Analytical and experimental studies on out-of-plane dynamic instability of shallow circular arch based on parametric resonance

Abstract

Little research about the out-of-plane dynamic stability of arches under in-plane loading has been reported in the literature hitherto. This paper presents analytical and experimental investigations of the out-of-plane dynamic instability of elastic shallow circular arches under an in-plane central concentrated periodic load owing to parametric resonance. Differential equations of out-of-plane motion of shallow arches are established using the Hamilton principle by accounting for the effects of geometric nonlinearity, additional concentrated weights and damping. The analytical solutions of the critical excitation frequencies of the concentrated periodic load for out-of-plane dynamic instability of arches are obtained. The corresponding experimental investigations are also carried out to verify the analytical solutions. Agreements between the analytical and experimental results are very good. In addition, the effects of the central concentrated weight and the in-plane excitation amplitude on out-of-plane dynamic instability of arches are investigated. It is found that as the weight increases, the bandwidth of the critical in-plane excitation frequencies for out-of-plane dynamic instability of the arch decreases. It is also found that the bandwidth of critical frequencies increases with an increase in the excitation amplitude. Furthermore, the nonlinear inertial force is derived, which is essential in determining the out-of-plane parametric resonance. It is shown that the curve of the excitation frequency versus amplitude of out-of-plane vibration bends toward the low-frequency region and that the “traction” out-of-plane instability may occur owing to “amplitude” perturbation. To authors’ knowledge, the analytical solutions and experimental investigations for out-of-plane dynamic instability of arches owing to parametric resonance presented in the paper are first time reported in the literature. The new findings in the paper can provide an in-depth understanding of out-of-plane dynamic instability behavior of arches under a periodic load.

Appendices

Appendix 1: Algebraic expressions of energies and works

Substituting Eqs. (7) and (8) into Eqs. (1)–(3) and performing the integrations lead to algebraic expressions for T, U, and V in terms of $u_{1}(t)$ and $\theta _{1}(t)$. The expression for kinetic energy T of out-of-plane motion of the system can be expressed as

$$\begin{aligned}&T(\dot{u}_1 (t),\dot{\theta }_1 (t))=\left( {\frac{3\alpha Rm}{4}+\frac{M_s }{2}} \right) \nonumber \\&\times \,\left[ {\left( {\dot{u}_1 (t)} \right) ^{2}+r_0^2 \left( {\dot{\theta }_1 (t)} \right) ^{2}} \right] . \end{aligned}$$

(46)

The expression for strain energy U of out-of-plane deformation of the system is given by

$$\begin{aligned}&U(u_1 (t),\theta _1 (t))\nonumber \\&\quad =\frac{EI_y }{4R^{3}}\left[ {\frac{16\pi ^{4}u_1 (t)^{2}}{\alpha ^{3}}-\frac{8\pi ^{2}Ru_1 (t)\theta _1 (t)}{\alpha }+3\alpha R^{2}\theta _1 (t)^{2}} \right] \nonumber \\&\qquad +\frac{GT\pi ^{2}}{R^{3}\alpha }\left[ {u_1 (t)^{2}-2u_1 (t)\theta _1 (t)R+\theta _1 (t)^{2}\hbox {R}^{2}} \right] \end{aligned}$$

(47)

The expression for the work V done by the internal bending moment and axial force and by the external load is given by

$$\begin{aligned}&V(u_1 (t),\theta _1 (t))=P(t)\left[ {k_{uu} u_1 (t)^{2}} \right. \nonumber \\&\quad \left. {\;+\,2k_{u\theta } u_1 (t)\theta _1 (t)+\left( k_{\theta \theta } +\frac{y_p }{2}\right) \theta _1 (t)^{2}} \right] . \end{aligned}$$

(48)

where

$$\begin{aligned} k_{uu}= & {} -\frac{2\pi ^{2}}{R^{2}\alpha ^{2}}\left( {1+\frac{r_0^2 }{R^{2}}} \right) \int _0^\alpha {\Xi _N \left( \varphi \right) \sin ^{2}\left( {{2\pi \varphi }/\alpha } \right) } d\varphi \nonumber \\ \end{aligned}$$

(49)

$$\begin{aligned} k_{u\theta }= & {} \frac{4\pi ^{2}}{\alpha ^{2}}\int _0^\alpha {\Xi _M \left( \varphi \right) \cos \left( {2{\pi \varphi }/\alpha } \right) \sin ^{2}\left( {{\pi \varphi }/\alpha } \right) } d\varphi \nonumber \\&+\frac{\pi ^{2}r_0^2 }{R^{2}\alpha ^{2}}\int _0^\alpha {\Xi _N \left( \varphi \right) \sin ^{2}\left( {{2\pi \varphi }/\alpha } \right) } d\varphi \end{aligned}$$

(50)

$$\begin{aligned} k_{\theta \theta }= & {} \int _0^\alpha {2\Xi _M R} \sin ^{4}(\pi \varphi /\alpha )\hbox {d}\varphi \nonumber \\&-\int _0^\alpha {2\Xi _N \frac{r_0^2 \pi ^{2}}{R^{2}\alpha ^{2}}} \sin ^{2}(2\pi \varphi /\alpha )\hbox {d}\varphi \end{aligned}$$

(51)

Appendix 2. Matrices $\hbox {A}_{11}, \hbox {A}_{12}, \hbox {A}_{21}, \hbox {A}_{22}$

Matrices $\mathbf{A}_{11}, \mathbf{A}_{12}, \mathbf{A}_{21}$, and $\mathbf{A}_{22 }$ in Eq. (28) are given by

$$\begin{aligned} \mathbf{A}_{11} =\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \cdots &{} {\mathbf{I}-\mathbf{B}_i \vartheta ^{2}}&{} \cdots &{} \mathbf{0}&{} \mathbf{0} \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \cdots &{} \mathbf{0}&{} \cdots &{} {\mathbf{I}-\mathbf{B}_2 \vartheta ^{2}}&{} {-{\varvec{\Lambda }}} \\ \cdots &{} \mathbf{0}&{} \cdots &{} {-{\varvec{\Lambda }}}&{} {\mathbf{I}+{\varvec{\Lambda }}-\mathbf{B}_1 \vartheta ^{2}} \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(52)

with $\mathbf{B}_i =\frac{\left( {2i-1} \right) ^{2}}{4}({\varvec{\Omega }}^{2})^{-1}$,

$$\begin{aligned} \mathbf{A}_{12} =\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \mathbf{0}&{} \mathbf{0}&{} \cdots &{} {-\mathbf{C}_i \vartheta }&{} \cdots \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \mathbf{0}&{} {-\mathbf{C}_2 \vartheta }&{} \cdots &{} \mathbf{0}&{} \cdots \\ {-\mathbf{C}_1 \vartheta }&{} \mathbf{0}&{} \cdots &{} \mathbf{0}&{} \cdots \\ \end{array} }} \right] \end{aligned}$$

(53)

$\hbox {with }{} \mathbf{C}_i =(2i-1)\zeta \mathbf{I}$,

$$\begin{aligned} \mathbf{A}_{21} =\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \cdots &{} \mathbf{0}&{} \cdots &{} \mathbf{0}&{} \mathbf{C}_1 \vartheta \\ \cdots &{} \mathbf{0}&{} \cdots &{} \mathbf{C}_2 \vartheta &{} \mathbf{0} \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \cdots &{} \mathbf{C}_i \vartheta &{} \cdots &{} \mathbf{0}&{} \mathbf{0} \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \end{array}}} \right] , \end{aligned}$$

(54)

and

$$\begin{aligned} \mathbf{A}_{22} =\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\mathbf{I}-{\varvec{\Lambda }}-\mathbf{B}_1 \vartheta ^{2}}&{} {-{\varvec{\Lambda }}}&{} \cdots &{} \mathbf{0}&{} \cdots \\ {-{\varvec{\Lambda }}}&{} {\mathbf{I}-\mathbf{B}_2 \vartheta ^{2}}&{} \cdots &{} \mathbf{0}&{} \cdots \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \mathbf{0}&{} \mathbf{0}&{} \cdots &{} {\mathbf{I}-\mathbf{B}_i \vartheta ^{2}}&{} \cdots \\ \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(55)

where 0 is the null matrix and I is the identity matrix.

Appendix 3: Matrices $\hbox {A}_{1}, \hbox {A}_{2}$, and $\hbox {A}_{3}$

Matrices $\mathbf{A}_{1}, \mathbf{A}_{2}$, and $\mathbf{A}_{3}$ in Eq. (29) are given by

$$\begin{aligned} \mathbf{A}_1= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{I}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{I}&{} {-{\varvec{\Lambda }}}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} {-{\varvec{\Lambda }}}&{} {\mathbf{I}+{\varvec{\Lambda }}}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} {\mathbf{I}-{\varvec{\Lambda }}}&{} {-{\varvec{\Lambda }}}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} {-{\varvec{\Lambda }}}&{} \mathbf{I}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{I}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(56)

$$\begin{aligned} \mathbf{A}_2= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} {-\mathbf{C}_i }&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {-\mathbf{C}_2 }&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} {-\mathbf{C}_1 }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\mathbf{C}_1 }&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} {\mathbf{C}_2 }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} {\mathbf{C}_i }&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ \end{array} }} \right] \nonumber \\ \end{aligned}$$

(57)

and

$$\begin{aligned} \mathbf{A}_3 =\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} {\mathbf{B}_i }&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} {\mathbf{B}_2 }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\mathbf{B}_1 }&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} {\mathbf{B}_1 }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\mathbf{B}_2 }&{} {\ldots }&{} \mathbf{0}&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ {\ldots }&{} \mathbf{0}&{} {\ldots }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} {\ldots }&{} {\mathbf{B}_i }&{} {\ldots } \\ {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots }&{} {\ldots } \\ \end{array} }} \right] \end{aligned}$$

(58)

Appendix 4: Nonlinear inertial force

The out-of-plane deformations can produce second-order in-plane displacements $\Delta v(\varphi , t)$, which will induce nonlinear inertia in association with the mass of the arch and additional central weight. It can be shown [24] that the second-order in-plane curvature produced by the out-of-plane deformations can be expressed as

$$\begin{aligned} \frac{\partial ^{2}\Delta v(\varphi ,t)}{\partial \varphi ^{2}}\approx \frac{\partial ^{2}u(\varphi ,t)}{\partial \varphi ^{2}}\theta (\varphi ,t). \end{aligned}$$

(59)

By integrating Eq. (46) twice, the second-order in-plane radial displacement $\Delta v(\varphi , t)$ can be obtained as

$$\begin{aligned} \Delta v(\varphi ,t)= & {} \int \int {\frac{\partial ^{2}u(\varphi ,t)}{\partial \varphi ^{2}}} \theta (\varphi ,t)\hbox {d}\varphi \hbox {d}\varphi _1 \nonumber \\&+\int \int {C\varphi } \hbox {d}\varphi \hbox {d}\varphi _1 +\int \int {C_1 } \hbox {d}\varphi \hbox {d}\varphi _1.\nonumber \\ \end{aligned}$$

(60)

Substituting the lateral and torsional displacements $u(\varphi , t)$ and $\theta (\varphi , t)$ given by Eqs. (7) and (8) into Eq. (47) and considering the fixed boundary conditions at both ends of the arch lead to

$$\begin{aligned} \Delta v(\varphi ,t)\!=\!\frac{u_1 (t)\theta _1 (t)}{8}\left[ {7-8\cos \frac{2\pi \varphi }{\alpha }\!+\!\cos \frac{4\pi \varphi }{\alpha }} \right] .\nonumber \\ \end{aligned}$$

(61)

Subsequently, the second-order radial displacement at the crown of the arch $(\varphi = \alpha /2)$ can be obtained as

$$\begin{aligned} \Delta v(\alpha /2,t)=2u_1 (t)\theta _1 (t). \end{aligned}$$

(62)

The nonlinear inertia force of the arch and central concentrated weight can then be obtained as

$$\begin{aligned} \Delta P(t)= & {} -M_s \Delta \ddot{v}(\alpha /2,t)-\int _0^\alpha m \Delta \ddot{v}(\varphi ,t)R\hbox {d}\varphi \nonumber \\= & {} -2M_s \left( {\ddot{u}_1 (t)\theta (t)+2\dot{u}_1 (t)\dot{\theta }_1 (t)+u_1 (t)\ddot{\theta }_1 (t)} \right) \nonumber \\&+\frac{7R\alpha m}{8}\left( {\ddot{u}_1 (t)\theta (t)+2\dot{u}_1 (t)\dot{\theta }_1 (t)+u_1 (t)\ddot{\theta }_1 (t)} \right) .\nonumber \\ \end{aligned}$$

(63)