Abstract
This chapter discusses methods for modelling the nonlinear vibration of beams. The starting point is to consider the physics of beams, for both small and large deflections. The resulting partial differential equations are then decomposed, using the techniques discussed in Chap. 5, to give a set of ordinary differential equations which can be analysed. Large deflections lead to nonlinear governing equations for the beam vibrations. Another important case in practice is when the beam is axially loaded, which also leads to nonlinearities in the governing expressions. In the final part of the chapter, control of beam vibrations using modal control is discussed.
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Notes
- 1.
Note that more complex formulations exist, such as the Timoshenko beam equation, but these are not considered here.
- 2.
In fact both these methods are part of a wider class called weighted residual methods, which includes the Rayleigh-Ritz method and finite element methods. See Hagedorn and DasGupta (2007) for an introduction to how these methods can be applied to beams and other continuous structures.
- 3.
For beams which cannot be considered as slender a Timonshenko beam analysis could be used (Timoshenko et al. 1974).
- 4.
Note that we derive the equations of motion for a beam aligned in the vertical position. An equivalent derivation can also be carried out for a horizontal beam, see for example Clough and Penzien (1993).
- 5.
Moving loads are not considered here, but a good overview is given by Ouyang (2011).
- 6.
Sometimes also called the eigenfunctions, modal basis or linear modes.
- 7.
- 8.
Forms of non-viscous damping often appear in vibration problems, see for example Jones (2001).
- 9.
Note that it is equally valid to give the functions \(h(t)\) and \(g(x)\) other units provided the resulting \(F(x,\ t)\) has units of force per length.
- 10.
See Blevins (1979), p. 455, for further details.
- 11.
See Fanson and Caughey (1990) for more details on modelling moments generated by piezoelectric actuators.
- 12.
If normal linear beam modes are not used, collocation can still be applied providing the fourth derivative of the shape function \(\phi _{j}\) can be computed for each collocation point.
- 13.
This assumes that the system is not numerically stiff, see Press et al. (1994) for more details.
- 14.
See Stronge (2000) for a detailed description of impact problems and definitions of the coefficient of restitution.
- 15.
Often known as Kirchhoff’s dynamical analogy after G.R. Kirchhoff.
- 16.
After S.P. Timoshenko (1878–1972).
- 17.
Note that deflection \(w\) now replaces \(z\), which was used earlier when statics were being considered.
- 18.
Note that the Dirac-Delta function has arbitrary units, in this case they are length\(^{-1}\).
- 19.
- 20.
Note that as the beam stretches, the cross-sectional area of the beam will decrease due to the Poisson’s ratio effect. This effect is not considered in the current analysis.
- 21.
Disturbance just means any unwanted, and usually unknown, signal.
- 22.
See Leo (2007) for a physical explanation of how this can be done using piezoelectric materials.
- 23.
These data are taken from the modal identification of an aluminium beam with a piezo-actuator attached, see Chap. 3 of Malik (2009).
- 24.
These data are taken from the modal identification of an aluminium beam with a piezo-actuator attached, see Chap. 3 of Malik (2009).
- 25.
Theoretically linearisation is possible.
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Wagg, D., Neild, S. (2015). Beams. In: Nonlinear Vibration with Control. Solid Mechanics and Its Applications, vol 218. Springer, Cham. https://doi.org/10.1007/978-3-319-10644-1_6
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