Vibraciones

Chapter 19 MECHANICAL VIBRATIONS
Consider the free vibration of a particle, i.e., the motion of a particle P subjected to a restoring force proportional to the displacement of the particle - such as the force exerted by a spring. If the displacement x of the particle P is measured from its equilibrium position O, the resultant F of the forces acting on P (including its weight) has a magnitude kxand is directed toward O. Applying Newton’s .. second law (F = ma) with a = x, the differential equation of motion is

-xm

O
Equilibrium

x

P

+xm

+

mx + kx = 0

..

mx + kx = 0
-xm setting ωn 2 = k/m

..

..
O
Equilibrium

x + ωn2x = 0
x P The motion defined by this expression is called simple harmonic motion. The solution of this equation, which represents thedisplacement of the particle P is expressed as +xm

x = xm sin (ωnt + φ)
where xm = amplitude of the vibration ωn = k/m = natural circular frequency φ = phase angle

+

-xm

x + ωn2x = 0 x = xm sin (ωnt + φ)
The period of the vibration (i.e., the time required for a full cycle) and its frequency (i.e., the number of cycles per second) are expressed as

..

O
Equilibrium

x

P+xm

+

2π Period = τn = ω n ωn 1 Frequency = fn = = 2π τn vm = xm ωn am = xm ωn2

The velocity and acceleration of the particle are obtained by differentiating x, and their maximum values are

O

xm
ωn t Q

φ QO

x P

a v x

am= xmωn2

ωn t + φ vm= xmωn

The oscillatory motion of the particle P may be represented by the projection on the x axis of the motion of a point Qdescribing an auxiliary circle of radius xm with the constant angular velocity ωn . The instantaneous values of the velocity and acceleration of P may then be obtained by projecting on the x axis the vectors vm and am representing, respectively, the velocity and acceleration of Q.

While the motion of a simple pendulum is not truely a simple harmonic motion, the formulas given above may be usedwith ωn 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 by choosing an appropriate variable, such as a distance x or an angle θ , to define the position of the body, drawing a diagram expressing the equivalence of the external and effective forces, and writing an equation relating the selectedvariable and its second derivative. If the equation obtained is of the form

..

x + ωn2x = 0

or

θ + ωn2θ = 0

..

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

The principle of conservation of energy may be used as an alternative method for the determination of the period and frequency of the simple harmonic motion of a particle orrigid body. Choosing an appropriate variable, such as θ, to define the position of the system, we express that the total energy of the system is conserved, T1 + V1 = T2 + V2 , between the position of maximum displacement (θ1 = θm) and the position . . of maximum velocity (θ 2 = θm). If the motion considered is simple harmonic, the two members of the equation obtained . consist of homogeneousquadratic expressions in θm and θm , . respectively. Substituting θm = θm ωn in this equation, we may factor out θm2 and solve for the circular frequency ωn .

x
Equilibrium

The forced vibration of a mechanical system occurs when the system is subjected to a periodic force or when it is elastically connected to a support which has an alternating motion. The differential δm sin ωf t equationdescribing δm each system is presented ωf t = 0 ωf t below.

P = Pm sin ωf t

mx + kx = Pm sin ωf t mx + kx = kδm sin ωf t
..
x
Equilibrium

..

mx + kx = Pm sin ωf t mx + kx = kδm sin ωf t
x
Equilibrium

..

δm ωf t = 0

δm sin ωf t ωf t

..

The general solution of these equations is obtained by adding a particular solution of the form

P = Pm sin ωf t

xpart = xm sin ωf t...