Journal o f Sound and Vibration (1974) 37(2), 273-279

LETTERS TO THE EDITOR

NOTES ON THE SOLUTION OF FORCED OSCILLATIONS OF A THIRD-ORDER NON-LINEAR SYSTEM

I. INTRODUCTION

In their paper [l] H. R. Srirangarajan and P. Srinivasan present a solution of steady-state forced oscillations of a non-linear oscillatory system described by a non-linear third-order differential equation. It is the aim of this contribution to point out the advantages which the Use of so-called skeleton (backbone) curves and limit envelopes has to offer by way of making of the analysis of the system and the evaluation of the effect of various parameters a straight- forward and rapid process.

2. ANALYTICAL SOLUTION

The system analyzed in reference [1] is described by the differential equation

+ a$ + bYc+ x + p f ( x ) = cos wT, (1)

where a, b and p are positive constants, andf(x) = x a. Let us approximate the solution by the form

x = A cos ( w T - q~), (2)

where A is the amplitude and tp is the angle of phase displacement. To simplify the solution let us introduce the displacement of the origin of time, To = tp/w (by writing T = t + tp/w). The right-hand side of equation (1) then takes on the form cos(wt + q~) and equation (2) the form

x = A cos wt. (2a)

On substituting expression (2a) into equation (I) (with the modified right-hand side) and comparing the coefficients of coswt and sinwt (harmonic balance method) we have the following equations for the determination of A and q~:

A(I + �88 2 - aw 2) = cos 9, (3)

Aw(b - w') = sin tp.

For the determination of A equations (3) and (4) give the equation

A2[(I + �88 2 - aw2) 2 + w~(b - w2) 2] = I,

and for tp, the equation

tan 9 = w(b - w2)/(l + :~pA" - aw2).

(4)

(5)

Equation (5) from which we determine the dependence A = A(w) is wholly identical with the equation derived in reference [I]. In the subsequent discussion we shall make use of a procedure analogous to that described in references [2] and [3]. For the case without excitation

273

(6)

274 LETTERS TO THE EDITOR

and simultaneously for costp = 0 equation (3) implies that

w, = [(1 + ]pA2,)/a] u2, (7a)

or

A , = 2[(aw~, - ] ) 13# ]u2 : (7b)

i.e., the equation of the so-called skeleton curve (to differentiate we have labelled the respective values of A and w with the subscript s).

As equation (4) implies the condition

Awlb - w2l .<< 1 (4a)

must hold. For the equality sign we get the equation of the so-called limit envelope,

AL = (wlb - w=l) - ' , (s)

with the letter L used for its symbol. For w -+ 0 and w -+ b u2 the value of AL grows beyond all bounds.

We can draw the following conclusion from the above exposition. All points of the curve A = A(w) must lie between the w-axis and the limit envelope which

the curve A(w) can only touch, and do so at the points where the skeleton curve intersects the limit envelope. At those points the tangent to the curve A = A(w) is identical with the respective tangent to the limit envelope AL = AL(w) (see reference [3] for the proofof th is statement.) For # > 0 and ab > 1 the skeleton curve will intersect the limit envelope either in three points or in one point. For the highest value of the co-ordinate of the point of intersection the condition that w > b tt2 holds. Provided that there exist three points of intersection, the co-ordinate w of the remaining two lies in the interval (0, bU2).

It further follows from equation (3) that for A = A, and for

and for

w < w, then cos tp > 0

w > w, then cos 9 < 0:

=0% 360 ~

270 ~ <~< ~

. ~ 9 0 ~

As/ ~=180 ~

/ !

W Figure 1. Division of the plane ot",4 (oscillation amplitude) and w (excitation frequency) into four regions

for various intervals of the angle of phase displacement .r

LETTERS TO THE EDITOR 275

.e., for the points of the curve A = A(w) which lie to the left of the skeleton curve cosq~ will ~e greater than zero, and for those to the right, it will be smaller than zero. For points lying )n the skeleton curve it naturally holds that cosq, = 0. What equation (4) implies is that for

and for

w < b '/2 then sin q, > 0

w > b uz then sinq, < 0.

The foregoing consideratioris make it easier to determine the course of the dependence q, = tp(w). The skeleton curve and the straight line w = b t/2 divide the plane A into four regions. As shown schematically in Figure 1 (which naturally applies also to q~ + 360 ~ there then belongs to the points of the amplitude-frequency curve A = A(w), lyingin a certain region, a definite interval of the angle of phase displacement q~.

3. EXAMPLES

We shall solve equation (1) using, similarly as in reference [1], a = 3, b = 1 and various values of p. The limit envelope curve AL(w) which is independent of p (solid line) and the skeleton curves (dot-and-dash.lihes) for # = 0.05, 0.07, 0.1, 0.5 and 1, pertaining to our case,

,> /,

/, /

/ ./A s (/.z -- 0.07)

I./ys I~=O.I)

~-A L

[/]/ ~ A (~=o.51 $ > . /

! / . / ~/ . / /

/ . /

!S y ,[ , J

0 05 I I'5

W

Figure 2. Diagram showing the dependence of the limit envelope on excitation frequency, AL = AL(w); diagrams of the skeleton (backbone) curves, A, = A=(w), for non-lincarity coefficient ~ = 0"05, 0.07, 0.1,0-5, 1.

are shown in Figure 2. For the first three values of / t we get three points of intersection while for the two highest values of it only one point of intersection of the skeleton curve and the limit envelope. The point of intersection of the skeleton curve and the branch of the limit envelope for w > b = I gives also, approximately, the peak of the resonance curve.

The case of one point of intersection of the skeleton curve and the limit envelope is not as

2 7 6 LETTERS TO THE EDITOR

o

i , i

/

0.5 1.5

W

Figure 3. Diagram showing the dependence of oscillation amplitude A on excitation frequency w (stable solution corresponds to the portion drawn in heavy solid line, unstable solution to that drawn in dash line); diagram of the limit envelope AL = At(w) and the skeleton curve A, = A,(w) for/z = 0"07.

interesting as that with three points of intersection. In the latter case we may expect that the resonance curve will become divided into two branches. This is evidenced by the diagram of the relationship A = A(w) for p = 0.07 (heavy solid line) in Figure 3 where we have also drawn the limit envelope AL = AL(w) (light solid line) and the skeleton curve (dot-and-dash line). Because of the limited scope of this contribution we have not examined the stability of the solution but have assumed the application of the rule of vertical tangents. The expected

360

270

90

0 15

! ! I I I

0 5 I

W

Figure 4. Diagram showing the dependence of the angle ofphase displacement ~ on excitation frequency w for/z = 0 " 0 7 .

L E T T E R S T O T H E E D I T O R 277

unstable solution is marked out by the dash-line portion of the curve A = A(w) in the diagram plotting the dependence of the amplitude A on the excitation frequency w. The relationship between the angle of phase displacement (p and the excitation frequency w is clear to see in Figure 4.

5

/

(a) ~. ' / I

3'61 I

t.--,~s ,i

0..5

.3.4

/ /

/

/

5.2

3.0 (b)

t t t 2 . 8 v - , I : , , ~ J - , ~ , ~ - 0 t 1.5 0-7 0 .8 0-85

/ 0-75

/ / / /

/ I +

/ /

I I F t

Figure 5. Diagram showing the dependence of oscillation amplitude A on excitation frequency w; diagram of the limit envelope AL = AL(w) and the skeleton curve A, = ,4,(w) for # = 0" 1. Figure 5(b) shows an enlarged section of Figure 5(a).

90

-90

180

-270 0 iS

I I I

I I 0 5 I

/,r

Figure 6. Diagram showing the dependence of the angle of phase displacement r on excitation frequenc'y w for/~ = 0"l.

278 LETTERS TO TIlE EDITOR

Figures 5 and 6 show analogous diagrams for the case of l t = 0.1, when the skeleton curve, though intersecting the limit envelope at three points, comes very close to it in the region between two points of intersection (for w < l); the result of this is a mere contraction of the resonance curve in that region. To make this point clear enough we show in Figure 5(b) an enlarged section of that region (crosses on the curve A = A(w) denote the points obtained by calculation from equation (5) while circles refer to the points of intersection of the skeleton curve and the limit envelope). In the interv.hl of the values of the amplitude A corresponding to the points of intersection of the skeleton curve and the limit envelope, the points of the curve A = A(w) must naturally lie to the right of the skeleton curve.

In the case where the amplitude dependence d(w) becomes divided into two branches (see Figure 3), it is not possible (e.g., by analogue solutions) to realize all stable solutions by merely raising or lowering the excitation frequency, as is usually done in most conventional non-linear systems.'[')

The stable solution corresponding to the separated portion of the curve A(w) can be effected by specifying certain initial conditions. In the interval of excitation frequency values where there exist two locally stable solutions (i.e., solutions which are stable for small disturbances from the equilibrium state), the conditions can be divided into two domains ofattraction with the separatrix forming the boundary between them. In view of the fact that the problem in question is specified by three initial conditions, the separatrix i s a surface in the three- dimensional space. This can be ascertained by the procedure described in reference [4] in which an analysis was made of a system defined by a set of differential equations, one of the first, the other of the second order.

4. CONCLUSION

The use of the skeleton curves and limit envelopes considerably facilitates analyses of steady-state forced oscillations of non-linear systems. It has been demonstrated in this note that these curves can be employed to advantage not only for systems described by a second- order differential equation but also for those described by a third-order differential equation.

ACKNOWLEDGMENT

The author gratefully acknowledges the help of his colleague Mr S. Mil~i~ek who carried out most of the numerical computations.

National Research Institute for Machflte Design, A. TONDL Bgchovice, Czechoslocakia (Receh;ed 19 ~Iarch 1974)

REFERENCES

1. H. R. SRIRANGARAJAN and P. SRINIVASAN 1974 Journal of Sound and Vibration 36, 513-519. Application of ultraspherical polynomials to forced oscillations of a third-order non-linear system.

2. A. TONDL 1971 Journal of Sound and Vibration 17, 429-434. Notes on the paper "Effects of non- linearity due to large deflections in the resonance testing structures".

"1 The authors could have carried out their analogue solution not only for a few points as they have done; by using track-store circuits they could have recorded the dependence of the extreme deflection x on excitation frequency by slowly varying the latter, with possible time delays in the transition from a resonant to a non- resonant solution and, vice versa, jump phenomena.

LETTERS TO TIlE EDITOR 279

3. A. TONDL 1973 Acta technica ~SA V 18, 166--179. Some properties of non-linear system character- istics and their application to damping identification.

4. A. TONDt. 1970 Monographs and AIemoranda, National Research Institute for Atachhte Design, B~chovice, No. 8. Domains of attraction for non-linear systems.

REPLY TO THE NOTES BY DR A. TONDL

We thank Dr A. Tondl for his interest in our paper and for his interesting note. As our intention was to show that the method of ultraspherical polynomial approximation is applicable to forced oscillations of a third-order non-linear system rather than a detailed study of the system, sample analog solutions were obtained for comparison. However, we agree that the analog solutions obtained are inadequate. The question of stability was not con- sidered. Regarding the solution in the neighbourhood of A = 3 in Figure 1 of reference [1] (~ = 0.1), we thank Dr Tondl for the information.

Department of Mechanical Engineering, H . R . SRIRANOARAJAN Indian Institute of Science, P. SRINIVASAN Bangalore 560012, lndia

(Received 15 May 1974)

REFERENCE

1. H. R. SRIRANGARAJAN and P. SRINIVASAN 1974 Journal of Sotmd and Vibration 36, 513-519. Application of uitraspherical polynomials to forced oscillations of a third-order non-linear system.