A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load
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摘要
The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.
The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.
The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.
引文
Abu-Hilal M(2006),"Dynamic Response of a Double Euler-Bernoulli Beam Due to a Moving Constant Load,"Journal of Sound and Vibration, 297:477-491.
Arani AG,Shiravand A, Rahi M and Kolahchi R(2012),"Nonlocal Vibration of Coupled DLGS Systems Embedded on Visco-Pasternak Foundation,"Physica B:Condensed Matter, 407(21):4123-4131
Basu D and Rao NSV(2013),"Analytical Solutions for Euler-Bernoulli Beam on Visco-Elastic Foundation Subjected to Moving Load,"International Journal for Numerical and Analytical Methods in Geomechanics,37(8):945-960.
Guan XF, Yu HT and Tian X(2016),"A Stochastic Second-order and Two-scale Thermo-mechanical Model for Strength Prediction of Concrete Materials,"International Journal for Numerical Methods in Engineering, 108(8):885-901.
Gurgoze M and Erol H(2004),"On Laterally Vibrating Beams Carrying Tip Masses, Coupled by Several Double Spring-Mass Systems,"Journal of Sound and Vibration,269:431-438.
Hamada TR, Nakayama H and Hayashi K(1983),"Free and Forced Vibration of Elastically Connected Double-beam Systems,"Bulletin of the Japan Society of Mechanics Engineers, 1936-1942.
Jang TS(2013),"A New Semi-analytical Approach to Large Deflections of Bernoulli-Euler-v. Karman Beams on a Linear Elastic Foundation:Nonlinear Analysis of Infinite Beams,"International Journal of Mechanical Sciences, 66:22-32.
Jang TS(2017),"A New Dispersion-relation Preserving Method for Integrating the Classical Boussinesq Equation,"Communications in Nonlinear Science and Numerical Simulation, 43:118-138.
Kenney JT(1954),"Steady-State Vibrations of Beam on Elastic Foundation for Moving Load,"Journal of Applied Mechanics, 21(4):359-364.
Kim SM and Roesset JM(2003),"Dynamic Responseof a Beam on a Frequency-Independent Damped Elastic Foundation to Moving Load,"Canadian Journal of Civil Engineering, 30(2):460-467.
Li MG, Yu HT, Wang JH, Xia XH and Chen JJ(2015),"A Multiscale Coupling Approach Between Discrete Element Method and Finite Difference Method for Dynamic Analysis,"International Journal for Numerical Methods in Engineering, 102(1):1-21.
Mathews PM(1958),"Vibration of a Beam on Elastic Foundation,"Journal of Applied Mathematics and Mechanics, 38(3-4):105-115.
Mathews PM(1959),"Vibration of a Beam on Elastic Foundation,"Journal of Applied Mathematics and Mechanics, 39(1-2):13-19.
Oniszczuk Z(2000),"Free Transverse Vibrations of Elastically Connected Simply Supported Double-Beams Complex System,"Journal of Sound and Vibration, 232:387-403.
Oniszczuk Z(2003),"Forced Transverse Vibrations of an Elastically Connected Complex Simply Supported Double-Beam System,"Journal of Sound and Vibration,264:273-286.
Palmeri A and Adhikari S(2011),"A Galerkin-Type State-Space Approach for Transverse Vibrations of Slender Double-beam Systems with Viscoelastic Inner Layer,"Journal of Sound Vibration, 330:6372-6386.
Saito J and Terasawa T(1980),"Steady-State Vibrations of a Beam on a Pasternak Foundation for Moving Loads,"Journal of Applied Mechanics, 47(4):879-883.
Sun L(2001),"A Closed-Form Solution of a BernoulliEuler Beam on a Viscoelastic Foundation under Harmonic Line Loads,"Journal of Sound and Vibration,242(4):619-627.
Sun L(2002),"A Closed-Form Solution of Beam on Viscoelastic Subgrade Subjected to Moving Loads,"Computer and Structure, 80(1):1-8.
Sun L(2003),"An Explicit Representation of Steady State Response of a Beam on an Elastic Foundation to Moving Harmonic Line Loads,"International Journal for Numerical and Analytical Methods in Geomechanics,27(1):69-84.
Vu HV,Ordonez AM and Karnopp BH(2000),"Vibration of a Double-Beam System,"Journal of Sound and Vibration, 229:807-822.
Wu YX and Gao YF(2015),"Analytical Solutions for Simply Supported Viscously Damped Double-Beam System under Moving Harmonic Loads,"Journal of Engineering Mechanics, ASCE, 141(7):04015004.
Wu YX and Gao YF(2016),"Dynamic Response of a Simply Supported Viscously Damped Double-Beam System under the Moving Oscillator,"Journal of Sound and Vibration, 384:194-209.
Yan X, Yu HT, Yuan Y and Yuan JY(2015),"Multi-Point Shaking Table Test of the Free Field under Non-uniform Earthquake Excitation,"Soils and Foundations, 55(5):985-1000.
Yuan Y,Yu HT. Li C,Yan X and Yuan JY(2018),"Multi-Point Shaking Table Test for Long Tunnels Subjected to Non-Uniform Seismic Loadings-PartⅠ:Theory and Validation,"Soil Dynamics and Earthquake Engineering, 108:177-186.
Yu HT,Cai C,Guan XF and Yuan Y(2016),"Analytical Solution for Long Lined Tunnels Subjected to Travelling Loads,"Tunnelling and Underground Space Technology,58:209-215.
Yu HT,Cai C,Yuan Y and Jia MC(2017),"Analytical Solutions for Euler-Bernoulli Beam on Pasternak Foundation Subjected to Arbitrary Dynamic Loads,"International Journal for Numerical and Analytical Methods in Geomechanics, DOI:10.1002/nag.2672.
Yu H and Yuan Y(2014),"Analytical Solution foran Infinite Euler-Bernoulli Beam on a Viscoelastic Foundation Subjected to Arbitrary Dynamics Loads,"Journal of Engineering Mechanics, 140(3):542-551.Yu HT,Yuan Y and Bobet A(2013a),"Multiscale Method for Long Tunnels Subjected to Seismic Loading,"International Journal for Numerical and Analytical Methods in Geomechanics, 37(4):374-398.
Yu HT,Yuan Y,Qiao ZZ,Gu Y,Yang ZH and Li XD(2013b),"Seismic Analysis of a Long Tunnel Based on Multi-Scale Method,"Engineering Structures, 49:572-587.
Yu HT,Yuan Y,Xu GP,Su QK,Yan X and Li C(2018),"Multi-Point Shaking Table Test for Long Tunnels Subjected to Non-Uniform Seismic Loadings-PartⅡ:Application to the HZM Immersed Tunnel,"Soil Dynamics and Earthquake Engineering, 108:187-195.
Arani AG,Shiravand A, Rahi M and Kolahchi R(2012),"Nonlocal Vibration of Coupled DLGS Systems Embedded on Visco-Pasternak Foundation,"Physica B:Condensed Matter, 407(21):4123-4131
Basu D and Rao NSV(2013),"Analytical Solutions for Euler-Bernoulli Beam on Visco-Elastic Foundation Subjected to Moving Load,"International Journal for Numerical and Analytical Methods in Geomechanics,37(8):945-960.
Guan XF, Yu HT and Tian X(2016),"A Stochastic Second-order and Two-scale Thermo-mechanical Model for Strength Prediction of Concrete Materials,"International Journal for Numerical Methods in Engineering, 108(8):885-901.
Gurgoze M and Erol H(2004),"On Laterally Vibrating Beams Carrying Tip Masses, Coupled by Several Double Spring-Mass Systems,"Journal of Sound and Vibration,269:431-438.
Hamada TR, Nakayama H and Hayashi K(1983),"Free and Forced Vibration of Elastically Connected Double-beam Systems,"Bulletin of the Japan Society of Mechanics Engineers, 1936-1942.
Jang TS(2013),"A New Semi-analytical Approach to Large Deflections of Bernoulli-Euler-v. Karman Beams on a Linear Elastic Foundation:Nonlinear Analysis of Infinite Beams,"International Journal of Mechanical Sciences, 66:22-32.
Jang TS(2017),"A New Dispersion-relation Preserving Method for Integrating the Classical Boussinesq Equation,"Communications in Nonlinear Science and Numerical Simulation, 43:118-138.
Kenney JT(1954),"Steady-State Vibrations of Beam on Elastic Foundation for Moving Load,"Journal of Applied Mechanics, 21(4):359-364.
Kim SM and Roesset JM(2003),"Dynamic Responseof a Beam on a Frequency-Independent Damped Elastic Foundation to Moving Load,"Canadian Journal of Civil Engineering, 30(2):460-467.
Li MG, Yu HT, Wang JH, Xia XH and Chen JJ(2015),"A Multiscale Coupling Approach Between Discrete Element Method and Finite Difference Method for Dynamic Analysis,"International Journal for Numerical Methods in Engineering, 102(1):1-21.
Mathews PM(1958),"Vibration of a Beam on Elastic Foundation,"Journal of Applied Mathematics and Mechanics, 38(3-4):105-115.
Mathews PM(1959),"Vibration of a Beam on Elastic Foundation,"Journal of Applied Mathematics and Mechanics, 39(1-2):13-19.
Oniszczuk Z(2000),"Free Transverse Vibrations of Elastically Connected Simply Supported Double-Beams Complex System,"Journal of Sound and Vibration, 232:387-403.
Oniszczuk Z(2003),"Forced Transverse Vibrations of an Elastically Connected Complex Simply Supported Double-Beam System,"Journal of Sound and Vibration,264:273-286.
Palmeri A and Adhikari S(2011),"A Galerkin-Type State-Space Approach for Transverse Vibrations of Slender Double-beam Systems with Viscoelastic Inner Layer,"Journal of Sound Vibration, 330:6372-6386.
Saito J and Terasawa T(1980),"Steady-State Vibrations of a Beam on a Pasternak Foundation for Moving Loads,"Journal of Applied Mechanics, 47(4):879-883.
Sun L(2001),"A Closed-Form Solution of a BernoulliEuler Beam on a Viscoelastic Foundation under Harmonic Line Loads,"Journal of Sound and Vibration,242(4):619-627.
Sun L(2002),"A Closed-Form Solution of Beam on Viscoelastic Subgrade Subjected to Moving Loads,"Computer and Structure, 80(1):1-8.
Sun L(2003),"An Explicit Representation of Steady State Response of a Beam on an Elastic Foundation to Moving Harmonic Line Loads,"International Journal for Numerical and Analytical Methods in Geomechanics,27(1):69-84.
Vu HV,Ordonez AM and Karnopp BH(2000),"Vibration of a Double-Beam System,"Journal of Sound and Vibration, 229:807-822.
Wu YX and Gao YF(2015),"Analytical Solutions for Simply Supported Viscously Damped Double-Beam System under Moving Harmonic Loads,"Journal of Engineering Mechanics, ASCE, 141(7):04015004.
Wu YX and Gao YF(2016),"Dynamic Response of a Simply Supported Viscously Damped Double-Beam System under the Moving Oscillator,"Journal of Sound and Vibration, 384:194-209.
Yan X, Yu HT, Yuan Y and Yuan JY(2015),"Multi-Point Shaking Table Test of the Free Field under Non-uniform Earthquake Excitation,"Soils and Foundations, 55(5):985-1000.
Yuan Y,Yu HT. Li C,Yan X and Yuan JY(2018),"Multi-Point Shaking Table Test for Long Tunnels Subjected to Non-Uniform Seismic Loadings-PartⅠ:Theory and Validation,"Soil Dynamics and Earthquake Engineering, 108:177-186.
Yu HT,Cai C,Guan XF and Yuan Y(2016),"Analytical Solution for Long Lined Tunnels Subjected to Travelling Loads,"Tunnelling and Underground Space Technology,58:209-215.
Yu HT,Cai C,Yuan Y and Jia MC(2017),"Analytical Solutions for Euler-Bernoulli Beam on Pasternak Foundation Subjected to Arbitrary Dynamic Loads,"International Journal for Numerical and Analytical Methods in Geomechanics, DOI:10.1002/nag.2672.
Yu H and Yuan Y(2014),"Analytical Solution foran Infinite Euler-Bernoulli Beam on a Viscoelastic Foundation Subjected to Arbitrary Dynamics Loads,"Journal of Engineering Mechanics, 140(3):542-551.Yu HT,Yuan Y and Bobet A(2013a),"Multiscale Method for Long Tunnels Subjected to Seismic Loading,"International Journal for Numerical and Analytical Methods in Geomechanics, 37(4):374-398.
Yu HT,Yuan Y,Qiao ZZ,Gu Y,Yang ZH and Li XD(2013b),"Seismic Analysis of a Long Tunnel Based on Multi-Scale Method,"Engineering Structures, 49:572-587.
Yu HT,Yuan Y,Xu GP,Su QK,Yan X and Li C(2018),"Multi-Point Shaking Table Test for Long Tunnels Subjected to Non-Uniform Seismic Loadings-PartⅡ:Application to the HZM Immersed Tunnel,"Soil Dynamics and Earthquake Engineering, 108:187-195.