Wuhan University of TechnologyChapter9FormulationoftheMOoDFequations of motion9.1Selectionofthedegreesoffreedom9.2 Dynamic-equilibrium condition9.3EquationAxial-forceeffects9-1
9-1 Wuhan University of Technology9.1 Selection of the degrees of freedom 9.2 Dynamic-equilibrium condition 9.3 Equation Axial-force effects Chapter 9 Formulation of the MODF equations of motion
Wuhan University of Technology9.1 Selection of the degrees of freedomThequality of the result obtained with a SDOFapproximationdepends onmany factors, principally the spatial distribution and time variation of theloading and the stiffness and mass properties of the structure.If thephysical properties ofthe system constrain it tomovemost easilywiththe assumed shape, andif the loading is such as to excite a significant response inthis shape, the SDOF solution will probably be a good approximation; otherwise,thetrue behavior maybearlittle resemblance to the computed response.One of thegreatest disadvantages of the SDOF approximation is that it isdifficult to assess the reliability of the results obtained from it.9-2
9-2 Wuhan University of Technology 9.1 Selection of the degrees of freedom The quality of the result obtained with a SDOF approximation depends on many factors, principally the spatial distribution and time variation of the loading and the stiffness and mass properties of the structure. If the physical properties of the system constrain it to move most easily with the assumed shape, and if the loading is such as to excite a significant response in this shape, the SDOF solution will probably be a good approximation; otherwise, the true behavior may bear little resemblance to the computed response. One of the greatest disadvantages of the SDOF approximation is that it is difficult to assess the reliability of the results obtained from it
Wuhan Universityof Technology9.1 Selection of the degrees of freedomp(x,t)t2m(x)EI(x)tU2(t)u,(t)U.(tFIGURE9-1 Discretization ofa general beam-type structure.9-3
9-3 Wuhan University of Technology 9.1 Selection of the degrees of freedom FIGURE 9-1 Discretization of a general beam-type structure
Wuhan University of Technology9.2 Dynamic-equilibrium conditionThe equation of motion of the system of Fig.91 can be formulated by expressingtheequilibriumoftheeffectiveforcesassociatedwitheachof itsdegreesoffreedom. In general four types of forces will be involved at any point i: theexternallyappliedloadp;(t)andtheforcesresultingfromthemotion,thatisinertia fri, damping fpi, and elastic fsiThus for each of the several degrees offreedomthedynamicequilibriummaybeexpressedasfri+fDi+fsi=Pi(t)ff2+fp2+ fs2=P2(t)fr3+fD3+fs3=p3(t)fr+fp +fs =p(t)9-4
9-4 Wuhan University of Technology 9.2 Dynamic-equilibrium condition The equation of motion of the system of Fig. 91 can be formulated by expressing the equilibrium of the effective forces associated with each of its degrees of freedom. In general four types of forces will be involved at any point i: the externally applied load pi(t) and the forces resulting from the motion, that is, inertia fIi, damping fDi, and elastic fSi. Thus for each of the several degrees of freedom the dynamic equilibrium may be expressed as
Wuhan University of Technology9.2 Dynamic-equilibrium conditionfsi=k11U1+k12U2+k13U3+.+kiNUNfs2=k21U1+k22V2+k23V3+.+k2NUNfsi=kiiU1+ki2U2+ki3U3+..+kiNUNIntheseexpressionsithasbeentacitlyassumedthatthestructuralbehaviorislinear, so that the principle of superposition applies. The coefficients ki, are calledstiffnessinfluencecoefficients,definedasfollows:kij=forcecorrespondingtocoordinateiduetoa unit displacement of coordinate9-5
9-5 Wuhan University of Technology 9.2 Dynamic-equilibrium condition In these expressions it has been tacitly assumed that the structural behavior is linear, so that the principle of superposition applies. The coefficients kij are called stiffness influence coefficients, defined as follows: