Power ScrewIntroduction

Power Screws are used for providing linear motion in a smooth uniform manner. They are linear actuators that transform rotary motion into linear motion.   Power screws are are generally based on Acme , Square, and Buttress threads.   Ball screws are a type of power screw.  Efficiencies of between 30% and 70% are obtained with conventional power screws.   Ball screws have efficiencies of above 90%.Power Screws are used for the following three reasons

To obtain high mechanical advantage in order to move large loads with minimum effort. e.g Screw Jack.

To generate large forces e.g A compactor press. To obtain precise axial movements e.g. A machine tool lead

screw.

Square Form

This form is used for power/force transmission i.e. linear jacks, clamps.  The friction is low and there is no radial forces imposed on the mating nuts.  The square thread is the most efficient conventional power screw form.   It is the most difficult form to machine.   It is not very compatible for using split nuts-as used on certain machine tool system for withdrawing the tool carriers

Acme Form

Used for power transmission i.e. lathe lead screws.  Is easier to manufacture compared to a square thread.  It has superior root strength characteristics compared to a square thread.  The acme screw thread has been developed for machine tool drives.  They are easy to machine and can be used with split nuts.  The thread has an optimum efficiency of about 70% for helix angles between 25o and 65o. Outside this range the efficiency falls away.

Buttress Form

A strong low friction thread.  However it is designed only to take large loads in on direction.  For a given size this is the strongest of the thread forms. When taking heavy loads on the near vertical thread face this thread is almost as efficient as a square thread form.

Recirculating Ball Screw

This type of power screw is used for high speed high efficiency duties.   The ball screw is used for more and more applications previously completed by the conventional power screws.

The ball screw assembly is as shown below and includes a circular shaped groove cut in a helix on the shaft.   The ball nut also includes an internal circular shaped groove which matches the shaft groove.   The nut is retained in position on the shaft by balls moving within the groove.  When the nut rotates relative to the shaft the balls move in one direction along the groove supporting any axial load.   When the balls reach one end of the nut they are directed back to the other end via ball guides.   The balls are therefore being continuously recirculated.

The recirculated ball screw has the following advantages and disadvantages to the conventional threaded power screws: Advantages

High Efficiency - Over 90% Predictable life expectancy - Precise and repeatable movement No tendency for slip-stick Minimum thermal effects Easily preloaded to eliminate backlash-with minimum friction

penalty

Smoother movement over full travel range Smaller size for same load

Disadvantages

Requires higher levels of lubrication Tend to overhaul- Needs additional brakes if locking is

required Susceptible to contamination For the same capacity ball screws are not as rigid as

conventional power screw

Roller Screw

A recent high specification power screw option is the roller screw.  For this unit the nut includes a number of special threaded rollers arranged around arround the central screw.   The rollers each take a part of the load.  This system is efficient and can withstand high loads.  

Notation

θ = Thread angle ...(radians)η = Screw Efficiencydm = Mean screw dia...(m)dmc = Mean collar dia...(m)μc = coefficient of friction of the screw /thrust collar surfacesμs = coefficient of friction of the screw surfacesF = Force to rotate thread (Torque /Mean Radius)-(N)l = lead of thread = n.p...(m)n = number of threads.p = pitch between adjacent threads... (m)α = Helix /lead angle (radians) = tan-1 l/(π.dm ).rci = Collar inside radius (m)rco = Collar outside radius (m)rm = Mean radius of thread (m)W = Vertical force generated by screw-(N)rmc = Thrust collar mean radius = ( rci + rco ) /  2 ...(m)TR = Torque to raise load ...(Nm)TL = Torque to lower load ...(Nm)To = Overhaul torque resulting from load ...(Nm)

Torque/ Efficiency equations for Power Screws

Torque equations for Power Screws

Consider a Force F applied at a mean radius rm which causes the load to be raised. The nut is turning the screw is prevented from turning.

The sketch above identifies the reactive forces acting at point O on the screw thread surface.The reactive force Fn acting normal to the surface has the following components in the plane of interest ABDO.

OD = Ff which is the friction force opposing movement up the thread surface( Ff = μs Fn )

OA = Is equal and opposite to the force being lifted. (W) OB = Is the vector sum of OD and OA and forms an angle θn

with vector (OB = Fn cos θ n )

The sketch below illustrates the horizontal and vertical forces acting at a representative point at a radius r m in the plane normal to the radius.For equilibrium the sum of all vertical forces = 0 and the sum of all horizontal forces = 0

Summing the forces in the vertical direction results in.

Fn cos θncos α = W + Ff sin α

The coefficient of friction for the screw surface materials is μs   : Ff = μs.Fn and therefore.

Fn = W / ( cosθncos α - μs.sin α )..........Equation A

Summing the moment of the forces around the centerline of the screw to obtain TR , the torque to raise the load W up the incline of the screw.

T R   =   F.r m   =   r m.(F f cos α + F n cos θnsin α )    =  r m.(μs.F n. cos α + F n

cos θnsin α )

There is an additional friction torque resulting from the friction force on the thrust collar see top sketch above.  This friction force = μc. W. ( μc = coefficient of friction between the screw thrust surface and the collar surface.).  This friction torque is assumed to be acting at the thrust collar mean radius rmc

The total torque required to raise the load W is therefore equal to

T R   =  r m.(μ s.F n cos α + F n cos θnsin α ) + rmc.μc. W

Substituting for Fn.. see equation A above and replacing rm by dm/2 ..( and rmc by dmc /2 )

dividing the first term numerator and denominator by cos α results in..

.........Equation B

Tr = the torque in Nm to lift the load W (N)//

BC = AE = OA tan θ = (OB cosα). tanθ ..therefore tan θn = BC/OB = cos α. tan θ ..therefore θn = tan-1 [ cos α. tan θ]

For many applications the helix angle is small compared to the thread angle and therefore cos α is approximately equal to 1. e.g. For M20 2.5 pitch the value of cos α = 0.999

Therefore it is reasonable to let tan θn = tan θ and therefore θn = θ. ..[However for multi start screws or screws with a relatively course lead (pitch) it is necessary to use θn ]

For normal screws and fine pitch power screws the above equation for TR can be written as :

The torque to lower the load is written as follows

These equations can be expressed in terms of the lead by substituting the relationship tan α = l / (θ.dm )

For applications where the thrust is taken on ball or roller thrust bearing the value of μc is sufficiently low that it can be taken as approximately 0 and therefore the second term can be ignored.    The approximate equations reduce to..

Overhauling

Overhauling occurs when the screw helix angle is such that the load W would cause to screw to rotate when the rotating force F = zero i.e. the Force is not only required to raise the load - it is also required to statically support the load .

The overhauling torque To as calculated below will cause the screw to overhaul when To is less than zero.

If the thrust collar torque is assumed to be near zero then the helix angle which allows overhauling (To = < zero) can be solved.

tan α = < μs / cos θn

Screw Efficiency

The efficiency of a screw thread can be defined as follows

η = Torque to raise load / Torque to raise load (zero friction)

Using equation B above the value of TR resulting when μs = 0 =

Dividing this by equation B to provides an equation for the efficiency of the power screw thread.

If the collar friction is very low compared to the screw friction the equation reduces to

Springs

Springs are mechanical components designed to store mechanical energy, working on the principle of flexible deformation of material.   Springs belong to the most loaded machine components.  Applications for springs include:

Storing energy as in clock and watch springs Energy absorbers for drives and reciprocating devices Applying set forces as used in relief valves Maintaining the position of a linked mechanical item such as a

brake panel or door Shock absorbers in anti-vibration protection Indicating /controlling a load in a scale or instrument. Lifting devices-Used to reduce effort in manual hoists

Spring Rate

An important initial factor in spring design is the Spring Rate

When considering linear motion the spring rate is the load divided by the elastic deflection.

k = P / δ

P = Force (N)δ = deflection (mm or m)When considering angular (rotary) motion the spring rate is the Torque divided by the elastic angular deflection.

ka = T / θ

T = Torque (Nm)θ = Angular displacement (Radians)

Spring Class

Metal springs are generally fall into one of three classes of duty;

1. High Duty..Springs subject to rapidly reciprocating loads e.g. engine valve springs

2. General Duty..Springs that work infrequently for limited periods

3. Static Load Springs..Springs that are used to apply a fixed load throughout their life

Spring Energy Storage

Based on the deformation pattern, springs can be divided into the following three types:

1. springs with linear characteristics 2. springs with degressive characteristics 3. springs with progressive characteristics

The W area under the spring characteristic curve represents the deformation work (energy) of a spring performed by the spring during its loading.   Deformation energy of springs subjected to compression, tension or bending is specified by the formula:

For springs subjected to torsion the deformation energy is:

Spring State

At any point in a springs operating life it can be in one of a number of states

Free- The spring is unloaded Preloaded - The spring is loaded as and initial operating state Loaded - The spring is loaded to under some operating

condition Fully Loaded - The spring is subject to the maximum design

operational loading Limiting Loaded - The spring is exposed to the limit load as

defined by the strength or design condition

A limiting load as defined by strength may be considered as at the limit of elasticity or at yield.A limiting load as defined by design limitations is exampled by a compression spring with all coils in contact.

Solid springs made from elastomers are not covered on this page.   This page covers materials used for making metal springs which mainly include helical compression, tensile, and torsion springs.  Leaf springs and disc spring materials properties may be identified in the more general notes.  The notes also concentrate more on the carbon steel and alloy steel grades rather than the non-ferrous grades.  Future updates will include more comprehensive information

A wide range of materials are available for the manufacture of metal springs including

Carbon steels Alloy steels Corrosion resisting steels Corrosion resisting steels Phosphor bronze Spring brass

Beryllium copper Nickel alloy steels Titanium alloy steels

Springs are manufactured by hot or cold working processes.   The process depends on the section of the material, the spring index (C= D/d) and the properties required.

Pre-hardened wire should not be used if D/d < 4 or if d >6mm>

Reference Standard Notes

BS EN 10270-1:2001 ..Steel wire for mechanical springs. Patented cold drawn unalloyed spring steel wire BS EN 10270-2:2001 ..Steel wire for mechanical springs. Oil hardened and tempered spring steel wire BS EN 10270-3:2001 ..Steel wire for mechanical springs. Stainless spring steel wire

BS EN 10270-1:2001 ..Steel wire for mechanical springs. Patented cold drawn unalloyed spring steel wire Wire designated within this standard is allocated one of a number of grades.

SL Grade is low tensile strength on static duties SM Grade is medium tensile strength on static duties SH Grade is high tensile strength on static duties DM Grade is medium tensile strength on dynamic duties DH Grade is high tensile strength on dynamic duties

A typical wire designation would be "Spring wire BS EN 10270-1-SH -3,60 ph.Spring wire grade SH with a nominal diameter of 3,6mm phosphated.The grade would have a tensile strength (according to the standard) Rm = 1700-1970MPa

BS EN 10270-2:2001 ..Steel wire for mechanical springs. Oil hardened and tempered spring steel wire Wire designated within this standard is allocated one of nine grades.

Low Tensile grades - FDC (Static)... TDC(Medium Fatigue)...VDC (High Fatigue)

Medium tensile tensile grades- FDCrV (Static)... TDCrV(Medium Fatigue)...VDCrV (High Fatigue)

High tensile tensile grades- FDSiCr (Static)... TDSiCr(Medium Fatigue)...VDSiCr (High Fatigue)

The FD,FDCrV, and FDSiCr (Static) Grades have a size range of 0,5 to 17,00mmThe TDC,FDCrV, and TDSiCr (Medium Fatigue) Grades have a size range of 0,5 to 10,00mmThe VDC,VDCrV, and VDSiCr (High Fatigue) Grades have a size range of 0,5 to

10,00mm

A typical wire designation would be "Spring wire BS EN 10270-2-VDC -3,60 ".Spring wire grade VDC with a nominal diameter of 3,6mm .The grade would have a tensile strength (according to the standard) Rm = 1550-1650MPa

BS EN 10270-3:2001 ..Steel wire for mechanical springs. Stainless spring steel wire This standard includes information on three steel grades 1,4310 ( with a normal tensile strength (NS) and a high tensile strength (HS)) , 1,4401, and 1,4568.

A typical designation according to this standard would be "Spring Wire BS EN 10270-2 - 1.4310 - NS -3,60 Ni coatedSteel designation number 1,4310 with nominal strength level. Nominal dia 3,6mm . Nickel coatedThis steel has a nominal tensile Rm = 1500 MPa

Spring Material

Music Wire

This is the most widely used of all spring materials for small springs because it is the toughest.   It has the highest strength tensile and can withstand higher stresses under repeated loading conditions than any other spring material.  It can be obtained in diameters from 0,12 to 3mm.  It has a usable temperature range from 0 to 120oC

Oil-tempered Wire.     Music wire will contract under heat, and can be plated.

This is a general purpose spring material used for spings where the cost of music wire is prohibitive and for sizes outside the range of music wire.  This material is not suitable for shock or impact loading.  This material is available in diameters from 3 to 12mm.   The temperature range for this material is 0 to 180 oC..Will not generally change dimensions under heat.   Can be plated.   Also available in square and rectangular sections.

Hard-drawn wire

This is the cheapest general purpose spring steel and is should only be used where life, accuracy and deflection are not too important.  This material is available in sizes 0,8mm to 12mm.  It has an operating range 0 to 120oC

Chrome Vanadium wire

This is the most popular alloy spring steel for improved stress, fatigue, long endurance life conditions as compared to high carbon steel materials.  This material is also suitable for impact and shock loading conditions.  Is available in annealed and tempered sizes from 0,8mm to 12mm.  It can be used for temperatures up to 220 oC.   Will not generally change dimensions under heat. Can be plated.

Chrome-silicon wire

This an excellent spring material for highly-stressed springs requiring long life and/or shock loading resistance.  It is available in diameters 0,8mm to 12mmm and can be used from temperatures up to 250oC.   Will not generally change dimensions under heat. Can be plated.

Martensitic Stainless steel wire

This is a corrosion, resisting steel which is unsuitable for sub-zero conditions.

Austentic Stainless steel wire

A good corrosion, acid, heat resisting steel with good strength and moderate temperatures.  Has low stress relaxation.

Spring Brass

This is a low cost material which is convenient to form.  It is a high conductivity material.  This material has poor mechanical properties.  This metal is frequently used in electrical components because of its good electrical properties and resistance to corrosion.

Phosphor Bronze

Popular alloy .Withstands repeated flexures.  This metal is frequently used in electrical components because of its good electrical properties and resistance to corrosion.   Suitable to use in sub-zero temperatures.   They are much more costly than the more common stocks and cannot be plated.   Generally will not change dimensions under heat.

Beryllium Copper

High elastic and fatigue strength.  Hardenable.  They are much more costly than the more common stocks and cannot be plated.   Generally will not change dimensions under heat.

Nickel base alloys

These alloys are corrosion resistant.  They can withstand a wide temperature fluctuation.   The materials are suitable to use in precise instruments because of their non-magnetic characteristic. They also poses a high electrical resistance and should not be used as an electrical conductors.

Titanium

Used mainly in aerospace industry because of its extremely light weight and high strength.  This material is very expensive,  It is dangerous to work as titanium wire will shatter explosively under stress if its surface is scored.  Size range 0,8 to 12mm.   Generally will not change dimensions under heat.   Cannot be plated.

Spring Material Strength Values

Important Note..It is important to note that it is best to obtain springs from

specialists suppliers who can provide the correct material for the specific application.  If springs are being designed for specific applications then strength values should be obtained from the relevant standards as identified above.  Care should be taken to include for fatigue and adverse operating conditions.   The notes on this page are for rough spring designs.

The material structure , the manufacturing process, and the heat treatment all have an influence on the strength of the spring material.  The strength of spring materials vary significantly with the wire size such that the strength of a selected spring material cannot be determined without knowing the wire size.  The standards identified all list the material strengths against the wire sizes.

The tensile strength versus the wire diameter is almost a straight line when plotted on log-log paper .   The equation for this line is..

Sut = A / dm

The table below provides some typical values for the above variables..

Material Diameter Range(mm) Exponent m A (MPa)

Music Wire 0,1 to 6,5 0,145 2211

Oil-Tempered 0,5 to 12 0,187 1855

Hard Drawn 0,7 to 12 0,190 1783

Chrome_Vanadium 0,8 to 12 0,168 2005

Chrome_Silicon 1,6 to 10 0,108 1975

302-Stainless 0,3 to 2,5, 0,146 1867

302-Stainless 2,5 to 5 0,263 2065

302-Stainless 5 to 10 0,478 2911

Phos-Bros 0,1 to 0,6 0 1000

Phos-Bros 0,6 to 2,0 0,028 913

Phos-Bros 2,0 to 7,5 0,064 932

In calculating the spring parameters the torsional yield strength (S ys ) is used. The relationship between the torsional yield strength and the ultimate strength Sut can be approximated with a range as follows.

0,35 Sut =< S ys =< 0,52 Sut

Music wire and hard drawn steel wire have an approximate relationship S ys = 0,45 Sut

Valve spring, CR_Va, CR-Si, Hardened and Tempered Carbon steel wires have an approximate relationship S ys >= 0,50 Sut

Many Non-ferrous materials have an approximate relationship S ys >= 0,35 Sut

Modulus Of Rigitity values

Typical Values for The modulus of Rigidity for different Spring materials are listed below

MaterialModulus of Rigitity = G

- (x 10 3 N/mm 2 )Carbon Steel 78,6 316 Stainless

68,9

Brass 34,5 Phos Bros 41,4 Monel 65,5 Iconel 72,4 Berylium copper

50,0

A helical spring is a spiral wound wire with a constant coil diameter and uniform pitch.   The most common form of helical spring is the compression spring but tension springs are also widely used. .   Helical springs are generally made from round wire... it is comparatively rare for springs to be made from square or rectangular sections.  The strength of the steel used is one of the most important criteria to consider in designing springs.  Most helical springs are mass produced by specialists organisations.  It is not recommended that springs are made specifically for applications if off-the-shelf springs can be obtained to the job.

Compression Springs

Tension Springs

Nomenclature

C = Spring Index D/d d = wire diameter (m)D = Spring diameter =

L 0 = Free Length (m)L s = Solid Length (m)n t = Total number of coils

(Di+Do)/2 (m)Di = Spring inside diameter (m)Do = Spring outside diameter (m)Dil = Spring inside diameter (loaded ) (m)E = Young's Modulus (N/m2)F = Axial Force (N)Fi = Initial Axial Force (N)      (close coiled tension spring)G = Modulus of Rigidity (N/m2)K d = Traverse Shear Factor = (C + 0,5)/CK W = Wahl Factor = (4C-1)/(4C-4)+ (0,615/C)L = length (m)

n = Number of active coilsp = pitch (m)y = distance from neutral axis to outer fibre of wire (m)τ = shear stress (N/m2)τ i = initial spring stress (N/m2)τ max = Max shear stress (N/m2)θ = Deflection (radians) δ = linear deflection (mm)

Note: metres (m) have been shown as the units of length in all of the variables above for consistency.   In most practical calculations milli-metres will be more convenient.

Spring Index

The spring index (C) for helical springs in a measure of coil curvature ..

For most helical springs C is between 3 and 12

Spring Rate

Generally springs are designed to have a deflection proportional to the applied load (or torque -for torsion springs).   The "Spring Rate" is the Load per unit deflection.... Rate (N/mm) = F(N) / δ e(deflection=mm)

Spring Stress Values

For General purpose springs a maximum stress value of 40% of the steel tensile stress may be used. However the stress levels are related to the duty and material condition (ref to relevant Code/standard). Reference Webpage Spring

Materials

Compression Springs- Formulae

a)     Stress

A typical compression spring is shown below

Consider a compression spring under an axial force F.   If a section through a single wire is taken it can be seen that, to maintain equilibrium of forces, the wire is transmits a pure shear load F and also to a torque of Fr.  

The stress in the wire due to the applied load =

This equation is simplified by using a traverse shear distribution factor K d = (C+0,5)/C.... The above equation now becomes.

The curvature of the helical spring actually results in higher shear stresses on the inner surfaces of the spring than indicated by the formula above.  A curvature correction factor has been determined ( attributed to A.M.Wahl). This (Wahl) factor K w is shown as follows.

This factor includes the traverse shear distribution factor K d.. The formula for maximum shear stress now becomes.

A table relating KW to C is provided below

C 3 4 5 6 7 8 9 10 11 12 13 14 15 16Kw 1,58 1,4 1,31 1,25 1,21 1,18 1,16 1,14 1,13 1,12 1,11 1,1 1,1 1,09

b)     Deflection

The spring axial deflection is obtained as follows.

The force deflection relationship is most conventiently obtained using Castigliano's theorem. Which is stated as ... When forces act on elastic systems subject to small displacements, the displacement corresponding to any force collinear with the force is equal to the partial derivative to the total strain energy with respect to that force.

For the helical spring the strain energy includes that due to shear and that due to torsion. Referring to notes on strain energy Strain Energy

Replacing T= FD/2, l = πDn, A = πd2 /4 The formula becomes.

Using Castiglianos theorem to find the total strain energy....

Substituting the spring index C for D/d The formula becomes....

In practice the term (1 + 0,5/C2) which approximates to 1 can be ignored

c)    Spring Rate

The spring rate = Axial Force /Axial deflection

In practice the term (C2 /(C2 + 0,5)) which approximates to 1 can be ignored

Compression Spring End Designs

The figure below shows various end designs with different handing.   Each end design can be associated with any end design.  The plain ends are not desirable for springs which are highly loaded or for precise duties.

The table below shows some equations affected by the end designs...

Note: The results from these equations is not necessarily integers and the equations are not accurate.   The springmaking process involves a degree of variation...

Term PlainPlain and Ground

ClosedClosed and Ground

End Coils (n e ) 0 1 2 2

Total Coils (n t )

n n+1 n+2 n+2

Free Length (L 0 )

pn+d p(n+1) pn +3d pn +2d

Solid Length (L s )

d(n t +1)

dn t d(n t +1 dn t

Pitch(p )(L 0-d)/n

L 0/(n +1)(L 0-3d)/n

(L 0-2d)/n

Helical Extension Springs

The formulae provided for the compression springs generally also apply to extension springs.

An important design consideration for helical extensions springs is the shape of the ends which transfers the load to the the spring body.  These must be designed to transfer the load with minimum local stress concentration values caused by sharp bends.   The figures below show some end designs.. The third design C) design has relatively low stress concentration factors.

Extension Spring Initial Tension

An Extension spring is sometimes tightly wound such that it is prestressed with an initial stress τ i . This results in the spring having a property of an initial tension which must be exceeded before any deflection can take place.   When the load exceeds the initial tension the spring behaves according the the formulae above.  This relationship is illustrated in the figure below

The initial tension load can be calculated from the formula.... T i = π τ i d 3/ ( 8 D)

Best range of of Initial Stress (τ i) for a spring related to the Spring Index C = (D/d)

C = D/d

Best Initial Tension Stress range = τ i

(N/mm 2 )3 140 2054 120 1855 110 1656 95 1507 90 1408 80 1259 70 11010 60 10011 55 90

12 45 8513 40 7514 35 6515 30 6016 25 55

If the coils in a tension spring are not tightly wound, there is no initial tension and the relevant equations are identical to those for the spring under compression as identified above.

The equations for tension springs with initial tension are provided below

Helical Compression Springs (Rectangular Wire)

Spring Rate and Stress

Rate (N/mm) = K 2 G b t 3/ (n D 3)

Stress (N/mm 2) = K W .K 1 F D /( b t 2 )

D = Mean Diameter of spring(mm) b = Largest section dimension(mm) t = Smallest Section dimension(mm) n = Number of Active turns F = Axial Force on Spring K 1 = Shape Factor (see table) K 2 = Shape Factor (see table) K W = Wahl Factor (see table) C = Spring Index = D/(radial dimension = b or t)

b/t 1.0 1.5 1.75 2.0 2.5 3.0 4.0 6.0 8.0 10.0K 1 2.41 2.16 2.09 2.04 1.94 1.87 1.77 1.67 1.63 1.60K 2 0.18 0.25 0.272 0.292 0.317 0.335 0.385 0.381 0.391 0.399

Conical Helical Compression Springs

These are helical springs with coils progressively change in diameter to give increasing stiffness with increasing load.  This type of spring has the advantage that its compressed height can be relatively small.  A major user of conical springs is the upholstery industry for beds and settees.

D1 = Smaller Diameter D2 = Larger Diameter

Allowable Force on Spring...Fa = allowable force (N)..τ = allowable shear stress (N/m2)

Stiffness of Spring...