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Advanced Computer Graphics
Spring 2016
K. H. Ko
School of Mechanical Engineering
Gwangju Institute of Science and Technology
Today’s Topics

Basic Concepts from Physics
 Rigid
Body Classification
 Rigid Body Kinematics
 Newton’s Laws
 Forces
 Momenta
 Energy
2
Basic Concepts from Physics

We will review some of the basic concepts of
physics that are relevant to the analysis of
motion and interaction of rigid bodies.

An appropriate choice of coordinate system
is important in many applications.
 Cartesian,
polar, cylindrical and spherical
coordinate systems.
3
Rigid Body Classification

Characterized by the region that its
mass lives in.
 Single
particle
 Particle system
 Continuous material
Curve mass
 Surface mass
 Volume mass

4
Rigid Body Classification

Discrete Material
 Single

particle
Mass m that occupies a single location x.
 Particle

system
A collection of a finite number of particles

The I th particle having mass mi and located at xi.
p
Qtotal 
Q
i
i 1
Qi : some physical quantity associated with the ith particle.
5
Rigid Body Classification

Continuous Material
 It
consists of infinitely many particles that lie in a
bounded region of space, R. -> a continuum of
mass



Curve mass: The region R is a bounded segment of a
curve in one, two or three dimensions.
Surface mass: The region R is a bounded region in the
plane or a bounded portion of a surface in space.
Volume mass: R can be a solid occupying a bounded
region of space.

Qtotal  QdR
R
6
Rigid Body Classification

Continuous material
 Curve
mass: the integration is computed as
a line integral.
 Surface mass: the integration is computed
as a double integral.
 Volume mass: the integration is computed
as a triple integral.
7
Rigid Body Kinematics

Kinematics
 It
is a study of motion of objects without
considering the influence of external forces.
8
Rigid Body Kinematics

Planar Motion of Single Particle in Cartesian
Coordinates.





The position of a particle at time t is r(t) = x(t) i + y(t) j.
The velocity of the particle at time t is v(t) = x’(t) i + y’(t) j.
The acceleration of the particle at time t is a(t) = x’’(t) i + y’’(t)
j.
The unit length tangent vector is given by T(t) = v/|v| =
(cos(φ(t)),sin(φ(t))).
The unit length normal vector is chosen as
N(t) = (-sin(φ(t)),cos(φ(t))).
9
Rigid Body Kinematics

Planar Motion of Single Particle in
Cartesian Coordinates.
A
coordinate system at a point on the
curve is defined by origin r(t) and
coordinate axis directions T(t) and N(t).

Moving frame: {r(t); T(t), N(t)}.
 Velocity:

v = |v|T = s’T.: s is the arc length measured
along the curve. s’ is the speed .
10
Rigid Body Kinematics

Planar Motion of Single Particle in Cartesian
Coordinates.
 Curvature
of the curve at arc length s
dT d
d
 (cos  , sin  ) 
( sin  , cos )  N( s)
ds ds
ds
 Acceleration
d
dT
2 dT
a  v  ( sT)  sT  s
 sT  s
dt
dt
ds
or
a  sT  s 2 N
11
Rigid Body Kinematics

Planar Motion of Single Particle in Cartesian
Coordinates.
 Acceleration
sT : Tangent acceleration, the acceleration in the direction of motion
s 2 N :normal acceleration, centripetal acceleration
 The
rate of change of the tangent vector with
respect to arc length is related to the normal
vector
dN d
d
 ( sin  , cos ) 
( cos , sin  )  T
ds
ds
ds
12
Rigid Body Kinematics

Planar Motion of Single Particle in Cartesian
Coordinates.
 Summarizing
the s-derivatives in a format matrix
notation.
 dT 
 ds   0    T 
 dN   
 N 


0
 
  
 ds 
The coefficient matrix is
skew-symmetric.
13
Rigid Body Kinematics

Planar Motion of Single Particle in Polar
Coordinates
vector: R = r/|r| and r = |r| from the
origin -> r = rR.
 A unit vector perpendicular to R is P.
 Position

Rotated by п/2 counterclockwise.
 The
moving frame {r(t); R(t), P(t)} provides an
alternate coordinate system to the tangent-normal
one.
 dR 
 ds   0  R 
 
 dP    
    0   P 
 ds 
14
Rigid Body Kinematics

Planar Motion of Single Particle in Polar
Coordinates
 Velocity
d
  rR  rP
v  r  (rR )  rR  rR
dt
 Acceleration
  d (r)P  rP  (r  r 2 )R  (r  2r)P
a  v  rR  rR
dt
15
Rigid Body Kinematics

Spatial Motion in Cartesian Coordinates
 Position
of a particle at time t
r (t ) x(t )i  y (t ) j  z (t )k
 Velocity
at time t
v(t )  r  xi  yj  zk
 Acceleration
at time t
a(t )  v  r  xi  yj  zk
 Unit-length
tangent vector
v
T(t ) 
v
16
Rigid Body Kinematics

Spatial Motion in Cartesian Coordinates
 Normal
vector
dT
  ( s )N( s )
ds

The choice of normal should be made in an attempt to
maintain a continuous function N(s)
 Binormal
vector
dB
B  T  N,
 N
ds
17
Rigid Body Kinematics

Spatial Motion in Cartesian Coordinates
 Normal
vector
N  BT
dN
dT dB
 B

 T  T  B
ds
ds ds
 Frenet-Serret
 dT 
 ds 
0
 dN  
    
 ds  
 dB   0
 ds 
equation

0  T( s ) 
0   N( s )
  0  B( s) 
18
Rigid Body Kinematics

Spatial Motion in Cylindrical Coordinates
 Position
r (t ) rR  zk
 Velocity
v  r  rR  rP  zk
 Acceleration
a  v  (r  r 2 )R  (r  2r)P  zk
   0  0 R 
R
 
  
P



0
0
P 
  
 k   0 0 0  k 
  

19
Rigid Body Kinematics

Spatial Motion in Spherical Coordinates
 Position r (t ) R
  (  sin  )P  ( )Q  (  )R
v  R  R
 Acceleration a  ((   2 ) sin   2  cos  ) P
 Velocity
 (  ( 2 sin  cos   )  2 )Q
 (    ( 2   2 sin  )) R
 P   0
 cos 
  
0
Q     cos 

    sin 
R


  
  sin    P 

  Q
0  R 
20
Rigid Body Kinematics

Motion About a Fixed Axis: the position,
velocity and acceleration of a particle that is
rotating about a fixed axis and is a constant
distance from that axis.
R
= (cosθ)ξ + (sinθ)η on the plane
 Angular speed: σ(t) = θ’(t)
 Angular velocity: w(t) = σ(t)D
 Angular acceleration: α(t) = σ’(t)D
21
Rigid Body Kinematics

Motion About a Fixed Axis
 The position of a particle
 r(t) = r0R(t) + h0D
in motion about the axis
 The velocity
 v(t) = wⅹr
 The acceleration
 a(t) = -r0σ2R + αⅹr
 -r0σ2R: the centripetal acceleration of the particle
 αⅹr: the tangential acceleration of the particle.-> tangent
to the circle of motion.
22
Rigid Body Kinematics

Motion About a Moving Axis
 The
concept of angular velocity for a time-varying
axis with unit-length direction vector D(t) can be
established by studying motion about a fixed axis.
 Particle position: r(t) = r0(cosθ(t)ξ + sinθ(t)η)+h0D.



Initial position: r0=r(0)=r0ξ
Position at later times: r(t) = R(t)r0
R(t) = I + (sin θ(t))Skew(D) + (1- cosθ(t))Skew(D)2
 0
Skew(u )   u3
 u2
 u3
0
u1
u2 
 u1 , u  u1 u2
0 
u3 
23
Rigid Body Kinematics

Motion About a Moving Axis
 Linear
velocity: r’(t) = w(t)ⅹr(t) = Skew(w(t))r(t)
 R’(t)=Skew(w(t))R(t)
 If the unit direction vector D varies with time, i.e.
D(t)

R(t) = I + (sin θ(t))Skew(D(t)) + (1- cosθ(t))Skew(D(t))2
 Angular

velocity
w = θ’D + (sin θ)D + (cos θ-1)DⅹD
24
Rigid Body Kinematics

Particle Systems and Continuous Materials
 World Coordinates
 Position at time t in world coordinates : x(t;P)
 Body Coordinates
 Given the origin of body coordinate, x(t;C), the position
x(t;P) with respect to body coordinate is given by b(t;P).
 x(t;P)
= x(t;C) + R(t)b(t;P), R(t) =
[U0(t),U1(t),U2(t)], the body axis directions with
respect to world coordinates. Here R(t) is a
rotation matrix.
25
Rigid Body Kinematics

Particle Systems and Continuous Materials
 Consider
a time-varying vector written in the body
coordinate system ξ(t) = R(t)s(t), where s(t) is the
body coordinates varying with time.


What the world observer sees
dξ
ds
ds
Dξ
 R  R s  R  R RT ξ 
 wξ
dt
dt
dt
Dt
What the body observer sees
Dξ
ds
 R(t )
Dt
dt
26
Rigid Body Kinematics

Particle Systems and Continuous Materials


The body origin has world velocity vcen=dX(t;C)/dt and world
acceleration acen = dvcen/dt.
Then the world velocity vwor
v wor  v cen  R



db 
Dr
 Rb  v cen 
 wr
dt
dt
Vcen: the drag velocity, the velocity of the body origin relative to
the world coordinates
Dr/Dt, the velocity of P measured relative to the body
coordinates
wⅹr : the velocity due to rotation of the frame
27
Rigid Body Kinematics

Particle Systems and Continuous Materials
 Then
the world velocity awor=dvwor/dt
a wor  a cen 
a wor
d  Dr  d

  w  r 
dt  dt  dt
Dw
Dr D 2r
 a cen  w  (w  r ) 
 r  2w 
 2
Dt
Dt Dt
1. acen: the translational acceleration of the body origin relative
to the world coordinates
2. wⅹ(wⅹr): the centripetal acceleration due to rotation of the
frame
3. (Dw/dt) ⅹr: the tangential acceleration due to angular
acceleration
4. 2wⅹ(Dr/Dt): the Coriolis acceleration
5. D2r/Dt2: the acceleration of P relative to the body
28
Rigid Body Kinematics

Newton’s Law of Physics

In the absence of external forces, an object at rest will
remain at rest. If the object is in motion and no external
forces act on it, the object remains in motion with constant
velocity

For an object of constant mass over time, its acceleration a
is proportional to the force F and inversely proportional to
the mass m of the object. a = F/m. If the mass changes over
time, the more general statement of the law is F = d(mv)/dt
= ma + (dm/dt)v.


mv is the linear momentum of the object.
The application of an external force on an object causes a
change in the object’s momentum over time.
29
Newton’s Law

If a force is exerted on one object, there is a force of
equal magnitude but opposite direction on some
other body that interacts with it. (Action/reaction)

Inertial frame: position, velocity and acceleration are
measured with respect to some coordinate system. > Inertial frame.



The inertial frame can be fixed or have a constant linear
velocity and no rotation.
Kinetic energy must be measured in an inertial system.
Noninertial frame: any other frame of reference.
30
Forces

Gravitational Forces
GmM
Fgravity 
r2
 Approximation of the gravitational force on the
earth.(near the flat surface of the earth)


F = -mgU.
Spring Forces
F
= -cΔU: c>0 is the spring constant. (Hooke’s
law)
31
Forces

Friction and Other Dissipative Forces
A
dissipative force is one for which energy of the
system decreases when motion takes place.
 Fdissipative = c|v|n.

c is mostly a constant but can vary with time.
32
Forces

Friction and Other Dissipative Forces

Friction




A friction force between two objects in contact opposes the
sliding of one object over the surface of the adjacent one.
It is tangent to the surface of the adjacent object and opposite
in direction to the velocity of the moving object.
The magnitude of the frictional force is assumed to be
proportional to the magnitude of the normal force between
surfaces.
It is also assumed to be independent of the area of contact and
independent of the speed of the object once that object starts
to move.
v


c
, v0
 k
F
|v|

0, v  0
33
Forces

Friction and Other Dissipative Forces
 Viscosity



A viscous force has magnitude with n=1.
Typically, when an object is dragged through a thick fluid,
a viscous force can be experienced.
The force is modeled to have direction opposite to that
of the moving object.
v
v
F   Fdissipative
 (c | v |)
 cv
|v|
|v|
34
Forces

Torque
 Let
F be the applied force and r the position of
the particle relative to the origin

τ=rⅹF.
 For
a system of p particles located at position ri.
p
   ri  Fi
i 1
 For
a continuum of mass that occupies a region R,
   r  FdT
R
35
Forces

Equilibrium
 An


object is in equilibrium if
The sum of all external forces acting on the object must
be zero.
The torques on the object must sum to zero.
 An
object in equilibrium is not necessarily
stationary. It is possible that the inertial frame in
which the object is measured is moving with
constant velocity.
36
Momenta


Linear and Angular Momentum
Mass of an object
 Center of mass of an object
 A finite number of point masses (discrete)
 A solid body (continuous)

Moments and products of inertia.

These concepts are particularly important
when discussing motion of a rigid body.
37
Linear Momentum

Single particle: p = mv
 The
applied force and momentum are related by
F=dp/dt.
p


mi v i
A system of particle: p  
i 1
A continuum of mass: p   vdm  vdR
R

R
Conservation of momentum
 If
the net external force on a system of objects is
zero, the linear momentum is a constant.
38
Angular Momentum

Single particle: L=rⅹp=rⅹmv
 Torque
is the time derivative of angular momentum
p


A system of particle: L   r  m v
A continuum of mass: L   r  vdm   r  vdR
i 1
i
R

i
i
R
Conservation of momentum
 If
the net external force on a system of objects is
zero, the angular momentum is a constant.
39
Center of Mass

Discrete Mass in One Dimension

x

p
i 1
p
mi xi
i 1

mi
Continuous Mass in One Dimension
b
x
 x ( x)dx
a
b
  ( x)dx
a
40
Center of Mass

Discrete Mass in Two Dimension
p
p


m
(
x
,
y
)
m
x
m
y



i
i
i
i
i
i
i

( x , y )  i 1 p
  i p1
, i p1


m
m
m
i 1 i
i 1 i 
 i 1 i
p

Continuous Mass in One Dimension
(x, y) 
 ( x, y) ( x, y)dxdy
R
  ( x, y)dxdy
R
41
Center of Mass

Discrete Mass in Two Dimension
p
p


m
(
x
,
y
)
m
x
m
y



i
i
i
i
i
i
i

( x , y )  i 1 p
  i p1
, i p1


m
m
m
i 1 i
i 1 i 
 i 1 i
p

Continuous Mass in One Dimension
(x, y) 
 ( x, y) ( x, y)dxdy
R
  ( x, y)dxdy
R
42
Center of Mass

Discrete Mass in Three Dimension
p
p
p


m
(
x
,
y
,
z
)
m
x
m
y
m
z




i
i
i
i
i
i
i
i
i
i

( x , y )  i 1 p
  i p1
, i p1
, i p1


m
m
m
i 1 mi



i
i
i
i 1
i 1
i 1


p

Continuous Mass in Three Dimension
( x, y, z )dxdydz

(x, y) 
 dxdydz
V
V
43
Moments and Products of Inertia

Moment of inertia is another important quantity
in physics.

This is a measure of the rotational inertia of a
body about an axis.

The more difficult it is to set the object into
rotation, the larger the moment of inertia about
that axis.
44
Moments and Products of Inertia

Moment of Inertia in One Dimension
A

system of discrete particles
p
With respect to the origin of the real line
I 0   mi xi2
i 1
p

With respect to the center of mass I   mi xi  xi 2  I 0  mx 2
i 1
A

continuous mass
b
2
With respect to the origin of the real line I 0   x  ( x)dx
a
b

With respect to the center of mass I   ( x  x ) 2  ( x)dx  I 0  mx 2
a
45
Moments and Products of Inertia

Moment of Inertia in Two Dimensions
A

system of discrete particles
p
With respect to the origin of the real line I 0   mi ( xi2  yi2 )
i 1

With respect to the center of mass
p
I   mi ( xi , yi )  ( xi , yi )  I 0  m( x 2  y 2 )
2
i 1
A
continuous mass

2
2
With respect to the origin of the real line I 0  R ( x  y ) ( x)dxdy

With respect to the center of mass
b
I   (( x  x ) 2  ( y  y ) 2 ) ( x, y)dxdy  I 0  m( x 2  y 2 )
a
46
Moments and Products of Inertia

Moment of Inertia in Three Dimensions
A

system of discrete particles
Moments
of inertiap about the x,y pand z axes.
p
I xx   mi ( yi2  zi2 ), I yy   mi ( xi2  zi2 ), I zz   mi ( xi2  yi2 )
i 1

i 1
i 1
Moments of inertia about a line L through origin O. given
parametrically as O+tD with unit-length direction vector
D=(d1,d2,d3)
p
I L   mi ( ri  (D  ri ) 2 )
2
i 1
 d12 I xx  d 22 I yy  d 32 I zz  2d1d 2 I xy  2d1d3 I xz  2d 2 d 3 I yz
p
p
p
i 1
i 1
i 1
I xy   mi xi yi , I xz   mi xi zi , I yz   mi yi zi
47
Moments and Products of Inertia

Moment of Inertia in Three Dimensions
A

system of discrete particles
Inertia Tensor
 I xx

J   I yx
  I xz


 I xy
I yy
 I yz
 I xz 

 I yz 
I zz 
Moments and products of inertia for a continuum of
mass occupying a region R
I xx   y 2  z 2 dm, I yy   x2  z 2 dm, I zz   x2  y 2 dm,
R
R
R
I xy   xydm, I xz   xzdm, I yz   yzdm,
R
R
R
48
Moments and Products of Inertia

The angular momentum
L
= rⅹmv = mrⅹ (wⅹr) = m(|r|2 I -rrT)w = Jw


The linear momentum equation p = mv.
The torque equation
 τ=Jα:


α is the angular acceleration
Newton’s second law: F = ma.
Euler’s equations of motion
dL
dw

M
 w  ( Mw )
dt
dt
M: the diagonal matrix of principal moments
49
Moments and Products of Inertia

Mass and Inertia Tensor of a Solid Polyhedron
 Mirtich’s

Reduction from 3D -> 2D



method
The construction uses the Divergence theorem from calculus
for reducing volume integrals to surface integrals.
It is a reduction from three-dimensional integrals to twodimensional integrals.
Reduction from 2D -> 1D



The polyhedron surfaces are planar faces.
Projection of them onto coordinate planes are used to set up
yet another reduction in dimension.
Green’s Theorem is used to reduce the planar integrals to lin
e integrals around the boundaries of the projected faces.
50
Energy

Work and Kinetic Energy
 Total
work done by the force on the particle as it
travels over a time interval [t0,t1] with the velocity of
the particle v(t)
t1
dx
W   F  dt   F  vdt
t0
t0
dt
t1
51
Energy

Conservative Force
 The
work is independent of the path.
 It only depends on the starting and ending points.
 The total energy of the system is conserved.

Nonconservative Force
 Friction,
viscous drag when moving through a fluid
and many forces that have explicit dependence on
time and velocity.
52
Energy

Potential Energy
 The
work done by the forces in transferring the
system is referred to as the potential energy.
p
V    Fi  vi dt
i 1

1
0
For having a conservative force
dE d (T  V )
E  T V ,

 x  (mx  F )  0
dt
dt
53