resonant in case x = 2. We shall refer to the case x = 2 as the case of primary
parametric resonance
4 .
The case x % 2 (as opposed to x = 2) is referred to as near-resonance.
Near-resonant excitation causes small divisor terms to appear in the solution for h 1 ,
and small divisor terms are just as unacceptable as secular terms. To understand this
we consider the particular solution of the equation D 0
2 h 1 + h 1 = e
iðxÀ1ÞT 0 , which is
h 1 = (1 – (x – 1)
2 )
–1 e
iðxÀ1ÞT 0 . Hence, when x % 2 the divisor (1 – (x – 1)
2 ) will be
small, and h 1 will be correspondingly large. This violates the assumption of h 1
being small, as compared to h 0 . Consequently, small divisor terms should be
eliminated from the response, by requiring the sum of near-resonant excitation
terms to vanish identically.
As a strength of the method of multiple scales we note that the perturbation
equations, when set up, automatically reveal which special cases to consider. For
the pendulum example it might (dependent on experience) be obvious that there is
something special about the value x = 2 for the excitation frequency: When x = 2
the pendulum support oscillates up and down at twice the linear natural frequency
of free pendulum oscillations. By simple physical reasoning one realizes that this is
just the right frequency ratio for generating large responses using little effort (as
would be x = 1, if an oscillating torque was applied at the pendulum hinge
instead). For other problems the special case(s) to consider may not be similarly
obvious, but setting up the multiple scales perturbation equations would reveal
them anyway.
Thus, for the pendulum example there are two cases to consider below: the
non-resonant case (x away from 2), and the near-resonant case (x % 2).
3.6.3 The Non-resonant Case
When x is away from 2 the only resonant term of (3.114) is the one containing e
iT 0
as a factor. Requiring the coefficient to e
iT 0 to vanish the following solvability
condition is obtained:
i2A
0
þ i2bA þ 3cA
2 A ¼ 0:
ð3:115Þ
With this condition fulfilled, a particular solution to equation (3.114) is obtained
by adding in turn the responses to each of the remaining harmonic excitation-terms,
that is:
4
More generally, for linear parametrically excited systems with n degrees of freedom and distinct
natural frequencies x j , j = 1,n, one can show (Nayfeh & Mook 1979) that parametric resonance
occurs when the excitation frequency X ¼ jx i Ç x j j=n for i; j; n ¼ 1; 2; . . .; where n is the
order of the parametric resonance; this is an example of combination resonances. Typically the
first-order (n = 1) primary (i = j = 1) resonance is the one of main interest.
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3 Nonlinear Vibrations: Classical Local Theory
parametric resonance
4 .
The case x % 2 (as opposed to x = 2) is referred to as near-resonance.
Near-resonant excitation causes small divisor terms to appear in the solution for h 1 ,
and small divisor terms are just as unacceptable as secular terms. To understand this
we consider the particular solution of the equation D 0
2 h 1 + h 1 = e
iðxÀ1ÞT 0 , which is
h 1 = (1 – (x – 1)
2 )
–1 e
iðxÀ1ÞT 0 . Hence, when x % 2 the divisor (1 – (x – 1)
2 ) will be
small, and h 1 will be correspondingly large. This violates the assumption of h 1
being small, as compared to h 0 . Consequently, small divisor terms should be
eliminated from the response, by requiring the sum of near-resonant excitation
terms to vanish identically.
As a strength of the method of multiple scales we note that the perturbation
equations, when set up, automatically reveal which special cases to consider. For
the pendulum example it might (dependent on experience) be obvious that there is
something special about the value x = 2 for the excitation frequency: When x = 2
the pendulum support oscillates up and down at twice the linear natural frequency
of free pendulum oscillations. By simple physical reasoning one realizes that this is
just the right frequency ratio for generating large responses using little effort (as
would be x = 1, if an oscillating torque was applied at the pendulum hinge
instead). For other problems the special case(s) to consider may not be similarly
obvious, but setting up the multiple scales perturbation equations would reveal
them anyway.
Thus, for the pendulum example there are two cases to consider below: the
non-resonant case (x away from 2), and the near-resonant case (x % 2).
3.6.3 The Non-resonant Case
When x is away from 2 the only resonant term of (3.114) is the one containing e
iT 0
as a factor. Requiring the coefficient to e
iT 0 to vanish the following solvability
condition is obtained:
i2A
0
þ i2bA þ 3cA
2 A ¼ 0:
ð3:115Þ
With this condition fulfilled, a particular solution to equation (3.114) is obtained
by adding in turn the responses to each of the remaining harmonic excitation-terms,
that is:
4
More generally, for linear parametrically excited systems with n degrees of freedom and distinct
natural frequencies x j , j = 1,n, one can show (Nayfeh & Mook 1979) that parametric resonance
occurs when the excitation frequency X ¼ jx i Ç x j j=n for i; j; n ¼ 1; 2; . . .; where n is the
order of the parametric resonance; this is an example of combination resonances. Typically the
first-order (n = 1) primary (i = j = 1) resonance is the one of main interest.
132
3 Nonlinear Vibrations: Classical Local Theory