CHAPTER 1
Free and Forced Vibrations of Simple Systems
Mechanical, acoustical, or electrical vibrations are the sources of
sound in musical instruments. Some familiar examples are the vibrations
of strings (violin, guitar, piano, etc), bars or rods (xylophone,
glockenspeil, chimes, clarinet reed), membranes (drums, banjo), plates
or shells (cymbal, gong, bell), air in a tube (organ pipe, brass and
woodwind instruments, marimba resonator), and air in an enclosed
container (drum, violin, or guitar body).
In most instruments, sound production depends upon the collecive
behavior of several vibrators, which may be weakly or strongly coupled
together. This coupling, along with nonlinear feedback, may cause the
instrument as a whole to behave as a complex vibrating system, even
though the individual elements are relatively simple vibrators.
In the first seven chapters, we will discuss the physics of mechanical
and acoustical vibrators, the way in which they may be coupled together,
and the way in which they radiate sound. Since we are not discussing
electronic musical instruments, we will not deal with electrical
oscillators except as they help us, by analogy, to understand mechanical
and acoustical oscillators.
Many objects are capable of vibrating or oscillating. Mechanical
vibrations require that the object possess two basic properties: a
stiffness or springlike quality to provide a restoring force when
displaced and inertia, which causes the resulting motion to overshoot
the equilibrium position. From an energy standpoint, oscillators have a
means for storing kinetic energy (mass), and a means by which energy is
gradually lost (damper). Vibratory motion involves the alternating
transfer of energy between its kinetic and potential forms.
The inertial mass may be either concentrated in one location or
distributed throughout the vibrating object. If it is distributed, it
is usually the mass per unit length, area, or volume that is important.
Vibrations in distributed mass systems may be viewed as standing waves.
The restoring forces depend upon the elasticity or the compressibility
of some material. Most vibrating bodies obey Hooke's law; that is, the
restoring force is proportional to the displacement from equilibrium, at
least for small displacement.
1.1. Simple Harmonic Motion in One Dimension
The simplest kind of periodic motion is that experienced by a point mass
moving along a straight line with an acceleration directed toward a
fixed point and proportional to the distance from that point. This is
called simple harmonic motion, and it can be described by a sinusoidal
function of time t: x(t) = A * sin(2 * pi * f * t), where the amplitude
A describes the maximum extent of the motion, and the frequency f tells
us how often it repeats.
The period of the motion is given by
T = 1 / f. (1.1)
That is, each T seconds the motion repeats itself.
A simple example of a system that vibrates with simple harmonic motion
is the mass-spring system shown in Fig. 1.1. We assume that the amount
of stretch x is proportional to the restoring force F (which is true in
most springs if they are not stretched too far), and that the mass
slides freely without loss of energy. The equation of motion is easily
obtained by combining Hooke's law, F = -K * x, with Newton's second law,
F = m * a = m * x". Thus,
m * x" = -K * x
and
m * x" + K * x = 0,
where
x" = d^2(x)/dt^2.
The constant K is called the spring constant or stiffness of the
spring (expressed in N / m). We define a constant omega sub 0 = (K /
m)^(1 / 2), so that the equation of motion becomes
x" + omega sub 0^2 * x = 0. (1.2)
This well-known equation has these solutions:
x = A * cos(omega sub 0 * t + phi) (1.3)
<Full-width insertion: illustration above centered caption, the
illustration showing a mechanical schematic of a vertical fixed base on
the left connected through a spring to a suspended mass, m, on the
right, subjected to a rightwards force, x. The caption:
Fig. 1.1. Simple mass-spring vibrating system.>
<Full-width insertion: illustration above caption, the illustration
showing a graph of three ordinates, x, v, and a, over time. Three sine
waves of similar amplitude and period 2 * pi, but different phase, are
shown: a solid cosine wave, a dashed negative sine wave, and a dotted
negative cosine wave, representing each of the variables, respectively.
The time axis is labeled at intervals of pi / 2, a dashed vertical line
approximately 10% of the width from the origin at the extreme left
indicates an abcissa labeled t = 0 and its distance from the origin, the
arrow of time is labeled omega sub 0 * t + phi, and the ordinates
of the intersections of x and v are labeled x sub 0 and v sub 0. The
ordinate of the intersection of a and the vertical axis is labeled
-omega sub 0^2 * A, and the ordinate of the origin is labeled 0. An A
part-way up the amplitude axis identifies it as such, I suspect. The
caption:
Fig. 1.2. Relative phase of displacement x, velocity v, and acceleration
a of a simple vibrator.>
or
x = B * cos(omega sub 0 * t) + C * sin(omega sub 0 * t), (1.4)
from which we recognize omega sub 0 as the natural angular frequency of
the system.
<Why "natural" angular frequency?>
The natural frequency f sub 0 of our simple oscillator is given by f
sub 0 = (1 / (2 * pi)) * (K / M)^(1 / 2), and the amplitude by (B^2 +
C^2)^(1 / 2) or by A; phi is the initial phase of the motion.
<Why "natural" frequency?>
Differentiation of the displacement x with respect to time gives the
corresponding expressions for the velocity v and acceleration a:
v = x' = -omega sub 0 * A * sin(omega sub 0 * t + phi), (1.5)
and
a = x" = -omega sub 0^2 * A * cos(omega sub 0 * t + phi). (1.6)
The displacement, velocity, and acceleration are shown in Fig. 1.2.
Note that the velocity v leads the displacement by pi / 2 radians (90
deg), and the acceleration leads (or lags) by pi radians (180 deg).
Solutions to second-order differential equations have two arbitrary
constants. In Eq. (1.3) they are A and phi; in Eq. (1.4) they are B and
C. Another alternative is to describe the motion in terms of constants x
sub 0 and v sub 0, the displacement and velocity when t = 0. Setting t
= 0 in Eq. (1.3) gives x sub 0 = A * cos(phi), and setting t = 0 in
Eq. (1.5) gives v sub 0 = -omega sub 0 * A * sin(phi). From these we
can obtain expressions for A and phi in terms of x sub 0 and v sub 0:
A = (x sub 0^2 + (v sub 0 / omega sub 0)^2)^(1 / 2),
and (1.7)
phi = arctan(-v sub 0 / (omega sub 0 * x sub 0)).
Alternatively, we could have set t = 0 in Eq. (1.4) and its derivative
to obtain B = x sub 0 and C = v sub 0 / omega sub 0 from which
x = x sub 0 * cos(omega sub 0 * t)
+ (v sub 0 / omega sub 0) * sin(omega sub 0 * t). (1.8)
1.2. Complex Amplitudes
Another approach to solving linear differential equations is to use
exponential functions and complex variables. In this description of the
motion, the amplitude and the phase of an oscillating quantity, such as
displacement or velocity, are expressed by a complex number; the
differential equation of motion is transformed into a linear algebraic
equation. The advantages of this formulation will become more apparent
when we consider driven oscillators.
This alternate approach is based on the mathematical identity e^(+-j *
omega sub 0 * t) = cos(omega sub 0 * t) +- j * sin(omega sub 0 * t),
where j = (-1)^(1 / 2). In these terms, cos(omega sub 0 * t) =
Re(e^(+-j * omega sub 0 * t)), where Re stands for the "real part of."
Equation (1.3) can be written
x = A * cos(omega sub 0 * t + phi)
= Re[A * e^(j * (omega sub 0 * t + phi))]
= Re(A * e^(j * phi) * e^(j * omega sub 0 * t))
= Re(~A * e^(j * omega sub 0 * t)). (1.9)
The quantity ~A = A * e^(j * phi) is called the complex amplitude of the
motion and represents the complex displacement at t = 0. The complex
displacement ~x is written
~x = ~A * e^(j * omega sub 0 * t). (1.10)
The complex velocity ~v and acceleration ~a become
~v = j * omega sub 0 * ~A * e^(j * omega sub 0 * t)
= j * omega sub 0 * ~x, (1.11)
and
~a = -omega sub 0^2 * ~A * e^(j * omega sub 0 * t)
= -omega sub 0^2 * ~x. (1.12)
Each of these complex quantities can be thought of as a rotating vector
or phasor rotating in the complex plane with angular velocity omega sub
0, as shown in Fig. 1.3. The real time dependence of each quantity can
be obtained from the projection on the real axis of the corresponding
complex quantities as they rotate with angular velocity omega sub 0.
<Full-width insertion: image over caption, the image a polar coordinate
representation of phasor A showing its direction as phi, of measure
omega sub 0 (approximately 40 deg), as well as two other directions,
omega sub 0 * A, 90 deg beyond that of the phasor, and omega sub 0^2 *
A, 90 deg beyond that. No measure of amplitude is given, but it is
constant, the phasor's path being illustrated by a circle centered on
the origin. The caption:
Fig. 1.3. Phasor representation of the complex displacement, velocity,
and acceleration of a linear oscillator.>
1.3. Superposition of Two Harmonic Motions in One Dimension
Frequently, the motion of a vibrating system can be described by a
linear combination of the vibrations induced by two or more separate
harmonic excitations. Provided we are dealing with a linear system, the
displacement at any time is the sum of the individual displacements
resulting from each of the harmonic excitations. This important
principle is known as the principle of linear superposition. A linear
system is one in which the presence of one vibration does not alter the
response of the system to other vibrations, or one in which doubling the
excitation doubles the response.
1.3.1. Two Harmonic Motions Having the Same Frequency
One case of interest is the superposition of two harmonic motions having
the same frequency. If the two individual displacements are
~x sub 1 = A sub 1 * e^(j * (omega * t + phi sub 1))
<Why has omega lost its subscript? Why did it have one?>
and
~x sub 2 = A sub 2 * e^(j * (omega * t + phi sub 2)),
their linear superposition results in a motion given by
~x sub 1 + ~x sub 2
= (A sub 1 * e^(j * phi sub 1) + A sub 2 * e^(j * phi sub 2))
* e^(j * omega * t)
= A * e^(j * (omega * t + phi)). (1.13)
The phasor representation of this motion is shown in Fig. 1.4.
Expressions for A and phi can easily be obtained by adding the phasors
A sub 1 * e^(j * omega * phi sub 1) and A sub 2 * e^(j * omega * phi sub
2) to obtain
A = ((A sub 1 * cos(phi sub 1) + A sub 2 * cos(phi sub 2))^2 + (A
sub 1 * sin(phi sub 1) + A sub 2 * sin(phi sub 2))^2)^(1 / 2), (1.14)
and
tan(phi) = (A sub 1 * sin(phi sub 1) + A sub 2 * sin(phi sub 2))
/ (A sub 1 * cos(phi sub 1) + A sub 2 * cos(phi sub 2)). (1.15)
<Full-width insertion: image with caption beneath, showing the first
quadrant of the graph in Fig. 1.3, augmented by showing two vectors
which result in A: A sub 1 and A sub 2, with their measures, and the
projections of A on the real and imaginary axes, expressed as the
denominator of Eq. (1.15) for the former, and its numerator for the
latter. Caption:
Fig. 1.4. Phasor representation of two simple harmonic motions having
the same frequency.>
What we have really done, of course, is to add the real and imaginary
parts of ~x sub 1 and ~x sub 2 to obtain the resulting complex
displacement ~x. The real displacement is
x = Re(~x) = A * cos(omega * t + phi). (1.16)
The linear combination of two simple harmonic vibrations with the same
frequency leads to another simple harmonic vibration at this same
frequency.
1.3.2. More Than Two Harmonic Motions Having the Same Frequency
The addition of more than two phasors is accomplished by drawing them in
a chain, head to tail, to obtain a single phasor that rotates with
angular velocity omega. This phasor has an amplitude given by
A = ((Sigma(An * cos phin))^2 + (Sigma(An * sin phin))^2)^(1/2)
(1.17)
[NOTE: The n's are subscripts. - GMP]
and a phase angle phi obtained from
tan phi = (Sigma(An * sin phin))/(Sigma(An * cos phin)) (1.18)
The real displacement is the projection of the resultant phasor on the
real axis, and this is equal to the sum of the real parts of all the
component phasors:
x = A * cos(omega * t + phi)
= Sigma(An * cos(omega * t + phin) (1.19)
1.3.3. Two Harmonic Motions with Different Frequencies: Beats
If two simple harmonic motions with frequencies f sub 1 and f sub 2 are
combined, the resultant expression is
~x = ~x sub 1 + ~x sub 2 = (1.20)
(A sub 1 * e ^ (j * (omega sub 1 * t + phi sub 1)) +
(A sub 2 * e ^ (j * (omega sub 2 * t + phi sub 2)),
where A, omega, and phi express the amplitude, the angular frequency,
and the phase of each simple harmonic vibration.
The resulting motion is not simple harmonic, so it cannot be
represented by a single phasor or expressed by a simple sine or cosine
function. If the ratio of omega sub 2 to omega sub 1 (or omega sub 1 to
omega sub 2) is a rational number, the motion is periodic with an
angular frequency given by the largest common divisor of omega sub 2 and
omega sub 1. Otherwise, the motion is a nonperiodic oscillation that
never repeats itself.
The linear superposition of two simple harmonic vibrations with nearly
the same frequency leads to periodic amplitude variations or beats. If
the angular frequency omega sub 2 is written as
omega sub 2 = omega sub 1 + Delta omega, (1.21)
the resulting displacement becomes
~x = A1 * e^(j * (omega1 * t + phi1)) (1.22)
+ A2 * e^(j * (omega1 * t + Delta omega * t + phi2))
= (A1 * e^(j * phi1) +
A2 * e^(j * (phi2 + Delta omega * t))) * e^(j * omega1 * t)
We can express this in terms of a time-dependent amplitude A(t) and a
time-dependent phase phi(t):
~x = A(t) * e^(j*(omega1 * t + phi(t))), (1.23)
where
A(t) = (1.24)
(A1^2 + A2^2 + 2 * A1 * A2 * cos(phi1 - phi2 - Delta omega * t))^(1/2),
and
tan phi(t) = (1.25)
(A1 * sin phi1 + A2 * sin(phi2 + Delta omega * t)) /
(A1 * cos phi1 + A2 * cos(phi2 + Delta omega * t).
The resulting vibration could be regarded as approximately simple
harmonic motion with angular frequency omega1 and with both amplitude
and phase varying slowly at frequency (Delta omega)/(2 * pi). The
amplitude varies between the limits A1 + A2 and |A1 - A2|.
In the special case where the amplitudes A1 and A2 are equal and phi1
and phi2 = 0, the amplitude equation [Eq. (1.24)] becomes
A(t) = A1 * (2 + 2 * cos (Delta omega1 * t))^(1/2) (1.26)
and the phase equation [Eq. (1.25)] becomes
tan phi(t) = (sin(Delta1 * t)) / (1 + cos(Delta1 * t)). (1.27)
Thus, the amplitude varies between 2 * A1 and 0, and the beating becomes
very pronounced.
The displacement waveform (the real part of ~x) is illustrated in
Fig. 1.5. This waveform resembles the waveform obtained by modulating
the amplitude of the vibration at a frequency (Delta omega)/(2 * pi),
but they are not the same. Amplitude modulation results from nonlinear
behavior in a system, which generates spectral components having
frequencies omega1 and omega1 +/- Delta omega. The spectrum of the
waveform in Fig. 1.5. has spectral components omega1 and omega1 + Delta
omega only.
Audible beats are heard whenever two sounds of nearly the same
frequency reach the ear. The perception of combination tones and beats
is discussed in Chapter 8 of Rossing(1982) and other introductory texts
on musical acoustics.
<Full-width insertion: image with caption beneath, the image showing
ten periods of a sine wave within a sine wave envelope. The first is
labeled with its wavelength, "(2 * pi) / omega1", and the second with
its wavelength, "(2 * pi) / Delta omega", about five times as long as
the first. The caption reads:
Fig. 1.5. Waveform resulting from linear superposition of simple
harmonic motions with angular frequencies omega1 and omega2.>
1.5. Energy
The potential energy E sub p of our mass-spring system is equal to the
work done in stretching or compressing the spring:
E sub p (1.28)
= -the definite integral with respect to x from 0 to x of F
= the definite integral with respect to x from 0 to x of (K * x)
= (1/2) * K * x^2.
Using the expression for x in Eq. (1.3) gives
E sub p (1.29)
= (1/2) * K * A^2 * cos^2(omega0 * t + phi).
The kinetic energy is E sub k = (1/2) * m * v^2, and using the
expression for v in Eq. (1.5) gives
E sub k (1.30)
= (1/2) * m * omega0^2 * A^2 * sin^2(omega0 * t + phi)
= (1/2) * K * A^2 * sin^2(omega0 * t + phi).
The total energy E is then
E = Ep + Ev = (1/2) * K * A^2 (1.31)
= (1/2) * m * omega0^2 * A^2
= (1/2) * m * U^2,
where U is the maximum velocity. The total energy in our loss-free
system is constant and is equal either to the maximum potential energy
(at maximum displacement) or the maximum kinetic energy (at the
midpoint).
1.5. Damped Oscillations
There are many different mechanisms that can contribute to the damping
of an oscillating system. Sliding friction is one example, and viscous
drag in a fluid is another. In the latter case, the drag force F sub r
is proportional to the velocity:
F sub r = -R * dotted x,
where R is the mechanical resistance. The drag force is added to the
equation of motion:
m * "x + R * dotted x + K * x = 0
or (1.32)
"x + 2 * alpha * dotted x + omega0^2 * x = 0,
where alpha = R / (2 * m) and omega0^2 = K / m.
We assume a complex solution ~x = ~A * e^(gamma * t) and substitute
into Eq. (1.32) to obtain
(1.33)
(gamma^2 + 2 * alpha * gamma + omega0^2) * ~A * e^(gamma * t) = 0.
This requires that gamma^2 + 2 * alpha * gamma + omega0^2 = 0 or that
gamma = -alpha +/- (alpha^2 - omega0^2)^(1/2) (1.34)
= -alpha +/- j * (omega0^2 - alpha^2) = alpha +/- j * omegad,
where omegad = (omega0^2 - alpha^2)^(1/2) is the natural frequency of
the damped oscillator (which is less than that of the same oscillator
without damping). The
<Full-width insertion: an illustration comprised of three parts arranged
vertically, the first part containing labels showing the signification
of each of the curves in the second part, a graph. The third part is a
caption. The labels indicate that the solid line represents alpha = 0,
the coarsely dashed line represents alpha/omega0 = 0.1, the finely
dashed line represents alpha/omega0 = 0.5, and the dotted line
represents alpha/omega0 = 1.0. The graph shows superimposed cosine
waves of similar period and phase. The solid one is of constant
amplitude = x sub 0. The dashed ones are damped, beginning at x sub 0,
but with the amplitude decreasing exponentially, the finely-dashed one
twice as fast as the other. The dotted line starts out like the others,
but its curvature decreases rapidly as it approaches one
half-wavelength, where it ends at x = 0. The caption is:
Fig. 1.6. Displacement of a harmonic oscillator with v sub 0 = 0 for
different values of damping. The relaxation time is given by 1/alpha.
Critical damping occurs when alpha = omega0.>
general solution is a sum of terms constructed by using each of the two
values of gamma:
~x = (1.35)
e^(-alpha * t) * (~A1 * e^(j * omegad * t) + ~A2 * e^(-j * omegad * t).
The real part of this solution, which gives the time history of the
displacement, can be written in several different ways as in the
loss-free case. The expressions that correspond to Eqs. (1.3) and (1.4)
are
x = A * e^(-alpha * t) * cos(omegad + phi), (1.36)
and
x = e^(-alpha * t) * (x0 * cos(omegad * t) + (1.37)
((v0 + alpha * x0) / omegad) * sin(omegad * t)).
Setting t = 0 in Eq. (1.37) and its derivatives gives the displacement
in terms of the initial displacement x0 and the initial velocity v0:
x = e^(-alpha * t) * (x0 * cos(omegad * t) + (1.38)
((v0 + alpha * x0) / omegad) * sin(omegad * t)).
Figure 1.6 shows a few cycles of the displacement for different values
of alpha when v0 = 0.
The amplitude of the damped oscillator is given by x0 * e^(-alpha *
t), and its motion is not strictly periodic. Nevertheless, the time
between zero crossings in the same direction remains constant and equal
to T sub d = 1/f sub d = 2 * pi / omegad, which is defined as the period
of the oscillation. The time interval between successive maxima is also
T sub d, but the maxima and minima are not exactly halfway between the
zeros.
One measure of the damping is the time required for the amplitude to
decrease to 1/e of its initial value x0. This time, tau, is called by
various names, such as decay time, lifetime, relaxation time, and
characteristic time; it is given by
tau = 1/alpha = 2 * m / R. (1.39)
When alpha >/= omega0, the system is no longer oscillatory. When the
mass is displaced, it returns asymptotically to its rest position. For
alpha = omega0, the system is critically damped, and the displacement is
x sub c = x0 * (1 + alpha * t) * e^(-alpha * t). (1.40)
For alpha > omega0, the system is overdamped and returns to its rest
position even more slowly.
It is quite obvious that the energy of a damped oscillator decreases
with time. The rate of energy loss can be found by taking the time
derivative of the total energy:
(d/dt)(Ep + EK) = (1.41)
(d/dt)((1/2) * K * x^2 + (1/2)(m * dotted x^2)) =
K * x * dotted x + m * dotted x * "x =
dotted x * (K * x + m * "x) =
dotted x * (-R * dotted x) =
-2 * alpha * m * dotted x^2,
where use has been made of Eq. (1.32). Equation (1.41) tells us that
the rate of energy loss is the friction force -R * dotted x times the
velocity dotted x.
Often a Q factor or quality factor is used to compare the spring force
to the damping force:
Q = (K * x0) / (R * omega0 * x0) = (1.42)
K / (R * omega0) = omega0 / (2 * alpha).
1.6. Other Simple Vibrating Systems
Besides the mass-spring system alread described, the following are
familiar examples of systems that vibrate in simple harmonic motion.
1.6.1. A Spring of Air
A piston of mass m, free to move in a cylinder of area-S and length-L
[see Fig. 1.7(a)], vibrates in much the same manner as a mass attached
to a spring. The spring constant of the confined air turns out to be K
= gamma * p sub a * S / L, so the
<Full-width insertion: four schematic images arranged horizontally with
a caption beneath. The images are titled beneath with "(a)" through
"(d)", and show respectively: (a) a horizontal cylinder labelled "p sub
a" with a tight-fitting piston protruding slightly from its right end,
the overall length of both labelled L, the side of the piston labelled
m, and the end of the piston labelled S; (b) a Helmholz resonator with
its volume indicated by V, the mass of air in its neck by m, the length
of its neck by L, and the area of its neck by S; (c) a right rectangular
prism of volume V, with a circle midway along its length shown by dashed
lines to be capable of moving inwards and outwards by an overall
distance of 2 * Delta L; and (d) a pendulum represented by a circle at
the end of a ray of length L displaced to the right by a distance x and
acted on by a downwards force of magnitude m * g. The caption reads:
Fig. 1.7. Simple vibrating systems: (a) piston in a cylinder; (b)
Helmholtz resonator with neck of length L; (c) Helmholtz resonator
without a neck; (d) simple pendulum.>
natural frequency is