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2. mathematics
3. algebra

Engineering Dynamics: A
Comprehensive Introduction—Errata
N. Jeremy Kasdin & Derek A. Paley
Last updated June 2, 2015
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
1
Chapter 2:
i) In tutorial 2.4, the solution for the critically damped case (starting at
the bottom of page 33) solves for λ1 , λ2 = −ζω0 . This is propagated
through to the solution (Eq. (2.26)), which makes it appear as if ζ is a
variable in the solution. This is misleading as ζ can only equal 1 in this
case. The two equation should be rewritten without ζ (3 instances).
ii) In problem 2.16, the downward acceleration should be g −
c 2
m y˙ .
Chapter 3:
i) In Section 3.4.1, the final equation on Page 68 defining the transformation matrix should be in terms of the scalar component magnitudes,
not the unit vectors. That is, the elements of the two matrices should
not be bold. It should thus read:
b1
cos θ sin θ
a1
=
b2 B
− sin θ cos θ
a2 A
ii) In Section 3.8.4 on page 93 describing the vector Taylor series the
vector (and components) were mistakenly written at t+a rather than t.
Thus, the sentence before Eq. (3.68), that equation, and the following
two equations should be written:
First, write the vector r(t) as components in a frame A = (O, a1 , a2 , a3 ):
r(t) = r1 (t)a1 + r2 (t)a2 + r3 (t)a3 .
(1)
Then take the Taylor series of the scalar magnitude of each component.
For i = 1, 2, 3, we have
1 d2 d (t − a) +
ri
(t − a)2 + . . .
ri (t) = ri (a) + ri dt t=a
2! dt2 t=a
Expanding and rearranging each coefficient in Eq. (3.68) yields
r(t) = r1 (a)a1 + r2 (a)a2 + r3 (a)a3
d d d + r1 (t − a)a1 + r2 (t − a)a2 + r3 (t − a)a3
dt t=a
dt t=a
dt t=a
1 d2 1 d2 1 d2 2
2
+
r1
(t − a) a1 +
r2
(t − a) a2 +
r3
2! dt2 2! dt2 2! dt2 t=a
t=a
+...
√
iii) In Problem 3.2, F3 = (5 − 4 6)ey N.
iv) Figure 3.45 is missing a downward gravity arrow labelled g.
t=a
(t − a)2 a3
2
Chapter 4:
i) In Example 4.1 on pgs. 115-116, the expressions for x(t) and x(t)
˙
should be written relative to the initial time t2 ,
−F¯P (t1 , t2 )
sin [ω0 (t − t2 )]
mP ω0
−F¯P (t1 , t2 )
x(t)
˙ =
cos [ω0 (t − t2 )]
mP
x(t) =
ii) On page 138, the first unnumbered equation should not have a minus
sign.
iii) In Tutorial 4.3, the second-to-last line on page 140 should say “The
magnitude of the specific angular impulse...” Also, the two moment expressions on page 141 should be divided by the satellite mass to correspond to the change in specific angular momentum, i.e., the left-handside of the two unnumbered equations should be kMA/O (t1 , t2 )k/mP
and kMB/O (t1 , t2 )k/mP , respectively.
iv) In Problem 4.11 on page 146, the feedback equation for the base motion
should be for the acceleration u
¨ of the base and not µ, i.e., the equation
should read
g sin θ + kθ + bθ˙
u
¨=−
cos θ
Chapter 5:
i) Top of page 155 should not be gray box, up to the start of Example
5.3.
ii) On page 163, in Example 5.8, the middle unnumbered equation has a
typo; the sign is wrong on the last term. It should read:
T b1 − µc N b1 + N b2 − mP g sin θb1 − mP g cos θb2 = 0
iii) On page 183, in Figure 5.18, right side, the length of the string should
be longer so that the height of the white particle is the same as the
height of the black particle.
iv) On page 183, the ellipse in Figure 5.19 should be taller so that it only
intersects the circle at point P, i.e., it completely encloses the circular
orbit of radius ρ.
v) In Problem 5.14, page 184, the value given for C in part (b) is actually
the square root of the correct value. C should equal 0.33, not -0.58.
3
Chapter 6:
(ext)
(ext)
i) Equation 6.10 on page 193 should use F
(t1 , t2 ) instead of F
.
(ext)
The following line should read “where F
(t1 , t2 ) is the total external
impulse acting on the system from t1 to t2 .”
ii) The second line of Equation 6.23 on page 203 should be
=
−k(m1 + m2 )
kr2/1 k − l0 ˆr2/1 .
m1 m2
iii) Figure 6.14(b) on page 214 is incorrect. It should be altered as below.
Figure 1 Corrected figure 6.14(b).
iv) On page 235, the unnumbered equation in the second bullet of the
(ext)
(ext)
Key Ideas should use F
(t1 , t2 ) instead of F
. The following line
(ext)
should read “where F
(t1 , t2 ) is the total external impulse from t1
to t2 .”
v) In Problem 6.7 on p. 239, the lower summation index should be i not
P
j, so that it reads N
i=1 .
vi) In Problem 6.23 on p. 244, the value for the exhaust velocity should
be 2.75 km/s, not 1 km/s.
vii) In Problem 6.24 on p. 244, the value for the spring rest length should
be l0 = 1 m, and the damper constant should be b = 25 N·s/m. The
second-to-last sentence should read “...in the rocket, starting with the
spring at its rest length.”
4
viii) In Problem 6.25 on p. 244, the second and third sentences should
read “Assume the model rocket weighs 2 kg and the fuel weighs 3 kg.
Assume a mass flow rate of 0.1 kg/s and an exhaust velocity of 500
m/s.”
Chapter 7:
i) In Example 7.4 on p. 254 there is a typo in the third equation (definition of the angular momentum of the center of mass about the origin).
The unnumbered equation reads “I hG/O = rG ×...” but should be
“I hG/O = rG/O ×...”.
ii) In Example 7.8 on p. 265 there is a typo in the expression for TG/O .
The square should be on y˙ G not outside the parenthesis, i.e.,
1
1
2
).
TG/O = mG kI vG/O k2 = mG (x˙ 2G + y˙ G
2
2
iii) In Example 7.11 on p. 273, the unit vectors of the polar frame have
been defined as er and eθ so the b1 and b2 are incorrect. The equations
for rP/G and I vP/G should be written
r
rP/G = − er
2
r˙
r˙
I
vP/G = − er − θe
θ.
2
2
Likewise, in the sentence that follows, the unit vector b1 should be
replaced by er .
iv) In Example 7.11 on p. 274, the final conditions in the last paragraph
˙
are not correct. The rotation rate at infinity should be θ(∞)
= 2hG /r02 .
The expression for the nonconservative work should be
mh2G m
4h2G
1
(nc,int)
2
W
= 2 −
r(0)
˙
+
− k(r(0) − r0 ).
2
4
r(0)
2
r0
v) On p. 274, penultimate paragraph there should be a leading Eqs. before (7.11) and (7.12).
vi) On p. 275, Figure 7.10(b), the depiction of θ is not correct. It should
be the angle measured counter-clockwise from the e1 axis to the er
axis. The figure should be corrected as annotated here.
vii) On p. 277 the final equation for the angular momentum relative to
the center of mass is incorrect. The correct equation is
m1 m2 2 ˙
I
hG =
r θe3 ,
m1 + m2
ANGULAR MOMENTUM AND ENERGY OF A MULTIPARTICLE SYSTEM
e2
G
r2/O
eθ
r1/O
G θ
m1
e1
e3
B
r1/2
rG/O
O
5
I
m2
I
275
r
(a)
er
(b)
Figure 7.10 Two bodies of masses m1 and m2 in orbit about their common center of
mass in the plane orthogonal to e3. (a) Inertial frame I = (O, e1, e2 , e3). (b) Polar frame
B = (G, er , eθ , e3).
Figure 2 Annotated correction to Figure 7.10(b)
viii)
ix)
also perhaps different from our intuitive image of the earth orbiting the sun. In fact,
the earth and sun orbit their common center of mass; because the sun is so much
larger than the earth, the center of mass of the two bodies is actually inside the sun.
Consequently,
thedescription
sun’s motion is very
small.3 (The
same
is true ofunnumbered
the earth’s motionequation
On p.
277 in the
following
the
second
I
when
it
is
orbited
by
a
small
satellite.)
Nevertheless,
it
is
very
common
useful
for hG , the specific angular momentum from Tutorialand4.2
should be
to find a description of the earth’s motion relative to the sun (or, to be more general,
hO rather
than
h
.
P
body 1 relative to body 2) given by r1/2(t).
Using the inertial frame I = (O, e1, e2, e3), we write the dynamics of body 1
relative
to body
2. We start with
of the center
of mass,
On p.
287,
everywhere
in the
thedefinition
last bullet
should
use EO instead of E
(four times).
m2
m1
r1/O +
r2/O ,
m1 + m 2
m1 + m 2
!
"#
$
!
"#
$
7.5, the last sentence before
rG/O =
x) On p. 289, Problem
the hint should read
!
!
=µ2
=µ1
“...when the baton falls, assuming
it starts
nearly upright.”
xi)
!
!
where we have introduced the dimensionless masses µ1 =
m1/(m1 + m2) and µ2 =
!
On p.
290,
Problem
the
text
problem
should
read
= mG
. Rearranging
and using
the “initial
+ µ2 =
1 and
m1 +ofm2the
m2/(m
2 ). Note µ17.7,
1+m
vector
r2/O =than
r2/1 + r“initial
1/O , we find
speed
v0 ”triad
rather
velocity v0 ”. Also, in part (a), it should
read “final speed
than
“final velocity vf ”. Part (b) is correct
f ”=rather
µ1rv1/O
rG/O − µ
2 r2/O = rG/O − µ2 (r2/1 + r1/O )
as written.
=r
+µ r −µ r ,
G/O
2 1/2
2 1/O
which means
Chapter
8:
r1/O = rG/O + µ2r1/2.
i) On p. 298, in Example 8.2, the equation at the bottom of the page
should be
3 Scientists searching for a planet about another star infer the presence of a planet by measuring the (small)
motion of the star. I
vO0 /O = −ωB e3 × (−Re2 ) = −RωB e1
Likewise, both of the b unit vectors in the preceeding paragraph should
be e unit vectors.
2010/12/9
6:33 p.In
275 the
(chap07)
Editorial
PCArespect
ZzTEX 13.9
ii) On
p. 315,
second equation, the velocityPrinceton
should
beAssoc.,
with
to frame B rather than I. The equation should thus be:
Kasdin third pages
B
aP/L = −
fl
ˆr
− 2ωr b3 × B vP/L − ωr2 b3 × (b3 × rP/L ).
mP P/L
6
iii) Kindle Edition Only Problem 8.1(a), I vP/O0 should be I vP/O .
iv) On p. 330, in Problem 8.3, the radius rB from the problem statement
is missing from Figure 8.24. It is the radius of the back gear, which
has center O00 .
v) p. 331, Problem 8.4, the problem does not constrain the initial orientation of the astronaut’s velocity vector with respect to the station’s
velocity vector. To remove ambiguity, the problem should state that
when the astronaut is at O0 (when r = 0), she starts moving in the
same direction as I vO0 /O .
vi) p. 331, Problem 8.7. Delete the word “massless” from the first line.
vii) p. 332, Problem 8.8, the first letters of itemized problem statements
(a) and (b) should be capitalized.
viii) p. 333, Problem 8.10, the first line should read “Consider a disk of
radius R = 1 m that is...”
Chapter 9:
i) On p. 365, in the middle of the page, the phrase “...fixed, implying that
Ia
I
P/O = 0 in Eq. (9.30)” should read “...fixed. Although aP/O 6= 0,
it is perpendicular to the ramp, implying rQ/G × I aQ/O = 0 in Eq.
(9.30), where Q is replaced by P here.”
ii) On p. 384, the sentence before Eq. (9.61) should read “...two unknowns
θ˙ and I ω B .”
iii) On p. 389, all of the expressions for the moment about the center
of mass, MG , are missing the unit vector in the b3 direction, which
should appear at the end of each equation (4 times).
iv) On p. 399, Problem 9.5. In figure, the distance l should go to the center
of the bar, not the edge. Also, the arrow on θ should be reversed.
v) On p. 401, Problem 9.7, the area moment of inertia should be J =
a4 /12.
vi) On p. 405, Problem 9.19 the equation of motion for the motor has an
incorrect sign on the motor torque. It should read:
IG θ¨ = −bθ˙ + ki.
Also, in text for part (b) it should read “the inductance L to 0.75 H”
rather than “the inductance h to 0.75 H”.
Chapter 10:
7
i) On p. 418, the second-to-last equation should start “β¨ − θ˙2 ...”
ii) p
On p. 422 the third unnumbered equation from the top should have
x2 + y 2 in the denominator instead of x2 + y 2 (two times).
iii) On p. 430, the paragraph following Eq. (10.37), in the second-to-last
B =b ·e
sentence, Cij = bi · ej should be I Cij
j
i
iv) On p. 439, in the first row of Eq. (10.50) and in the first of the following
unnumbered set of three equations, csc θ should be sec θ (two times).
v) On p. 448, the second and third unnumbered equations, and the line
between them, should read as follows:
I
B
= ei · bj
Cij
i, j = 1, 2, 3.
Using the fact that bj = ej prior to the rotation, we can substitute
for bj from Eq. (10.60) to obtain
I
B
Cij
= ei · (ej cos θ − (ej × k) sin θ + (ej · k)k(1 − cos θ)) .
vi) On p. 448, the last unnumbered equation should read
I
B
C11
= cos θ + k12 (1 − cos θ)
I B
C12 = −k3 sin θ + k1 k2 (1 − cos θ)
I B
C13 = k2 sin θ + k1 k3 (1 − cos θ)
I B
C21 = k3 sin θ + k1 k2 (1 − cos θ)
I B
C22 = cos θ + k22 (1 − cos θ)
I B
C23 = −k1 sin θ + k2 k3 (1 − cos θ)
I B
C31 = −k2 sin θ + k3 k1 (1 − cos θ)
I B
C32 = k1 sin θ + k3 k2 (1 − cos θ)
I B
C33 = cos θ + k32 (1 − cos θ).
vii) On p. 449, Eqs. (10.63)–(10.66) should read

q
B + I CB + I CB
1 + I C11
22
33

θ = 2arccos 
2
B
− I C23
2 sin θ
I CB − I CB
13
31
k2 =
2 sin θ
I CB − I CB
21
12
k3 =
.
2 sin θ
k1 =
I CB
32
8
viii) On p. 462, Problem 10.7, change “when it reaches the bottom of the
funnel” to “when it reaches a height of h/2. (Note that a solution
exists only for sufficiently small s0 .)”
ix) On p. 463, Problem 10.12 is worded incorrectly. The proper wording
is “...Sensors on the plane measure the position and velocity of the
released vehicle relative to the C-130, rP/O0 and B vP/O0 . The C-130
telemeters...”
Chapter 11:
i) On p. R467, in the second-to-last sentence ofR Example 11.1, “Because
m = p B dV ...” should be “Because m = ρ B dV ...”.
ii) On p. 471, the first equation at the top of the page has two sign errors.
It should read:
0 = −2mP l2 θ˙φ˙ cos φ − mP l2 θ¨ sin φ + hφ˙
iii) On p. 471, Eq. (11.5) has a sign error. It should read
θ¨ + 2θ˙φ˙ cot φ −
hφ˙
=0
mP l2 sin φ
iv) On p. 492, the first line after Eq. (11.43) should be ”As long as ωn2 > 0
. . . ”
v) On p. 494, Eq. (11.45), csc θ should be sec θ.
vi) On p. 496, the unnumbered equation before (11.53) should have dm
instead of i (four times).
vii) On p. 501, Example 11.13, in the first line of the long equation at the
bottom of the page, the leading I2 in the second row of the first matrix
d
(I2 ψ˙ sin θ + h).
should be deleted, so it reads dt
viii) On p. 509, Example 11.16, first equation on page should be
l2
h
rO/G2 = − c1 − l1 b1 − a1
2
2
ix) On p. 529, in Problem 11.11, Figure 11.30, arrow on rotary pendulum
for angle ψ should point the other way.
x) On p. 531, in Problem 11.17, Figure 11.34 makes it look as if the cord
is attached to an arbitrary point along the rod, whereas it should be
attached at the end of the rod (point A).
9
xi) On p. 533, in Problem 11.19, the value given for D is actually the
value of D/IT , so that the problem should read “...D/IT = 0.5...”
Chapter 12:
i) On p. 540, in the first unnumbered equation, all of the “cos ωd t” terms
should be replaced by “cos(ωd t)” (three times) and all of the “sin ωd t”
terms should be replaced by “sin(ωd t)” (three times).
ii) On p. 540, in the second line of the second unnumbered equation, the
“cos 2ωd t” term should be replaced by “cos(2ωd t)” and the “sin ωd t”
term should be replaced by “sin(2ωd t)”.
iii) On p. 541, the last equation on p. 541 should be
!
p
2 − gζ
2glω
g
g
p 0
x(t) = 2 + e−ζω0 t
sin(ωd t) − 2 cos(ωd t) .
ω0
ω0
ω02 1 − ζ 2
iv) On p. 553, the first of the two unnumbered equations before Eq. (12.19)
should be “y˙ 1 = y2 ”.
v) On p. 558, the final unnumbered equation and the preceding sentence
should be “...from the initial conditions y(0) and y(0):
˙
y(0) = c1 v1 + c2 v2
y(0)
˙
= λ1 c1 v1 + λ2 c2 v2 .”
Chapter 13:
i) On p. 583, bottom of the page, boxed equation should read
fk (r1/O , r2/O , . . . , rN/O , t) = 0,
k = 1, . . . , K.
ii) On p. 591, the right side of the first unnumbered equation should be
NC X
N
X
i=1 j=1
|
(act)
Fj
·
{z
I ∂r
j/O
∂qi
δqi ,
}
,Qi
Appendix A:
i) On p. 625, the sentence following Definition A.2 should read “Figure A.2 shows examples of continuous functions with continuous and
discontinuous derivatives.”
10
ii) On pgs. 628-630, Examples A.1, A.2, and A.3 should be in gray background like all other examples.
Appendix C:
i) On p. 647, Example C.1 should have gray background like all other
examples.
ii) On p. 658, fifth line of text, “TSPAN” should be “TSPAN”.
iii) On p. 658, seven lines from the bottom, lowercase “l” should be uppercase “L” (two times).