Stability and Limit Cycles of Parametrically Excited

E. M. Mockensturm Graduate Student Research Assistant. Department of Mectianical Engineering, University of California, Berl
where the eigenvalue, X = ito, and the eigenfunctions, (^ (or /3), are complex quantities which obey the following orthogonality properties

with boundary conjugates [pAMU,r

flf(x) = I3(x){k

- l)v"

(4)

Jo

where fj)n(x) = f,(x) + i^'„(x). The resulting traveling string eigenfunctions are Pn(x) = a„ sin n7rA:[cos (gymvx) + i sin

(gymrx)],

g = [\ + (I - K)y'r"\

(10)

with natural frequencies u}„ = ng-K(\ ~ Ky^) (defining a critical speed, Tc = l/v/c) and normalization factor a„ = l/wWl — Ky^. The complex eigenfunctions above have been shown to be a superior basis for the solution of linear response problems under free and externally excited conditions (Wickert and Mote, 1990). Presently, they will be used in a Galerkin discretization of both linear and nonlinear, parametrically excited response problems.

where primes denote partial differentiation with respect to x and dots denote partially differentiation with respect to t and X = X/L,

u = U/L,

P(t) = M ( 1 , t) - u(0, t), t = T(n/pA,Ly\

3

V = V/L, K = 1 - 77

y = c(A„p/7•*)"^ i; = EAJT*.

(5)

A discretization basis is found by evaluating the associated Unear, free-response problem for the axially translating (or traveling) string: V + 2yv' -h (/cy' - \)v" = 0,u(0) = D ( 1 ) = 0.

(6)

Linear Response and Stability Boundaries

A perturbation analysis of the linear, response problem is pursued and leads to solutions for the stability boundaries of the trivial (straight) belt state. Linearizing the equation of motion (4) about the trivial equilibrium, produces i; + 2yi)' + [Ky^ - 1 - C,P(t)\v" = 0

(11)

where a harmonic, time-dependent relative end displacement, P(t) = a cos Qt, is assumed. In canonical form this equation becomes

Following Wickert and Mote (1990), this problem can be cast in the canonical form Aw + Bw - e(cos D.t)Cw = 0,

Aw(.x:, t) + Bw(jc, r) = 0 1 A =

(1 -

0 e = flC = 8uEA„(LT*)-

0

Ky')

0 ,

0 (Ky^ -

I)

dx^

(Ky'~

ax

B

C

1)

where A, B and w are defined in Eq. (7). Note, e is a small parameter representing the magnitude of the dynamic tension divided by the static tension. Although the dimensionless parameter C, is large, the amplitude of the relative end displacement a = max [U(L, t) - U(0, t)]/L is assumed

dx^

t

dx^ (7)

with boundary conditions w(0, t) = w ( l , 0 = 0. Journal of Vibration and Acoustics

(12)

small such that e < 1. If e > 1, the belt would be in compression during part of the excitation cycle, an unlikely occurrence in serpentine drives. Since the translating string eigenvectors are not orthogonal with respect to the new operator matrix, C, the Galerkin procedure will lead, in general, to a set of coupled equations governJULY 1996, Vol. 1 1 8 / 3 4 7

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ing the modal coordinates. Principal parametric instability regions (Q i= 2uj„) are well described, to first order in e, by a oneterm, «*-mode discretization. A two-term discretization adds no new information to e '-order. Summation and difference parametric instabilities may also exist, and this possibility is examined using a two-term, first- and second-mode discretization. For the one-term, M^-mode discretization, w is approximated by

conclusions. General (n*-mode) primary parametric instability regions and the first summation and difference parametric instability regions are examined below. 3.1 Primary Parametric Instability Region: fl «< lot,,. In this case, the one-term (n*-mode) discretization is assessed upon introducing the detuning parameter, a, defined by w^ = 0,12 + ta. Elimination of secular terms yields e'-order amplitude state equations

^ ((c^J, C«^f) + (
c,j>l)) Pn

n ((
w « „(x)il/„(t) = 2 Re[ 1j' 0> 1=1

i/fj = Pj sin (ujjt) + qj cos (ujjt) + X e'u'jiiPj, %, t), (17) /=! i = 1, 2, where «,, are, as yet, undetermined functions. The amplitudes, Pj and qj, are slowly varying functions of time, determined by Pi = X £'Pji{Pi,P2, qu qz), 1=1

qj = X e'Qji(pi,p2, qi, ^2); ;• = 1, 2.

(18)

i=l

In general, the elimination of secular terms in the expansion leads to amplitude state equations that have stable, trivial solutions. However, when the excitation frequency, Q, is near 2w„, or uj„ ± a;„, extra secular terms may arise leading to different 348 / Vol. 118, JULY 1996

(20)

[nirg(l -

2e sin {ng-Ky) ngTxy{l — Ky^)

Ky^)y

(21)

(14)

For the two-term discretization, w is approximated by

+/'fj'A? -/^faiAa) = 0,

> 0

which leads to first-order approximations to the stability boundaries

-{IJ'„ + LO„,p^„ ~ 2£(cos nt)({'„, C