Research On Crack Identification Method Of A Circular Cross Section Beam By Modal Characteristics

The beam having circular cross section is widely used in the field of mechanical equipment,civil engineering,bridge,aviation and so on.During the opertation of these equipments or structures,the circular cross section beam must be working in the enviorment of dynamic load,vibration or other harsh conditions.Ignored cracks or cracks which are not identified may result in fatigue failure.In this thesis,a circular cross section beam having cracks is used as the study object.Based on the contour line method,two crack identification methods are proposed by using modal characteristics.The main contributions of this paper are as follows:Firstly,the equation of stress intensity factor of a cracked circular section beam under the load of pure tension,pure bending and pure shear were derived respective acoording to the Euler Bernoulli elastic beam theory and linear fracture mechanics.Through the Castigliano's theorem and the finite element method,the crack element in the beam was replaced by a spring structure.And the local flexibility coefficients of the cracked element and uncracked element were calculated.the local stiffness matrixes of the cracked element and the uncracked element were derived based on the static balance condition,The effects of material parameters and crack parameters on the stiffness matrixes of cracked elements were discussed.Then,the motion equation of the circular section cantilever beam is established,and the effects of the ratio of crack position,the ratio of crack depth and the ratio of distance between the position of cracks on the modal frequencies,bending modes and rotational modes of the circular cross section beam having a crack or having double cracks were investigated.Thereafter,a crack identification method of circular cross section beam by discontinuity of rotational mode are proposed.The principle of this identification method is as follows: arranging the first three dimensionless angular displacement of each node in the circular cross section beam by order to get an dimensionless angular displacement curve.After that,the first three response surfaces of the circular cross section beam with various crack conditions are obtained by combining the contour method with the value of discontinuity.The corresponding contour lines are obtained by intersecting the surface with three discontinuity values planes.Then,three contour lines would intersect at one point.The x coordinate of this point denotes the ratio of crack position.The y coordinate of this point denotes the ratio of crack depth.Results Indicates that this method can identify the number of cracks,crack location and crack depth of the circular cross section beam,but the identification errors increases with the increasing of crack numbers.Finally,a crack identification method of circular cross section beam by modal frequency contour method are proposed.The basic principle of this method is as follows: the first three modal frequency response surfaces of beam with different crack conditions are given,and the frequency contour lines are obtained by intersecting the surface with three planes of the specimen natural frequency subsequently.After that,three contour lines would intersect at one point.The x coordinate of this point denotes the crack position ratio,y coordinate of this point denotes the crack depth ratio.Due to the accuracy of modal testing,a improved identification method of frequency contour method is proposed.Results indicates that the improved modal frequency contour method decreases the requirements of accuracy of the modal frequencies testing.It is more accurate for the crack identification by this improved method.The minimum error is reduced to 3.78%.But,the maximum error reaches to 52.00% as crack is very small.