Kane Dynamics Lecture Notes PDF

Dynamics: Theory and Applications - Kane
Luke Sy
November 2017
1
Differentiation of Vectors
1.8
1.1
Vector Functions
At different reference frame, order is important. At similar
reference frame, order is not important.
A B
B A
δ δ
δ δ
6=
(r, s = 1, . . . , n)
(1.10)
δqs δqr
δqs δqr
δ δ
δ δ
=
(r, s = 1, . . . , n)
(1.11)
δqs δqr
δqs δqr
When the magnitude or direction of v is dependent on q
in frame A, v is called a function of q in A. Otherwise, we
say v is independent of q in A.
1.2
Several Reference Frames
v may be a function of q in frame A but not in frame B.
1.9
Second Derivatives
Total and Partial Derivatives
n
dv X A δv A δv
=
+
dt
δqr
δt
r=1
A
1.3
Scalar Functions
Given a reference frame A (3D)
v = v1 a1 + v2 a2 + v3 a3
(1.1)
vi ai is called the ai component of v, and vi is called
the ai measure number of v.
When a1 , a2 , a3 are mutually perpendicular, vi =
v · ai .
v = v · a1 a1 + v · a2 a2 + v · a3 a3
1.4
d δv
δ dv
=
dt δqr
δqr dt
2
(1.2)
• 1-5 rotational motion of a rigid body.
• 6-8 translational motion of a point
• 9-13 constraints
• 14-15 partial linear and angular velocity
If v is a vector function of n scalar variables q1 , . . . , qn in
frame A,
3
X
δv
δvi
,
ai
δqr
δq
r
i=1
1.5
, (r = 1, . . . , n)
2.1
Representation of Derivatives
A
1.6
A
A
Notation for Derivatives
ω B , b1 ·
Differentiation of Sums and Products 2.2
δ
δs
δv
(sv) =
v+s
δqr
δqr
δqr
δ
δv
δw
(v · w) =
·w+v·
δqr
δqr
δqr
δ
δv
δw
(v × w) =
×w+v×
δqr
δqr
δqr
If P = F1 F2 . . . FN , in general,
δP
δF1
δFN
=
F2 . . . FN + · · · + F1 F2 . . .
δqr
δqr
δqr
A
A
db2
db3
db1
· b3 + b2 ·
· b1 + b3 ·
· b2
dt
dt
dt
(2.1)
dβ
= A ωB × β
dt
β = any vector fixed in ref B
No mention of reference implies (i) any frame can be used
or (ii) all the subsequent eq. are in the same frame.
N
X
δvi
δ
v=
for (r = 1, . . . , n)
δqr
δq
r
i=1
Angular Velocity
Though abstract, angular velocity’s definition provides a
sound basis for the derivation of theorems used to solve
problems.
Let b1 , b2 , b3 form a right handed set of mutually perpendicular unit vectors fixed in a rigid body B moving in
a reference frame A. The angular velocity of B in A is
denoted by
(1.3)
Derivative in frame A is not necesarrily equal to derivative
in frame B.
1.7
(1.13)
Kinematics
First Derivatives
A
(1.12)
(2.2)
(2.3)
Simple Angular Velocity
When a rigid body B move in frame A in such a way that
(1.4) a unit vector k is independent of t in both A and B, then
B is said to have a simple angular velocity in A throughout this time interval. Note that B need not be mounted
in A for B to have a simple angular velocity in A.
(1.5)
A B
ω = ωk
(2.4)
(1.6)
ω , θ̇
(1.7)
2.3
(1.8)
Differentiation
Frames
in
(2.5)
Two
Reference
B
dv
dv A B
=
+ ω ×v
dt
dt
(2.6)
A
(1.9)
1
2.4
Auxiliary Reference Frames
2.11
2.12
Number of Generalized Coordinates
Generalized Speeds
Addition theorem for angular velocities.
A B
ω = A ω A1 + A1 ω A2 + · · · + An ω B
(2.7) Kinematical differential equations for S in A. Generalized
Specially useful if each ω are simple angular velocity. This speeds can be time-derivatives of the generalized coordinates and time, but this is not always the case.
has no angular acceleration counterpart.
n
X
ur ,
Yrs q̇s + Zr
(r = 1, . . . , n)
(2.18)
2.5
Angular Acceleration
s=1
where Yrs and Zr are functions of q1 , . . . , qn and t.
A A
B A B
d ωB
d ω
=
(2.8)
dt
dt
There is no angular acceleration counterpart for the addition theorem.
When B has a simple angular velocity in A, we have
the ff. where α is called the scalar angular acceleration.
A B
α = αk
(2.9)
dω
α=
(2.10)
dt
A
2.6
αB ,
2.13
Motion Constraints
• Nonholonomic constraint equations = equations expressing motion constraints.
• If S is not subject to motion constraints, then S is
said to be a simple holonomic system possessing n
degrees of freedom in A.
• If S is subject to motion constraints, then S is said
to be a nonholonomic system.
Velocity and Acceleration
ur ,
p
X
Ars · us + Br
(r = p + 1, . . . , n)
(2.19)
s=1
A
dp
dt
A A P
d v
A P
a ,
dt
A P
v ,
2.7
p,n−m
(2.20)
where Ars and Br are functions of q1 , . . . , qn and t.
(2.11)
(2.12)
2.14
Partial Angular Velocities, Partial
Velocities
Two points fixed on a Rigid Body
The angular velocity, ω, in A of rigid body B and the velocity, v, in A of particle P belonging to S, can be unique
If P and Q are two points fixed on a rigid body B having expressed as
n
an angular velocity A ω B in A,
X
A P
A Q
A B
ω r ur + ω t
(2.21)
ω
=
v = v + ω ×r
(2.13)
A P
a = A aQ + A ω B × (A ω B × r) + A αB × r
r = vector from point Q to P
2.8
r=1
(2.14)
(2.15)
v=
One point moving on a Rigid Body
ω=
If a point P is moving on a rigid body B while B is moving in a reference frame A, then we have the ff. where
2 A ω B × B v P is referred as the Coriolis acceleration.
A P
v = A v B̄ + B v P
A P
A B̄
B
P
B
B P
r=1
p
X
v r ur + v t
(2.22)
ω̃ r ur + ω̃ t
(2.23)
ṽ r ur + ṽ t
(2.24)
ωr ,
(2.17)
(2.25)
ωt ,
vr ,
Configuration Constraints
If subject S is affected by other bodies (e.g., contact), it
is subject to configuration constraints.
vt ,
1. Holonomic constraint equations = equations
expressing restrictions that is of the form
f (x1 , y1 , z1 , . . . , xv , yv , zv , t) = 0.
2. Rheonomic = holomic constraint equation is DEPENDENT on time t.
3. Scleronomic = holomic constraint equation is NOT
DEPENDENT on time t.
2.10
r=1
p
X
r=1
(2.16)
A
a = a + a +2 ω × v
2.9
v=
n
X
(2.26)
n
X
δp
Wsr , (r = 1, . . . , n)
δqs
s=1
n
X
δp
δqs
s=1
Xs +
ω̃r , ωr +
ω̃t , ωt +
ṽr , vr +
Generalized Coordinates
δp
δt
n
X
s=p+1
n
X
r=p+1
n
X
s=p+1
n
X
(2.27)
(2.28)
ωs Asr
(2.29)
ωr Br
(2.30)
vs Asr
(2.31)
• When a set S has v points subject to M Holonomic
constraint equations, it has n = 3v −M independent
ṽt , vt +
vr B r
(2.32)
equations.
r=p+1
• One can express xi , yi , zi (i = 1, . . . , v) as
q1 (t), . . . , qn (t). The values of q1 (t), . . . , qn (t) are
where ω r , v r , ω̃ r , ṽ r are the rth partial holonomic
called the generalized coordinates for S in A.
angular velocity, holonomic velocity, nonholonomic angu2
lar velocity, nonholonomic velocity, respectively, and are
functions of q1 , . . . , qn ; ω t ,v t , ω̃ t , ṽ t are functions of t.
Ia ,
v
X
mi pi × (na × pi )
(3.5)
i=1
2.15
Iab , Ia · nb = Iba
v
X
=
mi (pi × na ) · (pi × nb )
Acceleration and Partial Velocities
1 d δv 2
δv 2
vr · a =
−
(2.33)
2 dt δ q̇r
δqr
n δv 2
1 X d δv 2
−
Wsr
(2.34)
vr · a =
2 s=1 dt δ q̇s
δqs
1 d δv 2
δv 2
ṽ r · a =
−
+
(2.35)
2 dt δ q̇r
δqr
n
1 X
d δv 2
δv 2
−
Asr
(2.36)
2 s=p+1 dt δ q̇s
δqs
n n
X
d δv 2
1X
δv 2
ṽ r · a =
−
Wsr +
Wsk Akr
2 s=1
dt δ q̇s
δqs
Ia =
=
Mass Center
mi ri = 0
Pv
mi pi
p = Pi=1
v
i=1 mi
3.2
(3.9)
ρp × (na × p)dτ
Z
Iab , Ia · nb =
ρ(p × na ) · (p × nb )dτ
F
Z
Ia =
ρl2 dτ
(3.10)
(3.11)
(3.12)
F
Mutually Perpendicular Unit Vectors
Given inertia vectors I1 , I2 , I3 of a body B relative to
a point O for three mutually perpendicular unit vectors
n1 , n2 , n3 ,
Ia =
3
X
aj Ij
(3.13)
j=1
aj , na · nj for j = 1, 2, 3
Iab =
(3.1)
3 X
3
X
aj Ijk bk
(3.14)
(3.15)
j=1 k=1
i=1
∗
mi li2 = mka2
F
Let S be a set of particles P1 , . . . , Pv of masses m1 , . . . , mv ,
ri be the distance between the mass center S ∗ to
P1 , . . . , Pv , p∗ be the position vector from O to S ∗ .
v
X
(3.8)
Z
3.4
3.1
mi (pi × na )2
Ia ,
(2.37)
Mass Distribution
i=1
v
X
(3.7)
i=1
k=p+1
3
i=1
v
X
(3.6)
bk , nb · nk for k = 1, 2, 3
(3.2)
Curves, Surfaces, and Solids
3.5
Inertia Matrix, Inertia Dyadic
3.5.1
Inertia Matrix
(3.16)
Each inertia matrix is associated with a specific basis vecLet B ∗ be the mass center, ρ be the mass density, dτ be tor. Set S does not possess a unique inertia matrix relative
the length/area/volume of a differential element of figure to O.
F , p be the position vector from O to P , p∗ be the position
I11 I12 I13
vector from O to B ∗ .
I , I21 I22 I23
(3.17)
I31 I32 I33
Z
ρrdτ = 0
(3.3)
a , [a1 a2 a3 ]
(3.18)
F
R
b , [b1 b2 b3 ]
(3.19)
ρpdτ
T
p∗ = RF
(3.4)
Iab = aIb
(3.20)
ρdτ
F
3.5.2
3.3
Inertia Vector, Inertia Scalars
Inertia Dyadic
Basis independent.
Let S be a set of particles P1 , . . . , Pv of masses m1 , . . . , mv ,
pi be the position vector from a point O to Pi , na is a
unit vector.
u = w · ab + w · cd + . . .
v = ab · w + cd · w + . . .
(3.21)
(3.22)
Q , ab + cd + . . . dyadic
u = w · Q scalar premultiplication
v = Q · w scalar postmultiplication
(3.23)
(3.24)
(3.25)
• Ia = inertia vector of S relative to O for na
• Iab = inertia scalar of S relative to O for na and nb
U , a1 a1 + a2 a2 + a3 a3
(3.26)
• Ia = moment of inertia of S with respect to line La ,
v
=
v
·
U
=
U
·
v
(3.27)
where La is the line passing through point O and
parallel to na . (Iaa )
where a1 , a2 , a3 are mutually perpendicular unit vectors.
3
I,
v
X
mi (U p2i − pi pi )
Iz = Iz nz
Ix 0
I = 0 Iy
0 0
2Iab
tan(2θ) =
Ia − Ib
(3.28)
i=1
Z
2
ρ(U p − pp)dτ
,
=
F
3
X
Ij nj
(3.29)
(3.30)
Ix , Iy =
j=1
=
3
3 X
X
Ijk nj nk
3.5.3
(3.32)
(3.33)
Angular Momentum
A
H S/O ,
A
B/O
H
v
X
mi pi × A v Pi
i=1
B/O
=I
· A ωB
Ia − Ib
2
2
+ Iab
1/2
(3.45)
4
Generalized Forces
4.1
Moment about a point, bound vectors, resultant
M , p × v moment of v about point P
v
X
R,
vi
(3.35)
i=1
S/Q
+ rP Q × R
(4.1)
(4.2)
(4.3)
Parallel Axes Theorems
1. p = position vector from point P to any point on
line L
2. M S/P = sum of vi moments about point P = moment of S about P
4.2
I
S/O
I
S/O
=I
S/S∗
=I
S/S∗
S∗/O
dyadic
(3.36)
S∗/O
inertia matrix
(3.37)
+I
+I
inertia vector
+ I S∗/O
= I S/S∗
I S/O
a
a
a
S/O
S/S∗
S∗/O
Iab = Iab + Iab
products of inertia
S/O
S/S∗
S∗/O
Ia = Ia
+ Ia
moments of inertia
(3.38)
(3.39)
4.3
(3.40)
(a) by definition (eq. 3.7)
(b) central inertia scalar + parallel axis
(c) utilize inertia vector, matrix, or dyadic
2. continuous Iab
4.4
(a) use tables (Appendix I of kane dynamics)
(b) assume uniform mass + dyadic
(c) by definition is done as last resort
k=
dI
radius of gyration
dM
Equivalence, Replacement
• 2 sets of bound vectors are ”equivalent” when they
have equal resultants and equal moments about one
point. either set is called a ”replacement” of the
other.
• If 2 sets are equivalent, they have equal moments
about every point.
• A set S can replaced by a set S 0 consisting of T
equal to the moment of S about P and v equal to
the resultant of S.
Evaluation of Inertia Scalars
r
Couples, Torque
• couple = set of bound vectors whole resultant is zero.
• simple couple = only 2 vectors in set.
• a couple has the same moment about all points.
1. discrete Iab
3.8
(3.44)
Maximum and Minimum Moments of
Inertia
(3.34)
• Inertia dyadic I S/O of a set S of v particles relative
to a point O
• Central inertia dyadic I S/S∗
S/S∗
S/S∗
• Central inertia scalars Iab
and Ia
3.7
(3.43)
3.9
M S/P = M
3.6
0
0
Iz
(3.31)
j=1 k=1
Ia = na · I
Iab = na · I · nb
Ia + Ib
±
2
(3.42)
Generalized Active Forces
If u1 , . . . un are generalized speeds for a simple nonholonomic system S possessing p degrees of greedom in a reference frame A,
v
X
F̃r ,
ṽrPi · Ri
(r = 1, . . . , p) nonholonomic
(3.41)
i=1
(4.4)
Fr ,
Principal Moments of Inertia
v
X
vrPi · Ri
(r = 1, . . . , n) holonomic (4.5)
i=1
• principal axis of S for O: line Lz passing through O
such that nz is parallel to I z
• principal plane of S for O: plane Pz passing through
O normal to nz
• principal moment of inertia of S for O: moment of
inertia Iz with respect to Lz
• principal radius of gyration of S for O: radius of
gyration of S with respect to Lz
• if point O = S ∗ , then you add ‘central‘ to the name
F̃r = Fr +
n
X
Fs Asr
(4.6)
s=p+1
where v is the number of particle in set S, Pi is a
typical particle of S, ṽrPi and vrPi are the nonholonomic/holonomic partial velocity of Pi in A, and Ri is
the resultant of all contact forces (e.g., friction) and distance forces (e.g., gravity) acting on Pi .
4
4.5
Noncontributing Forces
4.10.2
Contribution to F̃r of:
dC = (nv + tvecτ )dA
(4.16)
|t| ≤ µn
(4.17)
|t| = µn
impending tangential motion
(4.18)
|t| = µ0 n
sliding
(4.19)
(4.20)
where n is called the pressure at point P , t is called the
shear at P .
• All contact forces exerted on particles of S across
smooth surfaces of rigid bodies vanishes.
• If B is a rigid body belonging to S, all contact and
distance forces exerted by all particles of B on each
other is equal to zero.
• When B rolls without slipping on a rigid body B 0
– all contact forces exerted on B by B 0 is equal
to zero if B 0 is not part of S.
– all contact forces exerted by B and B 0 on each
other equal to zero if B 0 is part of S.
4.6
4.11
Forces Acting on a Rigid Body
4.7
(r = 1, · · · , p)
Generalized Inertia Forces
F̃r∗ ,
If B is a rigid body belonging to a nonholonomic system
S possessing p DoF in reference frame A, and a set of contact/distance forces acting on B is equivalent to a couple
of torque T and force R on point Q of B, then (F̃r )B the
contribution of this set of forces to F̃r is
(F̃r )B = A ω̃r B · T + A ṽr Q · R
Rigid body B in contact with rigid body
C across area Ā
v
X
i=1
(4.21)
Fr∗ ,
v
X
There is contribution to F̃r if:
T∗ , −
• two particles of a system are not rigidly connected
to each other, the gravitational forces exerted by the
particles on each other can make such contributions.
• bodies connected to each other by certain energy
storage or energy dissipation devices.
(i = 1, . . . , v)
(F̃r )γ = M gk · ṽr∗
v
X
(F̃r )γ =
ṽrPi · Gi
R∗ , −M a∗
(4.8)
(4.10)
where M is the total mass of S and ṽr∗ is the rth partial
velocity of the mass center of S in A.
5
Generalized Forces
5.1
Potential Energy
ur , q˙r (r = 1, . . . , n)
δV
Fr = −
δqr
δV
0=
δt
Nonholonomic (general):
n
X
ur ,
Yrs q˙s + Zr
• Bring noncontributing force/torque of interest into
evidence through the introduction of a generalized
speed related to it.
• In effect, this permits points to have certain velocities or rigid bodies to have certain angular velocities
which they cannot possess.
• Original generalized speeds and associated generalized active forces remain unaltered.
q˙s =
4.10.1
Particle P in contact with rigid body C
(4.28)
Nonholonomic (simple):
Bridging Noncontributing Forces into
Evidence
Coulomb Friction Forces
(4.25)
(F̃r∗ )B = A ω̃r B · T ∗ + A ṽr∗ B∗ · R∗ , (r = 1, · · · , p) (4.29)
(4.9)
4.10
mi ri × ai
(4.24)
= − α · I B/B∗ − A ω B × I B/B∗ · A ω B (4.26)
= −[α1 I1 − ω2 ω3 (I2 − I3 )]c1
− [α2 I2 − ω3 ω1 (I3 − I1 )]c2
(4.27)
− [α3 I3 − ω1 ω2 (I1 − I2 )]c3
i=1
4.9
β
X
(4.23)
i=1
A B
Terrestrial Gravitational Forces
Gi = mi gk
vrPi · Ri∗ , (r = 1, . . . , n) holonomic (4.22)
i=1
∗
Ri , −mi ai , (i = 1, . . . , v)
n
X
F̃r∗ = Fr∗ +
Fs∗ Asr
s=p+1
(4.7)
Contributing Interaction Forces
4.8
ṽrPi · Ri∗ , (r = 1, . . . , p) nonholonomic
s=1
n
X
Wsr ur + Xs , (s = 1, . . . , n)
r=1
n
X
(4.11)
(4.12)
impending tangential motion (4.13)
sliding
(4.14)
(4.15)
where N is nonnegative, v is the vector from C to P , τ
is perpendicular to v, µ is the coefficient of static friction,
µ0 is the coefficient of kinetic friction.
0=
(5.2)
(5.3)
(5.4)
(5.5)
δV
Wsr
δqs
s=1
(5.6)
n
X
δV
δV
+
Xs
δt
δq
s
s=1
(5.7)
Fr = −
C = Nv + Tτ
|T | ≤ µN
|T | = µN
|T | = µ0 N
(5.1)
V̇ = −
n
X
Fr u r
(5.8)
r=1
δ δV
δ δV
=
δqs δqr
δqr δqs
5
(5.9)
Holonomic (simple):
ur , q˙r (r = 1, . . . , n)
p
X
q˙k =
Ckr q˙r + Dk , (k = p + 1, . . . , n)
v
1X
K,
mi (v Pi )2
2 i=1
(5.10)
(5.11)
KB = Kω + Kv
1
Kω = ω · I · ω
2
1
= Iω 2
2
3
3
1 XX
=
ωj Ijk ωk
2 j=1 i=1
r=1
F̃r = −(
n
X
δV
δV
+
Csr ), (r = 1, . . . , p)
δqr s=p+1 δqs
n
X
δV
δV
+
Ds
0=
δt
δqs
(5.12)
(5.13)
k=p+1
(5.14)
Holonomic (general):
p
X
uk ,
Akr ur + Bk , (k = p + 1, . . . , n)
q˙k =
r=1
p
X
F̃r = −
δV
(Wsr +
δq
s
s=1
(5.15)
Wsk Akr , (r = 1, . . . , p)
5.5
n
X
F˜r ur
(5.29)
1X
Ij ωj2
2 j=1
(5.30)
1
mv 2
2
(5.31)
Homogeneous Kinetic Energy Functions
K = K0 + K1 + K2
p
p
1 XX
mrs ur us
K2 =
2 r=1 s=1
(5.18)
k=p+1
V̇ = −
(5.28)
(5.32)
k=p+1
n
n
X
X
δV
δV
+
(Xs +
Wsr Br )
δt
δqs
s=1
(5.27)
(5.16)
(5.17)
0=
(5.26)
3
=
Kv =
Ckr q˙r + Dk , (k = p + 1, . . . , n)
r=1
n
n
X
X
(5.25)
(5.19)
r=1
mrs ,
Solving for V:
v
X
(5.33)
(5.34)
mi v˜r Pi v˜s Pi , (r, s = 1, . . . , p)
(5.35)
i=1
= msr
δV
1. fs−p , δqs for s = p + 1, . . . , n.
(5.36)
δV
2. Replace δqs with fs−p .
δ δV
3. Form δqj δqr
4. hRearrange to ZX
=
Y
δf1
δf1
δfm
. . . δqn . . . δq1 . . .
δq1
5. Get Reduced Row Echelon Form
6. Infer V
δV
δV
7. Substitute fs−p into δq1 . . . δqs
5.2
where
i
X
5.6
=
Kinetic Energy and Generalized Inertia Forces
δfm
δqn
K̇2 − K̇0 = −
p
X
F̃r∗ ur
(5.37)
=0
(5.38)
r=1
is satisfied iff.
v
X
Potential Energy Contributions
mi v Pi · ṽ˙
Pi
i=1
V , Vα + Vβ + . . .
∗
Vγ , −M gk · p gravity
Z x
Vσ ,
f (ζ)dζ spring (general)
(5.20) OR
(5.21)
(5.22)
n
δK X δK
+
δt
δqs
s=1
δ
δt
0
=
5.3
1 2
kx linear spring
2
Xs +
!
Wsr Br
=0
(5.39)
r=p+1
n
X
Xs +
!
Wsr Br
= 0, (s = 1, . . . , n)
r=p+1
(5.40)
δXs
+
δt
Dissipation Functions
F is called a dissipation function for set C.
δF
(F˜r )C = −
δur
5.4
(5.23)
n
X
n
X
δWsk
k=1
δt
uk =
δXs
+
δqr
n
X
δWsk
k=1
δqr
uk = 0, (r, s = 1, . . . , n)
(5.41)
If K is function of q1 , . . . , qn and q̇1 , . . . , q̇n , then


n n
X
X
d δK
δK 
Wsr +
Wsk Akr 
F̃r∗ = −
−
dt
δ
q̇
δq
s
s
s=1
(5.24)
k=p+1
Kinetic Energy
(5.42)
• KB = constribution of rigid body B to K of set S.
• Kω = rotational kinetic energy of B in A.
• Kv = translational kinetic energy of B in A.
Fr∗ = −
n X
s=1
6
d δK
δK
−
dt δ q̇s
δqs
Wsr
(5.43)
If ur = q̇r ,
#
"
n
X
δK
δK
d δK
d δK
∗
−
+
−
Csr
F̃r = −
dt δ q̇r
δqr s=p+1 dt δ q̇s
δqs
(5.44)
Fr∗ =
d δK
δK
−
dt δ q̇r
δqr
(5.45)
7