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FUNDAMENTALS OF ACOUSTICS 3

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Title: FUNDAMENTALS OF ACOUSTICS 3


1
FUNDAMENTALS OF ACOUSTICS (3)
  • Particles Vibrating Systems

2
REVIEW Simple harmonic motion
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1-2 DAMPED OSCILLATIONS
  • Whenever a real body is set into oscillation,
    dissipative (frictional)forces arise.These forces
    are of many types, depending on the particular
    oscillating system.but they will always result
    in a damping of the oscillations- a decrease in
    the amplitude of the free oscillations with time.

7
Let us first consider the effect of a viscous
frictional force
On a simple oscillator
  • Such a force is assumed to be proportional to
    the speed of the mass and to be directed so as to
    oppose the motion. It can be expressed as

Where Rm is a positive constant called the
mechanical resistance of the system.
8
  • It is evident that mechanical resistance has the
    units of nweton-second per meter (N.s/m)
  • If the effect of resistance is included ,the
    equation of motion of an oscillator becomes

9
  • Dividing through by m and recalling that

We have
10
  • This equation may be solved by the complex
    exponential method.Assume a solution of the form

µ1?µ2
Are two solutions of follow equation
11
We can obtain
1. When
,Rm24mD
If the mechanical resistance Rm is large enough,
the system is no longer oscillatory
12
  • In most cases of importance in acoustics ,the
    mechanical resistance Rm is small enough so that


lt
is complex. Now , µis given by
13
Defining a new constant Oby
  • Now µ is given by
  • And Ois seen to be the natural angular frequency
    of the damped oscillator.
  • Notice that Ois always less than the natural
    angular frequency ?0 of the same oscillator
    without damping.

14
The complete solution is the sum of the two
solutions obtained above,
  • One convenient form of this general solution is

Where A and fare real constants determined by the
initial conditions.
15
The amplitude of the damped oscillator, now
defined as
A(t) is on longer constant, but decreases
exponentially with time. As with the undamped
oscillator,the frequency is independent of the
amplitude of oscillation.
16
  • One measure of the rapidity with which the
    oscillations are damped by friction is the time
    required for the amplitude to decrease to 1/e of
    its initial value.This time is called decay
    modulus and is given by

The quantity d is called the resistance
coefficient .
17
Fig.displays the time history of the displacement
of a damped harmonic oscillator
X(t)
18
ENERGY OF VIBRATION
19
1.3 FORCED OSCILLATIONS
  • 1 . A simple oscillator, when driven by an
    externally applied force F, the differential
    equation for the motion becomes

20
If FFmcos?t, and it will be advantageous to
replace the real driving force Fmcos?t by its
equivalent complex driving force fF exp(j?t
)we can write in the form
?? ?0
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Initial conditions
For A?0,f0 0
22
This special pattern of motion is known as the
beating phenomenon. Sound waves of slightly
different frequencies will also give rise to
beats .
23
2.In a damping of the oscillations,the
differential equation for the motion becomes
(1-3)
24
  • O?0

When
Assume a solution of the form
And substitute into equation to obtain
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We obtain
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Zm is called the complex mechanical impedance ,Rm
is called the mechanical resistance Xm is
called the mechanical reactance
The mechanical impedance
Has magnitude
And phase angle
27
  • The dimensions of mechanical impedance are the
    same as those of mechanical resistance and are
    expressed in the same units, N.s/m, often defined
    as mechanical ohms.
  • It is to be emphasized that , although the
    mechanical ohm is analogous to the electrical
    ohm, these two quantities do not have the same
    units.
  • The electrical ohm has the dimensions of voltage
    divided by current
  • The mechanical ohm has the dimensions of force
    divided by speed.

28
The solution of equation is The sum of two parts
a transient term a
steady-state term
29
For the case of a sinusoidal driving force
f(t)Fmcos(?t) applied to the oscillator at some
initial time, the solution of (1-3) is the sum of
two parts a transient term containing two
arbitrary constants and a steady-state term which
depends of F and ? but does not contain any
arbitrary constants. The transient term is
obtained by setting F equal to zero. The
arbitrary constants are determined by applying
the initial conditions to the total solution.
30
After a sufficient time interval, The damping
term makes this portion of the solution
negligible. Leaving only the steady state term
whose angular frequency ? is that of the driving
force
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ENERGY OF VIBRATION
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Problems-Waves
  • 1-2 Prove each wave addition

34
Problem Simple Harmonic Motions
  • 1-3 A simple harmonic motion is given as
    x(t)10sin(10t-300), where x is measured in
    meters , t in seconds , and the phase angle in
    degrees. Find (1) the frequency and period of the
    motion, (2) the maximum displacement ,velocity
    and acceleration, (3) the displacement ,
    velocity and acceleration at t0 and t1 seconds.
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