# A geometric softening phenomenon of a rotating cantilever beam

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## Abstract

In this study, we reconstructed a dynamic model of a rotating cantilever beam for which the geometric stiffening term was obtained by accounting for the longitudinal shrinkage caused by the transverse deflection of the beam. Previous investigations focused on kinetic energy but neglected strain energy. For this study, we retained these strain energy coupling terms. We used Hamilton’s principle to derive the complete coupling model. Taking the effect of steady-state axial deformation into account, we obtained the transverse equation of motion and the coupling general characteristic equation. Unlike previous models, this model incorporates not only the geometric stiffening effect but also the geometric softening effect. In relevant numerical examples, as the angular velocity increases, the bending frequency gives rise to geometric stiffening in line with the results obtained in previous studies. When the angular velocity reaches and exceeds a critical value, the bending frequency produces a geometric softening phenomenon.

## Keywords

Rotating cantilever beam Foreshortening effect Geometric stiffening Geometric softening Bending frequency## List of symbols

*L*Length of beam in the undeformed configuration, m

*V*Volume of beam, \(\hbox {m}^{3}\)

*E*Young’s modulus, \(\hbox {N}/\hbox {m}^{2}\)

*A*Cross-sectional area, \(\hbox {m}^{2}\)

*I*Second moment of area of the cross section, \(\hbox {m}^{4}\)

- \(P_{0}\)
Arbitrary point of the beam in the undeformed configuration

*P*Point \(P_{0}\) in the deformed configuration

- \({\varvec{u}}\)
Deformation vector, m

- \({\varvec{r}}\)
Global position vector of the point

*P*, m- \({\varvec{B}}\)
Planar rotation matrix

*x*,*y*Horizontal/vertical component of \(P_{0}\), m

- \(u_{1}, u_{2}\)
Total longitudinal and transverse deformations, m

- \(w_{1}, w_{2}\)
Axial and bending deformations, m

- \(w_{1\mathrm{r}}\)
Longitudinal deformation caused by rotating of the cross section, m

- \(w_\mathrm{c}\)
Longitudinal shrinkage caused by the bending deformation, m

- \({\varvec{v}}\)
Global velocity vector of the point

*P*, m*T*Kinetic energy

*U*Strain energy

- \({\varvec{K}}_\mathrm{ccm}\)
Stiffness matrix

- \({\varvec{M}}_\mathrm{ccm}\)
Mass matrix

- \(\gamma \)
Mass per unit volume, \(\hbox {kg/m}^{3}\)

- \(\theta \)
Angle of rotation, rad

- \({{\dot{\theta }}}\)
Angular velocity, rad/s

- \({\varvec{\rho }}_{0}\)
Local position vector of an arbitrary point \(P_{0}\), m

- \(\varepsilon _{xx}\)
Longitudinal normal strain

- \(\nu \)
Poisson’s ratio

- \(\omega \)
Bending frequency, Hz

- \(\omega _{0}\)
First bending frequency without rotational motion

- \({\varvec{\beta }}\)
The mode shape without rotational motion

*i*An imaginary number

## Notes

### Acknowledgements

The authors gratefully acknowledge the support for this work from the National Natural Science Foundation of China under Grant Nos. 11372056 and 11432010.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interests.

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