Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by th...Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The unde- termined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.展开更多
On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:\$\$u\-\{tt\}+u\-\{xxxx\}+f(u)=0,\$\$where f(u)=u\++-1 .Here the bridge is seen as a vib...On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:\$\$u\-\{tt\}+u\-\{xxxx\}+f(u)=0,\$\$where f(u)=u\++-1 .Here the bridge is seen as a vibrating beam supported by cables,which are treated as a spring with a one\|sided restoring force.The existence of a traveling wave solution to the above piece\|wise linear equation has been proved by solving the equation explicitly (McKenna & Walter in 1990).Recently the result has been extended to a group of equations with more general nonlinearities such as f(u)=u\++-1+g(u) (Chen & McKenna,1997).However,the restrictions on g(u) do not allow the resulting restoring force function to increase faster than the linear function u-1 for u >1.Since an interesting “multiton” behavior,that is ,two traveling waves appear to emerge intact after interacting nonlinearly with each other,has been observed in numerical experiments for a fast\|increasing nonlinearity f(u)=e u-1 -1 ,it hints that the conclusion of the existence of a traveling wave solution with fast\|increasing nonlinearities shall be valid as well.\;In this paper,the restoring force function of the form f(u)=u·h(u)-1 is considered.It is shown that a traveling wave solution exists when h(u)≥1 for u≥1 (with other assumptions which will be detailed in the paper),and hence allows f to grow faster than u-1 .It is shown that a solution can be obtained as a saddle point in a variational formulation.It is also easy to construct such fast\|increasing f(u) 's for more numerical tests.展开更多
基金Project supported by the State Key Program of the National Natural Science Foundation of China(No.11232009)the National Natural Science Foundation of China(Nos.11372171 and 11422214)
文摘Under the 3：1 internal resonance condition, the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied. The viscoelastic behaviors of the traveling beam are described by the standard linear solid model, and the material time derivative is adopted in the viscoelastic constitutive relation. The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes. For the first time, the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam. The unde- termined coefficient method is used to approximately establish the real modal functions. The approximate analytical results are confirmed by the Galerkin truncation. Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses. To illustrate the effect of the internal resonance, the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.
基金Project supported by National Natural Science Foundation of China! (19701029) by Outstanding Young Teacher Foundation of Chi
文摘On studying traveling waves on a nonlinearly suspended bridge,the following partial differential equation has been considered:\$\$u\-\{tt\}+u\-\{xxxx\}+f(u)=0,\$\$where f(u)=u\++-1 .Here the bridge is seen as a vibrating beam supported by cables,which are treated as a spring with a one\|sided restoring force.The existence of a traveling wave solution to the above piece\|wise linear equation has been proved by solving the equation explicitly (McKenna & Walter in 1990).Recently the result has been extended to a group of equations with more general nonlinearities such as f(u)=u\++-1+g(u) (Chen & McKenna,1997).However,the restrictions on g(u) do not allow the resulting restoring force function to increase faster than the linear function u-1 for u >1.Since an interesting “multiton” behavior,that is ,two traveling waves appear to emerge intact after interacting nonlinearly with each other,has been observed in numerical experiments for a fast\|increasing nonlinearity f(u)=e u-1 -1 ,it hints that the conclusion of the existence of a traveling wave solution with fast\|increasing nonlinearities shall be valid as well.\;In this paper,the restoring force function of the form f(u)=u·h(u)-1 is considered.It is shown that a traveling wave solution exists when h(u)≥1 for u≥1 (with other assumptions which will be detailed in the paper),and hence allows f to grow faster than u-1 .It is shown that a solution can be obtained as a saddle point in a variational formulation.It is also easy to construct such fast\|increasing f(u) 's for more numerical tests.
