Wuhan University of Technology2.4Influence of gravitationalforcesmi(t)+ci(t)+ky(t) = p(t)my(t)+cy(t)+ ky(t) = p(t)It isobservedthatthe equationof motionexpressedwithreferenceto the static-equilibrium position of the dynamic system is not affectedby gravity forces.Forthisreason,displacementsinallfuturediscussionswillbereferenced fromthe static-eguilibriumpositionandwill bedenoted y(t)(i.e., without the overbar);Thedisplacementswhicharedetermined will representdynamicresponse.Therefore, total deflections, stresses, etc. are obtained by addingthe corresponding static quantities to the results of the dynamicanalysis.3-16
3-16 Wuhan University of Technology 2.4 Influence of gravitational forces It is observed that the equation of motion expressed with reference to the static-equilibrium position of the dynamic system is not affected by gravity forces. For this reason, displacements in all future discussions will be referenced from the static-equilibrium position and will be denoted y(t) (i.e., without the overbar); The displacements which are determined will represent dynamic response. Therefore, total deflections, stresses, etc. are obtained by adding the corresponding static quantities to the results of the dynamic analysis. my t cy t ky t p t () () () () my t cy t ky t p t () () () ()
Wuhan Universityof Technology2.5 Influence of support excitationDynamic stresses and deflections can be induced in astructure not only by a time-varying applied load but also bymotions of its support points;Important examples of suchexcitationarethemotions ofabuildingfoundationcausedbyanearthguakeormotionsofthe base support of a piece of equipment due to vibrationsof the building.3-17
3-17 Wuhan University of Technology 2.5 Influence of support excitation Dynamic stresses and deflections can be induced in a structure not only by a time-varying applied load but also by motions of its support points; Important examples of such excitation are the motions of a building foundation caused by an earthquake or motions of the base support of a piece of equipment due to vibrations of the building
Wuhan Universityof Technology2.5 Influence of support excitationAsimplifiedmodeloftheearthguakeexcitationproblemisshown in following Fig, in which the horizontal ground motioncaused by the earthquake is indicated by the displacement y.(t) ofthestructuresbaserelativetothefixed referenceaxis.y(t)2-22y.t3-18
3-18 Wuhan University of Technology A simplified model of the earthquake excitation problem is shown in following Fig, in which the horizontal ground motion caused by the earthquake is indicated by the displacement yg(t) of the structures base relative to the fixed reference axis. 2.5 Influence of support excitation c y t(t) y (t) m k 2 yg (t) Fixed reference axis k2 c y t(t) y (t) m k 2 yg (t) Fixed reference axis k2
Wuhan Universityof Technology2.5Influence of support excitationfi(t)s(t)fp(t)fs(t)22As shown in above Fig., the equilibrium of forces for this systemcan bewrittenasfi(t)+ fb(t)+ fs(t)= 03-19
3-19 Wuhan University of Technology 2.5 Influence of support excitation As shown in above Fig., the equilibrium of forces for this system can be written as () () () 0 IDS ft f t ft fI ( t ) fS ( t ) 2 fD ( t ) fS ( t ) 2 fI ( t ) fS ( t ) 2 fD ( t ) fS ( t ) 2
Wuhan University of Technology2.5 Influence of support excitationy'({)-y(t)k2kyg()Theinertial forceinthiscaseisgivenbyThetotaldisplacementf (t) = mj'(t)ofthemassfromthefixedreferenceaxisfb(t) = cy(t)AlsO,fs(t) = ky(t)3-20
3-20 Wuhan University of Technology 2.5 Influence of support excitation The inertial force in this case is given by Also, The total displacement of the mass from the fixed reference axis () () t I f t my t () () Df t c y t () () Sf t k y t c y t (t) y(t) m k 2 yg(t) Fixed reference axis k c 2 y t (t) y(t) m k 2 yg(t) Fixed reference axis k 2