The Spring Characteristic

The performance of a spring is characterised by the relationship between the loads ( F) applied to it and the deflections (δ) which result, deflections of a compression spring being reckoned from the unloaded free length as shown in the animation. The F- δ characteristic is approximately linear provided the spring is close- coiled and the material elastic. The slope of the characteristic is known as the stiffness of the spring k = F/ δ ( aka. spring ‘constant’, or ‘rate’, or ‘scale’ or ‘gradient’) and is determined by the spring geometry and modulus of rigidity as will be shown. The yield limit is usually arranged to exceed the solidity limit as illustrated, so that there is no possibility of yield and consequent non-linear behaviour even if the spring is solidified whilst assembling prior to operation. The largest working length of the spring should be appreciably less than the free length to avoid all possibility of contact being lost between spring and platen, with consequent shock when contact is re-established. In high frequency applications this may be satisfied by the design constraint Fhi /Flo d” 3.

than to sudden contact between all coils simultaneously. Any contact leads to impact and surface deterioration, and to an increase in stiffness. To avoid this, the working length of the spring should exceed the solid length by a clash allowance of at least 10% of the maximum working deflection - that is Fs - Fhi e” 0.1Fhi , though this allowance might need to be increased in the presence of high speeds and/or inertias. Stresses and Stiffness The free body ( a) of the lower end of a spring whose mean diameter is D : embraces the known upward load F applied externally and axially to the end coil of the spring, and cuts the wire transversely at a location which is remote from the irregularities associated with the end coil and where the stress resultant consists of an equilibrating force F and an equilibrating rotational moment FD/2.

The Wire axis is Inclined at the Helix Angle a.

at the free body boundary in the side view

b.

(Note that this is first angle projection). An enlarged view of the wire cut conceptually at this boundary

c.

Shows the force and moment triangles from which it is evident that the stress resultant on this cross-section comprises four components - a shear force (F cos á), a compressive force (F sin á), a torque (1/ 2FD cos á) and a bending moment (1/ 2FD sin á).

Assuming the helix inclination α to be small for close- coiled springs - then sin α ≈ 0 Cos α ≈ 1, and the significant loading reduces to torsion plus direct shear. The maximum shear stress at the inside of the coil will be the sum of these two component shears :

As the spring approaches solidity, small pitch differences between coils will lead to progressive coil- to- coil contact rather

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τ = τ torsion + τ direct = Tr/J + F/A

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LESSON 29: THE SPRING CHARACTERISTIC

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= (FD/2) (d/2)/(Ðd 4/32) + F/(Ðd 2/4) = (1 + 0.5d/D) 8 FD/Ðd 3 τ= K 8FC /Ðd 2

1.

in which the stress factor, K assumes one of three values, either ...K=1 when torsional stresses only are significant - ie. the spring behaves essentially as a torsion bar, or K = K s a” 1 + 0.5/C which accounts approximately for the relatively small direct shear component noted above, and is used in static applications where the effects of stress concentration can be neglected, or

Buckling Compression springs are no different from other members subject to compression in that they will buckle if the deflection (ie. the load) exceeds some critical value äcrit which depends upon the slenderness ratio Lo /D rather like Euler buckling of columns, thus : 3a. c1 d crit /Lo = 1 - √[ 1 - ( c2 D/ λLo )2 ] in which the constants are defined as c1 = ( 1 +2υ)/( 1 + υ) = 1.23 for steel ; c2 =Π√[( 1 + 2υ)/( 2 + υ) ] = 2.62 for steel The end support parameter λ reflects the method of support. If both ends are guided axially but are free to rotate (like a hinged column) then λ= 1. If both ends are guided and prevented from rotating then ë = 0.5. Other cases are covered in the literature. The plot of the critical deflection is very similar to that for Euler columns. A rearrangement of ( 3a) suitable for evaluating the critical free length for a given deflection is : Lo.crit = [ 1 + ( c2D/c1d? ) 2 ] c1d /2

K = K h H” ( C + 0.6)/( C - 0.67) which accounts for direct shear and also the effect of curvatureinduced stress concentration on the inside of the coil (similar to that in curved beams). K h should be used in fatigue applications; it is an approximation for the Henrici factor which follows from a more complex elastic analysis as reported in Wahl op cit. It is often approximated by the Wahl factor Kw = ( 4C - 1)/( 4C - 4) + 0.615/C. The factors increase with decreasing index as shown here :The deflection ä of the load F follows from Castigliano’s theorem. Neglecting small direct shear effects in the presence of torsion : d= ∂U/ ∂F = ∂/ ∂F (T2/2GJ) ds ]where T = FD/2 = +ƒ

length

(T/GJ) (∂T/∂F) ds

Example 28.1 Estimate the stiffness and maximum operating stress of the close coiled steel spring with squared and ground ends illustrated. The wire diameter d = 4 mm and the external diameter

= (T/GJ) (D/2)*(wire length) = (FD/2GJ) (D/2) n aÐD

which leads to

k = F/d = Gd / 8naC3

2.

Do = 30 mm, so the mean coil diameter D = Do - d = 26 mm and the index is C = D/d = 6.5

in which n a is the number of active coils Despite many simplifying assumptions, equation ( 2) tallies well with experiment provided that the correct value of rigidity modulus is incorporated, eg. G = 79 GPa for cold drawn carbon steel. Standard tolerance on wire diameters less than 0.8mm is 0.01mm, so the error of theoretical predictions for springs with small wires can be large due to the high exponents which appear in the equations. It must be appreciated also that flexible components such as springs cannot be manufactured to the tight tolerances normally associated with rigid components. The spring designer must allow for these peculiarities. Variations in length and number of active turns can be expected, so critical springs are often specified with a tolerance on stiffness rather than on coil diameter. The reader is referred to BS 1726 or AE11 for practical advice on tolerances.

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Stress factors from ( 1) are Ks = 1+0.5/C = 1.08; K h = (C+0.6)/(C-0.67) = 1.22 The total number of turns is now counted. This is a somewhat inaccurate process if the spring cannot be inspected physically. The leftmost coil here ends in a feather edge at the top, and so the end of the wire (imagined before grinding) must coincide with the vertical plane which contains the spring axis. Starting to count from this vertical plane - following the wire around towards the observer, then downwards, around the back and upwards to meet the plane again at the point ‘a’ illustrated - ‘a’

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The active turns are n a = n t -2 = 93/ 4; the solid length is Ls = n td = 47 mm, and the pitch is p = ( Lo -2d)/n a = ( 85 -8)/93/ 4 = 7.9 mm. The corresponding helix angle is á = arctan( p/D) = 5.5o -the spring is certainly close coiled. From ( 2) : k = Gd /8 n aC3 = 79E3 x 4 / 8 x 9.75 x 6.53 = 14.8 N/mm The solid deflection d s = Lo -Ls = 38 mm, so the solidification load is Fs = k d s = 14.8 x 38 = 560 N. Assuming that solidification is infrequent and outside the working range, then this load will be treated as static so that the direct shear stress factor K s applies in ( 1). The corresponding solid shear stress from ( 1) is therefore : d= K s 8 Fs C /Πd 2 = 1.08 x 8 x 560 x 6.5 / Π42 = 625 MPa. Assume that at the maximum working state the abovementioned 10% minimum clash allowance applies, so that ds - dhi ? 0.1 dhi ie. d hi ? ds/1.1 = 38/1.1 say dhi = 34 mm Proceeding as with the solid state, F hi = k dhi = 14.8 x 34 = 505 N and so : N and so :

dhi = K s 8 F hi C / ? d2 = 1.08 x 8 x 505 x 6.5 /? 42 = 563 MPa static conditions still assumed. Considering stability at this maximum state and assuming hinged ends (l = 1), then l Lo /c2D = 1 x 85 /2.62 x 26 = 1.25. Since this is greater than unity buckling could occur, so investigate via ( 3a): dc = { 1 - ?( 1 - 1/1.25 2 )} x 85/1.23 = 27 mm and since the maximum deflection d hi exceeds äc then buckling will occur unless the spring is supported by a rod or surrounding cylinder. Alternatively if end rotation is prevented ( λ = 0.5) then λ Lo /c2D < 1 which automatically guarantees stability. Example 28.2 A return spring of infinite life (ie. 107 cycles) is required for a cam follower which moves through 25 mm, 400 times a minute. The spring must fit over a F 15 mm shaft and inside a F 65 mm hole, and it must exert a force which varies between 300 and 600 N. Design a suitable spring with closed and ground ends, made from ASTM A229.

Although the material specified is not optimum for fatigue applications, the solution to the problem will be a useful benchmark for other solutions. A small safety factor of 1.1 will be selected for reasons given previously and presuming the consequences of failure are not severe. Completing the characteristic, given dhi - dlo = 25 mm, F lo = 300 N and Fhi = 600 N, it follows that the spring stiffness must be k = ( 600 -300)/25 = 12 N/mm, and further dlo = 25 mm, dhi = 50 mm Assuming a 12% clash allowance, then ds - dhi ? 0.12 dhi, so take ds = 56 mm and hence the solid load is F s = kd s = 672 N. This completes the characteristic as sketched. The mean and alternating load components are Fm = 450 N and Fa = 150N, so, from ( 5b) with Sus /Sut = 0.63 and Ses/Sut = 0.13 for A229 from Table 2, and taking forces in N a. Fe = ( 2 x 450/0.63) C K s + ( 2 x 150/0.13) C K h = C ( 1430Ks + 2310 K h ) - a function of C only, from ( 1). In order to obtain a ball-park estimate of the wire diameter necessary, assume C = 7.5 which lies midway in the usual recommended range. From ( a) it follows that Fe = 7.5 ( 1430 x 1.07 + 2310 x 1.19) = 32.1 kN, so that the tensile capacity of a wire suitable for this fatigue application is Fut = nFe = 1.1 x 32.1 = 35.3kN. From Table 3, a suitable wire diameter lies somewhere between 5 and 6.3 mm; so a table of candidate solutions is prepared for solutions based on wire diameters in this range from the R20 series of AS 2338. Succeeding rows of the table assess the practicability of each candidate wire diameter by addressing : Calculation of the maximum spring index via ( a) corresponding to the selected safety factor (1.1 in fatigue here), with subsequent checks for spring overall geometry - candidates A and E do not meet the present geometric constraints, so these candidates are dropped. Calculation of the corresponding number of turns necessary for the target spring stiffness (12 N/mm here). Verification that the candidate is close-coiled ( helix angle α ≤ 2o ) - all remaining candidates conform here. Determination of whether the candidate is absolutely stable, conditionally stable or unstable. Candidate B is unstable here, so is passed over. Computation of the candidate’s fundamental natural frequency which ideally should be at least 12 times the running frequency. In the present case, the ratio of natural / running frequencies for the two remaining candidates is over 14, so neither candidate should lead to resonance problems. Assessing whether the candidate is likely to yield if solidified during assembly. The shear stress of both remaining candidates when solid is much less than their shear yield, so yielding is unlikely - though the effect of tolerance on both wire and coil diameters would have to be taken into account in a real design. Two candidates solutions emerge here - C and D - however D is rejected in favour of C as it is less compact ( ?/ 4 Do 2 Lo ) and because its OD, being so close to the limit, may cause problems

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therefore corresponds to one complete turn. Repeating this, we arrive at point ‘b’ after 11 complete turns. Continuing from ‘b’, the bottom is reached (another half turn) without the wire yet ground to a feather edge. So the wire must end at about 3/4 turn after ‘b’, therefore n t H” 113/ 4 turns. Ends are not ground right down to feather edges in practice, as shown by this cadmium plated hydraulic valve return spring.

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when practical manufacturing tolerances are allowed for. Manufacturing dimensions of the chosen solution are therefore wire diameter 5.6 mm mean coil diameter 38.7 mm total number of turns 16.0 free length 146 mm ends squared and ground.

depends upon the cantilever’s geometry and elastic modulus, as predicted by elementary beam theory. Unlike the constant crosssection beam, the leaf spring shown on the right is stressed almost constantly along its length because the linear increase of bending moment from either simple support is matched by the beam’s widening - not by its deepening, as longitudinal shear cannot be transmitted between the leaves. Notes

Laminated Spring Leaf springs: Leaf springs start out as bars of flat stock, which are then heated red-hot and formed, either by machine or by hand.

Springs are unlike other machine/structure components in that they undergo significant deformation when loaded - their compliance enables them to store readily recoverable mechanical energy. In a vehicle suspension, when the wheel meets an obstacle, the springing allows movement of the wheel over the obstacle and thereafter returns the wheel to its normal position. Another common duty is in cam follower return - rather than complicate the cam to provide positive drive in both directions, positive drive is provided in one sense only, and the spring is used to return the follower to its original position. Springs are common also in force- displacement transducers, eg. in weighing scales, where an easily discerned displacement is a measure of a change in force. The simplest spring is the tension bar. This is an efficient energy store since all its elements are stressed identically, but its deformation is small if it is made of metal. Bicycle wheel spokes are the only common applications which come to mind

Beams form the essence of many springs. The deflection ä ?of the load F on the end of a cantilever can be appreciable - it

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