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Lecture 1

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

2 Pages
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Department
Civil Engineering
Course Code
CIVENG 2Q03
Professor
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
Position/Velocity/Acceleration Vectors
i
r is the position of mass particle i ( 1, 2, 3, ,in=K)
ii
=vr
&
iii
==avr
&&
&
RIGID BODIES
Position/Velocity/Acceleration Vectors
x is the position of a material point on the body
Between any two points on the body with positions
1
x and 2
x:
12 12
()
âˆ’
=Ã— âˆ’vv Ï‰xx
12 1 2 1 2
12 12
()[()]
âˆ’
=âˆ’=âˆ’=
=Ã— âˆ’ + Ã— Ã— âˆ’
aa vv xx
Î±xx Ï‰Ï‰ xx
&&&&&&
where Ï‰ and Î± are the angular velocity and
angular acceleration respectively.
Center of Mass
r is the position vector of the center of mass,
which is denoted as G.
11
1
nn
ii ii
ii
n
i
i
mm
m
m
==
=
==
âˆ‘âˆ‘
âˆ‘
rr
r, where
1
n
i
i
mm
=
=âˆ‘
=vr
&, ==avr
&&
&
Center of Mass
x is the position vector of the center of mass,
which is denoted as G.
dm dm
m
dm
==
âˆ«
âˆ«
âˆ«
BB
B
xx
x, where mdm=âˆ«
B
=
vx
&,
=
=avx
&&&
Linear Momentum
11
nn
iii
ii
m
==
==
âˆ‘âˆ‘
GG v
m=Gv
Linear Momentum
ddm==
âˆ«
âˆ«
BB
GGv
m
=
Gv
Angular Momentum
The angular moment of the system of particles
relative to a fixed point P is
11
() ()
nn
PiPiiPii
ii
m
==
=âˆ’Ã—=âˆ’Ã—
âˆ‘âˆ‘
HrrGrrv
If P is the origin,
11
nn
Oiiiii
ii
m
==
=Ã—=Ã—
âˆ‘âˆ‘
HrGrv
Angular Momentum
The angular moment of the rigid body relative to a
fixed point P is
() ()
PP P
ddm= âˆ’Ã—= âˆ’Ã—
âˆ«âˆ«
BB
HxxGxxv
If P is the origin,
Oddm=Ã— =Ã—
âˆ«âˆ«
BB
HxGxv
1
r
i
r
r
G
O
system
boundary
x
x
G
O
rigid body B

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Description
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 =Ã—1 2()x aaâˆ’v v=xâˆ’x= âˆ’ = 1 21 2 1 2 =Î± âˆ’](1 2[) Ï‰ âˆ’Ï‰ 1 2 xx where Ï‰ and Î± 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 âˆ‘âˆ‘ mrii i i n âˆ«xdm âˆ« dm i=1i =1 x = = B , where md= r= = n , wheremm= âˆ‘ i dm m âˆ« âˆ‘ mi m i=1 âˆ« B i=1 B & & && vx= &, av= x&= v= , av== Linear Momentum Linear Momentum n n GG= âˆ‘âˆ‘ i i i m GG mâˆ«d d âˆ« 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=âˆ’ âˆ’( ) ( ) m PPi ii Pii HxPxPâˆ« âˆ«)= âˆ’(Ã—P d d i=1i=1 B B If P is the origin, If P is the origin, n n GrO iiii m HxOvxÃ—âˆ«âˆ« Ã—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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