Wuhan University of TechnologyC2.3 Equation of motion of the basic dynamicsystemIntheformerequation,thenegativesignsresultfromthefactthatthe associated forces act opposite tothe sense of the virtualdisplacement.Therefore,it has[-mj(t) -ci(t) - ky(t) + p(t)] · Sy = 0mj(t) +ci(t) + ky(t) = p(t)3-11
3-11 Wuhan University of Technology 2.3 Equation of motion of the basic dynamic system In the former equation, the negative signs result from the fact that the associated forces act opposite to the sense of the virtual displacement. Therefore, it has [ ( ) ( ) ( ) ( )] 0 my t cy t ky t p t y my t cy t ky t p t () () () ()
Wuhan University of Technology2.4 Influence of gravitational forcesConsidernowthesystemshownabove,which is the system of former Fig.rotatedkCthrough 90o so that the force of gravityacts inthe directionof thedisplacementInthiscase,thesystemofforcesmacting in the direction of the displacement(W)degree offreedomisthatset shown inv(t)right Fig. Using Eqs. (22), the equilibriump(tof these forces is given by3-12
3-12 Wuhan University of Technology 2.4 Influence of gravitational forces Consider now the system shown above, which is the system of former Fig. rotated through 90 o so that the force of gravity acts in the direction of the displacement. In this case, the system of forces acting in the direction of the displacement degree of freedom is that set shown in right Fig. Using Eqs. (22), the equilibrium of these forces is given by k c m ( W) p ( t ) y ( t ) k c m ( W) p ( t ) y ( t )
Wuhan University of Technology12.4 Influence of gravitational forcesTheequilibriumoftheseforcesisgivenbymi(t) +ci(t)+ ky(t) = p(t) + WIf the total displacement y(t) isfs(t) fo(t)expressed as the sum of thestaticdisplacementAstcausedbyfi (t)the weight W plus the additionalWdynamic displacement y(t) asStatic4displacementstshown in Figp(t)J(t)y(t) = △st + (t)3-13
3-13 Wuhan University of Technology The equilibrium of these forces is given by 2.4 Influence of gravitational forces If the total displacement y(t) is expressed as the sum of the static displacement Δst caused by the weight W plus the additional dynamic displacement y(t) as shown in Fig my t cy t ky t p t W () () () () () () st yt yt I = fS(t) fD(t) f (t) W Static st displacement p(t) y(t) I = fS(t) fD(t) f (t) W Static st displacement p(t) y(t)
Wuhan University of Technology-2.4 Influence of gravitational forcesThen,thespringforceisgivenbyfs(t) = ky(t) = k△st +k(t)We can obtain the following equation of motionmj(t) +ci(t) +k△, +ky(t) = p(t)+Wk△,=WNoting thatLeads tomj(t) +ci(t) + kj(t) = p(t)3-14
3-14 Wuhan University of Technology Then, the spring force is given by 2.4 Influence of gravitational forces We can obtain the following equation of motion Noting that Leads to () () () S st f t k y tk k y t () () () () m st y t c y tk k y t pt W st k W my t cy t ky t p t () () () ()
Wuhan University of Technology2.4 Influenceofgravitationalforces Noting that Ast does not vary with time, andj(t) =(t)j(t) = (t)my(t)+cy(t) + ky(t) = p(t)Leadsto3-15
3-15 Wuhan University of Technology 2.4 Influence of gravitational forces Leads to Noting that Δst does not vary with time, and yt yt () () yt yt () () my t cy t ky t p t () () () ()