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G. Chakraborty and N. Jani
(a) Softening behaviour
(b) Hardening behaviour
Fig. 2 Frequency–amplitude curves for forced excitation
a is the amplitude of oscillation. For a forced system, the response curve depends
on the magnitude of f 0 and c.
(b) The response amplitude can take any of the three possible values depending on
the initial conditions, when the excitation frequency is within a zone in between
ω 1 and ω 2 (shown in Fig. 3).
The non-uniqueness in the amplitude gives rise to ‘jump phenomena’ when the
frequency of excitation is quasistatically increased or decreased continuously.
The direction of the jump (upward or downward) as well as its amount depends
on the sign of α. For example, a hardening-type nonlinearity exhibits downward
jump in the frequency sweep up operations while a softening-type nonlinear
system shows upward jump. However, if the frequency is not changed quasistatically, then significant transient oscillation takes place before the amplitude sets
down to a new value.
Case 2: ω ≈ 3ω n . Although the predicted response is small according to linear analysis, the presence of cubic nonlinear term can make it large. The steady-state response
has two frequencies, namely, the excitation frequency ω and the natural frequency
ω n . The phenomena of large amplitude of oscillation is known as ‘subharmonic
resonance’.
Fig. 3 Bifurcation frequency in case of softening and hardening nonlinearities
G. Chakraborty and N. Jani
(a) Softening behaviour
(b) Hardening behaviour
Fig. 2 Frequency–amplitude curves for forced excitation
a is the amplitude of oscillation. For a forced system, the response curve depends
on the magnitude of f 0 and c.
(b) The response amplitude can take any of the three possible values depending on
the initial conditions, when the excitation frequency is within a zone in between
ω 1 and ω 2 (shown in Fig. 3).
The non-uniqueness in the amplitude gives rise to ‘jump phenomena’ when the
frequency of excitation is quasistatically increased or decreased continuously.
The direction of the jump (upward or downward) as well as its amount depends
on the sign of α. For example, a hardening-type nonlinearity exhibits downward
jump in the frequency sweep up operations while a softening-type nonlinear
system shows upward jump. However, if the frequency is not changed quasistatically, then significant transient oscillation takes place before the amplitude sets
down to a new value.
Case 2: ω ≈ 3ω n . Although the predicted response is small according to linear analysis, the presence of cubic nonlinear term can make it large. The steady-state response
has two frequencies, namely, the excitation frequency ω and the natural frequency
ω n . The phenomena of large amplitude of oscillation is known as ‘subharmonic
resonance’.
Fig. 3 Bifurcation frequency in case of softening and hardening nonlinearities