the growth of subharmonic oscillations

418 W. J. CUNNINGHAM

can occur with small changes in frequency in the vicinity of certain of the normal modes of free vibration of the

solid scatterer. These changes include the appearance of deep minima in the scattering pattern at certain angles and may include, for cylinders, a near-null in the sound scattered back toward the source.

VI. ACKNOWLEDGMENTS

The author is indebted to Professor F. V. Hunt for his

guidance and encouragement throughout this investiga- tion. The assistance of Dorothea Greene, who per- formed the laborious computations for Figs. 16 and 17, is gratefully acknowledged.

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 23, NUMBER 4 JULY, 1951

The Growth of Subharmonic Oscillations

W. J. C•N•neoaAa Yale University, New Haven, Connecticut

(Received January 14, 1951)

Subharmonic oscillations at one-half the frequency of excitation may appear in certain types of oscillating systems, among which is the direct-radiator loudspeaker. These oscillations occur at very nearly the resonant frequency of the system when the parameters of the system are made to vary at twice this frequency. The rate of growth of the subharmonic depends upon the amount of variation of the parameters relative to the dissipation in the system. If the dissipation is small, the rate of growth may be large. In the loudspeaker, conditions are such that the rate of growth is usually small for typical conditions of operation.

HE generation Of subharmonic oscillations by a direct-radiafor loudspeaker has often been ob-

served. •-4 Such oscillations usually occur at one-half the frequency of the current supplied to the' loudspeaker, and appear for only certain discrete frequencies near the center of the audio spectrum. In most cases, the subharmonic is not present unless the loudspeaker is being operated near its maximum power. When present, the subharmonic is easily audible, even though sound pressure measurements indicate the amplitude of the subharmonic is only a few percent relative to the funda- mental. The statement has been made that this subharmonic distortion is usually of little practical im- portance in the operation of the loudspeaker. 5 The reasoning is based on the observed fact that an ap- preciable length of time is required for the amplitude of the subharmonic to grow to its ultimate value. Since typical program material is of constantly changing nature, there is little opportunity for the subharmonic to build up. In the following rather simple discussion, the growth of the subharmonic oscillation is considered with the intent of determining what factors influence the rate of growth and why this rate is low for the loudspeaker.

Subharmonic oscillation at one-half the frequency of an exciting force may occur in oscillating systems having

• H. F. Olson, Acoustical Engineering (D. Van Nostrand Com- pany, Inc., New York, 1947), p. 167.

•' P.O. Pederson, J. Acoust. Soc. Am. 6, 227-238 (1935), and 7, 64-70 (1935).

a F. yon Schmoller, Telefunken Zeitung 67, 47-54 (June, 1934). 4 G. Schaffstein, Hochfrequenztechn. Elektroakust. 45, 204-213

(1935). 5 See reference 2. Also, H. S. Knowles, "Loudspeakers and room

acoustics," Sec. 22, Henney's Radio Engineering Handbook (McGraw-Hill Book Company, Inc., New York, 1941), p. 902.

a single degree of freedom. a. 7 For the subharmonic to appear, the quiescent resonant frequency of the system must be very nearly one-half the exciting frequency. Further, operation must be such that under excitation the resonant frequency of the system is caused to vary at the exciting frequency. This variation must take place in such a way that sufficient energy is being supplied to the system to replace that lost by dissipation. If more than this amount of energy is supplied, the amplitude of the subharmonic grows, in theory, without limit. Ultimately, in practical systems, some additional effect takes over and the amplitude achieves a steady value.

In order to give a simple example of this type of operation, an electric circuit will be considered in some detail. This circuit contains in series combination an

inductance L, a resistance R, and a capacitance C. If q is the instantaneous charge on the capacitance, the sum of voltages around the circuit is

z+tO+q/C=O,

where dots indicate time derivatives. In some way the capacitance is made to vary sinusoidally in time by an amount AC about the mean value Co. The instantaneous capacitance is

C= C0(1 q- a sin2to•t), (2)

where the angular frequency of the variation is taken as 2to•, and am AC/Co. Evidently a can never exceed unity. It is possible to show that such a variation in capacitance can add energy to the oscillating circuit. The resonant angular frequency of the circuit in its quiescent

0 N. Minorsky, Nonlinear Mechanics (Edwards Brothers, Inc., Ann Arbor, 1947), Chap. XIX.

7 N. W. McLachlan, Ordinary Nonlinear Differential Equations (Oxford Univerdty Press, London, 1950), Chap. VII.

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15:36:37

GROWTH OF SUBHARMONIC OSCILI_ATIONS 419

state is o•0=(LC0) -•, but the value changes at the angular frequency 2o•1 as the capacitance varies. If Eq. (2) is substituted into Eq. (1) and the symbol b= RCo is introduced, the result may be written as

(1-3- a sin2o•lt)•-+- bo•0•'(1-+- a sin2o•lt)•-+- o•0•'q = 0. (3)

Equation (3) has coefficients varying with time and therefore cannot be solved exactly as a simple linear equation with constant coefficients. If minor changes are made in the form of the equation, it may be converted into the Mathieu equation which has been studied ex- tensively and for which exact solutions are known. However, an approximate solution for the equation is obtained easily without using the more elaborate theory. Such an approximate solution is

q=ql sino•lt, (4)

where the amplitude ql is assumed to be a function of time. This approximate solution has the angular frequency o•l, one-half the frequency of variation of the capacitance, and represents the subharmonic oscillation. If Eq. (4) is substituted into Eq. (3), use made of the identities

sin2o•lt sino•lt- «(coso•lt- cos3o•lt), sin2o•lt coscolt= «(sino•lt+ sin3o•lt),

and terms collected, the result is

•«a01+ (2col+«abco0•')Ol+ (bCOo%1--«acolV')q1 -] COSt-0II + ["•1+ (aco1-]- b(.o02)•l q- (coo •'-- tol•'+ «abtoo%1)q1 -] sin•lt -- [-«aO1+ 2X--abtoo•'•1- «ao•1•'q1-[ cos3o•1t

+ [aco1•l+ «a&oo%1q1-] sin3o•1t = 0. (s)

If this equation is to hold at all times, the coefficients of the four sine and cosine terms must be identically zero. Since the solution in Eq. (4) is not correct, the coefficients of Eq. (5) will not vanish identically. As a first approximation, the conditions may be found for which the coefficients of the two sine and cosine terms of

angular frequency o•l will vanish. These terms represent the subharmonic oscillation.

The coefficient of the term in coso•lt in Eq. (5), set equal to zero, may be written as

q21- kq1--O, (6)

where, approximately,

k= (a- 2/Qo)co1/4 (7)

and Qo-= Ltoo/R= 1/RCotoo. A solution for Eq. (6) is

ql = q0 exp(kt) (8)

if q0 is the initial amplitude. It is evident that if the dissipation is small, k may be large and ql will increase rapidly with time. Several assumptions have been made in obtaining the above equations. First, the growth of ql, given by Eq. (8), is assumed slow enough that the change in amplitude of q in Eq. (4), measured per cycle of oscillation, is relatively small. A reasonable value

might be such that k_•to1/50, in which case the amplitude increases less than about thirteen percent each cycle. The growing oscillation at the left of Fig. 1 is drawn for this value of k. Under the assumption of slow growth, the term in •1 appearing in the coefficient of coso•lt of Eq. (5) is small compared with the terms in •1 and ql. The term in •1 is neglected in writing Eq. (6). Finally, in order to obtain the simple form for k given in Eq. (7), it is assumed that there is small dissipation so that 4>>a/Qo, and that the angular frequencies of oscillation o•1 and of resonance o•0 are nearly the same. This last approximation will be justified below. It is evident from Eq. (7) that the requirement for growth of the subharmonic is that a> 2/Qo. A subharmonic, once started, will decay if a<2/Qo. If a=0, Eq. (7) reduces to the correct form for a passive circuit. Since a can never exceed unity, there can be no subharmonic unless Q0 is at least two.

If the solution, Eq. (8), is substituted into the coefficient of the term in sincolt in Eq. (5), set equal to

! !

!

i I

Fro. 1. Experimentally observed growth of subharmonic oscillation. Only the positive half of the wave exp(kt)sino•t is shown, where k has been given the value o•/50. Limiting occurs at the right.

zero, the result is

mY'-] - (aco1-]- to1/Q0) k-I- (to0 v'- tolv'-] - «atoo•'/Qo) = 0. (9)

In the terms with Q0, the assumption of approximate equality of o•0 and o•x has been made. The resulting equation in o•1 may be solved and small terms neglected to give

•l=•o[-1-+-(a-2/po)(a-•a-+ - 1/16Q0)•. (10)

Since both a and 1/Qo are small quantities, o•l and o•0 are nearly the same. If a> 2/Qo, so that growth of the subharmonic occurs, then o•1> o•0. If a= 2/Qo, giving constant amplitude of the subharmonic, then o•x=o•0. If a=0, Eq. (10) reduces to the correct form for a passive circuit.

Equations (8) and (10) may be combined to give as the first approximation for the charge in the series circuit

q-qo exp(kt) sino•0t, (11)

where k is given by Eq. (7) and the capacitance varies at nearly the angular frequency 2o•0. Experimentally the


15:36:37


Cf

L

R

amplifier

FIO. 2. Experimental circuit giving the equivalent of a series LRC circuit with C variable in time, controlled by E,•.

voltage e, across the capacitance is observed more easily than the charge. This voltage is

e c= q/C= (q• sinco•t)/Co(1 q- a sin2co•t) .

or, if a<<l,

ec=(q•/Co)(sinco•t-«acosco•tq-«a cos3co•t). (12)

There is a component with frequency three times that of the subharmonic and having an amplitude «a times that of the subharmonic. The frequency of this component is ] times the frequency of variation of the capacitance. There is no component at the frequency of variation of the capacitance.

In order to check the theory just developed, an electric circuit operating in the manner assumed was set up in the laboratory. a The inductance and resistance were an air-core coil having a self-inductance of about 128 mh. The varying capacitance was obtained by using a single-stage amplifier with a capacitance C/connected between input and output terminals as shown in Fig. 2. The capacitance effective between input terminals may

,øø 1 $oo.

K

ZOO

IOO

0 I

Cf-'.001 Ff 5

.01

.z .$ .4 .5 0 • '6C/Co

Fro. 3. Experimental data for k vs a obtained with the circuit of Fig. 2 using indicated values of C/. End points are for k = co/50.

8 A somewhat similar experiment is described by W. L. Barrow, Proc. Inst. Radio Engrs. 22, 201-212 (1934).

be shown to be 9

Co=(l+A)C/, (13)

where A is the voltage amplification of the amplifier. The experimental amplifier •ø was arranged so that the amplification could be controlled by a voltage E•, but a balancing circuit prevented a signal due to E• from appearing at the output terminals. In other words, the amplifier acts much like a balanced modulator. If m is the degree of amplitude modulation produced by voltage E•, the factor a used in the preceding theory is given by

a= AC/Co= mA/(1 q- A). (14) ß

Thus, factor a and the frequency of variation of the capacitance are determined by the amplitude and frequency of the voltage E,•. The quiescent amplification A of the amplifier is about twenty.

If the frequency of voltage E,• is adjusted to twice the resonant frequency of the series circuit and the amplitude of Em is increased sufficiently, the voltage across the input terminals of the amplifier will vary as shown in Fig. 1. This voltage is easily observed with an oscilloscope. Only the positive half of the actual oscillation is drawn in the figure. The initial growth at the left of the figure is exponential as predicted by the theory. After the amplitude has increased sufficiently, the amplifier tends to overload and its amplification becomes smaller. This results in a reduction in effective capacitance at the input terminals and the addition of extra dissipation. Both effects limit further growth of the oscillation, giving the constant amplitude at the right of the figure. An exponentially shaped mask was used on the screen of the oscilloscope, and horizontal deflection was pro- vided with a linear sweep having known operating speeds. By initiating the sweep and the oscillation at the same time, the relation between constants k and a could be determined experimentally. Results of such measurements are shown in Fig. 3, where data for the various curves were obtained using indicated values of capacitance Ci. For clarity experimental points are not shown in Fig. 3, but such points fell near the curves with differences of only ten or fifteen percent, reasonable errors considering the relatively crude method of meas- urement. Most consistent data were obtained for small

values of capacitance C•. The curves are arbitrarily stopped at values of k-co•/50; beyond these points the curves tend to bend to the right somewhat, since the experimental situation no longer fits the simple theory. Intersections of the curves with the horizontal axis

occur at a--2/Qo, where Q0 is measured at the angular frequency of oscillation. These values of amin are plotted in Fig. 4 .as a function of the frequency of oscillation, i.e., the subharmonic frequency. Values of Q0 can be measured by conventional means and are found to check

9H. J. Reich, Theory and Applications of Electron Tubes (McGraw-Hill Book Company, Inc., New York, 1944), p. 214.

10 W. J. Cunningham. Am. J. Phys. 18, 314-318 (1950).


15:36:37

GROWTH OF SUBHARMONIC OSCILLATIONS 421

well with these data. Slopes of the curves of Fig. 3 are dk/da= c0•/4, as found from Eq. (7).

Once limiting has taken place and a steady amplitude is obtained, the presence of a third harmonic in the oscillation is easily observed. Because limiting is not considered in the simple theory, the relative amplitude of this third harmonic is predicted only approximately by Eq. (12).

The preceding analysis and experiment has considered only the simplest possible case of subharmonic generation in a system with lumped parameters. Sub- harmonics in a direct-radiator loudspeaker are usually generated as a result of vibration of the cone. Since the cone must be described in terms of a relatively complicated geometrical shape with distributed parameters, its analysis is much more difficult. Certain general ob- servations may be made, however. It is found experimentally that when the loudspeaker is generating a subharmonic at half the driving frequency, the cone itself is vibrating at the subharmonic frequency. There are concentric circular nodes near the voice coil and the

perimeter of the cone, and a concentric circular loop in the cone between these extremes. In accordance with the

simple theory, this vibration must represent a resonant frequency of the cone and this resonant frequency must be made to vary by the driving forces of the loudspeaker.

The cone is usually made of paper which is under some tension and has some stiffness. A generating element of the cone might be considered as intermediate between an ideal string having mass and tension but no stiffness, and an ideal bar having mass and stiffness but no tension. The oscillation of the string and bar are well known. The vibration of a dissipationless string is governed by the differential equation n

( 02y/ Ox •) - MO•y/ TOt •= O, (15)

where y is the deflection normal to the string, x is the distance along the string, t is time, T is tension in the string, and M is its mass per unit length. The lowest natural frequency of vibration of a string fixed at both ends may be shown to be dependent upon the factor (T/M)L If the tension T is made to vary at twice this natural frequency, it is evidently possible to obtain subharmonic oscillation at the natural frequency. The tension might be varied by a force applied at one of the end supports of the string, collinear with the string.

The analogous situation for a bar is somewhat more complicated. Corresponding to the force varying the tension in the string might be a force applied collinear with the bar. The vibration of a dissipationless bar, having such a force, is governed by the differential equation •

O4y ½F O:yq_M O2•Y=0 ' (16) Ox 4 EL Ox • EI Ot •

np. M. Morse, Vibration and Sound (McGraw-Hill Book Com- pany, Inc., New York, 1948), p. 83.

x•' J. Prescott, Applied Elasticity (Longmans Green, London, !924)•'p. 240.

where F is the collinear compressive force, E is Young's modulus for the material of the bar, and I is the moment of inertia of a cross section of the bar. The lowest natural

frequency of vibration of a bar of length l pivoted at both ends may be shown to be dependent upon the factor [(;r•EI/MI4)--F/MI•L If the force F is made to vary at twice this natural frequency, it is evidently possible to obtain subharmonic oscillation at the natural frequency.

In the loudspeaker cone, a varying force applied by the voice coil at the apex of the cone has a component collinear with a generating element of the cone. If one of the natural frequencies of oscillation of the cone is made to vary as a result of this force, as is the case with the string or bar, subharmonic oscillation should be possible. In accordance with the theory of the electric circuit, the subharmonic should not appear until the driving force is large enough to produce changes (i.e., factor a) in the parameters of the cone which govern its

.4 .04

f = .001 •f .2 .005 .0oz

0 , I i I I

0 I000 ZOO0 ;5000 4000

Fro. 4. Experimental data obtained with the circuit of Fig. 2. The minimum value of a which will just produce the subharmonic oscillation is plotted against the frequency of the subharmonic. The parameter is C•,.

resonant frequency. The amount of dissipation in the system determines how large the changes must be before the subharmonic is possible. The rate of growth of the subharmonic is greater for larger changes in the parameters. The nature of the cone is such that the stresses in it

must be fairly large before significant changes in the parameters occur. Large stresses will exist only when the driving force is large, the loudspeaker is operating near its maximum power rating, and a large amplitude of oscillation exists at the driving frequency. This situation is more complicated than that of the electric circuit, since in that case no oscillation existed at the driving frequency.

Data from experiments with a twelve-inch direct- radiator loudspeaker of rather poor quality are shown in Fig. 5. Strong subharmonics appeared for driving frequencies, f, of 1140 and 1370 cps; the subharmonic frequency, f/2, is half the driving frequency. Measure- ments were made of the time tx required for the sub-


15:36:37


,3 f• I$70 c

'4, I

o

0 .?- .4 Ic - amp 0 .0• .16 .$6 .&4 I.oo

Fro. 5. Experimental data obtained with a direct-radiator loudspeaker. The reciprocal of the time t• needed for the sub- harroonic to reach a steady state is plotted against the voice coil current. The driving frequency is indicated.

harmonic to reach its steady amplitude after a sudden application of current in the voice coil. The reciprocal 1/t• of this time is plotted in the figure as a function of the current Ic in the voice coil. The curves are similar to those of Fig. 3, obtained with the electric circuit. It is difficult to measure the actual growth of the subharmonic in the loudspeaker because of the presence of the large component at the driving frequency. A reasonable assumption might be that the time t• involves a total change of 1000 to 1 in amplitude of the subharmonic. On the basis of this assumption rough values may be calculated for some of the quantities. From such calcu- lations, approximately, k=7/t•, a=0.1Ic, and at f- 1370 cps, Q 0 = 50.

An experimental analysis of the relative amplitudes of the sound pressure produced by the loudspeaker in steady-state operation at f= 1370 cps with Ic- 0.4 amp is shown in Fig. 6. The subharmonic is only slightly over one percent of the driving frequency in amplitude. How- ever, the presence of additional components spaced f/2 along the frequency scale leads to the pitch sensation of the subharmonic frequency. The subharmonic is easily audible. The sound pressures change but slightly as the current in the voice coil is increased further, indicating the loudspeaker is operating near its maximum power.

I

> .01

o• .OOl

0 f/• • 3•/Z i•q: 5•/2 3t:

Fro. 6. Relative sound pressures observed experimentally for the loudspeaker of Fig. 5. The driving frequency f was 1370 cps with a voice-coil current of 0.4 amp.

The results of the preceding analysis might be sum- marized as follows. A subharmonic at one-half the exciting frequency may be produced when excitation is such as to vary the resonant frequency of the system at twice this frequency. The amplitude of this subharmonic grows as an exponential function of time until some limiting action takes place. The rate of growth is dependent upon the constant k=(a-2/Qo)o•/4. Since a_< 1, it is necessary that Q0_> 2 for a subharmonic to exist. If dissipation is small, so that Q0 is large, it is possible for k to become large. The limiting value is k-o•/4, too large for the simple theory to apply. How- ever, with this value of k, the amplitude of the oscillation would increase about five times for each cycle, corresponding to extremely rapid growth. In a direct- radiator loudspeaker, constant a representing the variation of parameters of the cone is small until stresses in the cone become large. This does not occur until the driving force approaches the maximum for which the loudspeaker is designed. Therefore, in normal operation of the loudspeaker, constant a never exceeds by much the minimum value allowing subharmonics, and the rate of growth is never large. There is no reason that the rate of growth should not become large if sufficient force can be developed by the voice coil and applied to the cone.


15:36:37

the growth of subharmonic oscillations

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