FIGURE 1.3. Phasor representation of the complex displacement, velocity, and acceleration of a linear oscillator.
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 wo.
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
Xl = A1ej(wt+c!>I)
and
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8 1. Free and Forced Vibrations of Simple Systems their linear superposition results in a motion given by
X1 + X2 = (A1d'Pl + A2ej¢2)ejwt = Aei(wt+¢)_ (1.13) The phasor representation of this motion is shown in Fig. 1.4.
Expressions for A and ¢ can easily be obtained by adding the phasors A1ejwc1>1 and A2eiwcl>2 to obtain
(1.14) and
(1.15) What we have really done, of course, is to add the real and imaginary parts of x1 and x2 to obtain the resulting complex displacement x. The real displacement is
x = Re(x) = Acos(wt + ¢). (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 w. This phasor has an amplitude given by
~ c
ã;;; ~ ..
+
~ c
ã;;;
(1.17)
FIGURE 1.4. Phasor representation of two simple harmonic motions hav.ing the same frequency.
1.3. Superposition of Two Harmonic Motions in One Dimension 9 and a phase angle ¢ obtained from
.-~, I: An sin¢n
tan'+'= I: An cos¢n . (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(wt + Â) =I: An cos(wt + Ân)ã
1.3.3 Two Harmonic Motions with Different Frequencies: Beats
(1.19)
If two simple harmonic motions with frequencies !1 and h are combined, the resultant expression is
(1.20) where A, w, and ¢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 w2 to WI (or WI to w2) is a rational number, the motion is periodic with an angular frequency given by the largest common divisor of w2 and WIã 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 vibrations or beats. If the angular frequency w2 is written as
W2 =WI + Llw, the resulting displacement becomes
x = Aiej(w1t+¢1) + A2 eJ(wlt+tJ.wt+¢2)
= [Aiej¢1 + A2ej(¢2+t.wt)]ejwlt.
(1.21)
(1.22) We can express this in terms of a time-dependent amplitude A(t) and a time-dependent phase ¢(t):
where
and
AI sin ¢I + A2 sin(¢2 + .6.wt) tan¢(t) =
AI cos ÂI+ A2 cos(Â2 + .6.wt) ã
(1.23)
(1.24)
(1.25)
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10 1. Free and Forced Vibrations of Simple Systems
FIGURE 1.5. Waveform resulting from linear superposition of simple harmonic motions with angular frequencies w1 and w2.
The resulting vibration could be regarded as approximately simple har- monic motion with angular frequency w1 and with both amplitude and phase varying slowly at frequency t1w j21f. The amplitude varies between the limits A1 + A2 and IA1 - A2lã
In the special case where the amplitudes A1 and A2 are equal and ¢1
and ¢2 = 0, the amplitude equation [Eq. (1.24)] becomes
A(t) = A1 y'2 + 2 cos t1w1t (1.26) and the phase equation [Eq. (1.25)] becomes
"'-( ) sin t1w1 t tan'+' t = ___ ,.::..-_
1 + cos t1w1 t (1.27)
Thus, the amplitude varies between 2A1 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 t1wj21r, but they are not the same. Amplitude modulation results from nonlinear behavior in a system, which generates spectral components having frequencies w1 and w1 ± t1w.
The spectrum of the waveform in Fig. 1.5. has spectral components w1 and w1 + t1w only.
Audible beats are heard whenever two sounds of nearly the same fre- quency 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.