Intro to Statics and Dynamics - A. Ruina, R. Pratap.pdf

(11184 KB) Pobierz
267202935 UNPDF
S TATICS
D YNAMICS
and
M s
F s
k
F 1
F 2
ı ˆ
N 1
N 2
Andy Ruina and Rudra Pratap
Pre-print for Oxford University Press, January 2002
Introduction to
ˆ
267202935.004.png 267202935.005.png 267202935.006.png
Summary of Mechanics
0) The laws of mechanics apply to any collection of material or ‘body.’ This body could be the overall system of study
or any part of it. In the equations below, the forces and moments are those that show on a free body diagram. Interacting
bodies cause equal and opposite forces and moments on each other.
I) Linear Momentum Balance (LMB)/Force Balance
Equation of Motion
F i = ˙ L
The total force on a body is equal
to its rate of change of linear
momentum.
(I)
t 2
F i · dt =
Impulse-momentum
(integrating in time)
L
Net impulse is equal to the change in
momentum.
(Ia)
t 1
˙ L = 0
Conservation of momentum
0 )
L = L 2 −
⇒
L 1 =
0
When there is no net force the linear
momentum does not change.
(Ib)
F i =
˙ L is negligible)
0
If the inertial terms are zero the
net force on system is zero.
(Ic)
II) Angular Momentum Balance (AMB)/Moment Balance
Equation of motion
M C =
˙ H C
The sum of moments is equal to the
rate of change of angular momentum.
(II)
Impulse-momentum (angular)
(integrating in time)
t 2
M C dt =
H C
The net angular impulse is equal to
the change in angular mo mentum.
(IIa)
t 1
Conservation of angular momentum
˙ H C =
˙
0
⇒
If there is no net moment about point
C then the angular momentum about
point C does not change.
(IIb)
0 )
H C =
H C2 −
H C1 =
0
˙ H C is negligible)
M C =
0
If the inertial terms are zero then the
total moment on the system is zero.
(IIc)
III) Power Balance (1st law of thermodynamics)
Equation of motion
Q + P = E K + E P + E in t
E
Heat flow plus mechanical power
into a system is equal to its change
in energy (kinetic + potential +
internal).
(III)
t 2
t 1 Qdt +
t 2
for finite time
Pdt = E
The net energy flow going in is equal
to the net change in energy.
(IIIa)
t 1
Conservation of Energy
(if
E
=
0
⇒
If no energy flows into a system,
then its energy does not change.
(IIIb)
Q
=
P
=
0)
E
=
E 2 −
E 1 =
0
Statics
(if
E K is negligible)
Q + P = E P + E int
If there is no change of kinetic energy
then the change of potential and
internal energy is due to mechanical
work and heat flow.
(IIIc)
Pure Mechanics
(if heat flow and dissipation
are negligible)
P = E K + E P
In a system well modeled as purely
mechanical the change of kinetic
and potential energy is due to mechanical
work.
(IIId)
(if F i =
Statics
(if
(if M C =
Statics
(if
267202935.007.png
Some Definitions
(Please also look at the tables inside the back cover.)
r or x
Position
r i / O is the position of a point
i relative to the origin, O)
e
.
g
.,
r i ≡
d r
dt
v
≡
Velocity
v i / O is the velocity of a point
i relative to O, measured in a non-rotating
reference frame)
e
.
g
.,
v i ≡
a
≡
d v
dt =
d 2 r
dt 2
Acceleration
., a i ≡ a i / O is the acceleration of a
point i relative to O, measured in a New-
tonian frame)
e
.
g
ω
Angular
velocity
A measure of rotational velocity of a rigid
body.
α
≡ ˙ ω
Angular acceleration
A measure of rotational acceleration of a
rigid body.
L
≡
m i v i discrete
v dm
Linear momentum
A measure of a system’s net translational
rate (weighted by mass).
continuous
=
m tot v cm
˙ L
≡
m i a i discrete
a dm
continuous
Rate of change of linear
momentum
The aspect of motion that balances the net
force on a system.
=
m tot a cm
H C
≡
r i / C × m i v i discrete
r
× v dm
continuous
Angular momentum about
point C
A measure of the rotational rate of a sys-
tem about a point C (weighted by mass
and distance from C).
/
C
˙ H C
≡
r i / C × m i a i discrete
r / C ×
a dm
continuous
Rate of change of angular mo-
mentum about point C
The aspect of motion that balances the net
torque on a system about a point C.
2 m i v
i
discrete
E K
≡
2 v
Kinetic energy
A scalar measure of net system motion.
1
2 dm continuous
E int =
(heat-like terms)
Internal energy
The non-kinetic non-potential part of a
system’s total energy.
F i ·
v i + M i ·
P
≡
ω i
Power of forces and torques
The mechanical energy flow into a sys-
tem. Also, P ≡ W , rate of work.
I cm
xx
I cm
xy
I cm
xz
[ I cm ]
≡
I cm
xy
I cm
yy
I cm
yz
Moment of inertia matrix about
cm
A measure of how mass is distributed in
a rigid body.
xz I cm
yz I cm
zz
.
.
.
1
I cm
267202935.001.png
Rudra Pratap and Andy Ruina, 1994-2002. All rights reserved. No part of this
book may be reproduced, stored in a retrieval system, or transmitted, in any form
or by any means, electronic, mechanical, photocopying, or otherwise, without prior
written permission of the authors.
c
This book is a pre-release version of a book in progress for Oxford University Press.
Acknowledgements. The following are amongst those who have helped with this
book as editors, artists, tex programmers, advisors, critics or suggestors and cre-
ators of content: Alexa Barnes, Joseph Burns, Jason Cortell, Ivan Dobrianov, Gabor
Domokos, Max Donelan, Thu Dong, Gail Fish, Mike Fox, John Gibson, Robert Ghrist,
Saptarsi Haldar, Dave Heimstra, Theresa Howley, Herbert Hui, Michael Marder,
Elaina McCartney, Horst Nowacki, Arthur Ogawa, Kalpana Pratap, Richard Rand,
Dane Quinn, Phoebus Rosakis, Les Schaeffer, Ishan Sharma, David Shipman, Jill
Startzell, Saskya van Nouhuys, Bill Zobrist. Mike Coleman worked extensively on
the text, wrote many of the examples and homework problems and created many of
the figures. David Ho has drawn or improved most of the computer art work. Some
of the homework problems are modifications from the Cornell’s Theoretical and Ap-
plied Mechanics archives and thus are due to T&AM faculty or their libraries in ways
that we do not know how to give proper attribution. Our editor Peter Gordon has
been patient and supportive for too many years. Many unlisted friends, colleagues,
relatives, students, and anonymous reviewers have also made helpful suggestions.
Software used to prepare this book includes TeXtures, BLUESKY’s implementation
of LaTeX , Adobe Illustrator , Adobe Streamline , and MATLAB .
Most recent text modifications on January 29, 2002.
S TATICS
D YNAMICS
and
M s
F s
k
F 1
F 2
ı ˆ
N 1
N 2
Andy Ruina and Rudra Pratap
Pre-print for Oxford University Press, January 2002
Introduction to
ˆ
267202935.002.png 267202935.003.png

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin