14.2D'Alembert's principleforparticle systemsThe external force system can be divided into the active forcesystem acting on the particle system and the external constraintreaction system. Let Fand N,be the resultant of the main forceacting on the i particle and the resultant of the externalconstraint reaction, respectively, and we get:ZF+ZN,+EF8=OEmo(F)+Zmo(N,)+Emo(F8)=0In other words, at any instant of the motion of a particle system.all the active force systems acting on the particle system, theconstrained reaction force system and the inertial force systemsupposedly added on the particle system constitute the formalequilibrium force system. This is the D'Alembert principle of theparticle system
The external force system can be divided into the active force system acting on the particle system and the external constraint reaction system. Let and be the resultant of the main force acting on the particle and the resultant of the external constraint reaction, respectively, and we get: Fi Ni i + + = 0 g Fi Ni Fi ( ) + ( ) + ( ) = 0 g mO Fi mO Ni mO Fi In other words, at any instant of the motion of a particle system, all the active force systems acting on the particle system, the constrained reaction force system and the inertial force system supposedly added on the particle system constitute the formal equilibrium force system. This is the D'Alembert principle of the particle system. 14.2 D'Alembert's principle for particle systems
14.3Simplificationof a rigid bodyinertial force systemI.Theprincipal vectorof the inertial force systemZm,r, = MrRigid body:mass M, accelerationof centerof mass isaParticlei: mass m, accelerationis aEm,a, = MacThentheprincipalvectoroftheinertialforce systemis:Fg =ZF =E(-m,a,)=-Zm,a,Fg=-MacIt shows that whatever motion does the rigid body do, theprincipal vector of the inertial force system is equal to the productof the mass of the rigid body and the acceleration of its center ofmass, and the direction is opposite to that of the acceleration of itscenter ofmass
Particle : i mass , acceleration is ai mi aC Rigid body: mass M, acceleration of center of mass is i i i i g i g F F m a m a = = (− ) = − C g F Ma = − It shows that whatever motion does the rigid body do, the principal vector of the inertial force system is equal to the product of the mass of the rigid body and the acceleration of its center of mass, and the direction is opposite to that of the acceleration of its center of mass. mi ai MaC = Ⅰ. The principal vector of the inertial force system Then the principal vector of the inertial force system is: i i C m r Mr = 14.3 Simplification of a rigid body inertial force system