The George W. Woodruff School of Mechanical Engineering ME ME 3180: Machine Design Helical Torsion Springs Lecture Notes

The George W. Woodruff School of Mechanical Engineering ME 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).

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d Number of Coils in Torsion Springs For straight ends, the contribution to equation 13.26b can be expressed as an equivalent number of coils N e : Active coils: Where N b 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)

The George W. Woodruff School of Mechanical Engineering ME Where: M is applied moment L w is length of wire E is Young’s modulus I is second moment of area for wire cross section about neutral axis

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d In specifying torsion spring, ends must be located relative to each other. Commercial tolerances on these relative positions are listed in Table 10-9.

The George W. Woodruff School of Mechanical Engineering ME 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 For analysis purpose nomenclature of Fig can be used. Communication with spring- maker is often in terms of the back-angle. Helical Torsion Springs – Cont’d

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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.

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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. Bending Stress

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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 K o is always less than unity, we shall use K i 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)

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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: (13.32b) (13.32c) (13.32a) Note: Next two slides are from Norton

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Spring - Cont’d For static failure (yielding) of torsion spring loaded to close its coils, compressive stress σ imax at inside of coil is of most concern For fatigue failure, which is a tensile-state phenomenon σ omax 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

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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. Deflection and Spring Rate (10-45)

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d Total angular deflection in radian is: Equivalent number of active turns N a 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)

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d Tests show that effect of friction between coils and arbor is such that constant 10.2 should be increased to 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)

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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)

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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 (from distortion-energy theory) to give (10-56) Static Strength (10-57)Music wire and cold-drawn carbon steels QQ&T (hardened & tempered) carbon and low-alloy steels Austenitic stainless steel and nonferrous alloys

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d Since spring wire is in bending, Sines equation is not applicable. The Sines model is in the presence of pure torsion. Since Zimmerli’s 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 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. Fatigue Strength (10-58)

The George W. Woodruff School of Mechanical Engineering ME Helical Torsion Springs – Cont’d 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 designer’s 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)