Abstract
The performance requirements of flexible structures are continually increasing. Often structures are required to have integrated control and sensor systems to carry out tasks such as limiting unwanted vibrations, detecting damage and in some cases changing the shape of the structure. These types of structures have become known as smart structures (sometimes called adaptive or intelligent structures). The ability to perform multiple tasks means that the smart structure is multifunctional. By their nature, these structures are typically highly flexible and are required to operate in a dynamic environment. As a result, the vibration behaviour of the structure is of critical importance. Not only is vibration important, it is often nonlinear, due to a range of effects which naturally arise in flexible structural dynamics. Applying control to the structure to limit unwanted vibration and to effect any shape changes also requires detailed knowledge of the vibration characteristics. This chapter introduces the basic ideas of nonlinear vibration and control, which will be used in later chapters to underpin the analysis of more complex structural elements.
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Notes
- 1.
Periodic is more general than harmonic, it has a repeating pattern but this repeating pattern is not necessarily limited to a single-frequency sine wave.
- 2.
Strictly speaking, this is only true in a vacuum. In low density fluids the behaviour is very close, but in liquids the effects of mass become significant, Neill et al. (2007).
- 3.
Amplitude is sometimes used to denote only the magnitude of displacements. Throughout this text it will be used to imply a magnitude of the quantity under discussion, be it force, velocity, acceleration or displacement.
- 4.
- 5.
In fact, if \(\dot{x}(0)\ne 0\) this assumption becomes invalid.
- 6.
In many texts on nonlinear vibration, \(\alpha \) is assumed to be small, in which case the square root term can be approximated using \((1+a)^{1/2}\approx (1+a/2)\) to give \(\omega _{r}\approx \omega _{n}[1+\frac{3\alpha X_{r}^{2}}{8\omega _{n}^{2}}]\).
- 7.
In fact this can be extended to include additional terms formed from combinations of \(M\) and \(K\), known as extended Rayleigh damping. See Clough and Penzien (1993) for a detailed discussion.
- 8.
- 9.
Sometimes known as setpoint or reference this is the desired system output.
- 10.
Note the negative feedback. As a general rule, positive feedback will cause instability.
- 11.
Feedforward control is a type of controller that does not use feedback from the system being controlled.
- 12.
Notice that there is a subtle difference between \({{\varvec{x}}}\), which is the \(2N\times 1\) state vector, and \(\mathbf x \), which is the \(N\times 1\) displacement vector. This is used to maintain (as far as possible) notation conventions from control engineering, nonlinear dynamics and structural vibration.
- 13.
To avoid confusion with the damping matrix, \(\bar{C}\) is used as the control output matrix.
- 14.
Note that the convention of writing the Laplace transform of a variable as a capital letter is used here.
- 15.
Poles are the complex roots of the denominator of \(G(s)\).
- 16.
In fact, for viscous damping, this is only approximately circular. See Ewins (2000) for a more detailed discussion of the properties of these functions.
- 17.
This is the method adopted in most derivations in this book. Other techniques such as Lagrange’s equation for energy or Hamilton’s principle can also be used in many cases.
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Wagg, D., Neild, S. (2015). Introduction to Nonlinear Vibration and Control. In: Nonlinear Vibration with Control. Solid Mechanics and Its Applications, vol 218. Springer, Cham. https://doi.org/10.1007/978-3-319-10644-1_1
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