Flywheel Problem

1
Flywheel Problem

• The Second Tutorial

2
Finding acceleration and velocity
y
r xexyey rer ercos?exsin?ey
z
er
e?
r
yey
v d/dt(rer)
ad/dt(d/dt(rer))
x
xex
3
Finding acceleration and velocity
v?r e?
vd/dt(rer)rd/dt (cos?exsin?ey)
v r(-sin?(d?/dt) excos ?(d?/dt)ey )
(1)
Now adv/dt d/dt(?r e? ) ?rd/dt(e?)

• Now recall that
• e? (cos?ex - sin ?ey)
• ?(d?/dt)

This becomes a- ?2r er
Substitution into (1) gives the form v?r e?

(2)
4
Kinetic Energy

• K(1/2)I?2 where I ?r2dm
• I ? ? r2dV ???? r2 dtdrd? (2p?tR4)/4
• d?2prdr
• ?dtt, thickness

Substitution gives K(1/4) p?tR4?2
5
Material consideration

• Yield- At high speeds centrifugal forces cause
  tensile strength
• Material needs to be designed resistant to these
  stresses
• Youngs modulus and Possions ratio must be
  accounted for in design

6
Stress-Strain relations

• Assuming plane stress tltltR

• Use Strain-Displacement relationships given
• uru(r), uz, u?r?t

errd/d?(ur), e??(1/r)d/d?(u?) ur/r, ezzd/dz
(uz)
e?r (1/2)(d/dz(u?) r-1d/d?(uz)-u?/r) e?z(1/2)(d
/dz(u?)r-1d/d?(uz)) erz(1/2)(d/dr(uz)d/dz(ur))
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Stress-Strain relations
Simplifying we find that
errd/d?(ur), e??ur/r, ezzd/dz (uz)
e?r (1/2)(d/dz(u?) r-1d/d?(uz)-u?/r)
0 e?z(1/2)(d/dz(u?)r-1d/d?(uz))
0 erz(1/2)(d/dr(uz)d/dz(ur)) 0
8
Stress-Strain relations
Using the governing Stress-Strain relation
sij (E/(1?) eij (v/(1-2v))(S ekk)dij
s?z srz sr? 0
szz(E/(1-v2))(errve??) 0 (from assumption)
srr(E/(1-v2))(errve??)
s??(E/(1-v2))(verre??)
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Equation of Motion for ur
Since looking for ur use the following Equation
of Motion
d/dr(srr)r-1d/d?(sr?)d/dz(srz)(1/r)(srr-
s??)?d2/dt2(ur) (ignores body force ?br)
Using the stresses derived earlier we can
simplify to the following
d/dr(srr)(1/r)(srr- s??)?d2/dt2(ur) -??2r
d2/dr2(ur)(1/r)(d/dr(ur))- (ur)/r2
-??2r(1-v2)/E (3)
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Equation of Motion for ur
d2/dr2(ur)(1/r)(d/dr(ur))- (ur)/r2
-??2r(1-v2)/E (3)
This equation will solve for ur. But first
re-write using reverse chain rule
d/dr(1/r)(d/dr(urr)) -??2r(1-v2)/E
Integrating yields the solution
urArBr-1-((1-v2)/8E)??2r3
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Stress field

• Using the stress-strain and the
  strain-displacement relations derived earlier

srr(E/(1-v2))(errve??)

s??(E/(1-v2))(verre??)
errd/d?(ur), e??ur/r, ezzd/dz (uz)
Plug into new equation for ur setting boundary
conditions
sr(R)0 (no mass rotating and
pulling on outside) ur(0)0 no
displacement in the r direction at center
12
Stress field

• After solving for A and B substitute to
• obtain final Stresses

srr??2(3v)/8(R2)(1-(r/R)2) s??(R2 ??2(3v)/8
)1-(r/R)2((13v)/(v3))

• When r0 you get the maximum stress

srrs??(R2 ??2(3v))/8
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Product bound
Plug in max stresses to mises yield condition
to obtain s(mises) s(max)
s(max)lts(yield)
?Rlt 2(2s(yield)/(?(3v))).5