Synchronous whirling of spinning homogeneous elastic cylinders: linear and weakly nonlinear analyses

Abstract

Stationary whirling of slender and homogeneous (continuous) elastic shafts rotating around their axis, with pin–pin boundary condition at the ends, is revisited by considering the complete deformations in the cross section of the shaft. The stability against a synchronous sinusoidal disturbance of any wavelength is investigated and the analytic expression of the buckling amplitude is derived in the weakly nonlinear regime by considering both geometric and material (hyper-elastic) nonlinearities. The bifurcation is supercritical in the long wavelength domain for any elastic constitutive law, and subcritical in the short wavelength limit for a limited range of nonlinear material parameters.

Appendix A: Expression of the functions defining the second-order correction to the displacement

The expressions of the functions g defined by Eqs. 34–37 are given in this appendix. These expressions are obtained by inserting Eqs. 30 and 31 into Eqs. 6–9 and in the boundary conditions Eqs. 10 at order $\varepsilon ^2$.

$$\begin{aligned}&g_{u1}(r)=\frac{\xi ^2}{r_0}\left\{ -\frac{r}{4r_0}(r_0 k)^2+\left( 4 \frac{r}{r}-\frac{r^3}{r^3_0}\right) \frac{(kr_0)^4}{32}\right. \\&\quad \left. +\,\left( 7 \frac{r}{r_0}-6 \frac{r^3}{r_0^3}+2 \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{128} +\cdots \right\} \\&g_{u3}(r)=\frac{\xi ^2}{r_0}\left\{ -\frac{r}{8r_0}( kr_0)^2\right. \\&\quad -\,\left( (2 \beta -9) \frac{r}{r_0}+3 \frac{r^3}{r^3_0}\right) \frac{(kr_0)^4}{32}\\&\quad +\,\left( (116 \beta -230) \frac{r}{r_0} +54 \frac{r^3}{r_0^3}\right. \\&\quad \left. \left. +\,(12 \beta -33) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1152} +\cdots \right\} \\&g_{u5}(r)=\frac{\xi ^2}{r_0}\left\{ \frac{r}{8r_0} (kr_0)^2\right. \\&\quad -\,\frac{r^3}{16r_0^3}(kr_0)^4+\left( (150 \beta -269) \frac{r}{r_0}\right. \\&\quad +\,(20 \beta +198) \frac{r^3}{r_0^3}\\&\quad \left. \left. +\,(-30 \beta -19) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1536} +\cdots \right\} \\&g_{u9}(r)=\frac{\xi ^2}{r_0}\left\{ \frac{r}{8r_0} (kr_0)^2-\left( 6 \frac{r}{r_0}-3 \frac{r^3}{r_0^3}\right) \frac{(kr_0)^4}{16}\right. \\&\quad +\,\left( (2 \beta +561) \frac{r}{r_0}+(60 \beta -422) \frac{r^3}{r_0^3}\right. \\&\quad \left. \left. +\,(-26 \beta +111) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1536} +\cdots \right\} \\ \end{aligned}$$

$$\begin{aligned}&g_{v5}(r)=\frac{\xi ^2}{r_0}\left\{ -\frac{r}{8r_0} (kr_0)^2-\left( (150 \beta -269) \frac{r}{r_0}\right. \right. \\&\quad \left. \left. +\,(40 \beta -36) \frac{r^3}{r_0^3}+(-90 \beta +75) \frac{r^5}{r^5_0}\right) \frac{(kr_0)^6}{1536} +\cdots \right\} \\&g_{v9}(r)=\frac{\xi ^2}{r_0}\left\{ -\frac{r}{8r_0} (kr_0)^2+\left( 3 \frac{r}{r_0}-\frac{r^3}{r^3_0}\right) \frac{(kr_0)^4}{8}\right. \\&\quad -\,\left( (2 \beta +561) \frac{r}{r_0}+(-8 \beta -348) \frac{r^3}{r_0^3}\right. \\&\left. \left. \quad +\,(-14 \beta +49)\frac{r^5}{r_0^5}\right) \frac{(kr_0)^6}{1536}+\cdots \right\} \\&g_{w3}(r)=\frac{\xi ^2}{r_0}\left\{ -\frac{kr_0}{8}+(2 \beta -5)\frac{(kr_0)^3}{32}\right. \\&\quad \left. -\,\left( (116 \beta -293)+(36 \beta -63) \frac{r^4}{r^4_0}\right) \frac{(kr_0)^5}{1152} +\cdots \right\} \\&g_{w9}(r)=\frac{\xi ^2}{r_0}\left\{ -\frac{r^2}{8r_0^2} (kr_0)^3\right. \\&\quad -\,\left( (16 \beta -44) \frac{r^2}{r_0^2}\right. \\&\quad \left. \left. +\,(-8 \beta +31) \frac{r^4}{r^4_0}\right) \frac{(kr_0)^5}{192} +\cdots \right\} \\&g_{q1}(r)=\xi ^2k^2\left\{ \frac{1}{4}-\left( 3-8(\beta -1 -6 \gamma ) \frac{r^2}{r_0^2}\right) \frac{(kr_0)^2}{16}\right. \\&\quad - \, \left( (-12 \beta +29+256\gamma )+(64 \beta -120-832\gamma ) \frac{r^2}{r_0^2}\right. \\&\quad \left. \left. +\,(-12 \beta +33+320\gamma ) \frac{r^4}{r_0^4}\right) \frac{(kr_0)^4}{128} +\cdots \right\} \\&g_{q3}(r)=\xi ^2k^2\left\{ \left( \beta +1+4(6 \gamma -\beta +1) \frac{r^2}{r_0^2}\right) \frac{(kr_0)^2}{8}\right. \\&\quad -\,\left( (-44 \beta +515+2304\gamma )\right. \\&\quad +\,(-720 \beta +252+2880\gamma )\frac{r^2}{r_0^2}\\&\quad \left. \left. +\,(180 \beta -153-576\gamma ) \frac{r^4}{r_0^4}\right) \frac{(kr_0)^4}{1152} +\cdots \right\} \\&g_{q5}(r)=\xi ^2 k^2\left\{ (24 \gamma -4 \beta +7) \frac{r^2}{r_0^2} \frac{(kr_0)^2}{8}\right. \\&\quad -\,\left( (-258 \beta +489+2496\gamma ) \frac{r^2}{r_0^2}\right. \\&\quad \left. \left. +\,(-24 \beta -74-960\gamma ) \frac{r^4}{r^4_0}\right) \frac{(kr_0)^4}{384} +\cdots \right\} \\&g_{q9}(r)=\xi ^2 k^2\left\{ -(24 \gamma -4 \beta +7) \frac{r^2}{r_0^2}\frac{(kr_0)^2}{8}\right. \\&\quad +\,\left( (-218 \beta +289+960\gamma ) \frac{r^2}{r_0^2}\right. \\&\quad \left. \left. +\,(72 \beta -106-192\gamma ) \frac{r^4}{r_0^4}\right) \frac{(kr_0)^4}{384} +\cdots \right\} \end{aligned}$$

with $\gamma =\gamma _{11}+\gamma _{12}+\gamma _{22}$. The other functions g defined in Eqs. 34–37 are equal to zero. Note that the boundary conditions Eq. 11 at $z=0$ and $z=L$ are fulfilled at order $\varepsilon ^2$.