Control of Nonlinear Vibrations

Abstract

In this chapter, methods which can be used to control nonlinear structural vibrations are discussed. Introductory examples showing the control of linear and nonlinear single-degree-of-freedom oscillators have already been discussed in Sect. 1.4 of Chap. 1. This chapter extends the ideas presented in these introductory examples to a range of controllers, which can be designed to control nonlinear vibrations. Control of structural vibrations is different from the majority of control problems, in that there are typically multiple lightly damped resonances in the system response. In addition, when an actuator is attached to the structure, its effect will be coupled to some resonances much more strongly than others. As a result, careful design is required to reduce particular resonant responses. Even with careful design, other resonances will exist which cannot be effectively controlled. Using feedback can induce instability in the system, and so ensuring the stability of any control design is of primary importance. The underlying ideas of stability for nonlinear systems have been introduced in Sect. 2.3, Chap. 2. In this chapter, these ideas are extended to include systems with feedback control, and the stability analysis is carried out using a particular type of potential function, called a Lyapunov function. The basic ideas of Lyapunov-based control design can be extended to a range of other approaches. The main control method described here is the effective linearisation of a system using feedback. Adaptive control, which can also be a useful method for nonlinear or uncertain systems is also discussed in the later part of the chapter.

Problems

• 3.1 Draw the control block diagram and analyse the stability of the closed loop transfer function of the system given by Eq. (3.19), using a similar approach to that of Example 1.4 in Chap. 1. What effect do $b$ and $\kappa$ have on the stability of the system?

• 3.2 For the mass-spring-damper system shown in Fig. 3.12, design a control law which uses feedback terms proportional to the acceleration or displacement instead of the velocity term derived in Example 3.4. What effect will this have on the resonance characteristics of the mass-spring-damper system?

• 3.3 Construct a Lyapunov function to assess the stability of the controlled Duffing equation, given by Eq. (3.21), using the potential energy plus the kinetic energy of the system. Assume the case when only a single equilibrium point at the origin exists in the system and $F_{e}$ is zero.

• 3.4 Choose a Lyapunov function to assess the stability of the equilibrium point at the origin for the system given by

  $$ \ddot{x}+\delta \dot{x}-x+\alpha x^{3}=0, $$

  where $\delta$ and $\alpha$ are both positive constants. Can the energy equation associated with this oscillator be used as a Lyapunov function?

• 3.5 Design a feedback linearisation controller for the following Duffing oscillator. Note that the underlying linear system has an unstable equilibrium point at the origin, and the control objective is to stabilize this equilibrium point.

  $$ \left[ \begin{array}{l} \dot{x}_{1}\\ \dot{x}_{2} \end{array}\right] = \left[ \begin{array}{cc} 0 &{} 1\\ \frac{k_{1}}{m} &{}-\frac{c}{m} \end{array}\right] \left[ \begin{array}{l} x_{1}\\ x_{2} \end{array}\right] \ +\ \left[ \begin{array}{c} 0\\ \frac{p}{m}u(t)-\frac{k_{3}}{m}x_{1}^{3} \end{array}\right] . $$

• 3.6 Use feedback linearisation to remove the nonlinear damping terms in the following nonlinear oscillator

  $$ m\ddot{x}+c\dot{x}(1+\delta x^{2})+kx=pu(t), $$

  where $u(t)$ is the control input. Is it possible to linearize the system and add more linear viscous damping simultaneously?

• 3.7 For the controlled Duffing oscillator given by

  $$\begin{aligned} \dot{x}_{1}&= x_{2},\\ \dot{x}_{2}&= -\frac{c}{m}x_{2}-\frac{k_{1}}{m}x_{1}-\frac{k_{3}}{m}x_{1}^{3}+\frac{p}{m}u(t), \end{aligned}$$

  with an output $y=x_{2}$, use input-output linearisation to design a control input $u$. How does this compare with the case when $y=x_{1}$?

• 3.8 For the oscillator

  $$ m\ddot{x}+c\dot{x}x+kx=pu(t), $$

  with an output $y=x_{1}+x_{2}$, use input-output linearisation to design a control input $u$. How does this compare with the case when (i) $y=x_{1}$ and (ii) $y=x_{2}$?

• 3.9 Use feedback linearisation control techniques to linearize the two-mode nonlinear system defined by

  $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \left[ \begin{array}{c} q_{1}\\ q_{2}\\ \dot{q}_{1}\\ \dot{q}_{2} \end{array}\right] =&\left[ \begin{array}{cccc} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1\\ -\omega _{n1}^{2} &{} 0 &{} -\zeta _{1}\omega _{n1} &{} 0\\ 0 &{} -\omega _{n2}^{2} &{} 0 &{} -\zeta _{2}\omega _{n2} \end{array}\right] \left[ \begin{array}{c} q_{1}\\ q_{2}\\ \dot{q}_{1}\\ \dot{q}_{2} \end{array}\right] - \left[ \begin{array}{c} 0\\ 0\\ \mu _{1}q_{1}^{2}+\delta _{1}q_{1}q_{2}\\ \mu _{2}q_{2}^{2}+\delta _{2}q_{2}q_{1} \end{array}\right] \\&+\left[ \begin{array}{c} 0\\ 0\\ \alpha _{1}p_{1}\\ 0 \end{array}\right] \ u_{1}+\ \left[ \begin{array}{c} 0\\ 0\\ 0\\ \beta _{2}p_{2} \end{array}\right] \ u_{2}, \end{aligned}$$

  where $\delta _{1}$ and $\delta _{2}$ are constant terms which determine the level of nonlinear cross-coupling between modes 1 and 2. Assume that both observation and control spillover are negligible and that the outputs are the modal displacements $y_{1}=q_{1}$ and $y_{2}=q_{2}$.

• 3.10 The vibration of a nonlinear beam can be approximated by the summation given in Eq. (3.37). If modes 2 and 3 need to be controlled, write down the equations of motion for the controlled modes in the system. Assuming the modes can be treated as effectively decoupled, suggest a feedback linearisation control scheme which would linearize the system.

• 3.11 Use adaptive feedback linearisation to linearize the following nonlinear oscillator

  $$ m\ddot{x}+c\dot{x}(1+\delta x^{2})+kx+\mu x_{1}^{3}=bu(t), $$

  where $u(t)$ is the control input and both $\delta$ and $\mu$ are uncertain parameters. Assume that the mass, $m=2$ kg, stiffness, $k=2\,\mathrm{N}/\mathrm{m}^{2}$ and damping, $c=0.2\,\mathrm{Ns}/\mathrm{m}$. The control gain has the value $b=1$. Both $\delta$ and $\mu$ have some uncertainty and initial estimated values can be assumed to be $\delta =1$ and $\mu =2$.