mechanical springs

Mechanical Springs

Springs – Parts made in particular configurations to provide a range of

force over a significant deflection and/or to store potential energy. Springs

designed to provide a push, a pull, or a twist force (torque), or to primarily

store energy.

Configurations of Springs.

Stresses in Helical Springs.

Helical Compression Spring Design for Static Service.

Critical Frequency of Helical Springs.

Fatigue Loading of Helical Compression Springs.

Torsion Bar Springs –

The basic stress, angular deflection and spring rate equations are:

T r

J

For a solid round rod of diameter d, these become:

TL

JG

JGK

L

3

16T

d

4

32TL

d G

4

32

d GK

L

2(1 )

EG

Springs can be categorized in different ways: based on load types or by the

spring’s physical configuration.

Configurations of Springs –

Figures below (a – i) show selection of spring configurations

(a) Helical compression springs, Push-wide load & deflection range-round or

rectangular wire. Standard has constant coil diameter, pitch, and rate. Barrel, hourglass,

and variable-pitch springs are used to minimize resonant surging and vibration. Conical

springs can be made with minimum solid height and with constant or increasing rate.

(b) Helical extension springs.

Pull-wide load and deflection

range-round or rectangular

wire, constant rate.

(c) Drawbar springs. Pull-uses

compression spring and

drawbars to provide extension

pull with fail-safe, positive

stop.

(d) Torsion springs. Twist-

round or rectangular wire-

constant rate.


(e) Spring washers. Push-Belleville has high loads and low deflections—choice of rates

(constant, increasing, or decreasing). Wave has light loads, low deflection, uses limited

radial space. Slotted has higher deflections than Belleville. Finger is used for axial

loading of bearings. Curved is used to absorb axial end play.

(f) Volute spring. Push-

may have an inherently

high friction damping.

(g) Beam springs. Push or Pull-wide

load but low deflection range-

rectangular or shaped cantilever or

simply supported.


(h) Power or motor springs. Twist-

exerts torque over many turns. Shown

in and removed from retainer.

(i) Constant Force. Pull-long deflection

at low or zero rate.

Helical Compression Spring (HCS) –

Sample springs and dimensional parameters for a standard helical compression

spring are shown,

d = the wire diameter.

D = the mean coil diameter.

𝐷i = the inside diameter of the coil.

𝐷o = the outside diameter of the coil.

Lf = the free length of the spring– the

overall length under unloaded condition.

P = the coil pitch.

Nt = the number of coils.

o Spring Lengths –

Compression springs have several lengths and deflections as shown in the next Figure:

o Spring Lengths –

Various Lengths of a Helical Compression Spring in Use. o Active Coils –

The total number of coils Nt may or may not contribute actively to the spring’s deflection,

depending on the end treatment. The number of active coils Na is needed for calculation

purposes. Four Styles of End-Coil Treatments for Helical Compression Springs are:

a. The four types of ends generally used for compression springs are also

illustrated in the Figure,

o Active Coils –

b. Table 10–1 shows how the type of end used affects the number of coils and the

spring length.

o Stability of HCSs – End Condition & Buckling of Compression Springs

the condition for absolute stability is:

the end-condition constant α depends upon how the ends of the spring are supported.

As with solid columns, the end constraints of the spring affect its tendency to

buckle.

Nonparallel ends. Parallel ends.

o Spring Materials –

Ideal spring material: high ultimate strength, high yield point, low E (to

provide maximum energy storage).

For dynamically loaded springs, the fatigue strength properties of material

are important.

High strength and yield points: Carbon alloys & steels.

Spring wire: round wire is the most common spring material.

Descriptions of the most commonly used steels will be found in Table 10–3.

Spring materials may be compared by examination of their tensile strength.

Wire size, materials and its processing have an effect on tensile strength.

Tensile strength vs. wire diameter almost straight line when plotted on log-

log papers.

A: intercept and m: slope can be found from Table 10-4.

Torsional yield strength: 0.35Sut ≤ Ssy ≤ 0.52Sut .

o Spring Materials –

o Spring Index –

The spring index C is the ratio of coil diameter D to wire diameter d,

4 ≤ 𝐶 ≤ 12 C < 4, the spring is difficult to manufacture,

at C > 12 prone to buckling and tangles easily when handled in bulk.

o Spring Deflection –

Forces & Torques on

the Coils of a HCS.

A HCS is a torsion bar wrapped into a helical form. The

deflection (y) of a round-wire HCS is:

G: shear modulus of the material.

A simplified model of this loading, neglecting the

curvature of the wire, is a torsion bar

o Spring Rates –

The equation for spring rate (K) is found by rearranging the deflection equation:

It is the slope of force-deflection curve of the spring. If the slop is constant, it is a

linear spring and 𝑘 can be defined as 𝑘=𝐹/𝑦.

The first & last few percent of its deflection have a

nonlinear rate. The spring rate K should be defined

between about 15% and 85% of its total deflection and

its working deflection range La – Lm kept in that region.

La (assembly) & Lm (minimum working).

when multiple springs are combined, the resulting spring rate depends on whether they

are combined in series or parallel.

have the same force passing through all springs and each contributes a part of the total deflection.

o Spring Rates –

1. Series combinations 2. Parallel combinations

all springs have the same deflection and the total force splits among the individual springs.

Stresses in Helical Compression Spring Coils –

The F.B.D. below shows two components of stress on any cross section of a coil: a

torsional shear stress from the torque T and a direct shear stress due to the force F.

These two shear stresses have the distributions across the

section as shown:

We can substitute the expression for spring index C:

where Ks is a shear-stress correction factor

o The previous equations are based on the wire being straight. The curvature

of the wire increases the stress of the inside of the spring.


Torsional stress in straight vs. curved torsional bars (note the increased stress on the inside surface of the curved bar.)

Wahl determined the stress-concentration factor for round

wire and defined a factor Kw (Wahl Factor) which includes

both direct shear effects & stress concentration due to

curvature (valid for round wire with C ≥ 1.2).

The combined stresses direct shear effects & stress concentration

The Curvature Effect


The Wahl Factor (Kw ):

The Bergsträsser Factor (KB):

or They differ by less than 1%.

The curvature correction factor can now be obtained by:

The Curvature Effect

Helical Compression Spring Design for Static Service –

Preferred range of spring index is 4 ≤ C ≤ 12, with the lower indexes being more difficult to form (because of the danger of surface cracking – a donut) and springs with higher indexes tending to tangle often enough to require individual packing.

The recommended range of active turns is 3 ≤ Na ≤ 15.

Maximum operating force should be limited to [Fmax ≤ 7/8 Fs]. Defining the fractional overrun to closure as ξ (robust linearity), where:

it is recommended (design condition) that ξ ≥ 0.15.

Also, ns is the factor of safety at closure (solid height), ns ≥ 1.2.

Spring design is an open-ended process. There are many decisions to be made, and many possible solution paths as well as solutions.

Helical Compression Spring Design for Static Service – Design Strategy –

Helical Compression Spring Design for Static Service – Design Strategy –

A music wire helical compression spring is needed to support an 89 N load

after being compressed 50.8 mm. Because of assembly considerations the

solid height cannot exceed 25.4 mm and the free length cannot be more than

101.6 mm. Design the spring.

o Solution: The a priori decisions are

1. Music wire, A228; from Table 10–4, A = 2211 MPa-mmm; m = 0.145;

from Table 10–5, E = 196.5 MPa, G = 81 GPa (expecting d > 1.61 mm).

2. Ends squared and ground.

3. Function: Fmax = 89 N, ymax = 50.8 mm.

4. Safety: use design factor at solid height of (ns)d = 1.2.

5. Robust linearity: ξ = 0.15.

6. Use as-wound spring (cheaper), Ssy = 0.45S ut from Table 10–6.

7. Decision variable: d = 2.03 mm, music wire gage #30, Table A–28.

o Example 10.2

Critical Frequency of Helical Springs –

Designer must be certain that the physical dimensions of the spring are not

such as to create a natural vibratory frequency close to the frequency of the

applied force; otherwise, resonance may result in damaging stresses.

The governing equation for the translational vibration of a spring is the wave

equation,

k = spring rate.

g = acceleration due to gravity.

l = length of spring.

W = weight of spring.

x = coordinate along length of spring.

u = motion of any particle at distance x.

The harmonic, natural, frequencies for a spring placed between two flat and

parallel plates, in radians per second, are:

where the fundamental frequency is found for m = 1, the second harmonic for

m = 2 and so on.

ω = 2πf

Critical Frequency of Helical Springs –

The frequency in cycles per second; since ω = 2πf,

assuming the spring ends are always in contact with the plates.

when one end is free, the frequency is

The weight of the active part of a helical spring is:

where γ is the specific weight

The fundamental critical frequency should be greater than 15 to 20 times the

frequency of the force in order to avoid resonance with the harmonics.

Fatigue Loading of Helical Compression Springs –

F. P. Zimmerli, “Human Failures in Spring Applications,” The Mainspring, no. 17, Associated Spring Corporation,

Bristol, Conn., August–September 1957.

o Zimmerli discovered that size, material & tensile strength have no effect on

the endurance limits (infinite life only) of spring steels in sizes under 10 mm.

o The corresponding endurance strength components for infinite life were

found to be

Unpeened:

Peened:

Peening: is a procedure used by manufacturers to increase the operating

capabilities of metals used in components. Shot peening is accomplished by

blasting metal surfaces with small particles that increase the material’s

strength and ability to withstand different types of damage.

o in constructing certain failure criteria on the designers’ torsional fatigue

diagram, the torsional modulus of rupture Ssu is:

o (Sut) see Table 10–04.

Fatigue Loading of Helical Compression Springs –

In shafts, fatigue loading in the form of fully reversed stresses.

But, helical springs are never used as both compression and extension

springs. In fact, they are usually assembled with a preload so that the

working load is additional.

The stress-time diagram (sinusoidal fluctuating stress) expresses the usual

condition for helical springs.

o Example 10.4

o Example 10.5

o Problem 1:

An as-wound HCS is made of music wire, has a wire size of d = 2 mm, an outside coil

diameter of D0 = 15 mm, a free length of L0 = 115 mm, number of active coil Na = 21

and both ends are squared & ground. The spring is unpeened. This spring is to be

assembled with a preload of 20 N and will operate with maximum load of 100 N.

i. Estimate the safety factor guarding against fatigue failure using a torsioanl Goodman

with Zimmerli data.

ii. Check the spring stability.

A helical coil spring with D = 50 mm and d = 5.5 mm is wound with a pitch

(distance between corresponding points of adjacent coils) of 10 mm. The

material is ASTM A227 cold-drawn carbon steel and Strength Criterion is

ferrous–without presetting. If the spring is compressed solid, would you

expect it to return to its original free length when the force is removed? Two

important assumptions must be made, briefly mention them. Considering

the curvature (stress concentration) factor for the inner surface KW, would

this spring return to its original length?

o Problem 2:

3 2

8 8w w

FD FK CK

d d

3 2

8 8s s

FD FK CK

d d

Values of , , , are plotted:w s w sK K K C K C

Stress correction factors for helical springs.

Stress and Strength Analysis for HCS: Static Loading

Beam Springs (Including Leaf Springs)

2

6FL

bh 2

6FL

bh 2

6FL

bh

3

3

6FL

Ebh

3

3

6FL

Ebh

3

3

12FL

Ebh

Beam Springs (Including Leaf Springs)

mechanical springs

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