Random Stuff

1. 1
BMED 3400 Equation Sheet
version 2.0 (May 3, 2016)
Rigid Body Model in Static Equilibrium
P ~
F = 0
P ~
M = 0
Models of Materials
ε = L
0
−L
L = 4x
x ε > 0 ⇒ tension
ν = −εtransverse
εaxial
σ = dF
dA
σ = Eε
τ = dF
dA
τ = Gγ
Simple Model of Axial Loading
σ = P
A P > 0 ⇒ tension P < 0 ⇒ compression
ε = P
AE
δ = P L
AE
Model of Axial Loading
´ L+
−
ω(x) dx = 0
P(x) =
´ x
−
ω(β) dβ
σ(x) = P (x)
A(x)
ε(x) = σ(x)
E(x)
δ(x) =
´ x
−
ε(β) dβ, δtotal = δ(x = L), δ(x = 0) = 0
εtherm(x) = α(x)4T(x)
δtherm(x) =
´ x
−
εtherm(β) dβ
Simple Model of Twisting (Torsion)
τ = T
J r
γ = T
GJ r
φ = T L
GJ
Model of Twisting (Torsion)
´ L+
−
ω(x) dx = 0
T(x) =
´ x
−
ω(β) dβ = −
´ L+
x
ω(β) dβ
T(x) =
´
shape(x)
rτ(x, r) dA
τ(x, r) = T (x)
J(x) r
γ(x, r) = τ(x,r)
G(x)
φ(x) =
´ x
−
1
r γ(β, r) dβ r 6= 0
Model of Bending
V (x) =
´ x
−
ωshear(β) dβ
M(x) =
´ x
−
ωmoment(β) + V (β) dβ
M(x) =
´
A
dM =
´
A
y dF =
´
A
yσ(x, y) dA
σ(x, y) = −M(x)y
I
ε(x, y) = −M(x)y
E(x)I
ε(x, y) = −y
ρ(x)
ρ(x) = E(x)I
M(x)
κ(x) = 1
ρ(x) =
d2v
dx2
1+(dv
dx )
2
3
2
≈ d2
v
dx2 if dv
dx 1
d2
v
dx2 = ν̈(x) = M(x)
E(x)I
ν(x) =
´ ´ M(x)
E(x)I dx
Geometric Quantities for Deformable Models
XCoM =
´
x ρ(x) dx
´
ρ(x) dx
Naxis =
´
A
y dA
´
A
dA
A =
´
A
dA
Acircle = πR2
J =
´
A
r2
dA
Jcircle = π
2 R4
Jring = JRout
− JRin
= π
2 (R4
out − R4
in)
I =
´
A
y2
dA
I/axis = I/centroid + Ad2
Irect = bh3
12
Icircle = π
4 R4
Point Mass Model with a Circular Trajectory
dθ
dt (t) = θ̇(t) = ω(t)
d2
θ
dt2 (t) = θ̈(t) = α(t)
~
x(t) = R(cos(θ(t))î + sin(θ(t))ĵ)
~
v(t) = v(t)êtan(t) = Rθ̇(t)êtan(t) = Rω(t)êtan(t)
k~
v(t)k = |v(t)| = ν(t) = R|θ̇(t)| = R|ω(t)|
∠~
v(t) =
θ(t) + π
2 if θ̇(t) 0
θ(t) − π
2 if θ̇(t) 0
~
a(t) = ~
atan(t) + ~
anorm(t)

2. 2
~
a(t) = v̇(t)êtan(t) + v(t)2
R ênorm(t)
~
a(t) = Rθ̈(t)êtan(t) + Rθ̇(t)2
ênorm(t)
k~
atan(t)k = |v̇(t)| = R|θ̈(t)| = R|α(t)|
k~
anorm(t)k = v(t)2
R = ν(t)2
R = Rθ̇(t)2
= Rω(t)2
~
F(t) = m~
a(t)
Kinematics of a Rigid Body Model
~
ω(t) = ω(t)k̂ ~
α(t) = α(t)k̂
~
xA/C(t) = ~
xB/C(t) + ~
xA/B(t) = arbitrary + circular
~
xA/C(t) = ~
xB/C(t) + R(cos(θ(t) + β)î + sin(θ(t) + β)ĵ)
~
vA/C(t) = ~
vB/C(t) + Rω(t)êtan(t)
~
vA/C(t) = ~
vB/C(t) + (~
ω(t) × ~
xA/B(t))
~
aA/C(t) = ~
aB/C(t) + Rα(t)êtan(t) + Rω(t)2
ênorm(t)
~
aA/C(t) = ~
aB/C(t) + (~
α(t) × ~
xA/B(t))
+ (~
ω(t) × (~
ω(t) × ~
xA/B(t)))
Simple Dynamics of a Rigid Body Model
~
Fcm, ~
M/cm : resultant force moment at center of mass (cm)
~
acm/C : center of mass accel. with respect to inertial frame (C)
~
Fcm = m~
acm/C
~
M/cm = I/cm~
α
Dynamics of a Rigid Body Model
~
FC, ~
M/C : resultant force moment at inertial frame (C)
pure rotational acceleration around C:
~
M/C = I/C ~
α
rotational and translational acceleration with respect to C:
~
acm/C = ~
aB/C + ~
acm/B
= ~
aB/C + ~
acm/B
tangent
+ ~
acm/B
normal
=
P
~
ai
±M/C = ±I/cmα +
P
±maidi
~
M/C = I/cm~
α + ~
xcm/C × m~
acm/C
= I/cm~
α + ~
xcm/C × m
P
~
ai
Rotational Inertia for Rigid Body Dynamics
I =
´
r2
dm =
´
r2
ρdV
I = I/cm + md2
sphere I/cm = 2
5
mR2
flat rectangle (⊥ to rectangle) I/cm = 1
12
m W2
+ H2
cylinder (k to central axis) I/cm = 1
2
mR2
cylinder (⊥ to central axis) I/cm = 1
12
m 3R2
+ L2
slender rod (⊥ to central axis) I/cm = 1
12
mL2
Dot Product and Cross Product Properties
d(~
p×~
q)
dt = ~
p × ˙
~
q + ˙
~
p × ~
q
d(~
p·~
q)
dt = ~
p · ˙
~
q + ˙
~
p · ~
q
î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ
~
a × (~
b + ~
c) = (~
a ×~
b) + (~
a × ~
c)
~
a ×~
b = −~
b × ~
a
~
p × ~
q = det




î ĵ k̂
px py pz
qx qy qz




= (pyqz − pzqy) î + (pzqx − pxqz) ĵ+
(pxqy − pyqx) k̂
= |~
p| |~
q| sin (θ) n̂
~
p · ~
q = |~
p| |~
q| cos (θ) = pxqx + pyqy + pzqz
Unit Circle for Trigonometric Calculations SI Prefixes
1
exa 1018
E
peta 1015
P
tera 1012
T
giga 109
G
mega 106
M
kilo 103
k
hecto 102
h
deca 101
da
deci 10−1
d
centi 10−2
c
milli 10−3
m
micro 10−6
µ
nano 10−9
n
pico 10−12
p
femto 10−15
f
atto 10−18
a
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