Accurate backbone curves and dynamic failure for large-amplitude axisymmetric vibrations of imperfect circular plates

Abstract

This paper deals with the solution of modified-Duffing ordinary differential equation for large-amplitude vibrations of imperfect circular plate. Four types of boundary conditions are considered as well as viscous damping. Lindstedt’s perturbation technique and Runge–Kutta method are applied. The solution from two methods are plotted and compared for a validity check. Lindstedt’s perturbation technique is proved to be accurate for sufficiently small vibration amplitude and imperfection. The results from Runge–Kutta method is plotted to form a backbone curve except for the case with clamped and zero radial stress boundary condition. Instead of expected softening–hardening process, the curve is only softening thus no longer backbone. More importantly, a dynamic failure is noticed when initial vibration amplitude grows under this boundary condition. A geometric imperfection will further help to trigger this failure at a smaller amplitude. This finding has served a new way to judge dynamic failure mode and is valuable for structure design concerning vibration.

Appendix: Lindstedt’s perturbation solution to Duffing equation with a quadratic term

The second order non-linear Duffing ordinary differential equation is

$$ {\text{w}}\left({\text{t}} \right),_{\text{tt}}\,+\,{\text{k}}\left[{{\text{w}}\left({\text{t}} \right) + \upepsilon{\text{a}}_{2} {\text{w}}\left({\text{t}} \right)^{2} + \text{}{\text{w}}\left({\text{t}} \right)^{3}} \right] = 0 $$

(30)

By changing the variable, $\uptau = {{\Omega}}{\rm t}$, Eq. (30) will turn into

$$ {{\Omega}}^{2} {\text{w}}\left({\uptau} \right),_{{{{\uptau\uptau}}}}\,+\,{\text{k}}\left[{{\text{w}}\left({\uptau} \right) + \upepsilon{\text{a}}_{2} {\text{w}}\left({\uptau} \right)^{2} + \upepsilon{\text{w}}\left({\uptau} \right)^{3}} \right] = 0 $$

(31)

The periodic solution ${\text{w}}\left( {\uptau} \right)$ and the associated frequency are assumed to be of the forms

$$ {\text{w}}\left({\uptau} \right) = {\text{w}}_{0} \left({\uptau} \right) + \upepsilon{}{\text{w}}_{1} \left({\uptau} \right) + \upepsilon{}^{2} {\text{w}}_{2} \left({\uptau} \right) + \cdots, $$

(32)

$$ { \Omega} = {\Omega}_{0} + \upepsilon{\Omega}_{1} + \upepsilon^{2} {\Omega}_{2} + \cdots. $$

(33)

Furthermore, the non-linear terms in Eq. (31) are also expanded in a power series in $\upepsilon$. Equating terms incolving $\upepsilon^0$, $\upepsilon$, and $\upepsilon^2$ and dividing through by $k$ (defined to be ${{\Omega }}_{0}^{2}$), one obtains

$$ {\text{w}}_{0} \left( {\uptau} \right),_{{{{\uptau\uptau}}}} + {\text{w}}_{0} \left( {\uptau} \right) = 0 $$

(34)

$$ \begin{aligned} {\text{w}}_{1} \left( {\uptau} \right),_{{{{\uptau\uptau}}}} + {\text{w}}_{1} \left( {\uptau} \right) & = - {\text{w}}_{0} \left( {\uptau} \right) - {\text{w}}_{0} \left( {\uptau} \right)^{3} - {\text{a}}_{2} {\text{w}}_{0} \left( {\uptau} \right)^{2} \\ & \quad - 2\left( {{{\Omega }}_{1} /{{\Omega }}_{0} } \right){\text{w}}_{0} \left( {\uptau} \right),_{{{{\uptau \uptau}}}} \\ \end{aligned} $$

(35)

$$ \begin{aligned} {\text{w}}_{2} \left( {\uptau} \right),_{{{{\uptau\uptau}}}} + {\text{w}}_{2} \left( {{\uptau }} \right) & = \left[ { - 2{\text{a}}_{2} {\text{w}}_{0} \left( {\uptau} \right) - 3{\text{w}}_{0} \left( {\uptau} \right)^{2} } \right]{\text{w}}_{1} \left( {\uptau} \right) \\ & \quad - \left[ {2\left( {{{\Omega }}_{2} /{{\Omega }}_{0} } \right) + \left( {{{\Omega }}_{1} /{{\Omega }}_{0} } \right)^{2} } \right]{\text{w}}_{0} \left( {\uptau} \right),_{{{{\uptau\uptau}}}} \\ & \quad - 2\left( {{{\Omega }}_{1} /{{\Omega }}_{0} } \right){\text{w}}_{1} \left( {\uptau} \right),_{{{{\uptau\uptau}}}} \ldots \\ \end{aligned} $$

(36)

and the initial conditions are

$$ {\text{w}}_{0} \left( {\uptau} \right),_{{{\uptau\uptau}}} = 0, \quad {\text{w}}_{1} \left( {\uptau} \right),_{{{\uptau\uptau}}} = 0,\quad {\text{w}}_{2} \left( {\uptau} \right),_{{{\uptau\uptau}}} = 0, \ldots $$

(37)

The solution of Eq. (34) is

$$ {\text{w}}_{0} \left( {\uptau} \right) = {\text{Acos}}\left( {\uptau} \right) $$

(38)

Substituting ${\text{w}}_{0} \left( {\uptau} \right)$ into Eq. (35), one obtains

$$ \begin{aligned} {\text{w}}_{1} \left( {\uptau} \right),_{{{\uptau\uptau}}}\,+\,\,{\text{w}}_{1} \left( {\uptau} \right) & = - \left( {{\text{a}}_{2} {\text{A}}^{2} /2} \right)\left[ {1 + \cos \left( {2{\uptau}} \right)} \right] \\ & \quad + \left[ {2\left( {{{\Omega }}_{1} /{{\Omega }}_{0} } \right){\text{A}} - \left( {3{\text{A}}^{3} /4} \right)} \right]{ \cos }\left( {\uptau} \right) \\ & \quad - \left( {{\text{A}}^{3} /4} \right){ \cos }\left( {3{\uptau}} \right) \\ \end{aligned} $$

(39)

In order to avoid secular terms, the coefficient of ${ \cos }\left( {\uptau} \right)$ is set to zero so that

$$ {{\Omega }}_{1} = \left( {3/8} \right){{\Omega }}_{0} {\text{A}}^{2} $$

(40)

Accordingly, the solution to the differential equation for ${\text{w}}_{1} \left( {\uptau} \right)$ is

$$ {\text{w}}_{1} \left( {\uptau} \right) = - \left( {{\text{a}}_{2} {\text{A}}^{2} /2} \right) + \left( {{\text{a}}_{2} {\text{A}}^{2} /6} \right)\cos \left( {2{\uptau}} \right) + \left( {{\text{A}}^{3} /32} \right){ \cos }\left( {3{\uptau}} \right) $$

(41)

Finally, substituting the known forms for ${\text{w}}_{0} \left( {\uptau} \right)$, ${\text{w}}_{1} \left( {\uptau} \right)$ and ${{\Omega }}_{1}$ into equation 35, one obtains

$$ \begin{aligned} &{\text{w}}_{2} \left( {\uptau} \right),_{{{\uptau\uptau}}} +\, {\text{w}}_{2} \left( {\uptau} \right) = \left[2\left( {{{\Omega }}_{2} /{{\Omega }}_{0} } \right){\text{A}} \right.\\ & \quad\left.+\, \left( {5/6} \right){\text{a}}_{2}^{2} {\text{A}}^{3} + \left( {15/128} \right){\text{A}}^{5} \right]{ \cos }\left( {\uptau} \right) \\ & \quad -\,\left[ {\left( {4/3} \right)\left( {{{\Omega }}_{1} /{{\Omega }}_{0} } \right){\text{a}}_{2} {\text{A}}^{2} - \left( {{\text{a}}_{2} /32} \right){\text{A}}^{4} } \right]{ \cos }\left( {2{\uptau}} \right) \\ & \quad + \left[ {\left( {21{\text{A}}^{5} /128} \right) - \left( {{\text{a}}_{2}^{2} {\text{A}}^{3} /6} \right)} \right]{ \cos }\left( {3{\uptau}} \right) \\ & \quad \times \left( {{\text{a}}_{2} {\text{A}}^{4} /32} \right){ \cos }\left( {4{\uptau}} \right) - \left( {3{\text{A}}^{5} /128} \right){ \cos }\left( {5{\uptau}} \right) \\ \end{aligned} $$

(42)

doneAgain, setting the coefficient of ${ \cos }\left( {\uptau} \right)$ to zero yields

$$ {{\Omega }}_{2} /{{\Omega }}_{0} = - \left[ {\left( {15{\text{A}}^{4} /256} \right) + \left( {5/12} \right){\text{a}}_{2}^{2} {\text{A}}^{2} } \right] $$

(43)

Furthermore, the solution to the differential equation for ${\text{w}}_{2} \left( {\uptau} \right)$ is

$$ \begin{aligned} {\text{w}}_{2} \left( {\uptau} \right) & = \left({ -1/3} \right)\left[ \left( {4/3} \right) \left({{{\Omega }}_{1} /{{\Omega }}_{0} } \right)\left( {{\text{a}}_{2} {\text{A}}^{2} } \right) \right.\\ &\quad\left.-\,\left( {{\text{a}}_{2} {\text{A}}^{4} /32} \right) \right]{ \cos }\left( {2{\uptau}} \right) \\ & \quad - \left( {1/8} \right)\left[ {\left( {21/128} \right){\text{A}}^{5} - \left( {{\text{a}}_{2}^{2} {\text{A}}^{3} /6} \right)} \right]{ \cos }\left( {3{\uptau}} \right) \\ & \quad + \left( {1/15} \right)\left( {{\text{a}}_{2} {\text{A}}^{4} /32} \right){ \cos }\left( {4{\uptau}} \right)\\ &\quad +\, \left( {{\text{A}}^{5} /10} \right){ \cos }\left( {5{\uptau}} \right) \\ \end{aligned} $$

(44)

The solution is obtained by assembling ${\text{w}}_{0} \left( {\uptau} \right)$, ${\text{w}}_{0} \left( {\uptau} \right)$ and ${\text{w}}_{0} \left( {\uptau} \right)$ in Eq. (32) and replacing $\uptau$ by ${{\Omega t}} + {{\upvarphi }}$. Thus, for a given set of initial conditions ${\text{w}}\left( {{\text{t}} = 0} \right)$ and ${\text{w}},_{\text{t}} \left( {{\text{t}} = 0} \right)$, the value of the amplitude A and the phase angle ${{\upvarphi }}$ can be found. Finally, substituting ${{\Omega }}_{1}$ and ${{\Omega }}_{2}$ into Eq. (33) shows that the ratio of the non-linear frequency to the linear frequency is related to the vibration amplitude by

$$ {{\Omega}}/{{\Omega}}_{0} = 1 + \left[{\left({3\upepsilon/8} \right) - \left({5/12} \right){\text{a}}_{2}^{2} \upepsilon^{2}} \right]{\text{A}}^{2} - \left({15\upepsilon^{2}/256} \right){\text{A}}^{4} $$

(45)