Transcript PDF#11 - Modeling & Simulation Lab.

Advanced Computer Graphics
Spring 2014
K. H. Ko
School of Mechatronics
Gwangju Institute of Science and Technology
Today’s Topics

Rigid Body Motion
 Newtonian
Dynamics
 Lagrangian Dynamics
 Euler’s Equations of Motion
2
Introduction

Lagrangian Dynamics
A
framework for setting up the equations of motion
for objects when constraints are present.
 The equations of motion are derived from the
kinetic energy function and naturally incorporate
the constraints.
 We could reduce the computational time of the
simulation compared to a general-purpose system
using Newtonian dynamics.
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Newtonian Dynamics

Dynamics describes how the particle must
move when external forces are acting on it.

We specify the acceleration and integrate to
obtain the velocity and position.
 This
is not always possible in a closed form. So
many problems require numerical methods to
approximate the solution.
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Lagrangian Dynamics

Inertial Frame
 Consider the motion of a mass over time
 F = ma = mv’ = mx’’.
 a: acceleration, v: velocity, x: position
 These quantities are measured with respect
to
some coordinate system, which is referred to as
the inertial frame.
 It can be fixed, can have a constant velocity and
no rotation.

Any other frame of reference is referred to as
a noninertial frame.
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Lagrangian Dynamics

The simplicity of Newton’s second law
can disguise the complexity of the
problem. F = ma
 We

must represent all relevant forces for F.
External forces and constraining forces that
apply to the mass.
 This
motivates what is called Lagrangian
dynamics.
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Lagrangian Dynamics

Equations of Motion for a Particle on a Curve
 Given
a curve x(q), q: parameter
 The Lagrangian equation of motion for a single
particle constrained to a curve x(q)
d  T  T

 
 Fq
dt  q  q
T  m x / 2, Fq  F  dx / dq
2
Fq is referred to as a generalized force.
The force term in the equation eliminates the constraining
forces
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Lagrangian Dynamics

Equations of Motion for a Particle on a Surface
 Given
a curve x(q1,q2)
 The Lagrangian equation of motion for a single
particle constrained to a curve x(q)
d  T  T

 
 Fqi
dt  qi  qi
T  m x / 2, Fqi  F  dx / dqi
2
I = 1,2
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Lagrangian Dynamics

Determining Constraint Forces

The degrees of freedom may very well be greater than three.





When constrained to a curve, you have two degrees of freedom
and two equations governing the motion.
A Lagrangian equation occurs for each degree of freedom.
The construction that led to the Lagrangian equations applies
equally well to additional parameters, even if those parameters
are not freely varying.
The generalized forces in these equations must include terms
from the forces of constraint.
These additional equations allow us to determine the actual
constraint forces.
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Lagrangian Dynamics

Time-Varying Frames or Constraints
 If
the frame of reference varies over time or if the
constraining curve or surface varies over time, the
Lagrangian equations of motion still apply.

The Lagrangian formulation is the natural
extension of Newton’s second law when the
motion is constrained to a manifold (curve or
surface).
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Lagrangian Dynamics

Equations of Motion for a System of Particles
 For
each particle, we find a Largrangian equation of
motion under each constraint of interest.

The Lagrangian equations of motion are
obtained by summing those for the individual
particles, leading to
d  T
dt  q j
p
p
 T
xi
1
2

 Fq j , T   mi xi , Fq j   Fi 
 q
2 i 1
q j
i 1
j

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Lagrangian Dynamics

Equations of Motion for a Continuum of Mass
 The
Lagrangian equations of motion are also valid
for a continuum of mass.
d  T
dt  q j
 T
1
x
2

 Fq j , T   vi , Fq j   Fi 
dR
 q
R
R
2
q j
j

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Lagrangian Dynamics

Equations of Motion for a Continuum of Mass
 Using

a transformation to local coordinates,
V = vcen + wⅹr

1
1
2
T  m vcen  112   2 22  3 32
2
2



The first term is the energy due to the linear velocity of the
center of mass.
The last terms are the energies due to the angular velocity
about principal direction lines through the center of mass.
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Euler’s Equations of Motion

Sometimes a physical application is more
naturally modeled in terms of rotation s about
axes in a coordinate system.
A


spinning top.
The top rotates about its axis of symmetry.
Simultaneously the entire top is rotating about a vertical
axis.
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Euler’s Equations of Motion

Coordinate Systems
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Euler’s Equations of Motion

The angular velocity in world coordinates is
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Euler’s Equations of Motion

The angular velocity in body coordinates is
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Euler’s Equations of Motion

Euler’s Equations of Motion
dw
τJ
 w  ( Jw )
dt
μi : the principal moments,
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