| The foundation beam model is widely widely used in civil engineering.In order to evaluate the dynamic performance of the foundation beam,avoid resonance and resonance correlation effects,improve the dynamic design efficiency of engineering structure and ensure the safety of engineering construction and normal use,the analysis of the dynamic performance of the foundation beam is an indispensable link in the in the related structural design.Furthermore,because the dynamic interaction mechanism between foundation and beam is very complex,the reverberation-ray matrix method with clear physical meaning,good numerical stability and high calculation accuracy is used study the natural vibration characteristics of foundation beam.In this paper,based on two viscoelastic foundation models and three beam theories,the natural vibration characteristics and modal orthogonal normalization conditions of single-span foundation beams under six boundary conditions are studied systematically in complex domain.Research and analyze the influence of factors such as moment of inertia,shear deformation and foundation parameters on the natural vibration characteristics of foundation beams.On this basis,the natural frequency equation,modal function and orthogonal normalization conditions of the multi-span continuous beam were further analyzed and deduced.The natural vibration characteristics of the two-span continuous foundation beam were studied as an example,and the comparison with the single-span foundation was analyzed.The connection and difference of the natural vibration characteristics of beams provide a reliable theoretical basis for practical engineering applications.The specific content of the study includes the following aspects:(1)Combining the viscoelastic foundation model with the Timoshenko beam theory,the frequency equations,modal functions and orthogonal normalization conditions under six boundary conditions are derived.The analytical solution of the natural vibration characteristics of the Timoshenko beam on the viscoelastic foundation under the boundary conditions of the two ends simply supported and the numerical solutions under other boundary conditions are obtained through specific calculation examples.The correctness of the lateral free vibration characteristics based on the reverberation-ray matrix method is verified,and comparisons are made.The natural vibration characteristics of Timoshenko beams on viscoelastic Pasternak and viscoelastic Winkler foundations are analyzed.(2)The influence of shear deformation of beam in the process of transverse vibration is not considered,Rayleigh beam theory and viscoelastic foundation model are combined,and its frequency equations,modal functions and their orthogonality under six classical boundary conditions The normalization conditions are derived.The analytical solution of the natural vibration characteristics of the Rayleigh beam on the viscoelastic foundation under the boundary condition of simply supported at both ends and the numerical solution under other boundary conditions are solved through specific calculation examples.The natural vibration characteristics of Rayleigh beams on viscoelastic Pasternak foundation and viscoelastic Winkler foundation are compared and analyzed.(3)Without considering the influence of shear deformation and moment of inertia of beam in the process of transverse vibration,Bernoulli-Euler beam theory and viscoelastic foundation model are combined.The frequency equation and model of the lateral free vibration of the Bernoulli-Euler beam on the viscoelastic foundation are derived.State function and orthogonal normalization conditions.Among them,the Bernoulli-Euler beam on the viscoelastic Winkler foundation due to the relationship between its two wave numbers,the analytical solution of the natural frequency under the boundary conditions of the two ends simply supported,the approximate analytical solution of the natural frequency under other boundary conditions and Analytical solution of attenuation coefficient.When the influence of the foundation is not considered,the results obtained in this paper are compared with the results of "Shock and Vibration Engineering Handbook" to further verify the correctness of the solution method in this paper.The natural vibration characteristics of Bernoulli-Euler beams on viscoelastic foundation are analyzed through specific calculation examples.(4)First,summarize the natural frequency equations,modal functions and orthogonal normalization conditions of various composite foundation beams under classical boundary conditions,and refine the natural frequencies,modal functions and orthogonal normalization conditions under various boundary conditions.Uniform column formula of conditions.Secondly,when various foundation beam models have the same foundation model,the influence of beam length on the natural vibration characteristics of foundation beams of different beam models is studied.Through the influence of the moment of inertia and shear deformation on the natural frequency and attenuation coefficient of different slenderness ratios In the study,the recommended applicable range of slenderness ratio of foundation beam models for different beam theories is given.(5)Using the combination of the viscoelastic Pasternak foundation model and the Timoshenko beam model,the natural frequency equation,modal function and orthogonal normalization conditions of the multi-span continuous foundation beam are derived.Taking the two-span continuous foundation beam as an example,the natural frequency,attenuation coefficient and modal curve of the two-span(unequal-span)continuous beam with six foundation beam models are solved,and the natural vibration characteristics of the two-span continuous beam are further analyzed.Relations and differences of the natural vibration characteristics of beams across foundations.Research shows:The reverberation-ray matrix method provides a dynamic analysis path with strong versatility,rigorous logic and reliable results for studying the natural vibration characteristics of foundation beams.This method is used to analyze the natural vibration characteristics of various foundation beam models to obtain the lateral natural frequency equations Compared with other research methods,it has the characteristics of uniform form,clear and concise expression,clear physical meaning,and easy to analyze and apply.Foundation parameters,boundary conditions,and shear deformation and moment of inertia of the beam during lateral vibration have a greater impact on the calculation results of the natural vibration characteristics of the foundation beam.When studying the lateral free vibration characteristics of the foundation beam,an appropriate calculation model should be selected to improve the calculation.The accuracy of the results and calculation efficiency have a significant impact. |
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