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.
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Abbreviations
- A :
-
Area of the cross section
- \(\mathbf{A}\) :
-
Coefficient matrix of Eq. (26)
- \(A_{t_i } \) :
-
Amplitudes of displacement decay curve at time \(t_{i}\)
- \(\mathbf{B}_{i}\) :
-
= \((2i-1)^{2}({\varvec{\Omega }}^{2})^{-1}/4\)
- \(\mathbf{C}\) :
-
Damping matrix
- \(\mathbf{C}_{i}\) :
-
= \((2i-1){\xi }{} \mathbf{I}\)
- D :
-
Damping dissipation energy
- \({EI}_{y}\) :
-
Out-of-plane bending stiffness
- GJ :
-
Torsional stiffness
- H :
-
Rise of the arch
- I :
-
Identity matrix
- \(\mathbf{K}_{e}\) :
-
Out-of-plane stiffness matrix
- \(\mathbf{K}_{g}\) :
-
Out-of-plane stability matrix
- L :
-
Span of the circular arch
- \(\mathscr {L}\) :
-
Lagrangian of the arch and load system
- M :
-
Out-of-plane mass matrix
- \(M_{s}\) :
-
Mass of the central concentrated weight
- \(M(\varphi , t)\) :
-
In-plane bending moment
- \(N(\varphi , t)\) :
-
Axial compressive force
- P(t):
-
Central periodic load
- \(P_\mathrm{cr}\) :
-
Static out-of-plane instability load of the arch
- R :
-
Radius of the circular arch
- T :
-
Kinetic energy
- U :
-
Strain energy
- V :
-
Work done or potential energy
- \(a_{un}, b_{un}\) :
-
Coefficients of the Fourier series
- \(a_{\theta n}, b_{\theta n}\) :
-
Coefficients of the Fourier series
- \(\mathbf{g}(t)\) :
-
Vector formed by \(u_{1}(t)\) and \(\theta _{1}(t)\)
- h :
-
Time step
- m :
-
Uniformly distributed mass of the arch
- \(r_{0}\) :
-
Polar radius of gyration of the cross section
- t :
-
Time
- \(u(\varphi , t)\) :
-
Lateral displacement
- \(u_{1}(t)\) :
-
Central lateral displacement
- \(v({\varphi , t})\) :
-
Radial displacement
- \(w({\varphi , t})\) :
-
Tangential (axial) displacement
- x :
-
Vector of coefficients of the Fourier series
- \(y_{p}\) :
-
Distance of the load above the shear center
- \(T_\mathrm{exc}\) :
-
Period of the in-plane central load
- \(\Delta P(t)\) :
-
Nonlinear inertial force
- \(\Delta v(\varphi , t)\) :
-
Second-order radial displacements
- \(\varvec{\Lambda }\) :
-
=\({\left[ {\beta _0 P_\mathrm{cr} (\mathbf{K}_e +\alpha _0 P_\mathrm{cr} \mathbf{K}_g )^{-1}{} \mathbf{K}_g } \right] }/2\)
- \(\varvec{\Omega }^{2}\) :
-
= \(\mathbf{M}^{-1} \mathbf{K}_e (\mathbf{I}+\alpha _0 P_\mathrm{cr} \mathbf{K}_e^{-1} \mathbf{K}_g )\)
- \(\alpha \) :
-
Included angle of the circular arch
- \(\alpha _{0}\) :
-
Static coefficient of the central load P(t)
- \(\beta \) :
-
Dimensionless excitation amplitude
- \(\beta _{0}\) :
-
Dynamic coefficient of P(t)
- \(\phi (t)\) :
-
Time function related to P(t)
- \(\varphi \) :
-
Angular coordinate
- \(\theta (\varphi , t)\) :
-
Twist torsion of the cross section
- \(\theta _{1}(t)\) :
-
Central twist torsion
- \(\vartheta \) :
-
Frequency of the in-plane central periodic load
- \(\zeta _{}\) :
-
Damping ratio
- \(\xi \) :
-
Mass-proportional damping coefficient
- \(\Xi _M \left( \varphi \right) \) :
-
Distribution of \(M(\varphi , t)\) along the arch
- \(\Xi _N \left( \varphi \right) \) :
-
Distribution of \(N(\varphi , t)\) along the arch
- 0 :
-
Null matrix
- \(()' = \hbox {d}()/\hbox {d}s\) :
-
Derivative with respect to arc length s
- \((^\cdot ) = \hbox {d}()/\hbox {d}t\) :
-
Derivative with respect to time t
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Acknowledgments
This investigation was sponsored by National Natural Science Foundation of China through a Research Project (No. 51578166) awarded to the first author, by Guangzhou government through a Yangcheng Fellowship (No. 1201541551), and by Technology Planning Project of Guangdong Province (No. 2016B050501004) awarded to the first author.
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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
The expression for strain energy U of out-of-plane deformation of the system is given by
The expression for the work V done by the internal bending moment and axial force and by the external load is given by
where
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
with \(\mathbf{B}_i =\frac{\left( {2i-1} \right) ^{2}}{4}({\varvec{\Omega }}^{2})^{-1}\),
\(\hbox {with }{} \mathbf{C}_i =(2i-1)\zeta \mathbf{I}\),
and
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
and
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
By integrating Eq. (46) twice, the second-order in-plane radial displacement \(\Delta v(\varphi , t)\) can be obtained as
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
Subsequently, the second-order radial displacement at the crown of the arch \((\varphi = \alpha /2)\) can be obtained as
The nonlinear inertia force of the arch and central concentrated weight can then be obtained as
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Liu, A., Lu, H., Fu, J. et al. Analytical and experimental studies on out-of-plane dynamic instability of shallow circular arch based on parametric resonance. Nonlinear Dyn 87, 677–694 (2017). https://doi.org/10.1007/s11071-016-3068-7
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DOI: https://doi.org/10.1007/s11071-016-3068-7