the system to grow unbounded with time. To examine the actual post-critical
behavior (which cannot be unbounded), we employed a multiple scales perturbation
analysis to the nonlinear system, written in first-order matrix form. This analysis
proved the zero solution to become unstable in favor of stable finite amplitude
periodic motion. Also, we showed that for some parameters the linearly stable zero
solution can be destabilized by a strong disturbance. Some observations have
appeared that hold generally:
• A linearized analysis of stability only reveals the consequences of applying a
small disturbance to a specific state of equilibrium.
• Linear analysis is generally incapable of predicting post-critical behavior.
• A linearized analysis predicts unstable solutions to grow exponentially in time.
This prediction is valid only initially, that is, to the time where motions have
become too large for the linearization to remain adequate.
4.6 Pendulum with a Sliding Disk
4.6.1 Introduction
Now we turn to studying nonlinear interactions between vibrating structures and
sliding solid bodies. As with internal resonance, the kind of interaction considered
is nonlinear. With linearization the interaction disappears, and the motions of
structure and solid body become completely independent.
To put focus on the nonlinear effects of sliding in a simple setting, we consider
in this section a vibrated pendulum with a disk sliding on its rod. Sections 4.7–4.8
then consider similar effects associated with flexible structures.
Fig. 4.13 Oscillation amplitudes of the double pendulum for a = 0.8, c = 0.1, m = 2 and varying
p. (——) Theory (Eq. (4.75)); ⃝, ⃞: numerical simulation (Eq. (4.50)). (Thomsen 1995)
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4 Nonlinear Multiple-DOF Systems: Local Analysis
behavior (which cannot be unbounded), we employed a multiple scales perturbation
analysis to the nonlinear system, written in first-order matrix form. This analysis
proved the zero solution to become unstable in favor of stable finite amplitude
periodic motion. Also, we showed that for some parameters the linearly stable zero
solution can be destabilized by a strong disturbance. Some observations have
appeared that hold generally:
• A linearized analysis of stability only reveals the consequences of applying a
small disturbance to a specific state of equilibrium.
• Linear analysis is generally incapable of predicting post-critical behavior.
• A linearized analysis predicts unstable solutions to grow exponentially in time.
This prediction is valid only initially, that is, to the time where motions have
become too large for the linearization to remain adequate.
4.6 Pendulum with a Sliding Disk
4.6.1 Introduction
Now we turn to studying nonlinear interactions between vibrating structures and
sliding solid bodies. As with internal resonance, the kind of interaction considered
is nonlinear. With linearization the interaction disappears, and the motions of
structure and solid body become completely independent.
To put focus on the nonlinear effects of sliding in a simple setting, we consider
in this section a vibrated pendulum with a disk sliding on its rod. Sections 4.7–4.8
then consider similar effects associated with flexible structures.
Fig. 4.13 Oscillation amplitudes of the double pendulum for a = 0.8, c = 0.1, m = 2 and varying
p. (——) Theory (Eq. (4.75)); ⃝, ⃞: numerical simulation (Eq. (4.50)). (Thomsen 1995)
238
4 Nonlinear Multiple-DOF Systems: Local Analysis