is defined by

Magnification factor = = xm

P m /k xmm

1

1 - ( f /n )2=

xEquilibrium

P = P m sin f t

f t = 0

mm sin f t

f t

xEquilibrium

mx + kx = P m sin f t ..

mx + kx = k m sin f t ..

x part = xm sin f t

Magnification factor = = xm

P m /k xmm

11 - ( f / n)2

=

The amplitude xm of the forced vibrationbecomes infinite when f = n , i.e., whenthe forced frequency is equal to thenatural frequency of the system . The

impressed force or impressed supportmovement is then said to be in resonancewith the system. Actually the amplitude of the vibration remains finite, due todamping forces.

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The equation of motion describing the damped free vibrationsof a system with viscous damping is

mx + cx + kx = 0.. .

where c is a constant called the coeficient of viscous damping .Defining the critical damping coefficient cc as

cc = 2 m = 2 mnk m

where n is the natural frequency of the system in the absenceof damping, we distinguish three different cases of damping,

namely, (1) heavy damping , when c > cc, (2) critical damping ,when c = cc, (3) light damping , when c < cc. In the first two cases,the system when disturbed tends to regain its equilibriumposition without oscillation. In the third case, the motion isvibratory with diminishing amplitude.

The damped forced vibrations of a mechanical system occurs

when a system with viscous damping is subjected to a periodicforce P of magnitude P = P m sin f t or when it is elasticallyconnected to a support with an alternating motion = m sin f t .In the first case the motion is defined by the differential equation

mx + cx + kx = P m sin f t .. .

mx + cx + kx = k m sin f t .. .

In the second case the motion is defined by the differentialequation

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The steady-state vibration of the system is represented by aparticular solution of mx + cx + kx = P m sin f t of the form

.. .

x part = xm sin ( f t - )

Dividing the amplitude xm of the steady-state vibration by P m/k inthe case of a periodic force, or by m in the case of an oscillatingsupport, the expression for the magnification factor is

xm P m/k

= = xmm

1[1 - ( f / n)2]2 + [2( c/cc)( f / n)]2

where n = k /m = natural circular frequency of undampedsystem

cc = 2m n = critical damping coefficientc/cc = damping factor

In addition, the phase difference between the impressed forceor support movement and the resulting steady-state vibration of the damped system is defined by the relationship

tan =2(c/cc) ( f / n)

1 - ( f / n)2

The vibrations of mechanical systems and the oscillations of electrical circuits are defined by the same differential equations.Electrical analogues of mechanical systems may therefore beused to study or predict the behavior of these systems.

xm P m/k

= = xmm1

[1 - ( f / n)2]2 + [2( c/cc)( f / n)]2

x part = xm sin ( f t - )