Vibrazioni
1.4 Vibrazioni
Béla I. Sandor with assistance by Stephen M. Birn
Vibrazioni in machines and structures should be analyzed and controlled
if they have undesirable effects such as noise, unpleasant motions,
or fatigue damage with potentially catastrophic consequences. Con-
versely, Vibrazioni are sometimes employed to useful purposes,
such as for compacting materials.
Undamped Free and Forced Vibrations
The simplest vibrating system has motion of one degree of freedom
(DOF) described by the coordinate x in Figure 1.4.1. (An analogous
approach is used for torsional Vibrazioni, with similar results.)
Figura 1.4.1 Model of a simple vibrating system.
Assuming that the spring has no mass and that there is no damping in
the system, the equation of motion for free vibration
(motion under internal forces only; F = 0) is
where 
= natural circular frequency in rad/sec.
The displacement x as a function of time t is
where C1 and C2 are constants depending on the initial conditions
of the motion. Alternatively,
where C1 = Acos?, C2 = Asin?, e ? is the phase angle, another
constant. A complete cycle of the motion occurs in time ?,
the period of simple harmonic motion,
The frequency in units of cycles per second (cps) or hertz (Hz) is f = 1/?.
The simplest case of forced vibration is modeled in Figure 1.4.1,
with the force F included. Using typical simplifying assumptions as above,
the equation of motion for a harmonic force of forcing
frequency ?,
Il Vibrazioni of a mass m may also be induced by the displacement
d = dosin?t of a foundation or another mass M to which m is attached
by a spring k. Using the same reference point and axis for both
x and d, the equation of motion for m is
where do is the amplitude of vibration of the moving support M,
e ? is its frequency of motion. The general solution of the
forced vibration in the steady state (after the initial, transient behavior)
is
where ? is the forcing frequency and ? is the natural circular
frequency of the system of m and k. Resonance. The amplitude of the
oscillations in forced Vibrazioni depends on the frequency ratio ?/?.
Without damping or physical constraints, the amplitude would become
in?nite at ? = ?, the condition of resonance. Dangerously large
amplitudes may occur at resonance and at other frequency ratios near
the resonant frequency. A magni?cation factor is de?ned as
Several special cases of this are noted:
1. Static loading: ? = 0, or ? ! ?; MF . 1.
2. Resonance: ? = ?; MF = ?.
3. High-frequency excitation: ? @ ?; MF . 0.
4. Phase relationships: The vibration is in phase for
? < ?, and it is 180? out of phase for ? > ?.
Damped Free and Forced Vibrations
un vibrating system of one degree of freedom and damping is modeled
in Figure 1.4.2. The equation of motion for damped free
Vibrazioni (F = 0) is
Figura 1.4.2 Model of a damped vibrating system.
The displacement x as a function of time t is
The value of the coef?cient of viscous damping c that makes the
radical zero is the critical damping coef?cient cc = 2m ?k /m = 2m?.
Three special cases of damped free Vibrazioni are noted:
1. Overdamped system: C > cc; the motion is nonvibratory or aperiodic.
2. Critically damped system: c = cc; this motion is also nonvibratory;
x decreases at the fastest rate possible without oscillation of the mass.
3. Underdamped system: C < cc; the roots ?1,2 are complex numbers;
the displacement is
where A and ? are constants depending on the initial conditions,
and the damped natural frequency is
The ratio c/cC is the damping factor ?. The damping in a system is
determined by measuring the rate of decay of free oscillations.
This is expressed by the logarithmic decrement ?, involving
any two successive amplitudes xi and x i+1,
The simplifying approximation for ? is valid for up to about 20%
damping (? . 0.2). The period of the damped vibration is ?d = 2?/?d.
It is a constant, but always larger than the period of the same system
without damping. In many real systems the damping is relatively small
(? < 0.2), where ?d . ? e ?d . ? can be used.
The equation of motion for damped forced Vibrazioni
(Figure 1.4.2; F ? 0) is
The solution for steady-state vibration of the system is
where the amplitude and phase angle are from
The magni?cation factor for the amplitude of the oscillations is
This quantity is sketched as a function of the frequency ratio ?/?
for several damping factors in Figure 1.4.3. Note that the amplitude of
vibration is reduced at all values of ?/? if the coef?cient of damping
c is increased in a particular system.
Figura 1.4.3 Magni?cation factor in damped forced vibration.
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