机械设计12_弹簧

School of Engineering Mechanical Engineering
TOPIC
ME 461
12
© Tulong Zhu, All rights reserved.
Design
Mechanical Springs
Types of Helical (Coil) Springs
Extension Springs
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Tension Spring
The cross section of a spring wire is exposed to a shear force and a torsion moment:
T FD 2
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Stress in Helical Springs
Internal forces Shear stress w/o Considering Curvature
m ax K s
8 FD
d 3
Ks
2C 1 2C
Ks: shear-stress correction factor
This result is derived based on the wire being straight.
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Need correction for curvature.
d 3
Kc
KW
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KW: Wahl factor
2C 4C 1 0.615 2C 1 4C 4 C 4C 1 0.615 4C 4 C
Significance
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Important for fatigue; can be neglected for static loading
Shear force, F and T orque T FD / 2
m ax
Tr F J A
(Assume uniform under shear)
FD d 4 d 2 d T , J , A , r 2 32 4 2
m ax
8FD
d 3

4F
d 2
C D/d
KB and KW
2.2 2 1.8
KW , KB
KB KB KW KW
Comparison Between K W and K B
1.6 1.4 1.2 1
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0
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5
C
10
15
Deflection of Helical Springs

Strain Energy
U UT U F T 2L F 2L 2GJ 2GA
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Stress in Helical Springs: Curvature Effect
Nom Stress Curvature Effect

Tr J
max K s 8FD /(d 3 )

Tr J
Ks (2C 1) / 2C

Tr J

Tr J
• The wire curvature increases the stress on the inside of the spring. This effect is characterized by a curvature correction factor, Kc Bergsträ sser Factor
Oppose extension
Compression Springs
Oppose compression
Torsional Springs
Oppose tosional motion
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Useful Resource
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www. acxesspring.com
Compression Spring Terminology
FD d 4 d 2 T , J , A , L πDN 2 32 4

Strain energy density
U
4 F 2 D3 N d G
4

2F 2 DN d 2G
C D/d
Deflection: Castigliano’s Theorem
U 8FD3 N 4 FDN y 2 4 F d G d G y 8FD3 N d G d 4G 8D N
L DN, Lt DNt
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Extension and Torsional Springs Terminology
Extension Springs
(GAP)
Wire Diameter, d
HOOK LENGTH FREE LENGTH
LOOP LENGTH
Torsional Springs
f 1 kg 2 W 1 kg 4 W
W AL 2d 2 DN / 4,
: specificweight
• One end against a flat plate and the other free
f
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Design
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The natural frequency should be greater than 15 to 20 times the frequency of the excitation.
m ax K c K s
8FD
d 3
KB
8FD
d 3
Kc
KB
2C (4C 2) (4C 3)(2C 1)
4C 2 4C 3
KB: Bergsträ sser factor Wahl Factor
m ax K c K s
8 FD
d 3
KW
8FD
源自文库
Hard-drawn A227
60-70
45-55
Oil-tempered A239 Valve spring A230 Chrome-vanadium A231 A232 Chrome-silicon A401 Stainless wire A313 Stainless wire 17-7PH Stainless wire 414 Stainless wire 420 Stainless wire 431 Phosphor-bronze B159 Beryllium B197
3 4
(1
1 2C
2
)
8FD3 N d 4G
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Spring rate, k k = F/y
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k
Stiffness of a spring
Critical Frequency of Helical Springs
Significance A natural frequency close to the excitation frequency will result in RESONANCE !!!
2u x
2
Equation of Motion

W 2u kgl2 t 2
x, u
W: k: l: g:
Weight of spring Spring rate Length of spring Gravity acceleration
Natural Frequency
• Spring ends are in contact with plates
Wire Diameter, d
Inside Diameter, Di
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Outside Diameter, Do
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Internal Forces in Helical Springs
Free Body Diagram
Compression Spring Internal Forces
Spring Materials: Ultimate Strength
Materials • Carbon steel, alloy steel, corrosion resisting steel, as well as nonferrous metals, etc. (Shigley Table10-3) Residual stress will exist after winding. Residual stress can be relived by a mild thermal treatment. Ultimate Strength • The ultimate strength of spring materials varies with wire size:
Do Di
p L0
• • • • • • •
D i: Do: D: C: d: p: L0 :
• Ls:
• N: • Nt: • L: • L t:
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Inside diameter Outside diameter Mean coil diameter Spring Index: D/d Wire diameter Pitch Free length: Overall length of a spring which is not under load Solid length: Length of a compression spring when deflected under sufficient load to bring all adjacent coils into contact. Number of active coils (coils which are free to deflect under load) Total number of coils Active wire length Total wire length
Yield Strength and Stiffness
Materials
Music wire A228 t (%) 65-75 p (%) 45-60
S y tSut , Ssy pSut
Msi 29.5 29.0 28.5 28.0 28.8 28.7 28.6 28.5 28.5 29.5 29.5 29.5 29.5 28 29.5 29 29 30 15 17 19 31 E GPa 203.4 200 196.5 193 198.6 197.9 197.2 196.5 196.5 203.4 203.4 203.4 203.4 193 208.4 200 200 206 103.4 117.2 131 213.7 Msi 12.0 11.85 11.75 11.6 11.7 11.6 11.5 11.4 11.2 11.2 11.2 11.2 11.2 10 11 11.2 11.2 11.5 6 6.5 7.3 11.2 G GPa 82.7 81.7 81.0 80.0 80.7 80.7 79.3 78.6 77.2 77.2 77.2 77.2 77.2 69.0 75.8 77.2 77.2 79.3 41.4 44.8 50.3 77.2
Sut A / d m (A and m are constants: Shigley Table 10-4 below)
3 3
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Relative 8FD N Diameter 1 8FD cost of A N Diameter A ASTM Exponent y ( 1 ) 4m Materials m in No. in MPa· mmm wire d 4G 2C 2 ksi· d G mm 0.004–0.256 0.1– 6.5 0.145 201 2211 2.6 Music wire A228 0.020–0.500 0.5–12.7 0.187 147 1855 1.3 Oil-tempered wire A229 0.028–0.500 0.7–12.7 0.190 140 1783 1.0 Hard-drawn A227 0.032–0.437 0.8–11.1 0.168 169 2005 3.1 Chrome-vanadium A232 0.063–0.375 1.6– 9.5 0.108 202 1974 4.0 Chrome-silicon A401 0.013–0.100 0.3– 2.5 0.146 169 1867 7.6–11 302 Stainless wire A313 0.100–0.200 2.5– 5.0 0.263 128 2065 0.200–0.400 5.0–10.0 0.478 90 2911 0.004–0.022 0.1– 0.6 0 145 1000 8.0 Phosphor-bronze B159 0.022–0.075 0.6– 2.0 0.028 121 913 0.075–0.300 2.0– 7.5 0.064 110 932 12 - 11