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Lecture 1

CIV ENG 2Q03 Lecture 1: System Particle VS Rigig Body Dynamics

2 Pages

Civil Engineering
Course Code
Dimitrios Konstantinidis

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McMaster University Department of Civil Engineering Winter 2012 CE 2Q03 (D. Konstantinidis) Useful Formulas for the Dynamics of Systems of Particles and Rigid Bodies SYSTEMS OF PARTICLES RIGID BODIES system rigid body B boundary G G ri r x 1 x O O Position/Velocity/Acceleration Vectors Position/Velocity/Acceleration Vectors i is the position of mass particlen1, 2,,K , ) x is the position of a material point on the body vi i& Between any two points on the body with positions av=r= && i i i x1 and x2: v1 2 =×1 2()x aa−v v=x−x= − = 1 21 2 1 2 =α −](1 2[) ω −ω 1 2 xx where ω and α are the angular velocity and angular acceleration respectively. Center of Mass Center of Mass r is the position vector of the center of mass, x is the position vector of the center of mass, which is denoted as G. which is denoted as G. n n ∑∑ mrii i i n ∫xdm ∫ dm i=1i =1 x = = B , where md= r= = n , wheremm= ∑ i dm m ∫ ∑ mi m i=1 ∫ B i=1 B & & && vx= &, av= x&= v= , av== Linear Momentum Linear Momentum n n GG= ∑∑ i i i m GG m∫d d ∫ i=1i =1 B B Gv= m Gv = m Angular Momentum Angular Momentum The angular moment of the system of particles The angular moment of the rigid body relative to a relative to a fixed point P is fixed point P is n n rrvrr=− −( ) ( ) m PPi ii Pii HxPxP∫ ∫)= −(×P d d i=1i=1 B B If P is the origin, If P is the origin, n n GrO iiii m HxOvx×∫∫ ×d d i=i =1 B B McMaster University Department of Civil Engineering Winter 2012 CE 2Q03 (D. Konstantinidis) Alternatively, Alternatively, HHP
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