S 15.2 Concept and calculation of virtual displacement2.Analytical methodThe analytic method is to find the relation between thevirtual displacements by variating the constraint equation orcoordinateexpression.Example: Elliptic gauge mechanismThe constrained equation of XB, YA isVx+y =1?yA(A,yA)Carry out variational operation onthe above formula:212xdB +2yAdA = 0@OB(XB,YB&x-_YA=-tgpSyA1 XB
2. Analytical method The analytic method is to find the relation between the virtual displacements by variating the constraint equation or coordinate expression. ( , ) A A A x y ( , ) B B B x y x y O A y B x l B A The constrained equation of is x , y 2 2 2 x y l B + A = Carry out variational operation on the above formula: 2xB xB + 2yA yA = 0 tg x y y x B A A B = − = − Example: Elliptic gauge mechanism §15.2 Concept and calculation of virtual displacement
S 15.2 Concept and calculation of virtual displacementOr we can represent X,yAastyfunctions of Φ, or we can figure outyA(XA,yA)the relationship between the virtualdisplacements:力yA=lsinpXB = lcosp0CB(XB,yB)Do the calculus of variationsdxg=-lsinpopA=lcosPop&xB =-tgSodysComparing the above two methods, it can be found thatthe geometric method is intuitive and relatively simple, whilethe analytical method is more standard
( , ) A A A x y ( , ) B B B x y x y O A y B x l Or we can represent as functions of , or we can figure out the relationship between the virtual displacements: B A x , y xB = l cos yA = lsin Do the calculus of variations xB = −lsin yA = l cos So tg y x A B = − Comparing the above two methods, it can be found that the geometric method is intuitive and relatively simple, while the analytical method is more standard. §15.2 Concept and calculation of virtual displacement
15.3Principleof virtual displacementI.Virtual workAs shown in the figure, if a particle is subjected toFforceF and a virtual displacement Sris given to the2MSrparticle, the work done by force F on the virtualdisplacement or is called virtual work.SW=F.r or sW=Fcos@&Obviously, the virtual work is also imaginary, and it is of thesame order as thevirtual displacement.If in any virtual displacement of the system of particles, thesum of the virtual work done by all the constraint forces isequal to zero, then this constraint is called an ideal constraint.ZSWN =EN,.8 = 0Common ideal constraints are: smooth fixed surface, smoothhinge, non-heavy rigid, non-retractable rope,no sliding rolling
M F r F As shown in the figure, if a particle is subjected to force and a virtual displacement is given to the particle, the work done by force on the virtual displacement is called virtual work. F r r W F r = or W = F cos r Obviously, the virtual work is also imaginary, and it is of the same order as the virtual displacement. If in any virtual displacement of the system of particles, the sum of the virtual work done by all the constraint forces is equal to zero, then this constraint is called an ideal constraint. WN = Ni ri = 0 Common ideal constraints are: smooth fixed surface, smooth hinge, non - heavy rigid, non - retractable rope, no sliding rolling Ⅰ. Virtual work §15.3 Principle of virtual displacement
15.3 Principle of virtual displacementII.Principle of virtual displacementThe necessary and sufficient conditions for a system of particleswith dual steady-state ideal constraints to bein equilibrium at acertain position are that the sum of the virtual work done by all theprincipalforces acting on the system of particles in any virtualdisplacement at thatpositionis equal tozero.ZF,.Sr, = 0OrZ F,Sr, cos αi = OCan be expressed as: E(X,&x, + Y,y, + Z,&,) = 0These three equations are called virtual work equationsThe virtual displacement principle is also called the virtualworkprinciple
The necessary and sufficient conditions for a system of particles with dual steady-state ideal constraints to be in equilibrium at a certain position are that the sum of the virtual work done by all the principal forces acting on the system of particles in any virtual displacement at that position is equal to zero. Fi ri = 0 Or Fi ri cosi = 0 Can be expressed as: (Xi xi +Yi yi + Zi zi ) = 0 These three equations are called virtual work equations. The virtual displacement principle is also called the virtual work principle. Ⅱ. Principle of virtual displacement §15.3 Principle of virtual displacement