mechanical springs shigley ch 10 lecture 21 · mechanical springs why do we need springs? •...
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Mechanical Springs
Shigley Ch 10
Lecture 21
Mechanical Springs Why do we need springs?
• Flexibility of structure
• Storing and releasing of energy
Standard spring types:
• Wire springs
- Helical springs of round or square wire, made to deflect under Compression, tension or torsion
- Can be used as a straight piece of wire for stiffer spring types (torsion bar suspensions springs)
• Flat springs
- Cantilever, elliptical, wound clock type (spiral), Flat Spring Washers (usually called Belleville springs), leaf springs, etc
• Special shape springs also exist for custom cases
� Most springs can be Linear or non linear, depending on geometry
Helical compression spring
Flat-leaf spring
Belleville spring
Torsion spring
Stresses in Helical Spring
23
max
48
d
F
d
FD
A
F
J
Tr
ππτ
τ
+=
+=
Torsion Shear force
Maximum shear equation
We define a term called the Spring index as
C-Usually ranges between 4 and 12
d
DC =
3
8
d
FDK s
πτ =
Next define a term Ks=shear stress correction factor
Then the stress equation becomes
C
CKs
2
12 +=
Spring design consideration
• Use round spring wire where possible, if space
is limited use nested round springs
• Springs with round wire are made
in large quantities, so for cost avoid
square/other shape sections
Valve spring in action, showing nested round springs
(Grey, red and white)
The Curvature effect
Helical springs are not straight but curved
Taking the curvature into account replace Ks
with KB (Bergsträsser factor)
Thus the largest shear stress is now given by
34
24
−
+=
C
CKB
3
8
d
FDK B
πτ =
C
CKs
2
12 +=
Deflection of Helical Springs
Strain energy for a helical spring is given by adding
the torsional-and shear-energy components
Substituting T, L, A and J gives
Where N=Na is the number of active coils
AG
lF
GJ
lTU
22
22
+=
Gd
DNF
Gd
NDFU
2
2
4
32 24+=
From Castigliano’s theorem the total deflection
Spring rate or also called “scale” of the spring is
given by
Gd
NFD
CGd
NFD
Gd
FDN
Gd
NFD
F
Uy
4
3
24
3
24
3
8
2
11
8
48
≅
+=
+=∂
∂=
ND
Gd
y
Fk
3
4
8≅=
Compression Springs, end conditions
End bent down to form 0° helix angle End ground flat to form flat
mounting surface for spring
End Conditions may differ check with manufacturer
or count and measure them
Stability
As long columns, long springs can buckle with
high loads
Where
Effective slenderness ratio
α End condition constant from table 10-2
Elastic constants
−−=
21
2
'
2'
1 11eff
ocr
CCLy
λ
D
L oeff
αλ =
)(2
'
1GE
EC
−=
EG
GEC
+
−=
2
)(2 2'
2
π
Table 10-2
End condition Constant α
Spring supported between flat parallel
surfaces
0.5
One end on flat surface perpendicular to
axis (fixed) other end hinged
0.707
Both ends hinged 1
One end clamped other end free 2
Absolute stability is given by
For steels
Squared and ground ends thus
21
02
)(2
+
−<
EG
GEDL
α
π
α
DL 63.20 <
5.0=α
DL 26.50 <
Spring Materials
Hot- or cold-working processes, depending on
size, spring index and properties desired
In general Pre Hardened material should not to
be used if
D/d < 4 or
d > 6mm
Table 10-3 shows most commonly used spring
materials
Table 10-3
Spring Materials
Graph of tensile strength vs. wire diameter is a
straight line when plotted on log-log paper
using the following equation:
Table 10-4 gives values for m and A
mutd
AS =
Table 10-4
Spring Calculations
Torsional yield strength is needed in designing springs butbecause tensile strength is easy to determine the torsionalyield strength is usually estimated from that
From DET the torsional yield strength is given by
The yield strength is between 60 and 90 percent of the ultimate tensile strength for steel, this approach results in the range
Table 10-5 gives the range of Ssy and table 10-6 gives the maximum percentage of tensile strength
ysy SS 577.0=
utsyut SSS 52.035.0 ≤≤
Table 10-5
• Set removal or presetting
– Make spring longer than required, close/press to
solid length to introduce residual stresses (10 to
30 percent of length is reduced this way)
– Stress at solid height is between 1.1 and 1.3 the
torsional yield strength
– Make spring stronger by inducing residual stresses
opposite those induced in service
– Not good for fatigue applications