Mechanics of rigid body
Statics Dynamics
Equilibrium
Galilei
Newton
Lagrange
Euler
Kinematics Kinetics
v=ds/dta=dv/dt
Σ F = 0
Σ F = m a
mechanics of rigid body
Statics Dynamics
12.1 Introduction
1.1. KinematicsKinematics Analysis of the geometric aspects of motion. Analysis of the geometric aspects of motion.
2.2. ParticleParticle A particle has a mass but negligible size and shape. A particle has a mass but negligible size and shape.
3.3. Rectilinear KinematicsRectilinear Kinematics Kinematics of objects moving along straight path and Kinematics of objects moving along straight path and characterized by objects position, velocity and characterized by objects position, velocity and acceleration.acceleration.
4.4. PositionPosition(1) position vector r(1) position vector r A vector used to specify the location of particle P at any A vector used to specify the location of particle P at any instant from origin O.instant from origin O.
12.2 Rectilinear Kinematics: Continuous motion
(2) position coordinate , S An algebraic scalar used to represent the position coordinate of particle P
rS
scalar vector
5. DisplacementChange in position of a particle , vector
(1) Displacement
rrr ' or SSS '
from O to P.
P’
P
r’
r
s
r
s
s’
o
(2) DistanceTotal length of path traversed by the particle.A positive scalar.
6. Velocity(1) Average velocity
t
rVavg
(2) Instantaneous Velocity
dt
rd
t
rV lim
0t
dt
ds
t
Slimv
0t
)(
speed = magnitude of velocity = | v |Average speed = Total distance/elapsed time = t
ST
(3) Speed
7. Acceleration
t
Vaavg
(2) (Instantaneous) acceleration
t
vd
t
vlima
0t
2
2
)(dt
Sd
dt
ds
dt
d
dt
dVa
(1) Average acceleration
)(
8. Relation involving a , s and vv=ds/dt, dt=ds/va=dv/dt, dt=dv/a
so, ds/v=dv/a vdv=ads
dt
dvac dtadv c dtadv c
atvv o
9. Constants acceleration a = ac
t
0c
v
vtav
0
dt
dsv
t
c
s
sdttavds
0 0 )(0
dsavdv c dsavdvs
s
c
v
v 00
s
sc
v
v
2
0
0
sav2
1 )ss(a2vv 0c
20
2
0
t
2
1t
So
SS tav
2
c0
tavs
2
c00 2
1tS
10. Analysis Procedure
(2) Kinematic Equations A. Know the relationship between any two of the
four variables a, v, a and t. B. Use the kinematic equations to determine the
unknown varaibles
(1)Coordinate System
A. Establish a position coordinate s along the path.
B. Specify the fixed origin and positive direction of the coordinate.
12.3 Rectilinear Kinematics : Erratic Motion
t t t
S V a
GivenGiven methodmethod Kinematics egnKinematics egn FindFind
S-t graphS-t graph Measure slopeMeasure slope V=ds/dtV=ds/dt V-t graphV-t graph
V-t graphV-t graph Measure slopeMeasure slope A=dv/dtA=dv/dt a-t grapha-t graph
A-t graphA-t graph Area integrationArea integration v-t graphv-t graph
v-t graphv-t graph Area integrationArea integration s-t graphs-t graph
A-s graphA-s graph Area integrationArea integration v-s graphv-s graph
v-s graphv-s graph Measure slopeMeasure slope A=v(dv/ds)A=v(dv/ds) a-s grapha-s graph
tadtvvv
00
tvdtsss
00
2
12
0 )2(0
vadsvs
s
1. Curvilinear motion1. Curvilinear motion
The particle moves along a curved path.The particle moves along a curved path.
Vector analysis will be used to formulate the
particle’s position, velocity and acceleration.
p
12-4 General Curvilinear Motion
s
= change in position of particle form p to p’
(t)r r
r
r'rr
o
p
r (t)
s
2. Position2. Position
3. Displacement3. Displacement
o
p
p’s
r
r ’
r
t
rVavg
td
rd
t
rV
t
lim0
= “tangent” to the curve at Pt .p= “tangent” to the path of motion
4. Velocity
(1) average velocity 平均
(2) Instantaneous velocity (2) Instantaneous velocity 瞬時瞬時
o
p
p’v
r
r ’
r
(3) Speedvv
dt
ds
t
s
t
rlimlim
0t0t
t
v
t
vva avg
dt
rd
dt
rd
dt
d
dt
vd
t
va
2
0tlim
(2) Instantaneous acceleration
which is not tangent to the curve of motion, but
tangent to the hodograph.
Hodographv
'vv
5. Acceleration
(1) Average acceleration(1) Average acceleration
= time rate of change of velocity vectors
Hodogragh is a curve of the locus of points for the arrowhead of velocity vector.
kzjyixr
xyz : fixed rectangular coordinate system
12-5 Curvilinear Motion : Rectangular components
x
y
z
θ
r
pathp
s
1. Position vector
rur
kzyx
zj
zyx
yi
zyx
xzyx
ktzjtyitxtr
222222222
222
)()()()(
Here
222 zyxrr = magnitude of r
r
ru r
= unit vector = direction of r
2. Velocity
v
zyx
uv
kvjviv
kdt
dzj
dt
dyi
dt
dxdt
kdzk
dt
dz
dt
jdyj
dt
dy
dt
idxi
dt
dx
kzjyixdt
ddt
)t(rd)t(v
0 0 0
222zyx vvvv
v
vu v
tangent to the path
3. Acceleration
a
au
aaaa
a
zyx
222
a
zyx
2
2
2
2
2
2
zyx
zvx
ua
kajaia
kdt
zdj
dt
ydi
dt
xd
kdt
dVj
dt
dVi
dt
dV
kVjViVdt
ddt
Vda
: initial velocity
: Constant downward acceleration
: velocity at any instant
0V
a
V
12.6 Motion of a projectile
y
x
v0
v0
a= - g j
v(v0)x
(v0)y
Position Vector (x,y components)
= x + y
initial position
= xo + yo
Velocity Vector
= = x + y = Vx + Vydt
vd
r
i
i
j
0r
j
V
i
j
j
i
Acceleration Vector
a = = + = -g = ax + ay
V0 = (Vx)o + (Vy)o (known)
dt
vd
idt
dVx j
dt
dVy j
i
j
i
j
1. Horizontal motion, 1. Horizontal motion, aaxx=0=0
0dt
dvx
)v(dt
dxv xx
One independent eqn
20x0x
20x
2x )v()xx(a2)v(v
Vx = (Vx)0 + axt = (Vx)0
X = X0 + (Vx)0t
Same as 1st Eq.
X = X0 + (Vx)0t
gt)v(ta)v(v 0yy0yy
two independent eqns
200 2
1)( tatvyy yy
)yy(g2)v(v 02
0y2
y
20y0 gt
2
1t)v(yy
2. Vertical motion, aayy=-g constant=-g constant
Can be derived from above two Eqs.
Path of motion of a particle is known.
1. Planar motion
12-7 Curvilinear Motion:Normal and Tangential components.
opath
o’n
ρ
un
uttp
Here: t (tangent axis ): axis tangent to the curve at P and positive in the direction of increasing S; ut: unit vectorn (normal axis ): axis perpendicular to t axis and directed from P
toward to the center of curvature o’; un: unit vectoro’ = center of curvature= radius of curvaturep = origin of coordinate system tn
s
(1) Path Function
(known) sts
(2) Velocity
dt
dsv tt utsuvv
(3) Acceleration
dt
udsusus
dt
d
dt
dva t
tt
o’ρ
un
ut
pds
dθ
ut’
un
dut
p
ut’
ut
dθ
ttt uduu '
dduud tt nt uud
//
tttt uduudud
dsd
nnt uu
dt
d
dt
ud
s
nt u
s
dt
ud
nnttn
2
t uauaus
usa
at: Change in magnitude of velocity
an: Change in direction of velocity
vdvdsat
n
2
tn
2
t uv
uds
dvvu
vuva
2
2
2/32
dxyd
dxdy1
If the path in y = f ( x )
1. Polar coordinates
u
r
pr ru
Reference line
r: radial coordinate , ru
: transverse coordinate , )..( wccu
(2) Position
rurr
(3) Velocity
rrr urururdt
d
dt
rdrv
)(
12-8 Curvilinear Motion: Cylindrical Components
(1) coordinates (r,)
o
1
1
duu
'ru
ruru
uuuu
uuu
rr
rr
'
utt
u r
utt
ulimlim
0t
r
0t
uu r
ururv r
uvuv rr
r
)r()r(v 22
rate of change of the length of the radial coordinate.
angular velocity (rad/s)
(4) Acceleration
vdt
vda
)uvuv( rr
rr u)u(u
uu
uu'u
uvuvuvuv rrrr
ruu
rr urururura )(
uaua
urrurr
rr
r
)2()( 2
22 )a()a(aa
r0t0t
u)t
(t
ulimlim
angular acceleration
2. Cylindrical coordinates
Position vector
zrp uzurr
Velocity
zrzrpp uzururuz)ur(rv
Acceleration
zrpp uzururva )(
zr uzurrurr )2()( 2
z
x
yr
pr
3D
ru
uzu
v
12.9 Absolute Dependent Motion Analysis of Two Particles
1. Absolute Dependent Motion The motion of one particle depends on the corresponding motion of another particle when they are interconnected by inextensible cords which are wrapped around pulleys.
A BA B
(1)position-coordinate equation
A. Specify the location of particles using position coordinates having their origin located at a fixed point or datum line.
B. Relate coordinates to the total length of card lT
(2)Time Derivatives
Take time derivatives of the position-coordinate equation to yield the required velocity and acceleration equations.
2.Analysis procedure
3. Example
A B
ASBS
DatumDatum
(1)position-coordinate equation
BAT SDCSl
(2) Time Derivatives
BA SS 0 BA VV BAT SDCSl
BA VV
BA aa
12.10 Relative-Motion Analysis of Two Particles
1. Translating frames of referenceA frame of reference whose axes do not rotate and are only permitted to translate relative to the fixed frame.
xyz:fixed framex’y’z’:translating frame moving with particle ArA、 rB : absolute positions of particle A & BrB/A : relative position of B with respect to A
x
o y
z
x’
y’
z’
B
A
rB/A
rB
rA
2. position vector
B
A
O
rA
rB
rB/A rA rB rB/A
3. velocity Vector
ABAB
ABAB
VVV
rrdt
d
dt
rd
/
/ )(
VB/A : relative velocity observed from the translating frame.
4. acceleration vector
A/BAB
A/BAB
aaa
)VV(dt
d)V(
dt
d
aB/A:acceleration of B as seen by an observer located at A and translating with x’y’z’ frame.
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