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
S 15.1 Constraints and their classificationII.Two- sided constraint and one- sided constraintThe constraint that simultaneously restricts the motion of a particle inone direction and the opposite direction is called a double-sided constraintConstraints that can only limit the motion of a particle in one directionand not in the opposite direction are called unilateral constraintsThe general form of the constraint equation isf,(xi,yi,zi,",xn,yn,zn)<0IV.Holonomic and nonholonomic constraintsThe motion constraints of which geometric constraints or theirconstraint equations can be integrated are called integrity constraintsIf the constraint equation contains the derivative of coordinateswith respect to time and cannot be integrated, the constraint iscalled nonholonomicIn this chapter, we only study the invariant two-sidedgeometric constraint problems
Ⅲ. Two - sided constraint and one - sided constraint ⚫The constraint that simultaneously restricts the motion of a particle in one direction and the opposite direction is called a double-sided constraint. ⚫Constraints that can only limit the motion of a particle in one direction and not in the opposite direction are called unilateral constraints. f r (x1 , y1 ,z1 , , xn , yn ,zn ) 0 Ⅳ. Holonomic and nonholonomic constraints ⚫The motion constraints of which geometric constraints or their constraint equations can be integrated are called integrity constraints. In this chapter, we only study the invariant two-sided geometric constraint problems. ⚫If the constraint equation contains the derivative of coordinates with respect to time and cannot be integrated, the constraint is called nonholonomic. The general form of the constraint equation is §15.1 Constraints and their classification
S 15.2 Concept and calculation of virtual displacementI .The concept of virtual displacementAt a certain instant, any tiny displacement that can berealized under the condition that the constraints allow is calledthe virtual displacement of the particle system. Such as:X1BrxM(x,y)TWhat is the difference between a virtual displacement and areal displacement?Common point:1) Both are constrained by the constraints, which are thedisplacements allowed by the constraints.2) The real displacement is one of several virtual displacementsunder constant constraint
Ⅰ. The concept of virtual displacement At a certain instant, any tiny displacement that can be realized under the condition that the constraints allow is called the virtual displacement of the particle system. Such as: O x y M (x, y) r O A B x y A r B r What is the difference between a virtual displacement and a real displacement? Common point: 1）Both are constrained by the constraints, which are the displacements allowed by the constraints. 2）The real displacement is one of several virtual displacements under constant constraint. §15.2 Concept and calculation of virtual displacement
S 15.2 Concept and calculation ofvirtual displacementDifferences:1) Thevirtualdisplacementis infinitesimal, while the realdisplacement can be infinitesimalor finite.2) Ittakes time to complete the realdisplacement, while thevirtualdisplacementcorrespondstotheinstantaneous.3) The realdisplacementis relatedto the force exerted on the system and theinitialconditions of motion, while the virtualdisplacementis purelyageometric concept, which neitherinvolves the actualmotion of the system northe action of forces, and has nothing to do with the time process and the initialconditionsofmotion.4) In the case ofa constantconstraint,a smallreal displacement must be one ofthe virtual displacements. In the case of unsteady constraint, the realdisplacementhas no relation with the virtualdisplacement.The representation method of virtual displacement:Sr,SxSGenerallinearangularExpressionsdisplacementdisplacement
Differences: 1）The virtual displacement is infinitesimal, while the real displacement can be infinitesimal or finite. 2）It takes time to complete the real displacement, while the virtual displacement corresponds to the instantaneous. 3）The real displacement is related to the force exerted on the system and the initial conditions of motion, while the virtual displacement is purely a geometric concept, which neither involves the actual motion of the system nor the action of forces, and has nothing to do with the time process and the initial conditions of motion. 4）In the case of a constant constraint, a small real displacement must be one of the virtual displacements. In the case of unsteady constraint, the real displacement has no relation with the virtual displacement. The representation method of virtual displacement: r x , , General Expressions linear displacement angular displacement §15.2 Concept and calculation of virtual displacement
S 15.2 Concept and calculation of virtual displacementI.Calculation of virtual displacement1.GeometricmethodOnly the case of time-invariant constraints is discussed.The real displacement under this condition is one of thevirtual displacements. We can use the method of realdisplacement to find the relationship between the virtualdisplacement of each particle. This method is also called virtualvelocitymethodSrB-VrotVBThat is:SraV.&tVA
Ⅱ. Calculation of virtual displacement 1. Geometric method The real displacement under this condition is one of the virtual displacements. We can use the method of real displacement to find the relationship between the virtual displacement of each particle. This method is also called virtual velocity method. A B A B A B v v v t v t r r = = That is: Only the case of time-invariant constraints is discussed. §15.2 Concept and calculation of virtual displacement
S 15.2 Concept and calculation of virtual displacementSince AB moves in the plane, the velocity projection theoremVs cos = VA cos[90° -(p + )]= V sin(β + 0)Srgsin(β+0)VBcosSrAVAOr, because C*is the instantaneous center ofBC*VBVBAB, soVABrSrAAC*xBC*AC*07VA0According to the sine theorem:BC*AC*AC*cosAsin(β+)ssin(90°-0)BC*Srgsin(β+)VBAlso get:AC*SrAcosOVA
O A B x y A r B r Since AB moves in the plane, the velocity projection theorem cos = cos90 − ( + )= sin( + ) B A A v v v cos sin( + ) = = A B A B v v r r Or, because is the instantaneous center of AB, so C C = = AC BC v v BC v AC v A A B ， B According to the sine theorem: sin( ) sin(90 ) cos = − = + BC AC AC Also get: cos sin( + ) = = = AC BC v v r r A B A B §15.2 Concept and calculation of virtual displacement