The Extinction of Predominantly Subharmonic Oscillations in Nonlinear SystemsCarl A. Ludeke and William Pong Citation: Journal of Applied Physics 24, 96 (1953); doi: 10.1063/1.1721142 View online: http://dx.doi.org/10.1063/1.1721142 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Subharmonics and ultraharmonics in the forced oscillations of weakly nonlinear systems Am. J. Phys. 44, 548 (1976); 10.1119/1.10394 Subharmonics, Resonant Systems, and Parametric Oscillations J. Acoust. Soc. Am. 46, 94 (1969); 10.1121/1.1973692 Subharmonic Oscillations in a Piecewise Linear System Am. J. Phys. 30, 500 (1962); 10.1119/1.1942083 Subharmonic Oscillations in Nonlinear Systems J. Appl. Phys. 24, 521 (1953); 10.1063/1.1721322 Predominantly Subharmonic Oscillations J. Appl. Phys. 22, 1321 (1951); 10.1063/1.1699858

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JOURNAL OF APPLIED PHYSICS VOLUME 24, NUMBER 1 JANUARY, 1953

The Extinction of Predominantly Subharmonic Oscillations in Nonlinear Systems

CARL A. LUDEKE AND WILLIAM PONG

Graduate School of Arts and Sciences, University of Cincinnati, Cincinnati, Ohio (Received September 24, 1952)

This paper describes the use of an electromechanical analog to examine the limiting values of various parameters needed to destroy a predominantly subharmonic oscillation of order one-third, existing in a nonlinear system. The results of the analog are then compared with the limiting values suggested by exist­ing theory.

INTRODUCTION

I N any oscillatory system having a nonlinear restoring force there is always the possibility of generating a

periodic oscillation whose frequency is a submultiple of the frequency of the forcing function. If the Fourier component having this subharmonic frequency has an amplitude larger than all other Fourier components we have the phenomena of predominantly subharmonic oscillations. Although some workl has been published on the conditions under which one can experimentally generate subharmonics little has been written on the experimental conditions by means of which these oscil­lations can be destroyed. Several theoretical conditions2.3 have been proposed and experimental investigation has been suggested. It is the purpose of this paper to pre­sent some experimental results on the extinction of pre­dominantly subharmonic oscillations in nonlinear sys­tems and then compare the experimental conditions

.9

.8

.7

~ .6 ii: Z C(

Q .5 II> ~ Q

~ ,4 :; IL Q .3 t-z ... (3 .2 if ... o " .1

O~--~5~O~--~IO~O--~15~O~--2~O~O~ CURRENT IN M.A.

FIG. 1. Calibration curve of damper.

1 C. A. Ludeke, J. App!. Phys. 22, 1321 (1951). 2]. J. Stoker, Nonlinear Vibrations (Interscience Publishers,

New York, 1950). 3 N. W. McLachlan, Ordinary Non-Linear Differential Equa­

tions (Oxford University Press, London, 1950).

96

under which extinction takes place with the conditions proposed by theory.

EXPERIMENTAL PROCEDURE

Because of the existing theoretical results it was de­cided to study a system represented by the equation

O+cO+aO+f3fJ3=H cO&.Jt-G sinwt, (1)

where 0= displacement (angular), c= coefficient of vis­cous damping, a= coefficient linear restoring force, f3 = coefficient cubic restoring force, w = angular frequency of forcing function, F=G2+.H2=magnitude of forcing function.

This equation was set up on an analog, previously described,4 to which had been added a calibrated elec­tromagnetic damper. The calibration curve of the damper is shown in Fig. 1. For no current in the damper the residual damping in the system equals only 0.03 of critical damping.

The first studies were made by generating the sub­harmonic of order I and then destroying it by in­creasing the value of c. Results were obtained for sys­tems having different values of a and for a whole set of values of F. The question then arose as to whether the point of extinction depended only upon the value of c or whether it was someway dependent upon the fact that the value of c was being changed during an oscillation rather than the value of F. To settle this question the experiments were repeated using a fixed value of c and lowering F until the subharmonic dis­appeared. The results showed that for a particular a and f3 there was a limiting relationship between c and F, and it mattered not whether c was increased to its upper limit or F was decreased to its lower limit. The data showed some differences of course, but itmust be kept in mind that the exact point of extinction was very difficult to locate because a short transition period occurred before the subharmonic ceased completely.

Finally, considerable data were taken by decreasing w until extinction occurred. The results shown in Table II were taken with no current through the electro­magnetic damper. Further experiments with current in the damper indicated that the lower limit of w was only slightly dependent on the value of c for the range of c being utilized.

4 C. A. Ludeke and C. L. Morrison, ]. App!. Phys. (to be published).

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PREDOMINANTLY SUBHARMONIC OSCILLATIONS 97

THEORY AND CONCLUSIONS

From the theory,S the interesting result with regard to the effect of viscous damping is the inequality

c<9{3AF/32wlX, (2)

in which A is the amplitude of the subharmonic, and c, lX, {3, w, and F are parameters of Eq. (1). The physical significance of this result is that it indicates an upper limit on the amount of viscous damping a system may have and still be able to have subharmonic solutions of order t. This inequality was derived, however, with the assumption that {3 was a small quantity. Most of the experimental results do not satisfy that assumption. It is unlikely therefore that the inequality should still be valid for large values of {3.

Before making such a comparison let us express inequality (2) completely in terms of the parameters of Eq. (1). In order to do this we make use of the ampli­tude-frequency relationship6

w2=9(a+0.92{3A2), (3)

which has been shown to be in good agreement with the experimental results on predominantly subharmonic os­cillations of order t. If the amplitude A is expressed in terms of lX, {3, and w, inequality (2) may be written as

c<_F [(1_9lX

)f3]i. 1O.22a w2

(4)

Thus for a particular value of a, {3, and w, we have a

TABLE I. Experimental results showing the limiting relationship between the extinction value of c and F.

Parameters Upper limit of c of Eq. 1 c from ineqnality ( ... fl.w) F (experimental) (4)

a=11.06 2.24 0.261 0.364 2.36 0.288 0.388

/1=843 3.02 0.493 0.497 3.23 0.545 0.531

w=12.57 3.96 0.635 0.651 (rad/sec)

a=5.84 0.87 0.261 0.284 /1=571 2.01 0.405 0.657 w=12.57 3.28 0.595 1.072

a=5.53 0.98 0.261 0.417 1.31 0.293 0.557

/1=843 1.38 0.332 0.587 1.56 0.412 0.664

w=12.57 2.11 0.493 0.887 2.24 0.545 0.953 2.99 0.635 1.271

a=5.53 0.97 0.261 0.561 /1= 1515 1.34 0.450 0.774 w=13.01 1.92 0.570 1.109

2.76 0.635 1.595

a=O 0.99 0.408 1.19 0.450

/1=843 1.27 0.460 1.43 0.490

w=12.57 1.71 0.595 2.21 0.635

• Reference 2, page 109. 6 Reference 1, Eqs. (18) and (19).

TABLE II. Experimental results showing the limiting rela-tionship between the angular frequency wand F for various values of a and /1.

w (experimental) 3[ .. +21/

... fl F rad/sec IOO4(BF'/"')]+

a=I1.06 2.37 9.73 10.32 /1=843 2.95 10.15 10.50

4.14 10.38 11.01

a=5.53 2.35 8.70 8.82 /1=843 2.79 8.85 9.46

3.39 9.72 10.44

a=5.53 1.23 8.87 7.97 /1= 1515 2.08 9.46 9.47

3.39 10.94 12.45

a=5.53 1.51 8.81 8.85 /1=2070 1.97 9.25 9.93

2.71 10.20 11.91

a=5.53 1.23 8.79 8.62 /1=2675 1.91 9.22 10.45

2.71 10.51 13.00

a=O 1.76 7.46 /1=843 2.36 8.69

2.77 9.26

limiting relationship between c and F. This does not however specify the minimum extinction value of c.

We note that the experimental results shown in Table I are in good agreement with inequality (4). For a= 11.06, {3= 846, and w= 12.57 rad/sec, the ex­perimental extinction value of c for a given value of F is in the neighborhood of the limited value specified by the inequality. However, the larger nonlinearity the value of c for which extinction takes place is approxi­mately one-half of the value given by relationship (4). In the case of a=O, the experimental results cannot be checked against inequality (4).

Turning now to the results shown in Table II, we find that the lower limit of w is a function of a, f3, and F. According to theory7 the subharmonic vibration of order t exists only for

w>3(a+21/16f3A2)i, (5)

in which A is the amplitude of the harmonic component and may be replaced by - F /8lX. In terms of the pa­rameters of Eq. (1) inequality (5) becomes

w>3[a+21/1004(f3F2/a2)Ji. (6)

Thus we have a theoretical lower limit of w expressed in terms of a, f3, and F, with the assumption that f3 is small.

A comparison of the results of experiment with in­equality (6) indicates a very good agreement in spite of the fact that {3 is not a small quantity. We note again that the theory does not account for the lower limit of w when a= O.

ACKNOWLEDGMENTS

It is a pleasure to thank the Research Corporation for the Frederick Gardner Cottrell Grant-in-Aid which a!'sisted this research.

7 Reference 2, page 106.

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