《结构动力学》课程教学课件（讲稿）08 Generalized SDOF systems

Wuhan University of TechnologyChapter8Generalized SDOFsystems8-1

8-1 Wuhan University of Technology Chapter 8 Generalized SDOF systems

Wuhan University of TechnologyContents8.1General comments on SDOFsystems8.2 Generalized properties: assemblages of rigid bodies8.3Generalizedproperties:distributedflexibility8.4Expressionsforgeneralizedsystemproperties8.5 Vibration analysis by Rayleigh's method8.6 Selection of the Rayleigh vibration shape8-2

8-2 Wuhan University of Technology 8.1 General comments on SDOF systems 8.2 Generalized properties: assemblages of rigid bodies 8.3 Generalized properties: distributed flexibility 8.4 Expressions for generalized system properties 8.5 Vibration analysis by Rayleigh's method 8.6 Selection of the Rayleigh vibration shape Contents

Wuhan Universityof Technology8.1 General comments on SDOF systemsInthischapterwewilldiscussthesegeneralizedSDOFsystems,andinformulatingtheir equationsof motion it is convenient to dividethem intotwocategories:(1) Assemblages of rigid bodies in which elastic deformations are limited tolocalizedweightlessspringelementsand(2)Systemshavingdistributedflexibilityinwhichthedeformationscanbecontinuousthroughoutthestructure,orwithinsomeofitscomponents.Inbothcategories,thestructureisforcedtobehavelikeaSDOFsystembythefactthatdisplacementsofonlyasingleformorshapearepermitted,andtheassumedsingledegreeoffreedomexpressestheamplitudeofthispermissibledisplacementconfiguration.8-3

8-3 Wuhan University of Technology 8.1 General comments on SDOF systems In this chapter we will discuss these generalized SDOF systems, and in formulating their equations of motion it is convenient to divide them into two categories: (1) Assemblages of rigid bodies in which elastic deformations are limited to localized weightless spring elements and (2) Systems having distributed flexibility in which the deformations can be continuous throughout the structure, or within some of its components. In both categories, the structure is forced to behave like a SDOF system by the fact that displacements of only a single form or shape are permitted, and the assumed single degree of freedom expresses the amplitude of this permissible displacement configuration

Wuhan Universityof Technology8.2 Generalized properties: assemblages ofrigid bodiesThetotal mass and the2212centroidalmassmomentm=mYa622ofinertiaofauniformmasmkngthrod and of uniformmassareaplatesofvarious shapes23aresummarized inFig.81.8-4

8-4 Wuhan University of Technology 8.2 Generalized properties: assemblages of rigid bodies The total mass and the centroidal mass moment of inertia of a uniform rod and of uniform plates of various shapes are summarized in Fig. 81

Wuhan UniversityofTechnology8.2 Generalized properties: assemblages ofrigid bodiesp(x,t)=p≤f(t)HingeWeightless, rigid bar EHm2,j2HNGEBDskak,C2aa2aFIGUREE8-1Example of arigid-body-assemblageSDOFsystem8-5

8-5 Wuhan University of Technology 8.2 Generalized properties: assemblages of rigid bodies FIGURE E8-1 Example of a rigid-body-assemblage SDOF system

Wuhan University of Technology8.2 Generalized properties: assemblages ofrigid bodiesExample E81. A representative example of a rigidbody assemblage,showninFig.E81,consistsoftworigidbarsconnectedbyahingeatEandsupported by a pivot at A and a roller at H. Dynamic excitation is provided by atransverse load p(x, t) varying linearly along the length of bar AB. In addition, aconstantaxialforceNactsthroughthesystem,andthemotionisconstrainedbydiscretespringsanddamperslocatedasshownalongthelengthsofthebarsThe mass is distributed uniformly through bar AB, and the weightless bar BCsupports a lumped mass m2 having a centroidal mass moment of inertia j28-6

8-6 Wuhan University of Technology Example E81. A representative example of a rigidbody assemblage, shown in Fig. E81, consists of two rigid bars connected by a hinge at E and supported by a pivot at A and a roller at H. Dynamic excitation is provided by a transverse load p(x, t) varying linearly along the length of bar AB. In addition, a constant axial force N acts through the system, and the motion is constrained by discrete springs and dampers located as shown along the lengths of the bars. The mass is distributed uniformly through bar AB, and the weightless bar BC supports a lumped mass m2 having a centroidal mass moment of inertia j2. 8.2 Generalized properties: assemblages of rigid bodies

Wuhan Universityof Technology8.2 Generalized properties: assemblages ofrigid bodiesp(x,n)=f(t)HingeWeightless,rigid barEHm2,j2HNJGERkBKJCa+2P(t)=8paf(t)8aE'3DZ(t)BMHIEBIDCCfo,(t)fs (t)(tfi(t)fs(t)FIGURE E8-2 SDOF displacements and resultant forces.8-7

8-7 Wuhan University of Technology 8.2 Generalized properties: assemblages of rigid bodies FIGURE E8-2 SDOF displacements and resultant forces

Wuhan University of Technology8.2 Generalized properties: assemblages ofrigid bodiesp(t)=8paf(0)8aE3D1Z(t)BMjVH1IEBIDCFCfb,(t)Js,(t)f,(t)fg,(t)fi(0)fo.(t)[% D'(]1z(t)Z(t)=而 LZ(t)=2a元Z(t)fp,(t) = c1fr,(t)=miC122L2mL142㎡2(t)fp,(t) = c2 Z(t)Z(t):Z(t) =Mi, (t) = j1124a4afs,(t) = ki [DD(t)] = h z()2(t)fi(t) = m2 31(t)fs,(t) = k2[GG(t)] = k2 z(t)Mis(t) = -j2 3a8-8

8-8 Wuhan University of Technology 8.2 Generalized properties: assemblages of rigid bodies

Wuhan University of Technology8.2 Generalized properties: assemblages ofrigid bodiesThe externally applied lateral load resultant isPi(t)=8pa f(t)Theequationof motionof thissystemmaybeestablished byequatingtozeroallworkdonebytheseforce components duringan arbitraryvirtual displacementZThevirtualdisplacementsthroughwhichtheforcecomponentsmoveareproportional to Z(t), as indicated in Fig.E82.Thus the total virtual work may bewritten22(0) 288Z4α2㎡z(t)8ZsW(t)=-2amZ(t)m213324a3Z(t)Z(t) z33Z(t)S722(t)8Z-SZ11244443a21Z(t) 8zoz=0k2+8paf(t)33138-9

8-9 Wuhan University of Technology 8.2 Generalized properties: assemblages of rigid bodies The externally applied lateral load resultant is The equation of motion of this system may be established by equating to zero all work done by these force components during an arbitrary virtual displacement Z. The virtual displacements through which the force components move are proportional to Z(t), as indicated in Fig. E82. Thus the total virtual work may be written

Wuhan University of Technology8.2 Generalized properties: assemblages ofrigid bodieswhichwhensimplifiedbecomes4j2amZ(t)amm29a29316916C8Z=0k1paf(t)t160thefinaleguationofmotionbecomes44J2Z(t)之(t)mam2299a21616k29Z(t)spaf(t)a168-10

8-10 Wuhan University of Technology 8.2 Generalized properties: assemblages of rigid bodies which when simplified becomes the final equation of motion becomes