Nonlinear Vibration Phenomena

Abstract

Nonlinear systems have a range of behaviour not seen in linear vibrating systems. In this chapter the phenomena associated with nonlinear vibrating systems are described in detail. In the absence of exact solutions, the analysis of nonlinear systems is usually undertaken using approximate analysis, numerical simulations and geometrical techniques. This form of analysis has become known as dynamical systems theory (or sometimes chaos theory) and is based on using a system state space. In this chapter the basic ideas of dynamical systems are applied to vibrating systems. Finally, the changes in system behaviour as one (or more) of the parameters is varied are discussed. Such changes are known as bifurcations, and they are highly significant for the understanding of nonlinear systems.

Problems

• 2.1 Derive the equation of motion for the system shown in Fig. 2.20a. Show that this equation can be approximated by the Duffing equation

  $$\begin{aligned} m\ddot{x}+c\dot{x}-\mu x+\alpha x^{3}=0, \end{aligned}$$

  and estimate when this might be a valid assumption.

• 2.2 The normal form of the Hopf bifurcation is usually written as

  $$\begin{aligned} \dot{x}&=\mu x+y-x(x^{2}+y^{2}), \nonumber \\ \dot{y}&=-x+\mu y-y(x^{2}+y^{2}). \end{aligned}$$

  (2.35)

  Show that this system can also be represented as

  $$\begin{aligned} \dot{r}&= r(\mu -r^{2}), \\ \dot{\theta }&= -1, \end{aligned}$$

  in polar coordinates. Examine the stability of the equilibrium point at the origin $(x=0,\ y=0)$ by finding the Jacobian of Eq. (2.35).

• 2.3 A nonlinear system is governed by the following set of first-order differential equations

  $$\begin{aligned} \dot{x}_{1}&=x_{2}, \nonumber \\ \dot{x}_{2}&=x_{1}-x_{1}^{2}-\mu x_{2}, \end{aligned}$$

  (2.36)

  where $\mu$ is a parameter which can be varied. Find the equilibrium points for the system when $0<\mu <\sqrt{4}$ and find the type and stability of each equilibrium point. Sketch typical trajectories in the system state space.

• 2.4 Consider the potential function⁴⁸ given by

  $$\begin{aligned} V=\frac{x_{2}^{2}}{2}-\frac{x_{1}^{2}}{2}+\frac{x_{1}^{3}}{3}. \end{aligned}$$

  Finding the time derivative of $V$ and substituting for $\dot{x}_{1}$ and $\dot{x}_{2}$ gives an indication of the stability of equilibrium points at the origin. For the case when $x_{1}$ and $x_{2}$ are small, use this function to determine the stability of the origin for the system given in Eq. (2.36). How does the sign of $\dot{V}$ relate to the stability?

• 2.5 For the system given in Eq. (2.36), when $\mu$ passes through zero a bifurcation occurs. Use local analysis to explain what happens at the bifurcation point. What type of bifurcation occurs?

• 2.6 The dynamics of a damped unforced pendulum can be modelled using the non-linear differential equation

  $$\begin{aligned} \ddot{\theta }+\delta \dot{\theta }+\omega ^{2}\sin \theta =0, \end{aligned}$$

  where $\theta$ is the angle of the pendulum from the downwards resting position, $\delta$ is the damping parameter and $\omega =\sqrt{\frac{g}{l}}$ is the natural frequency of the pendulum where $g$ is the force due to gravity and $l$ is the length of the pendulum. Find the equilibrium points for the pendulum in the range $2\pi \le \theta \le 2\pi$ when $\delta ^{2}<4\omega ^{2}$. Indicate the type and stability of each equilibrium point and sketch the pendulum trajectories in the $\theta ,\dot{\theta }$ plane.

• 2.7 For small angles the motion for a pendulum can be approximated by

  $$\begin{aligned} \ddot{\theta }+\delta \dot{\theta }+\omega ^{2}(\theta -\frac{\theta ^{3}}{3!})=0. \end{aligned}$$

  Use the potential (i.e. Lyapunov) function

  $$\begin{aligned} V=\frac{1}{2}\dot{\theta }^{2}+\frac{\omega ^{2}}{2}\theta ^{2}-\left( \frac{\omega ^{2}}{3!}\right) \frac{\theta ^{4}}{4} \end{aligned}$$

  to determine the stability of the point $\theta =0,\dot{\theta }=0$, by finding the sign of $\dot{V}$. Assume that $\omega =1$.

• 2.8 The logistic map is a single state mapping which is used to model population dynamics represented as

  $$\begin{aligned} x_{n+1} =\lambda x_n(1-x_n), \end{aligned}$$

  Identify the period one fixed points for the system and their stability. Which bifurcations occur at $\lambda =1$ and $\lambda =3$?