THEORYMECHANICS1902Chapter 15 Principle ofvirtual displacementCollege of Mechanical andVehicle Engineering王晓君
College of Mechanical and Vehicle Engineering 王晓君 THEORY MECHANICS Chapter 15 Principle of virtual displacement
Principle of virtualdisplacementAfew staticRigid-body andnon-deformableequilibriumnecessaryand sufficientconditionsrigid-bodyconditionsestablishedsystem balanceNecessaryIratherthansufficientconditionsnecessary and sufficientPrincipleofvirtualconditionsdisplacementEquilibriumproblem for any(analytic statics)system of particlesStatic mechanics (vector mechanics) : the relationship betweenthe principal force and the binding forceThe virtual displacement principle and D'Alembert principletogether form the basis of analytical dynamics
Equilibrium problem for any system of particles Principle of virtual displacement A few static equilibrium conditions established Rigid-body and non-deformable rigid-body system balance necessary and sufficient conditions Necessary rather than sufficient conditions necessary and sufficient conditions Principle of virtual displacement ➢ Static mechanics (vector mechanics) : the relationship between the principal force and the binding force ➢ The virtual displacement principle and D'Alembert principle together form the basis of analytical dynamics （analytic statics）
S 15.1 Constraints and their classificationThe conditions that limit the position and motion of eachparticle in a system of particles are called constraints. Theexpressions of these constraints are called constraint equationsConstraints can be of the following types:I . Geometric constraints and motion constraintsConstraints that limit a particle or a system of particles in thegeometry of space are called geometric constraints. Such as:XA(x,yA)B(pyB)xM(x,y)x? +y = r21(XB -xa)?+(yB -y)? = 1?x~+y?=lThe general form of geometric YB = Of.(Xi,y1,z1,,Xn, yn,zn) = 0constraint eguation is:
The conditions that limit the position and motion of each particle in a system of particles are called constraints. The expressions of these constraints are called constraint equations. Constraints can be of the following types: Ⅰ. Geometric constraints and motion constraints Constraints that limit a particle or a system of particles in the geometry of space are called geometric constraints. Such as: O x y M (x, y) l 2 2 2 x + y = l O ( , ) A A A x y ( , ) B B r B x y l x y 0 ( ) ( ) 2 2 2 2 2 2 = − + − = + = B B A B A A A y x x y y l x y r The general form of geometric constraint equation is: f r (x1 , y1 ,z1 , , xn , yn ,zn ) = 0 §15.1 Constraints and their classification
S15.1 Constraints and theirclassificationThe constraint that can limit not only the position of the particlesystem but also the velocity of each particle in the system is calledmotionconstrainttyB = r is the geometric constraint equationB(XB,YBXg -r@p= O is the motion constraint equation.1RX0CThe general form of the motion constraint equation isf,(x1,J1,Z1,",Xn,yn,zn,X1,j1,z1,",xn,jn,zn) =0
The constraint that can limit not only the position of the particle system but also the velocity of each particle in the system is called motion constraint. ( , ) B B B x y B v O x y C r y r B = is the geometric constraint equation. x B −r = 0 is the motion constraint equation. The general form of the motion constraint equation is f r (x1 , y1 ,z1 , , xn , yn ,zn , x 1 , y 1 ,z 1 , , x n , y n ,z n ) = 0 §15.1 Constraints and their classification
S 15.1 Constraints and their classificationII . Steady constraint and unsteady constraintA constraint that does not change with time is calledasteadyconstraintConstraints that constrain conditions to change overtime are called unsteady constraints. Such asXThe equation is: x? + y? = (lo ut)OM(x,y)The general form of the unsteady constraintVequation is: f,(x, 1,21,*,Xm, yn,2n,t)=0
Ⅱ.Steady constraint and unsteady constraint ⚫Constraints that constrain conditions to change over time are called unsteady constraints. Such as O x y M (x, y) u 0 l The equation is: 2 0 2 2 x + y = (l −ut) ⚫A constraint that does not change with time is called a steady constraint. The general form of the unsteady constraint equation is: f r (x1 , y1 ,z1 , , xn , yn ,zn ,t) = 0 §15.1 Constraints and their classification