Numerical simulations show that quasiperiodic and chaotic motions prevail in this
region (cf. Chap. 6).
4.4 Other Systems Possessing Internal Resonance
The below examples are meant to be just indicative of other areas where internal
resonance may dominate the response of a system.
A system consisting of two orthogonally clamped beams with a two-to-one
internal resonance have been studied analytically and experimentally in a number of
papers (e.g., Nayfeh and Zavodney 1988; Nayfeh and Balachandran 1989; Nayfeh
et al. 1989; Balachandran and Nayfeh 1991).
Cylindrical shells may exhibit nonlinear coupling between breathing and flexural
modes due to internal two-to-one resonances (Nayfeh and Raouf 1987; Nayfeh and
Balachandran 1989).
The dynamics of ships rolling in sea waves (Nayfeh and Khdeir 1986) are also
governed by coupled equations with quadratic nonlinearities, thus being a candidate
for complicated motions.
The nonlinear dynamics of the Golden Gate Bridge in San Francisco have been
considered with a special emphasis to the occurrence of internal resonance
(Rossikhin and Shitikova 1995).
Consideration to modal interactions caused by a three-to-one internal resonance
was included in a study of vibration-induced fluid flow in pipes (Jensen 1997).
Fig. 4.10 Stability of stationary solutions as a function of loading parameters X and q. Region A:
only the linear solution is stable; B: only the nonlinear solution is stable; C: neither solutions are
stable; D: both solutions are stable. •: Numerical simulation. (x = 2.44, j = 2.63, m = 3.32,
b = 0.03)
4.3 Nonlinear Mode-Coupling of Non-shallow Arches
227
region (cf. Chap. 6).
4.4 Other Systems Possessing Internal Resonance
The below examples are meant to be just indicative of other areas where internal
resonance may dominate the response of a system.
A system consisting of two orthogonally clamped beams with a two-to-one
internal resonance have been studied analytically and experimentally in a number of
papers (e.g., Nayfeh and Zavodney 1988; Nayfeh and Balachandran 1989; Nayfeh
et al. 1989; Balachandran and Nayfeh 1991).
Cylindrical shells may exhibit nonlinear coupling between breathing and flexural
modes due to internal two-to-one resonances (Nayfeh and Raouf 1987; Nayfeh and
Balachandran 1989).
The dynamics of ships rolling in sea waves (Nayfeh and Khdeir 1986) are also
governed by coupled equations with quadratic nonlinearities, thus being a candidate
for complicated motions.
The nonlinear dynamics of the Golden Gate Bridge in San Francisco have been
considered with a special emphasis to the occurrence of internal resonance
(Rossikhin and Shitikova 1995).
Consideration to modal interactions caused by a three-to-one internal resonance
was included in a study of vibration-induced fluid flow in pipes (Jensen 1997).
Fig. 4.10 Stability of stationary solutions as a function of loading parameters X and q. Region A:
only the linear solution is stable; B: only the nonlinear solution is stable; C: neither solutions are
stable; D: both solutions are stable. •: Numerical simulation. (x = 2.44, j = 2.63, m = 3.32,
b = 0.03)
4.3 Nonlinear Mode-Coupling of Non-shallow Arches
227