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