Study On The Modified Timoshenko Cracked Beam Based On Dynamic Stiffness Method

The beam was used extensively in many fields,such as civil,mechanical,aerospace engineering,etc.In pratical engineering applications,the presence of the cracks easily lead to tremendous accidents and huge costs to people live.Thus,it is significant to prior to analysis free vibration of damaged beam accurately and efficiently based on dynamic quantity damage detection method.The present paper proposes a novel method,which name is dynamic matrix method,to addresses the evaluation of natural frequencies and vibration mode of novel dynamic beam theory-modified Timoshenko beam with multicracks.This method is not only reliable and efficient but also exact and economical comparing results from traditional FEM.Based on exact dynamic stiffness matrix,the FFT technique can be used to solve dynamic response problem for cracked beam.It is convinent to other researchers to analysis this beam theory,because the general solution of governing eqation has been derived during deriving dynamic matrix,which has theory significant.1?The dynamic stiffness method which comes from undamped free vibration governing equation of modified Timoshenko beam,the relationship beween lateral displacement and rotation has been derived.The huge differences of shape function between modified Timoshenko beam and traditional Timoshenko has been showed as small depth-span ratio and higher test frequency.The well-established Wittrick-Williams algorithm and guided and guarded Newton method can be used to calculate frequency and mode elegantly.In the later,the effect of axial force has been considered.The tensile force lead natural frequency increasing,the pressure opposite.2 ? Study on the free vibration of cracked motified Timoshenko beam with assembling crack stiffness matrix into intact one to analysis seven massless bending spring crack model.The variations of effect for shear spring with depth-span ratio for decline natural frequencies has been studied in detail.The results show that main effct of bending spring model decrease frequency with small depth-sapn ratio,shear spring model with higher depth-span ratio.There are some analysis of influence of boundary conditions,crack strength and position on the natural frequencies and vibration mode.3?Study on two parameters Pasternak elastic foundation cracked beam and cracked frame structure.The gonverning eqation was introduced in Pasternak foundation.Thus,it has four types general solution according different elastic,shear layer parameters and test frequency.The influence of two parameters on free vibration cracked beam has been investigated.The numerical results demonstrate that the reduction depends not only on the number,depth and locations of the damage,but also on the elastic parameters and position of the foundation.The three different frame structure has been studied to analysis the effects of changing the crack position in frame natural frequencies with axial dynamic stiffness introduced.Givning a insight of inverse problem of crack detection.4?Finally,the dynamic response in modified Timoshenko cracked beam in different crack relative positions and depths under moving loads were analyzed.Since the dynamic stiffness matrix can also be regarded as the stiffness matrix in the frequency domain,the dynamic response of the structure can be solved in the frequency domain,then the solution in time domian can be obtained by numerical inverse FFT.The results show that the dynamic response in mid span of span length which is 8 meters will be lager in high level crack depth and higher velocity of moving load.It can provide some reference value for bridge engineers in analyzing cracked beam structure.Finally,the algorithm present in this paper is coded by MATLAB and Mathematica.The efficient,accuracy and reliable have been verified by compared with some previous work.