In-depth lectures notes covering all lecture material for MIT mechanical engineering course 2.003 Dynamics and Controls 1. The notes cover material from momentum principles, Euler-Lagrange, and vibrations.
2 003 Dynamics and Controls Lecture Notes
Lecture Generalized Coordinates Degree of Freedoms
S main units kinematics Newton's Laws Euler
Lagrange Equilibria Stabilit
Vibrations 2.004 Intro
Identifygoodand badsetsofgeneralized coordinates
Countdegrees offreedom
Dynamics the system
studyofthecggespinthestateota
metalized
coordinates
freedom
system a particle rigidobject many bodies
chemicalreaction
powergrid
the stock market
State representation numbers coordinates of the configuration of the system
generalized coordinates
Generalizedcoordinates must be
1 Complete locate all
parts of the system in all configurations
2 Independent N coordinates If I freeze N 1 the
system can still evolve
ÉampeeshpartideinZD
I p ix yl complete independent
r no complete independent
10 xlxiy.tl complete not independent
2 Particle On A Wire
a ix
yl complete not independent
x not complete not independent
IME s complete independent
3 RigidBody inx 2D
4 not complete independent
y
ix y
x
y yzl not
My.IM complete
mmpfge not independent
eetpytgtnihdependent
ReferenceFrames
an
I origin
Figg axes
xandy
unit length i jalongaxesalia
OA
ix T
,Ya Phasgeneralizedcoordinates ix
y
Fro Ey Frolpositionnectorlexityj
g X
i
0g
Changes Degrees of Freedom
CanthesystemgotromatoB
Whatforces are involved
Egfstafers
of freedom
Whatenergydoweneed
admissible set ofchanges
Degrees
we
studysystems number of generalized coordinates
a to
ich sx
Is Ify
Generalizedcoordinates x
y 3 GeneralizedCoordinates Nonholonomic
Degrees of Freedom Sx Syl 2 Degreesof Freedom systems
Holonomicsystem
Constraints mechanisms that eliminate degreesoffreedom
Crankand slider
Rigidbody in Zbhas3DoF
A 111111 y
3 rigidbodies 3 3 9 DOE
y 3 pinjoints 2 3 6 DOE
I sliding
joint 2 1 2 DOF
Rolling without slipping Ibof
3DOFitfree
To us Ifor vertical
I no
11111111 I DOF
sliding
rolling
no ship
, Lecture 2
Expressions for position velocity acceleration
Usingmovingreferenceframes to simplify
Derivative of a rotating vector
TypicalcomponentsofDynamicsProblem
Example Crankand slider
CC I DOF I GC
O
1 l 1 11 11
RigidBodies
t
want todescribethe motion ofthesecomponents
Kinematics
Constraints that reduce the numberofdegrees offreedom
Onlyneed 1 Gc to describe the motion of the entiresystem Oltl
We will want to know the behaviorof all bodies for this Q t
Oil I 0
I
0 42
s t
lo
Lawsof Motion
Motion will besubjecttophysicalprinciples
Newtonian Dynamics
Lagrangian Dynamics
Energyconservation
ex Fl O 01 0 relate motionsforcesenergies and will provide a differential
equation Equationof Motion
Initial Conditions
The initial stateof the system will determineits evolution
Needthe initial positionsandvelocitiesfor all generalizedcoordinates
a 014
17 0 E t o
Kinematics thegeometry of motion
It
Weknowthe degreesoffreedom
WechooseGcs
Goal is to expressposition velocity acceleration of all partsand bodies as a functio
q and their derivatives
, J Generalizedcoordinates x
y
Enola firmolt position Fmo XItyJ
velocity Flt
Fplolttot Iplo gyppio fiMormolftot
y
O E acceleration
jmo dqymo.fi ovpiolttotl vpiolt
of
Triolet t more insight intotrajectory
in.ie Ianolt1 EighaeYgni ggigsa
Are twoderivativesenough Yesbecause F mi
smoothmanipulator
de trio Jerk am control
drones
IfaTrio snap rockets
Example Ball On A Pole
What is the tension that the polefeels
Motion c Forces
T Free BodyDiagram
y Forcebalance on particle
If C
F
11111111114
mi Ting
ing F ing mi
Approach1 Longleartesian way
Overheadview
in K j t Rws0É Rsinaj gimpy
É j
É jqsy no dfFpio Reino I RWsO0j
ROI sinOI cos05
Into ddftpio ROL WSOOI sinOOJ
o g.INT
ROI WSOI sin 05
Approach 2
Framemoves with C
ÉY Fromabove
up Tis é É é
É
Reference frame Er épée
alwayspoints in the direction
fromrodeoparticle
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