Introduction to Acoustics

952 Part G Structural Acoustics and Noise
Part G 22.7
Example of a Quadratic Nonlinear Coupling
Here, the case of a quadratic nonlinear coupling between
two oscillators is investigated using the multiple-scales
method [22.57]. The equations are:
⎧
⎨
¨x1 + ω
⎩
2 1x1 = ε � �
− β12x1x2 − 2µ1 ˙x1 ,
¨x2 + ω2 2x2 = ε � − β21x2 1 − 2µ2 ˙x2 + P cos Ωt � ,
(22.416)
where x1 and x2 are the displacements, ω1 and ω2
are the eigenfrequencies of the oscillators in the linear
range. The perturbation terms are grouped on the
right-hand side of (22.416). The nondimensional parameter
ε ≪ 1 indicates that these terms are small. The
quadratic nonlinear coupling is due to the presence of
the terms β12x1x2 and β21x 2 1
. It is assumed that the sys-
tem has a so-called internal resonance in the sense that
ω2 = 2ω1 + εσ1, whereσ1 is the internal detuning parameter.
The forcing frequency Ω is close to ω2 so that
we can write Ω = ω2 + εσ2 where σ2 is the external detuning
parameter, and µ1 and µ2 are viscous damping
parameters.
Resolution with the Method of Multiple Scales
Solving the above equations involves calculating the amplitudes
a1 and a2 of both oscillators as a function of the
external detuning parameter σ2. Another goal is to determine
the values of the threshold in terms of amplitude
and frequency, at which the nonlinear set of oscillators
exhibit bifurcations and unstable behavior. The present
example contains most of the concepts and methods
used in the theory of multiple scales applied to nonlinear
oscillators. The main steps of the calculations are the
following:
1. Definition of time scales and general form of the
solution
2. Solvability conditions. Elimination of secular terms
3. Autonomous system and fixed points
4. Stability of the system
5. Amplitudes and phases of the solution
Definition of Time Scales and General Form of the
Solution. The time scales are defined as
Tj = ε j t with j ≥ 0 (22.417)
and the solutions are expanded into
⎧
⎨
x1(t) = x10(T0, T1) + εx11(T0, T1) + O(ε
⎩
2 )
x2(t) = x20(T0, T1) + εx21(T0, T1) + O(ε2 )
,
(22.418)
where the expansion is limited here to the first order in ε.
The differentiation operators can be written
⎧
∂
⎪⎨ ∂t
⎪⎩
∂2 ∂t2 = ∂
+ ε
∂T0
∂
,
∂T1
∂2
=
∂T 2 0
+ 2ε ∂
∂T0
∂
.
∂T1
(22.419)
In what follows, we write D j = ∂/∂Tj. Inserting
(22.419) into(22.416) and matching the coefficients of
terms with the same power in ε yields:
• for the zero-order term ε 0 = 1
D 2 0 x10 + ω 2 1 x10 = 0 ; D 2 0 x20 + ω 2 2 x20 = 0 ;
(22.420)
• for the first-order term ε:
⎧
D2 0x11 + ω2 1x11 ⎪⎨ =−2D0D1x10 − β12x10x20 − 2µ1 D0x10 ,
D2 0x21 + ω2 2x21 =−2D0D1x20 − β21x
⎪⎩
2 10
−2µ2 D0x20 + P cos Ωt .
The solutions of (22.420) can be written
(22.421)
x10(t) = A1(T1)e iω1t ∗
+ A1 (T1)e −iω1t
;
x20(t) = A2(T1)e iω2t ∗
+ A2 (T1)e −iω2t
, (22.422)
where the ( ∗ ) indicates the complex conjugate.
Solvability Conditions. In order to calculate the complex
terms A1(T1)andA2(T1), the expressions (22.422)
are inserted into (22.421). Then, conditions are determinedsothatnosecularterms,suchast
cos ωt, exist
in the solution. This leads to the so-called solvability
conditions:
⎧ � �
∂A1
−2iω1 + µ1 A1 − β12 A
⎪⎨ ∂T1
⎪⎩
∗ 1 A2 e iσ1T1 = 0 ,
� �
∂A2
−2iω2 + µ2 A2 − β21 A
∂T1
2 1 e−iσ1T1
+ P
2 eiσ2T1 = 0 .
(22.423)