Chapter 13: Springs
Outline
™ Spring
™ Helical
Functions & Types
Springs
¾Compression
¾Extension
¾Torsional
with several figures from:
MACHINE DESIGN - An Integrated Approach, 2ed by Robert L. Norton,
Prentice-Hall 2000
The Function(s) of Springs
Some Review
Most fundamentally: to STORE ENERGY
linear springs: k=F/y
F
k
Many springs can also:
push
pull
twist
nonlinear springs: k =
Parallel
y
Series
1
1 1
1
= +
+
ktotal k1 k 2 k3
Types of Springs
dF
dy
ktotal=k1+k2+k3
More Springs
Helical:
Washer Springs:
Beams:
Power springs:
Compression
Extension
Torsion
1
Helical Compression
Springs
d
D
Lf
p
Nt
Length Terminology
minimum of 10-15%
clash allowance
diameter of wire
mean coil diameter
free length
pitch
Total coils
Free Length
may also need:
Do and Di
Lf
Max Working
Load
La
Lm
F
Plain Ground
Spring Index
Typically:
Na =
Active Coils
τ max = K s
T
Square
Bottomed
Out
Ls
Stresses in Helical
Springs
End Conditions
Plain
Assembled
Length
Square Ground
F
8FD
πd 3
C=D/d
4 ≤ C ≤ 12
, where K s =
2C + 1
2C
F
T
F
Curvature Stress
Spring Deflection
Inner part of spring is a stress concentration
(see Chapter 4)
Kw includes both the direct shear factor and
the stress concentration factor
4C − 1 0.615
, where K w =
τ max = K w
+
C
4C − 4
πd 3
8 FD
™
™
y≈
8 FD 3 N a
d 4G
under static loading, local yielding eliminates stress
concentration, so use Ks
under dynamic loading, failure happens below Sy:
use Ks for mean, Kw for alternating
2
Spring Rate
y≈
Helical Springs
8 FD 3 N a
d 4G
k=F/y
k≈
d 4G
8D 3 N
™ Torsion
a
iterative
¾Some values must be set to calculate
stresses, deflections, etc.
™ Truly
Design
¾there is not one “correct” answer
¾must synthesize (a little bit) in addition to
analyze
Spring/Material
Treatments
™
¾Nomenclature
¾Stress
¾Deflection and Spring Constant
¾Static Design
¾Fatigue Design
™ Extension
Static Spring Design
™ Inherently
™ Compression
Setting
¾ overstress material in same direction as applied
load
» increase static load capacity 45-65%
» increase energy storage by 100%
Material Properties
™ Sut
ultimate tensile strength
¾Figure 13-3
¾Table 13-4 with Sut=Adb
™ Sys
torsional yield strength
¾Table 13-6 – a function of Sut and set
What are You Designing?
Given
Find
F, y
k, y
k
F
¾ use Ks, not Kw (stress concentration relieved)
Load Reversal with Springs
™ Shot Peening
™
¾ What type of failure would this be most effective
against?
+
d, C, D*, Lf*, Na*, clash
allowance (α)**, material**
design variables
Such that:
Safety factor is > 1
Spring will not buckle
Spring will fit in hole, over pin, within vertical space
* - often can calculate from given
** - often given/defined
3
Static Design: Wire
Diameter
Static Spring Flow Chart
if GIVEN F,y, then find k; If GIVEN k, y, then find F
Na , α
d, C
D, Ks, Kw
material strengths
STRESSES
DEFLECTION
Ns=Sys/τ
Lf, yshut, Fshut
for shut spring if possible
if not, for max working load
material
Three things to know:
• effect of d
• shortcut to finding d
• how to check buckling
τ max = K s
8 FD
y≈
πd 3
8FD 3 N a
d 4G
Based on Ns=Ssy/τ and above equation for τ:
1 ( 2 +b )
CHECK
ITERATE?
buckling, Nshut, Di, Do
Nshut=Sys/τshut
8 N (C + 0.5)[Fwork (1 + α ) − Finitial (α )]
d = s

πK m A


Three things to know:
• effect of d
• shortcut to finding d
• how to check buckling
Buckling
use Table 13-2 to select standard d near
calculated d
K =S /S
m
ys
ut
*maintain units (in. or mm) for A, b
Helical Springs
™ Compression
S .R. =
y′ =
¾Nomenclature
¾Stress
¾Deflection and Spring Constant
¾Static Design
¾Fatigue Design
Lf
D
yinit + y working
Lf
Three things to know:
• effect of d
• shortcut to finding d
• how to check buckling
™ Torsion
In general; S .R. =
Lf
D
< 4 for safe design
Modified Goodman for
Springs
Material Properties
™
Sus
ultimate shear strength
Sfw´
torsional fatigue strength
Sew´
torsional endurance limit
¾ Sus≈0.67 Sut
™
¾ Table 13-7
-- function of Sut, # of cycles
¾ repeated, room temp, 50% reliability, no corrosion
™
¾ for steel, d < 10mm
¾ see page 816 (=45 ksi (310 MPa) if unpeened,
=67.5 ksi (465 MPa) if peened)
¾ repeated, room temp., 50% reliability, no corrosion
™
τa
Sfs
0.5 Sfw
Sfw, Sew are for torsional strengths, so von
Mises not used
C
ep
R
d
te
ea
S fs = 0.5
B
S fw S us
(
S us − 0.5 S fw
)
A
0.5 Sfw
Sus
τm
4
Fatigue Safety Factor
τa
Fi=Fmin
Fa=(Fmax-Fmin)/2
Fm=(Fmax+Fmin)/2
τa
lo a
dl
ine
Sfs
S
N fs = a
What are you Designing?
0.5 Sfw
Given
Find
Fmax,Fmin, ∆y
k, ∆ y
k
F
d, C, D*, Lf*, Na*, clash
allowance (α)**, material**
mload
Sa
design variables
τa
mgood
0.5 Sfw
τi τm
Sus
N fs =
τa,load = τa,good at intersection
τm
S fs (S us − τ i )
S fs (τ m − τ i ) + S usτ a
Such that:
Fatigue Safety Factor is > 1
Shut Static Safety Factor is > 1
Spring will not buckle
Spring is well below natural frequency
Spring will fit in hole, over pin, within vertical space
* - often can calculate from Given
** - often given/defined
…on page 628
Fatigue Spring Design
Strategy
Fatigue Design:Wire
Diameter
if GIVEN F,y, then find k; If GIVEN k, y, then find F
Na , α
d, C
D, Ks, Kw
material strengths
+
DEFLECTION
STRESSES
S fs (S us − τ i )
S fs (τ m − τ i ) + S usτ a
N fs =
Lf, yshut, Fshut
material
as before, you can iterate to find d, or you can use an equation
derived from relationships that we already know:
1 ( 2+b )
 8CN fs 
 

N fs − 1
Ad b 
− 1 K w Fa  
d =
K s Fmin + 1.34
 K s Fm −


N fs
S fw
 
 0.67πA 


use Table 13-2 to select standard d near
calculated d
Two things to know:
• shortcut to finding d
• how to check frequency
ITERATE?
CHECK
buckling, frequency,
Nshut, Di, Do
Nshut=Sys/τshut
Natural Frequency: Surge
Two things to know:
• shortcut to finding d
• how to check frequency
*maintain units (in. or mm) for A, b
Review of Design Strategy
Surge == longitudinal resonance
for fixed/fixed end conditions:
ITERATIVE
1 kg
fn =
2 Wa
(Hz)
ideally, fn will be at least 13x more than fforcing…
it should definitely be multiple times bigger
Two things to know:
• shortcut to finding d
• how to check frequency
USING d EQUATION
Find Loading
Select C, d
Find Loading
Select C, safety factor
Find stresses
Determine material properties
Find safety factor
Solve for d, pick standard d
Find stresses
Determine material properties
Check safety factor
…see pages 814-815 for more
5
Strategy Review
Continued
Consider the Following:
Find spring constant, Na, Nt
Find FSHUT (must find lengths and y’s to do this)
Find static shut shear stress and safety factor
Check Buckling
Check Surge
Check Di, Do if pin to fit over, hole to fit in
Torsion Springs
Helical Springs
Deflection & Spring Rate
™ Compression
θ rev =
¾Nomenclature
¾Stress
¾Deflection and Spring Constant
¾Static Design
¾Fatigue Design
where,
Lw = length of wire = πDN a
πd 4
I=
64
θ rev,roundwire = 10.2
θ rev,roundwire = 10.8
™ Torsion
k=
Stresses
MDN a
d 4E
MDN a
d 4E
(if we account for
Friction)
M
θ rev
Materials
(1) Static -Compressive is Max –(Kbi > Kbo) – Inside of Coil
σ imax
1 MLw
,
2π EI
32 M max
M
c
= K bi max = K bi
I
πd 3
see Tables 13-13 and 13-14, page 850
follow book on Sewb=Sew/0.577… for now
2
4C − C − 1
4C (C − 1)
(2) Fatigue – (since fatigue is a tensile stress phenomenon) – Outside of Coil
K bi =
σ omax = K bo
32 M max
πd 3
σ omin = K bo
K bo =
32 M min
πd 3
2
4C + C − 1
4C (C + 1)
6
Strategy
θ
M
Helical Springs
Select C, d
K
• fit over pin (if there is one)
• don’t exceed stresses
™ Compression
¾Nomenclature
¾Stress
¾Deflection and Spring Constant
¾Static Design
¾Fatigue Design
™ Torsion
7