A comprehensive analysis of hardening/softening behaviour of shearable planar beams with whatever axial boundary constraint

Abstract

The free nonlinear oscillations of a planar elastic beam are investigated based on a comprehensive asymptotic treatment of the exact equations of motion. With the aim of investigating the behaviour also for low slenderness, shear deformations and rotational inertia are taken into account. Attention is payed to the influence of the geometrical and mechanical parameters, and of the boundary conditions in changing the nonlinear behaviour from softening to hardening. An axial linear spring is added to one end of the beam, and it is shown how the behaviour changes qualitatively on passing from the hinged-hinged (commonly hardening) to the hinged-supported (commonly softening) case. Some interesting, and partially unexpected, results are obtained also for values of the slenderness moderately low but still in the realm of practical applications.

Appendix

The nonlinear frequency correction $\omega _2$ is given by [14]:

$$\begin{aligned} \omega _2 = U_a^2 \frac{c_1 \omega _{2a} + c_2 \sin \left( \frac{2L \omega _0 \sqrt{\rho B}}{\sqrt{E A}} \right) \omega _{2b} + \omega _{2c} }{\omega _{2d}}, \end{aligned}$$

(43)

where the expressions of $\omega _{2a}$, $\omega _{2b}$, $\omega _{2c}$ and $\omega _{2d}$ are reported in the following. Note that $\omega _{2d}$ does not vanish for the considered values of $\omega _0$.

Using the dimensional expressions in (45)–(48) it is possible to rewrite (43) in the form

$$\begin{aligned} \omega _2 = U_a^2 \frac{1}{L^4} \sqrt{\frac{EJ}{\rho A}} \,\, \frac{c_1 \bar{\omega }_{2a} + c_2 \sin \left( \frac{2 \bar{\omega }_0 \sqrt{x}}{l} \right) \bar{\omega }_{2b} + \bar{\omega }_{2c} }{\bar{\omega }_{2d}}, \end{aligned}$$

(44)

from which one gets (29), with the associated expression of the dimensionless quantity $\bar{\omega }_2$.

$$\begin{aligned} \omega _{2a}&= {} 32 EA \pi ^2 n^2 \left( EA \pi ^2 n^2 - \omega _0^2 L^2 \rho B\right) \nonumber \\&\quad\times \left[ EA L^2 - EJ \pi ^2 n^2 \alpha _1^2 +GA L^2 \left( \alpha _1^2-1\right) \right] = \frac{(EJ)^3}{L^4} \bar{\omega }_{2a}, \nonumber \\ \bar{\omega }_{2a}&= {} - 32 \, l^2 \pi ^2 n^2 \left( \pi ^2 n^2 \alpha _1^2 - \alpha _1^2 l^2 z + l^2 z - l^2\right) \left( \pi ^2 n^2 l^2 - x \bar{\omega }_0^2\right), \end{aligned}$$

(45)

$$\begin{aligned} \omega _{2b}&= {} 16 EA \pi ^2 n^2 GA L^2 \left( \alpha _1^2-1\right) \left( 2 \rho B \omega _0^2 L^2- EA \pi ^2 n^2\right) \nonumber \\&+ 16 (EA)^2 \pi ^2 n^2 \left( EJ \alpha _1^2 \pi ^4 n^4 - EA \pi ^2 n^2 L^2 + 2 \rho B \omega _0^2 L^4\right) = \frac{(EJ)^3}{L^4} \bar{\omega }_{2b}, \nonumber \\ \bar{\omega }_{2b}&= {} 16 \, l^4 \pi ^2 n^2 \left[ \pi ^4 n^4 \alpha _1^2 - \left( \alpha _1^2 z - z + 1 \right) \left( \pi ^2 n^2 l^2 - 2 x \bar{\omega }_0^2\right) \right], \end{aligned}$$

(46)

$$\begin{aligned} \omega _{2c}&= {} 6 \pi ^6 n^6 L^2 (EA)^3 \nonumber \\&\quad- \pi ^4 n^4 (EA)^2 \left[ -6 \pi ^2 n^2 L^2 \left( \alpha _1^2-1\right) GA + 6 \pi ^4 n^4 \alpha _1^2 EJ + 7 \rho B \omega _0^2 L^4 \right] \nonumber \\&\quad+ EA \left\{ \pi ^6 n^6 \alpha _1^2 \left[ -6 \pi ^2 n^2 \left( \alpha _1-1\right) GA + 5 L^2 \rho B \omega _0^2 \right] EJ \right. \nonumber \\&\left.\quad -\,\pi ^4 n^4 L^2 GA \left( \alpha _1-1\right) \left[ 6 n^2 \pi ^2 \left( \alpha _1^2-1\right) GA + \omega _0^2 L^2 \rho B \left( 7 \alpha _1+9\right) \right] \right\} \nonumber \\&\quad+\,4 \rho B \left( \alpha _1-1\right) L^2 GA \pi ^4 n^4 \omega _0^2 \left[ \left( \alpha _1^2-1\right) L^2 GA + n^2 \pi ^2 \alpha _1^2 EJ \right] = \frac{(EJ)^3}{L^4} \bar{\omega }_{2c}, \nonumber \\ \bar{\omega }_{2c}&= {} - \pi ^4 n^4 l^2 \{ 6 \pi ^2 n^2 \left( \alpha _1^3 z^2 - \alpha _1^2 z^2 - \alpha _1^2 z - \alpha _1 z^2 + z^2 + z - 1 \right) l^4 \nonumber \\&\quad+ \left[ - x \left( 4 \alpha _1^3 z^2 - 4 \alpha _1^2 z^2 - 7 \alpha _1^2 z - 4 \alpha _1 z^2 - 2 \alpha _1 z + 4 z^2 + 9 z - 7 \right) \bar{\omega }_0^2 \right. \nonumber \\&\left.\quad +\,6 \pi ^4 n^4 \alpha _1^2 \left( \alpha _1 z - z + 1 \right) \right] l^2 - \pi ^2 n^2 \alpha _1^2 x \bar{\omega }_0^2 \left( 4 \alpha _1 z - 4 z + 5 \right) \}, \end{aligned}$$

(47)

$$\begin{aligned} \omega _{2d}&= {} 64 EA L^4 \omega _0 \left( EA \pi ^2 n^2 - \omega _0^2 L^2 \rho B\right) \left( \rho A L^2 + \pi ^2 n^2 \alpha _1^2 \rho J\right) = (EJ)^2 \sqrt{EJ \rho A} \,\, \bar{\omega }_{2d}, \nonumber \\ \bar{\omega }_{2d}&= {} 64 \, \bar{\omega }_0 \left( \pi ^2 n^2 l^2 - x \bar{\omega }_0^2\right) \left( \pi ^2 n^2 \alpha _1^2 y + l^2\right). \end{aligned}$$

(48)