Study On The Cold Rolled Strip Buckling And Vibration Based On Incompatible Deformation Theory

Cold rolled strip is one of the most important products in iron and steel industry,and widely used in automobiles,home appliances and electric power industries.Flatness is an important regulation to judge the quality of cold rolled strip.If the flatness defect is higher than a certain level,the waviness will be observed in strip,and adversely affect the production process.The mechanical essence of flatness defect is residual stress,while waviness is large deflection essentially.The research of relationship between residual stress,or the difference between.tension distribution and its integral mean value,and post-buckling deflection is a significant subject.On the other hand,the flatness control of cold rolled strip closely depends on the detection of residual stress.For the recently widely used flow induced contactless flatness measurement system SI-FLAT,the research of relationship between residual stress and amplitude of its fluid-structure interaction vibration,or mathematical model to detecting principle of SI-FLAT,is another significant subject.Aiming at the two significant subjects mentioned above,the incompatible deformation theory was induced to study the buckling and vibration of cold rolled strip.Further,this thesis tried to sort out and analyze this engineering problem from the angle of mathematics and mechanics.The research method was mainly theoretical analysis,with comparison and experimental verification as supplements.The emphasis of this work was to design solving methods of post-buckling problem and fluid-structure interaction forced vibration problem of cold rolled strip,and to summarize some general and innovative conclusion.The main research work and achievements were as follows:(1)The residual stress of cold rolled strip was boiled down to incompatible deformation.Further,based on incompatible deformation theory,the buckling and vibration of cold rolled strip was studied uniformly.The incompatible Foppl-von Karman equations with inertia term were established,after separation variables of time and coordinate along rolling direction,the buckling and vibration were boiled down to a fourth-order ordinary differential equation with eigenvalues.An orthogonal family of function was obtained by solving the eigenvalue problem mentioned above,which established the foundation for further mathematical analysis.(2)To post-buckling problem of thin cold rolled strip,since the pre-buckling problem is only a special case of the post-buckling problem,three kinds of semi-analytical and semi-numerical solving method were designed only for post-buckling problem,included complete method,simplified method and boundary layer method.The deflection of complete method was composed of a linear combination of orthogonal family of function,which strictly satisfied boundary conditions,and undetermined coefficients of the linear combination were determined by Ritz method;The deflection form of simplified method was obtained by analyzing the magnitude of involved physical quantities,which could not always satisfy boundary conditions,and wave length was determined by Ritz method;Based on simplified method,the deflection form was modified according to boundary layer solution method,which approximately satisfied boundary conditions,while wave length and positions of boundary layers were determined by Ritz method.For any kinds of method,the residual stress components released by strip’s buckling could also be calculated.Finally,the accuracy of latter two solving method were verified by actual measured data,while the calculation results were compared with other literature.Considering the anisotropic and variable thickness of cold rolled strip,the boundary value problem of non-uniform orthotropic strip’s buckling and vibration was obtained,which was closer to physical reality,while the corresponding simplified method was applied to analyze the post-buckling problem.(3)To vibration problem of thin cold rolled strip,focusing on the issue of fluid-structure interaction forced vibration,the detecting principle of SI-FLAT flatness measurement was analyzed.While simultaneous equations consisted of incompatible Foppl-von Karman equations with inertia term,fluid loading term as well as ideal incompressible fluid equations.Further,the basic governing equations of fluid-structure interaction vibration were established,while the boundary conditions and the connection conditions between fluid and structure were built based on practical engineering background,then the whole boundary value problem was established.This boundary value problem was solved by the expansion using orthogonal family of function after separating time variable,while the fluid-structure interaction vibration model of detecting principle to SI-FLAT was established.The accuracy of the model was verified by practical measured data indirectly,and several parameters which significantly affect the strip’s amplitude were analyzed.Finally.combining fluid-structure interaction vibration model and thermal stress analysis,the basic target curve for SI-FLAT was calculated.A tentative plan was that,with consideration of orthotropy and thickness variation of strip,the contactless flatness-profile measurement system was proposed,and the corresponding mathematical model to detecting principle was also established.