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Behaviour of an oscillator with even non-linear damping
BEHAVIOUR
OF AN OSCILLATOR WITH EVEN NON-LINEAR DAMPING P. J. HOLMES
Institute
of Sound & Vibration
Research,
(Received 25 April
‘The University,
1976 tmd it1 revisedjitrm
Southampton
3 Mc~rc~h 1977)
Abstract---In this short note we prove two theorems on the behnviour oscillator with linear stitTness and even non-linear damping terms. EFFECTS
Consider
the ordinary
OF
differential
EVEN
NON-LINEAR
of :I single degree of freedom
DAMPING
equation
.U+sc.\:+f(i)+fix Without loss of generality on the phase-plane R2 :
SO9 SNt1, Engkmd
= 0;
p > 0.
(1)
and rewrite (1) as a pair of first order equations
we can normalise i, =x2
(2)
ix = -x1 -ctxz-f(x,). We study the case whereS(x,)
is a purely even non-linear
“ox,, =
function
of x2 given by
2 UZnX2Y II=1
In fact we could generalise f”‘(x,) = 0 only at isolated (. )’ E d*/dx, ; i.e.f(xz) must is one to one. Straightforward analysis = (01. The linearisation of
to consider a wider class of even functions provided that,f’(x,), points, and when,f’(x,) = O,f”(x,) # 0 except at x2 = 0, where be non-degenerllte everywhere except at x2 = 0. Thus x2 +f(.y2 1 reveals that (2) possesses a unique (2) near CO),
fixed point at the origin (0,O)
(4) possesses a matrix with eigenvalues >L,,2 = +(-cI+~=) thus when c( > 0; = 0; < 0; Re(i, ,*) < 0; = 0; > 0 respectively. Hence for u > 0; = 0; < 0 (0) is a stablefocus or sink; a non-hyperbolic centre; and an unstublefocus or source respectively (cf. [ 11, chaps. 5, 9 $1). After these preliminaries we can state the main theorems. Theorem
1
When M= 0 the uniqueJixed closed orbits. Theorem
point at {O} is a centre and the phase plane is denselyJilled
with
2
When c( > O(resp. < 0) the uniyuejxed point crt (0) is u sink (resp. source) Lmd ~111 orbits spiral around (0) und towards it us t + + co(resp. t + - co). ProoJs Theorem 1. We define the four curves y,, g2, h ,, h, and their complement R* (Fig. 1) 91 = 1x1 =
--f(x2)lx2
‘0)
92
-f(~z)lXz
<
=
ix,
=
0)
h, = {x2 = O/x, > 0} h, = {x2 = 0(x, < O}. 323
A + B + C + D in
324
P. .I. HOLMES
Fig. I
Note that on .(I, u gl, .+, = .Y?; 4, = Oand on h = h, u/r?, .t, = 0,1, = -.Y, ; furthermore the vectorfields in regions A, B, C, D, take the signs indicated. At [O}, of course, 1 r = i2 = 0. We prove the theorem by the following propositions. Proposition
I
All solution trujectories sturting on g, meet h I ; all trujectories starting on hi meet g, ; t/l/ trujectories sttrrtiny on g, meet h 2 ; all trujectories startiny on h, meet g,. Trujectories therefiwe move clockwise about (0) 11st increases. Proof: Consider the “quadrant” y, u A v 11,. On 61,. .<, = .Y~> 0 and .qr = 0, any trajectory starting on y, must therefore enter A. Once in A, 1 i > 0, .q2 < 0 and since .?,, .t2 do not change sign the trajectory is thus forced “downward” towards IT,. As x2 + O+, so ii + O+, 1, --f -x ,, and the trajectory must therefore meet and cross h, with vertical tangent at h,. Considerations of symmetry and allowing time to run backwards establish that trajectories starting on 11, meet gz and similar arguments suftice to prove the remainder of proposition 1 (cf. [l], pp. 219-220). Proposition
2. The phtrse plane is filled Mith periodic orbits
Proqf We define a “time 1” or Poincare map P:h, --, h, (Fig. 2) and prove that for all pEhL, P(p) = p. It then follows ([ 11, pp. 220-225 ; 28 l-285) that p lies on a closed orbit and that the plane is then filled with closed orbits. In addition detine P, : /I, + h,, P, : h, -+ h,, so that P = P, o P,, P, composed with P,. Let P, carry a point p E h i to y E IT,. Now consider time run backwards, so that a trajectory starting at p moves anticlockwise to meet h, at y’ where y’ = P, ’ (p). However, if ~1= P,(p), the retlectional symmetry about h, Uh, ensures that y’ = P, ‘(p) = cl also. Hence P,(p) = P; ‘(p) and thus P(p) = P, o P,(p) = P, oP; ‘(p) = p. Proposition 2 is thus established and the theorem proved.
Fig. 2.
Behaviour of an oscillator with even non-linear damping
325
Theorem 2 Consider the energy function E = 3(x: +xz) for the oscillator (2). The time derivative of E along trajectories of (2) is thus -CIX~ -xZf(.x2). Set LX= 0 in which case B takes the signs inc!icated in Fig. 3a, with E = 0 on x2 = 0, and x2 = + ci where +ci, - ci are the intersections of g,, y2 with x, = 0; i = 1,2 ,.... Consider a closed orbit y. We know that for (periodic) oscillations to be maintained i, I? = constant, where {, indicates integration once around y. Hence s; t? = 0 = s,. - x,f(x,).‘Th’ IS merely states the intuitively obvious fact that energy “supplied” to the system while p E y lies in regions for which & > 0 is exactly balanced by energy “subtracted” in regions where E < 0; consider the symmetry of y and equipartition of R” into regions of 6 > 0, E < 0. Now let CI> 0 and suppose that P again follows y (it is constrained to) (Fig. 3b). In this caseJ,, l? = J> (-LXX: - xJ’(xz )) = J; - axi = -as,, 5:. This integral is clearly < 0 for all x2 # 0 and hence energy decreases on y and, unless it is supplied by external forces, p must in fact follow an orbit of decreasing energy y’. Thus all solutions spiral towards the sink at f0). Physically, the regions of positive and negative energy rate B now no longer balance. The case a < 0 can be proved similarly, by letting time run backwards.
iL.0
iL0
i>O iC0
(0)
CL
(b) OCZO
0
Fig. 3.
Remurk (I ). In the above proof we are essentially using the fact that equation (l), with CI z 0, remains invariant under reversals of time (in which we replace .i by - 1 while x, .;i remain the same).
Remwks (2). A surprising corollary to Theorem 2 is that the addition of even non-linear damping terms to a linear damped oscillator does not adversely affect the stability characteristics in a qutrlitlrtiue sense; global stability is maintained regardless of the sign of the coefficients (I~, c/4,(I,, . . in equation (3). The particular case of an oscillator occurring in model\ of wind-induced oscillation of buildings: j;, +2[,o,j, -tr#t +o:!., = 0, then, is /lot destabilised by the negative -LI& term. Note, however, that this term does make the attraction towards [O} n,eoker than in the linear case for large values of 1.~1= IJml. As 1.~1+ -/v so the damping terms -ux2 -f(x*) +f(x,) and behaviour tends to that of case c1 = 0 (Theorem 1). (3) If’f(x,) is a non-even function of x2 the situation is far more complicated. In particular, if,f’(r’) = ~1~~x2(I ) becomes the van der Pal oscillator and it is well known that this can possess a unique closed orbit surrounding [O). If r < 0, ~1~> 0 this orbit is asymptotically stable, if c( > 0, ~1~< 0 it is unstable (cf. [l], chap. 10: [2]). It is not difficult to construct odd functions with alternating signs to the terms a3, a5, u7 which possess nests of alternately stable and unstable limit cycles. In [3] for example, Novak discusses a further situation ofinterest in
326
P. J. HOLMES
wind engineering form
in which the motions
of a bluff body are modelled
by an oscillator
of the
Here ~7, /i,, . , /I, are real positive constants and I/ 2 0 is the (non-dimensional) wind velocity. As Novak shows [3, Fig. 31, (5) can possess up to three limit cycles in addition to the fixed point (s,i) = iOi_. In particular, application of the Krylov-Bogoliubov averaging Theorem [3] shows that the even terms (P2,f14,flC,) d o not appear in the final amplitude expression and that odd terms alone are important. In classifying the generic bifurcations of systems possessing multiple limit cycles, Takens [4] has described the ways in which limit cycles can be created and annihilated in a more general class of planar systems for which the non-linear terms are odd (non-analytic but infinitely differentiable) functions. Takens thus shows how specific equations such as (5) fit into the general framework of bifurcation theory. (4) The validity of Theorems 1 and 2 can be checked in specific cases by use of Den Hartog’s concept of “equivalent linearised damping” ([S], pp. 60; 354; 359). In this method the oscillator is supposed to be sinusoidally forced and the response (for small u2i) is assumed to be almost sinusoidal. The energy dissipated over one cycle of motion is then integrated and equated to an equivalent quantity for a linear oscillator. In the case thatf(a) is even it can be checked that the integrals are zero, and that the non-linear terms consequently have no effect. Note, however, that the proof here is more general, since it admits my periodic response motion. Similarly, the use of the Krylov-Bogoliubov averaging Theorem [3] also predicts that even damping terms have no net effect but again the parameter range for which the Theorem is valid is limited. REFERENCES M. W. Hirsch and S. Smale, Di@rerztitrl Equutions, Dynumictrl S_vstems trnd Linear A!qehra. Academic Press, New York (1974). C. Hayashi. Vorl-lirxetrr Ov.i/hriorls irl Ph,nictr/ S~cwn. McGraw-Hill, New York (1964). M. Novak, Aeroelastic galloping of prismatic bodies. Proc. A.S.C.E. J. Eny. Mech., 115-142 (1969). F. Takens, Unfoldings of certain singularities of vectorfields: generalised Hopf bifurcations, J. II@ Eqns 14, 476-493 (1913).
J. P. Den HartofT 'ClrchurCctrl Vibrcltiorls. McGraw-Hill, New York (1934).
Dans cette courte note nous d6montrons deux thgor'emes sur le comportement d'un oscillateur 'a un degre' de libertg avec une rigidit lineaire et des termes d'amortissement non lingaires de degrC pair.
Zusammenfassunq: In dieser Kurzmitteilung beweisen wir zwei Theoreme ijber das Verhalten eines Schwingers mit einem Freiheitsgrad und mit linearen Steifigkeitsgliedern und geraden, nichtlinearen Dimpfungsgliedern.
OF AN OSCILLATOR WITH EVEN NON-LINEAR DAMPING P. J. HOLMES
Institute
of Sound & Vibration
Research,
(Received 25 April
‘The University,
1976 tmd it1 revisedjitrm
Southampton
3 Mc~rc~h 1977)
Abstract---In this short note we prove two theorems on the behnviour oscillator with linear stitTness and even non-linear damping terms. EFFECTS
Consider
the ordinary
OF
differential
EVEN
NON-LINEAR
of :I single degree of freedom
DAMPING
equation
.U+sc.\:+f(i)+fix Without loss of generality on the phase-plane R2 :
SO9 SNt1, Engkmd
= 0;
p > 0.
(1)
and rewrite (1) as a pair of first order equations
we can normalise i, =x2
(2)
ix = -x1 -ctxz-f(x,). We study the case whereS(x,)
is a purely even non-linear
“ox,, =
function
of x2 given by
2 UZnX2Y II=1
In fact we could generalise f”‘(x,) = 0 only at isolated (. )’ E d*/dx, ; i.e.f(xz) must is one to one. Straightforward analysis = (01. The linearisation of
to consider a wider class of even functions provided that,f’(x,), points, and when,f’(x,) = O,f”(x,) # 0 except at x2 = 0, where be non-degenerllte everywhere except at x2 = 0. Thus x2 +f(.y2 1 reveals that (2) possesses a unique (2) near CO),
fixed point at the origin (0,O)
(4) possesses a matrix with eigenvalues >L,,2 = +(-cI+~=) thus when c( > 0; = 0; < 0; Re(i, ,*) < 0; = 0; > 0 respectively. Hence for u > 0; = 0; < 0 (0) is a stablefocus or sink; a non-hyperbolic centre; and an unstublefocus or source respectively (cf. [ 11, chaps. 5, 9 $1). After these preliminaries we can state the main theorems. Theorem
1
When M= 0 the uniqueJixed closed orbits. Theorem
point at {O} is a centre and the phase plane is denselyJilled
with
2
When c( > O(resp. < 0) the uniyuejxed point crt (0) is u sink (resp. source) Lmd ~111 orbits spiral around (0) und towards it us t + + co(resp. t + - co). ProoJs Theorem 1. We define the four curves y,, g2, h ,, h, and their complement R* (Fig. 1) 91 = 1x1 =
--f(x2)lx2
‘0)
92
-f(~z)lXz
<
=
ix,
=
0)
h, = {x2 = O/x, > 0} h, = {x2 = 0(x, < O}. 323
A + B + C + D in
324
P. .I. HOLMES
Fig. I
Note that on .(I, u gl, .+, = .Y?; 4, = Oand on h = h, u/r?, .t, = 0,1, = -.Y, ; furthermore the vectorfields in regions A, B, C, D, take the signs indicated. At [O}, of course, 1 r = i2 = 0. We prove the theorem by the following propositions. Proposition
I
All solution trujectories sturting on g, meet h I ; all trujectories starting on hi meet g, ; t/l/ trujectories sttrrtiny on g, meet h 2 ; all trujectories startiny on h, meet g,. Trujectories therefiwe move clockwise about (0) 11st increases. Proof: Consider the “quadrant” y, u A v 11,. On 61,. .<, = .Y~> 0 and .qr = 0, any trajectory starting on y, must therefore enter A. Once in A, 1 i > 0, .q2 < 0 and since .?,, .t2 do not change sign the trajectory is thus forced “downward” towards IT,. As x2 + O+, so ii + O+, 1, --f -x ,, and the trajectory must therefore meet and cross h, with vertical tangent at h,. Considerations of symmetry and allowing time to run backwards establish that trajectories starting on 11, meet gz and similar arguments suftice to prove the remainder of proposition 1 (cf. [l], pp. 219-220). Proposition
2. The phtrse plane is filled Mith periodic orbits
Proqf We define a “time 1” or Poincare map P:h, --, h, (Fig. 2) and prove that for all pEhL, P(p) = p. It then follows ([ 11, pp. 220-225 ; 28 l-285) that p lies on a closed orbit and that the plane is then filled with closed orbits. In addition detine P, : /I, + h,, P, : h, -+ h,, so that P = P, o P,, P, composed with P,. Let P, carry a point p E h i to y E IT,. Now consider time run backwards, so that a trajectory starting at p moves anticlockwise to meet h, at y’ where y’ = P, ’ (p). However, if ~1= P,(p), the retlectional symmetry about h, Uh, ensures that y’ = P, ‘(p) = cl also. Hence P,(p) = P; ‘(p) and thus P(p) = P, o P,(p) = P, oP; ‘(p) = p. Proposition 2 is thus established and the theorem proved.
Fig. 2.
Behaviour of an oscillator with even non-linear damping
325
Theorem 2 Consider the energy function E = 3(x: +xz) for the oscillator (2). The time derivative of E along trajectories of (2) is thus -CIX~ -xZf(.x2). Set LX= 0 in which case B takes the signs inc!icated in Fig. 3a, with E = 0 on x2 = 0, and x2 = + ci where +ci, - ci are the intersections of g,, y2 with x, = 0; i = 1,2 ,.... Consider a closed orbit y. We know that for (periodic) oscillations to be maintained i, I? = constant, where {, indicates integration once around y. Hence s; t? = 0 = s,. - x,f(x,).‘Th’ IS merely states the intuitively obvious fact that energy “supplied” to the system while p E y lies in regions for which & > 0 is exactly balanced by energy “subtracted” in regions where E < 0; consider the symmetry of y and equipartition of R” into regions of 6 > 0, E < 0. Now let CI> 0 and suppose that P again follows y (it is constrained to) (Fig. 3b). In this caseJ,, l? = J> (-LXX: - xJ’(xz )) = J; - axi = -as,, 5:. This integral is clearly < 0 for all x2 # 0 and hence energy decreases on y and, unless it is supplied by external forces, p must in fact follow an orbit of decreasing energy y’. Thus all solutions spiral towards the sink at f0). Physically, the regions of positive and negative energy rate B now no longer balance. The case a < 0 can be proved similarly, by letting time run backwards.
iL.0
iL0
i>O iC0
(0)
CL
(b) OCZO
0
Fig. 3.
Remurk (I ). In the above proof we are essentially using the fact that equation (l), with CI z 0, remains invariant under reversals of time (in which we replace .i by - 1 while x, .;i remain the same).
Remwks (2). A surprising corollary to Theorem 2 is that the addition of even non-linear damping terms to a linear damped oscillator does not adversely affect the stability characteristics in a qutrlitlrtiue sense; global stability is maintained regardless of the sign of the coefficients (I~, c/4,(I,, . . in equation (3). The particular case of an oscillator occurring in model\ of wind-induced oscillation of buildings: j;, +2[,o,j, -tr#t +o:!., = 0, then, is /lot destabilised by the negative -LI& term. Note, however, that this term does make the attraction towards [O} n,eoker than in the linear case for large values of 1.~1= IJml. As 1.~1+ -/v so the damping terms -ux2 -f(x*) +f(x,) and behaviour tends to that of case c1 = 0 (Theorem 1). (3) If’f(x,) is a non-even function of x2 the situation is far more complicated. In particular, if,f’(r’) = ~1~~x2(I ) becomes the van der Pal oscillator and it is well known that this can possess a unique closed orbit surrounding [O). If r < 0, ~1~> 0 this orbit is asymptotically stable, if c( > 0, ~1~< 0 it is unstable (cf. [l], chap. 10: [2]). It is not difficult to construct odd functions with alternating signs to the terms a3, a5, u7 which possess nests of alternately stable and unstable limit cycles. In [3] for example, Novak discusses a further situation ofinterest in
326
P. J. HOLMES
wind engineering form
in which the motions
of a bluff body are modelled
by an oscillator
of the
Here ~7, /i,, . , /I, are real positive constants and I/ 2 0 is the (non-dimensional) wind velocity. As Novak shows [3, Fig. 31, (5) can possess up to three limit cycles in addition to the fixed point (s,i) = iOi_. In particular, application of the Krylov-Bogoliubov averaging Theorem [3] shows that the even terms (P2,f14,flC,) d o not appear in the final amplitude expression and that odd terms alone are important. In classifying the generic bifurcations of systems possessing multiple limit cycles, Takens [4] has described the ways in which limit cycles can be created and annihilated in a more general class of planar systems for which the non-linear terms are odd (non-analytic but infinitely differentiable) functions. Takens thus shows how specific equations such as (5) fit into the general framework of bifurcation theory. (4) The validity of Theorems 1 and 2 can be checked in specific cases by use of Den Hartog’s concept of “equivalent linearised damping” ([S], pp. 60; 354; 359). In this method the oscillator is supposed to be sinusoidally forced and the response (for small u2i) is assumed to be almost sinusoidal. The energy dissipated over one cycle of motion is then integrated and equated to an equivalent quantity for a linear oscillator. In the case thatf(a) is even it can be checked that the integrals are zero, and that the non-linear terms consequently have no effect. Note, however, that the proof here is more general, since it admits my periodic response motion. Similarly, the use of the Krylov-Bogoliubov averaging Theorem [3] also predicts that even damping terms have no net effect but again the parameter range for which the Theorem is valid is limited. REFERENCES M. W. Hirsch and S. Smale, Di@rerztitrl Equutions, Dynumictrl S_vstems trnd Linear A!qehra. Academic Press, New York (1974). C. Hayashi. Vorl-lirxetrr Ov.i/hriorls irl Ph,nictr/ S~cwn. McGraw-Hill, New York (1964). M. Novak, Aeroelastic galloping of prismatic bodies. Proc. A.S.C.E. J. Eny. Mech., 115-142 (1969). F. Takens, Unfoldings of certain singularities of vectorfields: generalised Hopf bifurcations, J. II@ Eqns 14, 476-493 (1913).
J. P. Den HartofT 'ClrchurCctrl Vibrcltiorls. McGraw-Hill, New York (1934).
Dans cette courte note nous d6montrons deux thgor'emes sur le comportement d'un oscillateur 'a un degre' de libertg avec une rigidit lineaire et des termes d'amortissement non lingaires de degrC pair.
Zusammenfassunq: In dieser Kurzmitteilung beweisen wir zwei Theoreme ijber das Verhalten eines Schwingers mit einem Freiheitsgrad und mit linearen Steifigkeitsgliedern und geraden, nichtlinearen Dimpfungsgliedern.