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Nonlinear periodic oscillations in a suspension bridge system under periodic external aerodynamic forces. (English) Zbl 1029.35023

From the text: The following suspension bridge model has been proposed by A. C. Lazar and P. J. McKenna [SIAM Rev. 32, 537-578 (1990; Zbl 0725.73057)] \[ \begin{aligned} m_cu_{tt} & -Qu_{xx}-K(w-u)^+=m_cg+ f_1 (x,t),\;0<x<L,\;t>0,\\ m_bw_{tt} & +EIw_{xxxx}+ K(w-u)^+= m_bg+f_2(x,t),\;0<x< L,\;t>0,\\ u(0,t) & =u(L,t)=0,\\ w(0,t) & =w(L,t)=0,\;w_{xx}(0,t)= w_{xx}(L,T) =0,\end{aligned} \] where \((w-u)^+= \max\{w-u,0\}\); \(m_c\) and \(m_b\) are the mass densities of the cable and the roadbed, respectively; \(Q\) is the coefficient of cable tensile strength; \(EI\) is the roadbed flexural rigidity; \(K\) is Hooke’s constant of the stays; \(f_1\) and \(f_2\) represent the external aerodynamic forces. We are interested in nonlinear periodic oscillations, which are symmetric about \(x=L/2\), \[ \begin{aligned} u(x,t+T)= u(x,t),\quad & w(x,t+T)= w(x,t),\;0\leq x\leq L,\;t>0,\\ u(x,t)=u(L-x,t),\quad & w(x,t)=w(L-x,t),\;0\leq x\leq L,\;t>0, \end{aligned} \] where \(T\) is the period of periodic oscillations. Let the external aerodynamic forces \(\{f_1,f_2\}\) be given by \(f_1(x,t)= \varepsilon h_1(x,t)\), \(f_2(x,t)= \varepsilon h_2(x,t)\), where \(h_1(x,t)\) and \(h_2(x,t)\) are \(\pi\)-periodic functions in \(t\) with \(h_1(-x,t)= h_1(x,t)\) and \(h_2(-x,t)= h_2(x,t)\) for \(-\pi/2<x <\pi/2\), and \(\varepsilon\) is a parameter. By applying the Mountain Pass Theorem to a dual variational formulation, we prove that, under certain assumptions, the system has at least two periodic solutions: one is a near-equilibrium oscillation; and the other is a nonlinear oscillation.

MSC:

35B10 Periodic solutions to PDEs
35L55 Higher-order hyperbolic systems
35L70 Second-order nonlinear hyperbolic equations
74H45 Vibrations in dynamical problems in solid mechanics
74K05 Strings

Citations:

Zbl 0725.73057
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Full Text: DOI

References:

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