ME 3180: Machine Design

1
ME 3180 Machine Design

• Helical Torsion Springs
• Lecture Notes

2
Helical Torsion Springs

• Can be loaded in torsion instead of compression
  or tension
• Ends are extended tangentially to provide lever
  arms on which to apply moment load
• Ends come in variety of shapes to suit
  application
• Coils are close wound like extension springs (but
  do not have any initial tension), but in few
  cases are wound with spacing like compression
  spring (this will avoid friction between coils)
• Applied moment should always be arranged to close
  coils rather than open them because residual
  stresses from coil-winding are favorable against
  a closing moment (i.e., residual stresses oppose
  working stresses).

3
Helical Torsion Springs
4
Helical Torsion Springs Contd

• Dynamic loading should be repeated or fluctuating
  with stress ratio R ? 0
• Applied moment should never be reversed in
  service
• Normal stresses are produced in torsion springs
• Load should be defined at angle ? between tangent
  ends in loaded position rather than at deflection
  from free position
• Rectangular wire is more efficient (because load
  is in bending) in terms of stiffness per unit
  volume (larger I for same dimension)
• However, most helical torsion springs are made
  from round wire because of its lower cost and
  larger variety of available sizes and materials
• Torsion springs are used in door hinges, rat
  traps, automobile starters, finger exercisers,
  garage doors and etc

5
Helical Torsion Springs Contd

• Number of Coils in Torsion Springs
• For straight ends, the contribution to equation
  13.26b can be expressed as an equivalent number
  of coils Ne
• Active coils
• Where Nb is number of coils in spring body
• Deflection
• Angular deflections of coil-end is normally
  expressed in radians, but is often converted to
  revolutions. Revolutions will be used.

(13.26a)
(13.26b)
(13.27a)
6

• Where M is applied moment
• Lw is length of wire
• E is Youngs modulus
• I is second moment of area for wire cross
  section about neutral axis

7
Helical Torsion Springs Contd

• In specifying torsion spring, ends must be
  located relative to each other. Commercial
  tolerances on these relative positions are listed
  in Table 10-9.

8
Helical Torsion Springs Contd

• Simplest scheme for expressing initial unloaded
  location of one end with respect to the other is
  in terms of angle defining partial turn
  present in coil body as ,
  as shown in Fig. 10-10. For analysis purpose
  nomenclature of Fig. 10-10 can be used.
  Communication with spring-maker is often in terms
  of the back-angle .

9
Helical Torsion Springs Contd

• Number of body turns is number of turns in
  free spring body by count.
• Body-turn count is related to the initial
  position angle by
• where is number of partial turns.
• The above equation means that takes on
  non-integer, discrete values such as 5.3, 6.3,
  7.3,, with successive differences of 1 as
  possibilities in designing a specific spring.
  This consideration will be discussed later.

10
Helical Torsion Springs Contd
Bending Stress

• Torsion spring has bending induced in coils,
  rather than torsion.
• Means that residual stresses built in during
  winding are in same direction but of opposite
  sign to working stresses that occur during use.
• Strain-strengthening locks in residual stresses
  opposing working stresses provided load is always
  applied in winding sense.
• Torsion springs can operate at bending stresses
  exceeding yield strength of wire from which it
  was wound.
• Bending stress can be obtained from curved-beam
  theory expressed in form shown below
• where K is stress-correction factor.

11
Helical Torsion Springs Contd

• Value of K depends on shape of wire cross section
  and whether stress is sought at inner or outer
  fiber. Wahl analytically determined values of K
  to be, for round wire,
• where C is spring index and subscript i and
  o refer to inner and outer fibers, respectively.
• In view of fact that Ko is always less than
  unity, we shall use Ki to estimate the stresses.
  When bending moment is M Fr and section modulus
  , we express bending
  equations as
• which gives the bending stress for a
  round-wire torsion spring.

(10-43)
(10-44)
12
Helical Torsion Springs Contd
Note Next two slides are from Norton

• Maximum compressive bending stress at inside coil
  diameter of round wire helical torsion spring,
  loaded to close its coils is
• Tensile bending stresses at the outside of the
  coil

(13.32a)
(13.32b)
(13.32c)
13
Helical Torsion Spring - Contd

• For static failure (yielding) of torsion spring
  loaded to close its coils, compressive stress
  simax at inside of coil is of most concern
• For fatigue failure, which is a tensile-state
  phenomenon somax at outside of coils is of
  concern
• Alternating and mean stresses are calculated at
  outside of coil
• Since closely spaced coils prevent shot from
  impacting inside diameter of coil, shot peening
  may not be effective in torsion springs

14
Helical Torsion Springs Contd
Deflection and Spring Rate

• For torsion springs, angular deflection can be
  expressed in radians or revolutions (turns). If
  term contains revolution units, term will be
  expressed with a prime sign.
• The spring rate is expressed in units of
  torque/revolution (lbf. in/rev or N. mm/rev) and
  moment is proportional to angle , expressed
  in turns rather than radians.
• Spring rate, if linear, can be expressed as
• where the moment M can be expressed as
  or .

(10-45)
15
Helical Torsion Springs Contd

• Total angular deflection in radian is
• Equivalent number of active turns Na is expressed
  as
• Spring rate k in torque per radian is
• Spring rate may also be expressed as torque per
  turn. Expression for this is obtained by
  multiplying Eq. (10-49) by rad/turn. Thus
  spring rate (units torque/turn) is

(10-47)
(10-48)
(10-49)
(10-50)
16
Helical Torsion Springs Contd

• Tests show that effect of friction between coils
  and arbor is such that constant 10.2 should be
  increased to 10.8. The equation above becomes
• (unit torque per turn). Equation(10-51)gives
  better results. Also Eq. (10-47) becomes
• Torsion springs are frequently used over round
  bar or pin. When load is applied to torsion
  spring, spring winds up, causing decrease in
  inside diameter of coil body.
• It is necessary to ensure that inside diameter of
  coil never becomes equal to or less than diameter
  of pin, in which case loss of spring function
  would ensure.

(10-51)
(10-52)
17
Helical Torsion Springs Contd

• Helix diameter of coil becomes
• where is angular deflection of body
  of coil in number of turns, given by
• New inside diameter
  makes diametral clearance between body coil
  and pin of diameter equal to

(10-53)
(10-54)
(10-55)
18
Helical Torsion Springs Contd

• Equation(10-55) solved for is
• which gives the number of body turns
  corresponding to a specified diametral clearance
  of arbor.
• This angle may not be in agreement with necessary
  partial-turn reminder. Thus, diametral clearance
  may be exceeded but not equaled
• First column entries in Table 10-6 can be divided
  by 0.577 (from distortion-energy theory) to give

(10-56)
Static Strength
(10-57)
Music wire and cold-drawn carbon steels
QQT (hardened tempered) carbon and low-alloy
steels
Austenitic stainless steel and nonferrous alloys
19
Helical Torsion Springs Contd
20
Helical Torsion Springs Contd
Fatigue Strength

• Since spring wire is in bending, Sines equation
  is not applicable. The Sines model is in the
  presence of pure torsion. Since Zimmerlis
  results were for compression springs (wire in
  pure torsion), we will use the repeated bending
  stress (R 0) values provided by Associated
  Spring in Table 10-10.
• As in Eq. (10-40) we will use the Gerber
  fatigue-failure criterion incorporating the
  Associated Spring R 0 fatigue strength
• Value of (and ) has been corrected
  for size, surface condition, and type of loading,
  but not for temperature or miscellaneous effects.

(10-58)
21
Helical Torsion Springs Contd

• Gerber fatigue criterion is now defined.
  Strength-amplitude component is given by Table
  6-7, p. 307, as
• where slope of load line is
  . Load line is radial through origin of
  designers fatigue diagram. Factor of safety
  guarding against fatigue failure is
• Alternatively, we can find directly by
  using Table 6-7, p. 307

(10-59)
(10-60)
(10-61)