Nonlinear vibration analysis of a shallow arch coupled with an elastically constrained rigid body

Abstract

Nonlinear vibration of shallow arch-rigid body (elastically supported) coupled system is asymptotically investigated, for refined modeling of dynamic interactions involved with a complex arch structure built in weak soils (implying non-ideal foundation). Through simplifying the weakly moving arch boundary (in weak soils) as a rigid body supported by elastic springs, an arch-rigid body coupled model is proposed in a direct perturbation formulation. The arch-rigid body two-way coupling coefficients are asymptotically derived. Two important factors turn out to be quantitative measures of coupling intensity, i.e., the moment of inertia and height of rigid body, which play key roles in arch-rigid body coupled dynamics. Furthermore, nonlinear arch-rigid body primarily resonant responses are fully investigated with stability/bifurcation characteristics determined, including both hardening and softening cases. A comparative study between the coupled model and ideal (i.e., uncoupled) arch dynamics demonstrates effectiveness of the newly proposed coupled model. Besides, an arch-foundation (boundary oscillator) coupled model, as a degenerate case of the arch-rigid body coupled model, is also investigated for comparison, partially unveiling limits of the previous arch-foundation (oscillator) model.

Appendices

Appendix A

The arch-rigid body coupled equations can be established by employing Newton's law (or extended Hamilton principle). Explicitly, for the rigid body we have:

$$ \tilde{J}\ddot{\tilde{\varphi }} + 2\tilde{\nu }\dot{\tilde{\varphi }} + \tilde{K}_{\varphi } \tilde{\varphi } = - \frac{EA}{{\tilde{l}}}\left[ {\tilde{U}\left( {\tilde{t}} \right) + \int_{0}^{{\tilde{l}}} {\tilde{y}^{\prime}\tilde{w}^{\prime}{\text{d}}\tilde{x} + \int_{0}^{{\tilde{l}}} {\frac{1}{2}\tilde{w}^{{\prime}{2}} {\text{d}}\tilde{x}} } } \right] \cdot \tilde{h} $$

(50)

After introducing scaling rules Eq. (4), and substituting them into Eq. (50), we have

$$ \begin{aligned} & J \cdot \rho Al^{3} \cdot \ddot{\varphi } \cdot \frac{{r^{2} }}{{l^{2} }} \cdot \frac{EI}{{\rho Al^{4} }} + 2J \cdot \nu \cdot \sqrt {\rho AEIl^{2} } \cdot \dot{\varphi } \cdot \frac{{r^{2} }}{{l^{2} }} \cdot \sqrt {\frac{EI}{{\rho Al^{4} }}} + K_{\varphi } \cdot \frac{EI}{l} \cdot \varphi \cdot \frac{{r^{2} }}{{l^{2} }} \\ & \quad = - \frac{EA}{l} \cdot \left[ {U \cdot \frac{{r^{2} }}{l} + \int_{0}^{1} {\left( {\frac{{ry^{\prime}}}{l} \cdot \frac{{rw^{\prime}}}{l} \cdot l} \right){\text{d}}x + \int_{0}^{1} {\left( {\frac{{rw^{\prime}}}{l} \cdot \frac{{rw^{\prime}}}{l} \cdot l} \right){\text{d}}x} } } \right] \cdot hl \\ \end{aligned} $$

(51)

leading to rigid body's dimensionless equation (i.e., Eq. (7))

$$ \ddot{\varphi } + 2\nu \dot{\varphi } + \frac{{K_{\varphi } }}{J}\varphi = - \frac{h\lambda }{J}\left[ {U + \int_{0}^{1} {\left( {y^{\prime}w^{\prime}} \right){\text{d}}x + \int_{0}^{1} {\frac{1}{2}w^{{\prime}{2}} {\text{d}}x} } } \right], \, U\left( t \right) = \varphi \left( t \right)h $$

(52)

Appendix B

The two second-order shape functions in Eq. (15), i.e., R₁(x) and R₂(x), are governed by the following boundary value problems (BVPs)

$$ - 4\omega_{n}^{2} R_{1} \left( x \right) + R_{1}^{(4)} \left( x \right) - y^{\prime\prime}\int_{0}^{1} {y^{\prime}} R^{\prime}_{1} \left( x \right){\text{d}}x = \Pi_{1} \left( x \right) $$

(53)

$$ R_{2}^{(4)} \left( x \right) - y^{\prime\prime}\int_{0}^{1} {y^{\prime}} R^{\prime}_{2} \left( x \right){\text{d}}x = \Pi_{2} \left( x \right) $$

(54)

with homogeneous boundary condition $R_{n} \left( 0 \right) = R_{n} \left( 1 \right) = 0$ and $R^{\prime}_{n} \left( 0 \right) = R^{\prime}_{n} \left( 1 \right) = 0$. The nonlinear coefficients are $\Pi_{1} \left( x \right) = \Pi_{2} \left( x \right) = \phi^{\prime\prime}_{n} \left( x \right)\int_{0}^{1} {y^{\prime}} \phi^{\prime}_{n} \left( x \right){\text{d}}x + \frac{1}{2}y^{\prime\prime}\int_{0}^{1} {\phi_{n}^{\prime 2} \left( x \right)} {\text{d}}x$. The shape functions R₁(x) and R₂(x) in Eqs. (53) and (54) are then solved by modal expansion method, namely

$$ R_{1} \left( x \right) = \sum\limits_{k = 1}^{\infty } {\frac{{\phi_{k} \left( x \right)\int_{0}^{1} {\Pi_{1} \left( x \right)\phi_{k} \left( x \right){\text{d}}x} }}{{ - 4\omega_{n}^{2} + \omega_{k}^{2} }}} ,\quad R_{2} \left( x \right) = \sum\limits_{k = 1}^{\infty } {\frac{{\phi_{k} \left( x \right)\int_{0}^{1} {\Pi_{2} \left( x \right)\phi_{k} \left( x \right){\text{d}}x} }}{{\omega_{k}^{2} }}} $$

(55)

The first three linear modal frequencies and mode shape functions of arches with $b = 1.0$ and $b = 2.0$ are presented in Fig. 

Fig. 18

Mode shapes, frequencies and the shape functions of typical arches with b = 1.0 and b = 2.0

18. Furthermore, typical second-order shape functions R₁(x) and R₂(x) are also depicted in Fig. 18. The shape functions R₁(x) and R₂(x) have been used in Eqs. 53 and 54.

Appendix C

The nonlinear function used in Eq. (16) is defined by

$$ \begin{aligned} \Upsilon_{1} \left( x \right) & = \frac{3}{2}\phi^{\prime\prime}_{n} \int_{0}^{1} {\phi_{n}^{\prime 2} } {\text{d}}x + \phi^{\prime\prime}_{n} \int_{0}^{1} {y^{\prime}R^{\prime}_{1} } {\text{d}}x + y^{\prime\prime}\int_{0}^{1} {\phi^{\prime}_{n} R^{\prime}_{1} } {\text{d}}x + R^{\prime\prime}_{1} \int_{0}^{1} {\phi^{\prime}_{n} y^{\prime}} {\text{d}}x \\ & \quad + 2\phi^{\prime\prime}_{n} \int_{0}^{1} {y^{\prime}R^{\prime}_{2} } {\text{d}}x + 2y^{\prime\prime}\int_{0}^{1} {\phi^{\prime}_{n} R^{\prime}_{2} } {\text{d}}x + 2R^{\prime\prime}_{2} \int_{0}^{1} {\phi^{\prime}_{n} y^{\prime}} {\text{d}}x \\ \end{aligned} $$

(56)