《结构动力学》课程教学课件（讲稿）09 Formulation of the MODF equations of motion

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:

Wuhan Universityof Technology9.2 Dynamic-eguilibrium conditionInmatrixform,thecompletesetofelasticforcerelationshipsmaybewritten[k11k12k13k1ifs1k1NU1k21k22fs2k23k2ik2NV2ki1ki2ki3kifsikiNUifs=kv9-6

9-6 Wuhan University of Technology 9.2 Dynamic-equilibrium condition In matrix form, the complete set of elasticforce relationships may be written

Wuhan Universityof Technology9.2 Dynamic-eguilibrium conditionIf it is assumed that the damping depends on the velocity, that is, the viscoustype, the damping forces corresponding to the selected degrees of freedom maybe expressed bymeans of damping influence coefficients in similarfashion.Byanalogy with Eq.(95),the complete set of dampingforces isgivenby[C1101C12C13CliCiND1fp202C21C22C23C2NC2iviCilCi2Ci3CiiCiNDCij=forcecorresponding to coordinateiduetounitvelocity of coordinatejfp=cv9-7

9-7 Wuhan University of Technology 9.2 Dynamic-equilibrium condition If it is assumed that the damping depends on the velocity, that is, the viscous type, the damping forces corresponding to the selected degrees of freedom may be expressed by means of damping influence coefficients in similar fashion. By analogy with Eq. (95), the complete set of damping forces is given by

Wuhan University of Technology9.2 Dynamic-eguilibrium conditionTheinertialforcesmaybeexpressedsimilarlybyasetof influencecoefficientscalledthemasscoefficients.Theserepresenttherelationshipbetweentheaccelerations ofthedegrees offreedomandtheresulting inertial forces;byanalogywithEq.(95),theinertialforcesmaybeexpressedas01fnm11m12m13miimINf12i2m21m22m23m2im2NuiTmi1mi2mimiNmi3mi=forcecorrespondingtocoordinateiduetounitaccelerationofcoordinatejf, =m v9-8

9-8 Wuhan University of Technology 9.2 Dynamic-equilibrium condition The inertial forces may be expressed similarly by a set of influence coefficients called the mass coefficients. These represent the relationship between the accelerations of the degrees of freedom and the resulting inertial forces; by analogy with Eq. (95), the inertial forces may be expressed as

Wuhan University of Technologs-9.2 Dynamic-equilibrium conditionSubstituting Eqs. (96), (99), and (912) into Eq. (92) gives the complete dynamicequilibriumofthestructure,consideringalldegreesoffreedommv(t)+cv(t)+kv(t)=p(t)ThisequationistheMDOFequivalentofEq.(23);eachtermoftheSDOFequationisrepresentedbyamatrix inEg.(913),theorderofthematrixcorrespondingtothenumberofdegreesoffreedomusedindescribingthedisplacementsofthestructure. Thus, Eq. (913) expresses the N equations of motion which serve todefinetheresponseoftheMDOFsystem.9-9

9-9 Wuhan University of Technology 9.2 Dynamic-equilibrium condition Substituting Eqs. (96), (99), and (912) into Eq. (92) gives the complete dynamic equilibrium of the structure, considering all degrees of freedom: This equation is the MDOF equivalent of Eq. (23); each term of the SDOF equation is represented by a matrix in Eq. (913), the order of the matrix corresponding to the number of degrees of freedom used in describing the displacements of the structure. Thus, Eq. (913) expresses the N equations of motion which serve to define the response of the MDOF system

WuhanUniversityof Technology9.3 Axial-force effectsItwasobservedinthediscussionofSDOFsystemsthataxialforcesoranyloadwhich maytend to cause bucklingof a structure mayhaveasignificant effect onthe stiffness of the structure. Similar effects may be observed in MDOF systems;the force component acting parallel to the original axis of the members leads toadditional loadcomponentswhichact inthedirection(and sense)ofthenodaldisplacementsandwhichwillbedenotedbyfG:Whentheseforcesareincluded,the dynamicequilibrium expression, Eq. (92), becomesfr +fp +fs -fg =p(t)9-10

9-10 Wuhan University of Technology 9.3 Axial-force effects It was observed in the discussion of SDOF systems that axial forces or any load which may tend to cause buckling of a structure may have a significant effect on the stiffness of the structure. Similar effects may be observed in MDOF systems; the force component acting parallel to the original axis of the members leads to additional load components which act in the direction (and sense) of the nodal displacements and which will be denoted by fG. When these forces are included, the dynamicequilibrium expression, Eq. (92), becomes