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COIL SPRING

A Coil spring, also known as a helical spring, is a mechanical device, which is typically used to store energy and subsequently release it, to absorb shock, or to maintain a force between contacting surfaces. They are made of an elastic material formed into the shape of a helix which returns to its natural length when unloaded.

Coil springs are a special type of torsion spring: the material of the spring acts in torsion when the spring is compressed or extended.

Metal coil springs are made by winding a wire around a shaped former - a cylinder is used to form cylindrical coil springs.

[edit] Variants

The two usual types of coil spring are:

• Tension coil springs, designed to resist stretching. They usually have a hook or eye form at each end for attachment.

• Compression coil springs, designed to resist being compressed. A typical use for compression coil springs is in car suspension systems.

• expansion coil spring.

[edit] Degradation

Many types of coil spring are wound in an annealed (soft) condition and then tempered to achieve their strength as a spring. Over time, this tempering can be lost[citation needed] and the spring will sag because it can no longer withstand the loads applied. Such springs can be re-set by annealing, returning to their original length (or deliberately setting them to a different length) and then re-tempering. Damage to springs, such as using oxy-acetylene to cut the end off a car suspension spring to lower a vehicle's ride height, can destroy the tempering in localised areas of the spring.

SPRING

A spring is an elastic object used to store mechanical energy. Springs are usually made out of hardened steel. Small springs can be wound from pre-hardened stock, while larger ones are made A spring is an elastic object used to store mechanical energy. Springs are usually made out of hardened steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication. Some non-ferrous metals are also used including phosphor bronze and titanium for parts requiring corrosion resistance and beryllium copper for springs carrying electrical current (because of its low electrical resistance).

When a spring is compressed or stretched, the force it exerts is proportional to its change in length. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve. An extension or compression spring has units of force divided by distance, for example lbf/in or N/m. Torsion springs have units of force multiplied by distance divided by angle, such as N·m/rad or ft·lbf/degree. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. The stiffness (or rate) of springs in parallel is additive, as is the compliance of springs in series.

Depending on the design and required operating environment, any material can be used to construct a spring, so long as the material has the required combination of rigidity and elasticity: technically, a wooden bow is a form of spring.

|Contents |

|[hide] |

|1 History |

|2 Types |

|3 Physics |

|3.1 Hooke's law |

|3.2 Simple harmonic motion |

|4 Theory |

|5 Zero-length springs |

|6 Uses |

|7 References |

|8 External links |

[edit] History

Simple non-coiled springs were used throughout human history e.g. the bow (and arrow). In the Bronze Age more sophisticated spring devices were used, as shown by the spread of tweezers in many cultures. Ctesibius of Alexandria developed a method for making bronze with spring-like characteristics by producing an alloy of bronze with an increased proportion of tin, and then hardening it by hammering after it is cast.

Coiled springs appeared early in the 15th century,[1] in door locks.[2] The first spring powered-clocks appeared in that century[2][3][4] and evolved into the first large watches by the 16th century.

In 1676 British physicist Robert Hooke discovered the principle behind springs' action, that the force it exerts is proportional to its extension, now called Hooke's law.

[edit] Types

[pic]

[pic]

A spiral torsion spring, or hairspring, in an alarm clock.

[pic]

[pic]

A volute spring. Under compression the coils slide over each other, so affording longer travel.

[pic]

[pic]

Vertical volute springs of Stuart tank

[pic]

[pic]

Tension springs in a folded line reverberation device.

[pic]

[pic]

A torsion bar twisted under load

[pic]

[pic]

Leaf spring on a truck

Springs can be classified depending on how the load force is applied to them:

• Tension/Extension spring - the spring is designed to operate with a tension load, so the spring stretches as the load is applied to it.

• Compression spring - is designed to operate with a compression load, so the spring gets shorter as the load is applied to it.

• Torsion spring - unlike the above types in which the load is an axial force, the load applied to a torsion spring is a torque or twisting force, and the end of the spring rotates through an angle as the load is applied.

They can also be classified based on their shape:

• Coil spring - this type is made of a coil or helix of wire

• Flat spring - this type is made of a flat or conical shaped piece of metal.

The most common types of spring are:

• Cantilever spring - a spring which is fixed only at one end.

• Coil spring or helical spring - a spring (made by winding a wire around a cylinder) and the conical spring - these are types of torsion spring, because the wire itself is twisted when the spring is compressed or stretched. These are in turn of two types:

o Compression springs are designed to become shorter when loaded. Their turns (loops) are not touching in the unloaded position, and they need no attachment points.

▪ A volute spring is a compression spring in the form of a cone, designed so that under compression the coils are not forced against each other, thus permitting longer travel.

o Tension or extension springs are designed to become longer under load. Their turns (loops) are normally touching in the unloaded position, and they have a hook, eye or some other means of attachment at each end.

• Hairspring or balance spring - a delicate spiral torsion spring used in watches, galvanometers, and places where electricity must be carried to partially-rotating devices such as steering wheels without hindering the rotation.

• Leaf spring - a flat springy sheet, used in vehicle suspensions, electrical switches, bows.

• V-spring - used in antique firearm mechanisms such as the wheellock, flintlock and percussion cap locks.

Other types include:

• Belleville washer or Belleville spring - a disc shaped spring commonly used to apply tension to a bolt (and also in the initiation mechanism of pressure-activated landmines).

• Constant-force spring — a tightly rolled ribbon that exerts a nearly constant force as it is unrolled.

• Gas spring - a volume of gas which is compressed.

• Ideal Spring - the notional spring used in physics: it has no weight, mass, or damping losses.

• Mainspring - a spiral ribbon shaped spring used as a power source in watches, clocks, music boxes, windup toys, and mechanically powered flashlights

• Negator spring - a thin metal band slightly concave in cross-section. When coiled it adopts a flat cross-section but when unrolled it returns to its former curve, thus producing a constant force throughout the displacement and negating any tendency to re-wind. The commonest application is the retracting steel tape rule.[5]

• Progressive rate coil springs - A coil spring with a variable rate, usually achieved by having unequal pitch so that as the spring is compressed one or more coils rests against its neighbour.

• Rubber band - a tension spring where energy is stored by stretching the material.

• Spring washer - used to apply a constant tensile force along the axis of a fastener.

• Torsion spring - any spring designed to be twisted rather than compressed or extended. Used in torsion bar vehicle suspension systems.

• Wave spring - a thin spring-washer into which waves have been pressed.[6]

[edit] Physics

[pic]

[pic]

Two springs attached to a wall and a mass. In a situation like this, the two springs can be replaced by one with a spring constant of keq=k1+k2.

[edit] Hooke's law

Main article: Hooke's law

Most springs (not stretched or compressed beyond the elastic limit) obey Hooke's law, which states that the force with which the spring pushes back is linearly proportional to the distance from its equilibrium length:

[pic]

where

x is the displacement vector - the distance and direction in which the spring is deformed

F is the resulting force vector - the magnitude and direction of the restoring force the spring exerts

k is the spring constant or force constant of the spring.

Coil springs and other common springs typically obey Hooke's law. There are useful springs that don't: springs based on beam bending can for example produce forces that vary nonlinearly with displacement.

[edit] Simple harmonic motion

Main article: Harmonic oscillator

Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like:

[pic]

[pic]

[pic]

The displacement, x, as a function of time. The amount of time that passes between peaks is called the period.

The mass of the spring is assumed small in comparison to the mass of the attached mass and is ignored. Since acceleration is simply the second derivative of x with respect to time,

[pic]

This is a second order linear differential equation for the displacement x as a function of time. Rearranging:

[pic]

the solution of which is the sum of a sine and cosine:

[pic]

A and B are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with B = 0 (zero initial position with some positive initial velocity) is displayed in the image on the right.

[edit] Theory

In classical physics, a spring can be seen as a device that stores potential energy, specifically elastic potential energy, by straining the bonds between the atoms of an elastic material.

Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension, the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension).

This law actually holds only approximately, and only when the deformation (extension or contraction) is small compared to the rod's overall length. For deformations beyond the elastic limit, atomic bonds get broken or rearranged, and a spring may snap, buckle, or permanently deform. Many materials have no clearly defined elastic limit, and Hooke's law can not be meaningfully applied to these materials. Moreover, for the superelastic materials, the linear relationship between force and displacement is appropriate only in the low-strain region.

Hooke's law is a mathematical consequence of the fact that the potential energy of the rod is a minimum when it has its relaxed length. Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.

Force of fully compressed spring

[pic]

where

E - Young's modulus

d - spring wire diameter

L - free length of spring

n - number of active windings

ν - Poisson ratio

D - spring outer diameter

[edit] Zero-length springs

"Zero-length spring" is a term for a specially-designed coil spring that would exert zero force if it had zero length. That is, in a line graph of the spring's force versus its length, the line passes through the origin. Obviously a coil spring cannot contract to zero length because at some point the coils will touch each other and the spring will not be able to shorten any more. Zero length springs are made by manufacturing a coil spring with built-in tension, so if it could contract further, the equilibrium point of the spring, the point at which its restoring force is zero, occurs at a length of zero. In practice, zero length springs are made by combining a "negative length" spring, made with even more tension so its equilibrium point would be at a "negative" length, with a piece of inelastic material of the proper length so the zero force point would occur at zero length.

A spring with zero length can be attached to a mass on a hinged boom in such a way that the force on the mass is almost exactly balanced by the vertical component of the force from the spring, whatever the position of the boom. This creates a pendulum with very long period. Long-period pendulums enable seismometers to sense the slowest waves from earthquakes. The LaCoste suspension with zero-length springs is also used in gravimeters because it is very sensitive to changes in gravity. Springs for closing doors are often made to have roughly zero length so that they will exert force even when the door is almost closed, so it will close firmly.

from annealed steel and hardened after fabrication. Some non-ferrous metals are also used including phosphor bronze and titanium for parts requiring corrosion resistance and beryllium copper for springs carrying electrical current (because of its low electrical resistance).

When a spring is compressed or stretched, the force it exerts is proportional to its change in length. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve. An extension or compression spring has units of force divided by distance, for example lbf/in or N/m. Torsion springs have units of force multiplied by distance divided by angle, such as N·m/rad or ft·lbf/degree. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. The stiffness (or rate) of springs in parallel is additive, as is the compliance of springs in series.

Depending on the design and required operating environment, any material can be used to construct a spring, so long as the material has the required combination of rigidity and elasticity: technically, a wooden bow is a form of spring.

|Contents |

|[hide] |

|1 History |

|2 Types |

|3 Physics |

|3.1 Hooke's law |

|3.2 Simple harmonic motion |

|4 Theory |

|5 Zero-length springs |

|6 Uses |

|7 References |

|8 External links |

[edit] History

Simple non-coiled springs were used throughout human history e.g. the bow (and arrow). In the Bronze Age more sophisticated spring devices were used, as shown by the spread of tweezers in many cultures. Ctesibius of Alexandria developed a method for making bronze with spring-like characteristics by producing an alloy of bronze with an increased proportion of tin, and then hardening it by hammering after it is cast.

Coiled springs appeared early in the 15th century,[1] in door locks.[2] The first spring powered-clocks appeared in that century[2][3][4] and evolved into the first large watches by the 16th century.

In 1676 British physicist Robert Hooke discovered the principle behind springs' action, that the force it exerts is proportional to its extension, now called Hooke's law.

[edit] Types

[pic]

[pic]

A spiral torsion spring, or hairspring, in an alarm clock.

[pic]

[pic]

A volute spring. Under compression the coils slide over each other, so affording longer travel.

[pic]

[pic]

Vertical volute springs of Stuart tank

[pic]

[pic]

Tension springs in a folded line reverberation device.

[pic]

[pic]

A torsion bar twisted under load

[pic]

[pic]

Leaf spring on a truck

Springs can be classified depending on how the load force is applied to them:

• Tension/Extension spring - the spring is designed to operate with a tension load, so the spring stretches as the load is applied to it.

• Compression spring - is designed to operate with a compression load, so the spring gets shorter as the load is applied to it.

• Torsion spring - unlike the above types in which the load is an axial force, the load applied to a torsion spring is a torque or twisting force, and the end of the spring rotates through an angle as the load is applied.

They can also be classified based on their shape:

• Coil spring - this type is made of a coil or helix of wire

• Flat spring - this type is made of a flat or conical shaped piece of metal.

The most common types of spring are:

• Cantilever spring - a spring which is fixed only at one end.

• Coil spring or helical spring - a spring (made by winding a wire around a cylinder) and the conical spring - these are types of torsion spring, because the wire itself is twisted when the spring is compressed or stretched. These are in turn of two types:

o Compression springs are designed to become shorter when loaded. Their turns (loops) are not touching in the unloaded position, and they need no attachment points.

▪ A volute spring is a compression spring in the form of a cone, designed so that under compression the coils are not forced against each other, thus permitting longer travel.

o Tension or extension springs are designed to become longer under load. Their turns (loops) are normally touching in the unloaded position, and they have a hook, eye or some other means of attachment at each end.

• Hairspring or balance spring - a delicate spiral torsion spring used in watches, galvanometers, and places where electricity must be carried to partially-rotating devices such as steering wheels without hindering the rotation.

• Leaf spring - a flat springy sheet, used in vehicle suspensions, electrical switches, bows.

• V-spring - used in antique firearm mechanisms such as the wheellock, flintlock and percussion cap locks.

Other types include:

• Belleville washer or Belleville spring - a disc shaped spring commonly used to apply tension to a bolt (and also in the initiation mechanism of pressure-activated landmines).

• Constant-force spring — a tightly rolled ribbon that exerts a nearly constant force as it is unrolled.

• Gas spring - a volume of gas which is compressed.

• Ideal Spring - the notional spring used in physics: it has no weight, mass, or damping losses.

• Mainspring - a spiral ribbon shaped spring used as a power source in watches, clocks, music boxes, windup toys, and mechanically powered flashlights

• Negator spring - a thin metal band slightly concave in cross-section. When coiled it adopts a flat cross-section but when unrolled it returns to its former curve, thus producing a constant force throughout the displacement and negating any tendency to re-wind. The commonest application is the retracting steel tape rule.[5]

• Progressive rate coil springs - A coil spring with a variable rate, usually achieved by having unequal pitch so that as the spring is compressed one or more coils rests against its neighbour.

• Rubber band - a tension spring where energy is stored by stretching the material.

• Spring washer - used to apply a constant tensile force along the axis of a fastener.

• Torsion spring - any spring designed to be twisted rather than compressed or extended. Used in torsion bar vehicle suspension systems.

• Wave spring - a thin spring-washer into which waves have been pressed.[6]

[edit] Physics

[pic]

[pic]

Two springs attached to a wall and a mass. In a situation like this, the two springs can be replaced by one with a spring constant of keq=k1+k2.

[edit] Hooke's law

Main article: Hooke's law

Most springs (not stretched or compressed beyond the elastic limit) obey Hooke's law, which states that the force with which the spring pushes back is linearly proportional to the distance from its equilibrium length:

[pic]

where

x is the displacement vector - the distance and direction in which the spring is deformed

F is the resulting force vector - the magnitude and direction of the restoring force the spring exerts

k is the spring constant or force constant of the spring.

Coil springs and other common springs typically obey Hooke's law. There are useful springs that don't: springs based on beam bending can for example produce forces that vary nonlinearly with displacement.

[edit] Simple harmonic motion

Main article: Harmonic oscillator

Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like:

[pic]

[pic]

[pic]

The displacement, x, as a function of time. The amount of time that passes between peaks is called the period.

The mass of the spring is assumed small in comparison to the mass of the attached mass and is ignored. Since acceleration is simply the second derivative of x with respect to time,

[pic]

This is a second order linear differential equation for the displacement x as a function of time. Rearranging:

[pic]

the solution of which is the sum of a sine and cosine:

[pic]

A and B are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with B = 0 (zero initial position with some positive initial velocity) is displayed in the image on the right.

[edit] Theory

In classical physics, a spring can be seen as a device that stores potential energy, specifically elastic potential energy, by straining the bonds between the atoms of an elastic material.

Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension, the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension).

This law actually holds only approximately, and only when the deformation (extension or contraction) is small compared to the rod's overall length. For deformations beyond the elastic limit, atomic bonds get broken or rearranged, and a spring may snap, buckle, or permanently deform. Many materials have no clearly defined elastic limit, and Hooke's law can not be meaningfully applied to these materials. Moreover, for the superelastic materials, the linear relationship between force and displacement is appropriate only in the low-strain region.

Hooke's law is a mathematical consequence of the fact that the potential energy of the rod is a minimum when it has its relaxed length. Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.

Force of fully compressed spring

[pic]

where

E - Young's modulus

d - spring wire diameter

L - free length of spring

n - number of active windings

ν - Poisson ratio

D - spring outer diameter

[edit] Zero-length springs

"Zero-length spring" is a term for a specially-designed coil spring that would exert zero force if it had zero length. That is, in a line graph of the spring's force versus its length, the line passes through the origin. Obviously a coil spring cannot contract to zero length because at some point the coils will touch each other and the spring will not be able to shorten any more. Zero length springs are made by manufacturing a coil spring with built-in tension, so if it could contract further, the equilibrium point of the spring, the point at which its restoring force is zero, occurs at a length of zero. In practice, zero length springs are made by combining a "negative length" spring, made with even more tension so its equilibrium point would be at a "negative" length, with a piece of inelastic material of the proper length so the zero force point would occur at zero length.

A spring with zero length can be attached to a mass on a hinged boom in such a way that the force on the mass is almost exactly balanced by the vertical component of the force from the spring, whatever the position of the boom. This creates a pendulum with very long period. Long-period pendulums enable seismometers to sense the slowest waves from earthquakes. The LaCoste suspension with zero-length springs is also used in gravimeters because it is very sensitive to changes in gravity. Springs for closing doors are often made to have roughly zero length so that they will exert force even when the door is almost closed, so it will close firmly

.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

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[pic]

This section is from the book "Handy Man's Workshop And Laboratory", by A. Russell Bond. Also available from Amazon: Handy Man's Workshop And Laboratory.

How To Wind A Spaced Coil Spring

When it is desired to wind a spring with the coils spaced a uniform distance apart, a simple method is to use a former or guide, made of wire which is as thick as the space desired, between the coils. The accompanying cut shows how this is done. The guide consists of a few coils which are spaced the requisite distance apart, and one end of the wire projects outward tangentially. The spring is then wound on the arbor between the turns of the guide. As the wire is fed on the arbor it is crossed over the extending end of the guide, in the manner shown, so that it presses inward against the coils of the former. As the arbor is turned the guide is automatically fed along the arbor, and the coils of the spring are uniformly spaced by the coils of the former.

[pic]

Fig. 77 - How to wind a spaced coil spring.

Another Method Of Making A Coil Spring

Get a metal rod the same diameter as the spring desired; drill a hole near the end to admit the end of the wire. Give the wire two or three turns around rod, spacing the turns according to the desired pitch. Clamp it between two blocks of hard wood in a vise, having the rod in the direction of the grain of the wood. Revolve the rod by means of a monkey wrench fitted on the flattened end of the rod. The wire will follow in and wind a spring as true and perfect as though it had been wound with a lathe.

[pic]

Fig. 78 - A simple method for making a coiled spring.

HOW TO MAKE SPRINGS

[pic]

There are three things you'll need to read before you get started. First, the DISCLAIMER:

|This document is designed to provide information in regard to the subject matter being covered. Every effort has|

|been made to ensure the accuracy of its contents. However there may be mistakes, both typographical and in |

|content. Additionally, work in the metal trades implies an acceptance of the risk of injury, loss, or damage, |

|the cause of which is clearly beyond the control of the writer of a work on the subject. Therefore, the author |

|of this document accepts no responsibility or liability whatsoever for any injury, loss or damage sustained by a|

|reader who, having read this material, then seeks to apply what he or she has learned therein. |

| |

|Please read the Terms of Use. |

Second, before you start to work with spring wire, read the section on safety. When you have read that section, read it again. No kidding.

And third, about this document: I've tried to write for the benefit of someone who has (or can gain access to) basic hand and power tools. The sections of the document are arranged in logical order presuming a minimal knowledge of the metalworking trades in general or of springmaking in particular, and cross-linked to provide a forward path that leads from this point through the entire manufacturing process. There's a glossary of spring terminology and an addendum, which should help you to define terms and find additional resources. Where possible, I've indicated where to find additional information in the main body of the text. Most of the current material relating to the subject is written for mechanical engineers, but there are some other writeups I've heard of, too — see the addendum for links to these.

In its first incarnation, this site was made in frames. In the second, I did away with the frames for the sake of design simplicity. Since then I've added a thing or two like for instance the site map, which will give you a bird's-eye view of the whole shebang. This is now the third incarnation and except for freshening the links from time to time, I consider it done. If you want to have the whole as a handy reference, spiral-bound so it lays flat on your workbench, feel free to buy your very own copy of the print version. Lastly, the text itself is of very limited use without the graphics, and there is no 'text-only' version of this site. So if you've got your graphics turned off, turn 'em on, OK?

Any comments or suggestions for improvement should be made to yours truly.

INTRODUCTION

This section will give you some basic information about springs, what they look like, what their parts are, and how they work.

If you already know about springs and want to get right to it, be my guest.

There are three basic types of springs:

|[pic] |Compression springs can be found in ballpoint pens, pogo sticks, and the valve assemblies of gasoline engines.|

| |When you put a load on the spring, making it shorter, it pushes back against the load and tries to get back to|

| |its original length. |

|[pic] |Extension springs are found in garage door assemblies, vise-grip pliers, and carburetors. They are attached at|

| |both ends, and when the things they are attached to move apart, the spring tries to bring them together again.|

|[pic] |Torsion springs can be found on clipboards, underneath swing-down tailgates, and, again, in car engines. The |

| |ends of torsion springs are attached to other things, and when those things rotate around the center of the |

| |spring, the spring tries to push them back to their original position. |

See the Glossary for detailed diagrams of these types of springs.

SPRING DESIGN

If you're trying to make a spring to replace a broken one, you don't need to know a whole lot about design. On the other hand, if you're making a prototype of a machine, for instance, and you don't know exactly what you want, then this page is for you. Here you'll learn some basic data about spring design, which is what you'll need to know to make exactly the spring you want.

|[pic] |General principles |

| |Mathematics |

| |Design limitations |

| |Buying design |

[pic]

General Principles

There are three basic principles in spring design:

• The heavier the wire, the stronger the spring.

• The smaller the coil, the stronger the spring.

• The more active coils, the less load you will have to apply in order to get it to move a certain distance.

Based on these general principles, you now know what to do to change the properties of a spring you already have. For instance, if you want to make automotive valve springs a little stronger than stock, you can a) go to a slightly heavier wire and keep the dimensions and coil count the same, b) decrease the diameter of the spring, keeping the wire size and coil count the same, or c) decrease the number of active coils, keeping the wire size and spring diameter the same. Naturally, you can also go to a stronger material to achieve the same result.

Now, what if you're making a spring from scratch, with nothing to go on in the way of a sample? You can engineer your own design (see the next section of this page for the math), coil a spring, and then test it. If it's what you want, fine. If it's, let's say, a skosh too strong, then you can a) go to a lighter wire, b) open up the coil diameter, or c) increase the number of active coils to get a slightly weaker spring.

Or, if you want to make things really simple, go to the Addendum, where you'll find a few websites that offer online design!

[pic]

Mathematics

Naturally, spring design software is available — you can find out where to get it in the Addendum. For the purists (or those who don't want to pay for a program), here's a very short summary of the mathematics of spring design. These equations, by the way, are taken from The New American Machinist's Handbook, published by McGraw-Hill Book Company, Inc.in 1955. I don't pretend to understand them.

[pic]

There's a lot more in the way of engineering that goes into spring design: these are only the basic equations. If you're interested, you can contact someone who makes spring design software or (gasp!) find it in the library under Dewey classification number 621.824.

You can also contact the Spring Manufacturers' Institute: they make a handy-dandy spring calculator, suitable for simple design work, that anyone can learn to use. They also have spring design software, training classes, and a bunch more stuff. Dave sez, “Check it out.”

[pic]

Design Limitations

Depending on what kind of spring you want to design, and depending on where it will be used, your design will be limited:

For all springs:

• A spring under load is stressed. If you put too much stress on a spring, its shape will deform and it will not return to its original dimensions.

• The material from which the spring is made will have an effect on the strength of the spring: it will also have an effect on how much stress the spring will withstand. The section on spring materials will tell you more about this.

• When you heat spring wire (which you always do), it may change its dimensions. Again, the section on materials will tell you more about this.

For compression springs:

• If the spring will set solid (compress all the way, so that all the coils touch each other) at the limit of its travel, the diameter of the wire times the number of coils cannot be greater than the space allowed, unless you want the spring itself to act as a mechanical stop to the motion.

• Springs that operate in a high-temperature environment (like for instance inside an engine) will need to be made slightly longer to compensate for the fact that the heat may have an effect on the length of the spring. The section on finishing will tell you more about this.

• As a compression spring assumes a load and shortens, the diameter of the active coils will increase. This is only a problem when the spring has to work in a confined space.

For extension springs:

• There should be some mechanical limit on how far the spring will extend, or the spring will lose its shape and not return to its initial condition with all coils closed.

• Extension springs operating in a high-temperature environment may have to be coiled extra-tight, as the heat will tend to weaken the spring. The section on extension springs will tell you more about this.

For torsion springs:

• When a torsion springs assumes a load, the diameter of the coil body will decrease. If the spring has something inside the coil, it will act as a mechanical stop to the action of the spring.

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Buying Design

If you want to have a mechanical engineer design your spring, your best bet is to call a spring shop. You can find spring shops in the phone book. If your phone book doesn't list any, go to the library: they should have phone books for major cities where spring factories are -- try Detroit or Los Angeles if there are none in your area.

A spring shop will generally do the design work for you for a small charge. They will also try to get you to let them make the spring for you, which you may or may not want.

The section on spring shops will tell you more about how their business operates. The addendum will give you links to spring shops, suppliers, people who make spring design software, and a whole slew of other stuff.

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SPRING MATERIALS

This section will tell you about the different kinds of material that springs are made out of. It will also tell you where to get your wire -- make sure you read the Safety section so you know how to handle it safely once you've got it.

|[pic] |Types of wire |

| |Buying wire |

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Types of Wire

Springs are usually made from alloys of steel. The most common spring steels are music wire, oil tempered wire, chrome silicon, chrome vanadium, and 302 and 17-7 stainless. Other materials can also be formed into springs, depending on the characteristics needed. Some of the more common of these exotic metals include beryllium copper, phosphor bronze, Inconel, Monel, and titanium. The following table summarizes the more important properties of each material:

|Material |Common Sizes |Properties and Uses |

|Music Wire |.003-.250 |A high-carbon steel wire used primarily for applications demanding high strength, medium |

| | |price, and uniformly high quality. Guitar and piano strings are made from this material, as |

| | |are most small springs. Music wire will contract under heat, and can be plated. |

|Oil Tempered Wire (OT) |.010-.625 |This is the workhorse steel spring wire, being used for many applications in which superior |

| | |strength or uniformity is not crucial. Will not generally change dimensions under heat. Can be|

| | |plated. Also available in square and rectangular sections. |

|Chrome Silicon, Chrome |.010-.500 |These are higher quality, higher strength versions of Oil Tempered wire, used in |

|Vanadium | |high-temperature applications such as automotive valve springs. Will not generally change |

| | |dimensions under heat. Can be plated. |

|Stainless Steel |.005-.500 |Stainless steels will not rust, making them ideal for the food industry and other environments|

| | |containing water or steam. 302 series stainless will expand slightly under heat: 17-7 will |

| | |usually not change. Cannot be plated. |

|Inconel, Monel, Beryllium |.010-.125 |These specialty alloys are sometimes made into springs which are designed to work in extremely|

|Copper, Phosphor Bronze | |high-temperature environments, where magnetic fields present a problem, or where corrosion |

| | |resistance is needed in a high-temperature working environment. They are much more costly than|

| | |the more common stocks and cannot be plated. Generally will not change dimensions under heat. |

|Titanium |.032-.500 |Used primarily in air- and spacecraft because of its extremely light weight and high strength,|

| | |titanium is also extremely expensive and dangerous to work with as well: titanium wire will |

| | |shatter explosively under stress if its surface is scored. Generally will not change |

| | |dimensions under heat. Cannot be plated. |

Titanium is the strongest material, but it is very expensive. Next come chrome vanadium and chrome silicon, then music wire, and then oil tempered wire. The stainless and exotic materials are all weaker than the rest.

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Buying wire

Spring wire is made in common sizes (see table above) and in special sizes to order. The common sizes that are manufactured are available within the ranges specified at intervals ranging from a couple of thousandths of an inch (for the smaller sizes) to sixteenths of inches (for the larger sizes). Metric-measure sizes are available outside the US.

These spring wire materials may be bought from steel suppliers in two forms: coils and straightened- and-cut bars. Unless you are dealing with extremely close tolerances, exotic materials, or need a stock size that is not commonly manufactured, you'll probably find it most economical to buy your stock in coils.

Bought in coils, spring steel is generally sold by the pound: the coils range in size from about 6 inches (for wire under .005') to 7 feet (for wire in the .437-.500' range) in diameter. The smaller coils are generally shipped UPS, while the larger sizes require truck transport as well as special unloading and storage facilities.

Finding a source of supply is as easy as looking in the phone book: if you're in a rural area, try the local library which will have Yellow Pages for the major metropolitan areas -- try Detroit or Los Angeles for starters. You can also contact the Spring Manufacturers' Institute and ask them for a copy of Springs magazine, which is filled with suppliers' advertisements (as well as technically interesting articles). The Addendum also lists some wire manufacturers and suppliers. By the way, this link will take you to a guide of how many feet of wire you'l get per pound of material ordered (thanks to Jeff for this).

One caution: you should not order straightened-and-cut wire until you're SURE you know what you want. Once you get your material, you'll find it impossible to return if the bars are an inch too short.

And one note: when spring wire is made, it develops what's known as a 'cast' from being tied into round coils. If you strip wire from a coil, it will likely not be perfectly straight: the 'natural' curvature of the wire is 'cast'. The cast of the wire will introduce an extremely small variance in the physical dimensions of the springs made from the wire -- it's only a problem when you're working with very close dimensional tolerances. Cast is why wire is also available in straightened-and-cut bars.

General Safety

Springs under load want to return to their original shape. The same goes for spring wire. Spring wire will try to straighten itself out if given the chance: don't let your body get in its way.

Small wire

Small wire (diameter less than about .025") will not hurt you if it hits you. On the other hand, small wire is nothing more than an edge, waiting for something to cut. Don't use your hand to try to stop wire that's moving, especially if it's moving under power (like being pulled by a lathe). Instead, wait till it stops moving. Gloves are an excellent idea, too.

Medium wire

Medium wire (diameter from about .025" - .312") is too wide to act as an edge, and usually not massive enough to break bones, but it can raise quite a knot if you get in its way. Again, always keep track of where the ends of the wire are, and if they start to move, get out of the way.

Heavy wire

Heavy wire (diameter greater than about .312") needs respect. If it gets loose, it can EASILY break bones, or worse.

Stainless steel

Stainless steel is a lot softer than other types of wire. When cut, the end of the wire is like a knife edge. Always keep track of where the end of the wire is, and keep your hands away from it while it's moving.

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Handling Wire

The two most dangerous times are when you're breaking open a coil of wire and when you're actually winding a spring.

Breaking open coils

Once you have your wire, you'll need to take it out of its coil. The coil may be wrapped in paper — take that off first.

Under the paper, the wire will be tied. Light wire will be tied with string. Medium wire will be tied with tie wire. Large wire will be tied with metal bands. Whatever size wire you have, remember that the coil should have only two ends. One will be on the inside of the coil, and the other will be on the outside. You'll normally use wire from the inside, to avoid tangling. Always make a hook on the “inside” end so it's easy to find again:

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Handling Small Wire in Coils

To break open a coil of small (up to about .125") wire, cut all the ties except two. Don't cut the closest tie holding the outside end of the wire, and the tie most directly opposite to that one.

To remove wire from the coil, start with the end on the inside of the coil: this will keep the coil from tangling. Grab the end of the wire and cut off the hook. Pull it slightly, until you can see the gap between it and the rest of the coil. Grab the wire at the gap and pull the end free from the tie holding it. Repeat this process, working around the coil, until you have the length you need.

Medium-sized wire

(.125 - .312") can be handled the same way, except that you should keep three ties instead of two. When uncoiling wire larger than .250", you should lay the coil flat on the ground and always stand in the center of the coil, for safety.

Large wire

(.312 - .625") needs special handling. First of all, you'll probably be using a hoist or forklift to move the coil, because of the weight. Lay the coil on top of something (a 2x4 or a pipe works great) to keep one end off the ground so that you can pick it up when you're done. Stand inside the coil from now on!

Then, take a length of tie wire and double it over. Loop it twice around the coil, right next to the second tie holding the inside end of the wire. Pull it tight and twist it so that you have a 'pigtail' and the tie wire is too tight to move by hand. Then, cut the first two original ties. Grab the end of the wire and flip it over the coil, so that it sticks out.

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Go to the next tie and repeat this process, working your way around the coil until you have the length you need. You can use heavy bolt cutters or an acetylene torch to cut the wire.

If heavy wire gets away from you and starts to come undone all by itself, the very best thing to do is

• Run like hell, and

• Pray it doesn't hit you.

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Coiling

If you're using a lathe to make your springs, you'll be standing there, letting the lathe pull the wire. The lathe will do what you want, but it will not know to stop if things get out of control. So, before you start the lathe, figure out what you're going to do if things go haywire. Know how to stop the lathe, and know which way you can safely run.

Never reach over the wire to get to your lathe controls, especially when working with heavy wire. Reach under it and avoid injury if your wire guide breaks.

Keep the lathe speed DEAD SLOW: with heavy wire, 10 rpm is about right.

Don't grab onto wire that's being fed into the lathe. Stop the lathe and back it off until there's no tension in the wire before you put your hands near.

NEVER try to guide wire by hand. Use tooling.

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When you're done

After you've removed wire from a coil, the coil will be looser than it was before. Before you put it away, retie it so that it doesn't tangle up or uncoil by itself. For light wire, use string. For medium size wire, use tie wire. For heavy wire, use tie wire doubled over, looped around the coil twice, and tied in a “pigtail”:

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Lastly, make a hook in the “inside” end of the wire, so you can find it again easily when you need to.

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Storing Wire

Safety first: always store your wire someplace where kids can't get at it.

Common sense second: keep your wire dry. Steel wire will rust if it gets wet.

More common sense: keep your wire tight. When you're finished working with a coil of wire, make a hook in the inside end (so you can find it again easily) and tie the coil securely. Be especially careful with heavy wire, which should be tied with at least six doubled strands of tie wire, each looped around the coil twice and cinched tight.

Still more common sense: if you live in an area that has earthquakes, tornadoes, hurricanes, etc., be sure that you chock your coils of wire so that they don't get loose and start to move around when mother nature starts acting up.

SPRINGS

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   δ of the load   F on the end of a cantilever can be appreciable - it depends upon the cantilever's geometry and elastic modulus, as predicted by elementary beam theory. Unlike the constant cross- section 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.

The shortcoming of most metal springs is that they rely on either bending or torsion to obtain significant deformations; the stress therefore varies throughout the material so that the material does not all contribute uniformly to energy storage. The wire of a   helical compression spring - such as shown on the left - is loaded mainly in torsion and is therefore usually of circular cross- section. This type of spring is the most common and we shall focus on it.

The   (ex)tension spring is similar to the compression spring however it requires special ends to permit application of the load - these ends assume many forms but they are all potential sources of weakness not present in compression springs. Rigorous duties thus usually call for compression rather than tension springs.

A tension spring can be wound with initial pre-load so that it deforms only after the load reaches a certain minimum value. Springs which are loaded both in tension and in compression are rare and restricted to light duty.

A   Belleville washer or 'disc spring' is a washer manufactured to a conical surface as shown by the cross- sections in this photograph. Belleville washers are characterised by short axial lengths and relatively small deformations, and are often used in stacks as illustrated. Their geometry can be engineered to produce highly non-linear characteristics which may display negative stiffness.

All the abovementioned springs are essentially translatory in that forces and linear deflections are involved. Rotary springs involve torque and angular deflection. The simplest of these is the torsion bar in which loading is pure torque; its analysis is based upon the simple torsion equation. Torsion bars are stiff compared to other forms of rotary spring, however they do have many practical applications such as in vehicle suspensions.

Torsion springs which are more compliant than the torsion bar include the clock- or   spiral torsion spring (left) and the   helical torsion spring (right). These rely on bending for their action, as a simple free body will quickly demonstrate. The helical torsion spring is similar to the helical tension spring in requiring specially formed ends to transmit the load.

The   constant force spring is not unlike a self- retracting tape measure and is used where large relative displacements are required - the spring motors used in sliding door closers is one application. There exists also a large variety of non-metallic springs often applied to shock absorption and based on rubber blocks loaded in shear. Springs utilising gas compressibility also find some use.

Close-coiled round wire helical compression springs

The close-coiled round wire helical compression spring is the type of spring most frequently encountered, and it alone is examined below. It is made from wire of diameter   d wound into a helix of mean diameter   D, helix angle   α, pitch   p, and total number of turns   nt. This last is the number of wire coils prior to end treatment (see Table 1 below) - in the spring illustrated   nt ≈ 8 1/2.

Close-coiled requires a small helix angle, say   α ≤ 12o, where   tanα = p/π D from the developed helix.

Various wire diameters are obtainable, but the availability of new springs is enhanced by specifying wire diameters from the R20 series of AS 2338 whose most frequently used decade is

. . .   0.8   0.9   1.0   1.12   1.25   1.4   1.6   1.8   2   2.24   2.5   2.8   3.15   3.55   4   4.5   5   5.6   6.3   7.1   8   9   10   11.2   12.5   . . . mm

The ratio of mean coil diameter to wire diameter is known as the   spring index,   C = D/d. Portions of two springs which have the same mean coil diameter but different wire diameters and hence different indices are compared here. It is clear that low indices result in difficulty with spring manufacture and in stress concentrations induced by curvature. Springs in the range 5 ≤ C ≤ 10 are prefered, while indices less than 3 are generally impracticable.

Loads are transferred into a spring by means of   platens, which are usually just flat surfaces bearing on the spring ends. Various end treatments are shown in Table 1.   Plain ends - when the wire is just cropped off to length - are suitable only for large index, light duty applications unless shaped platens or coil guides are employed, because each spring end contacts its platen at a point offset from the spring axis and this leads to bending of the spring and uncertain performance.

Ground ends distribute the load into the spring more uniformly than do plain ends, but the contact region on a flat platen will be very much less than 360o which is ideal for concentricity of bearing surface and spring axis. One or more turns at the end of a spring may be wound with zero pitch, this is called a   squared or   closed end. Subsequent grinding produces a seating best suited for uniform load transfer, and so   squared and ground ends are invariably specified when the duty is appreciable. Grinding the ends becomes difficult when the spring index exceeds 10, and is obviously inappropriate for small wire sizes - say under 0.5 mm.

The   active turns   na are the coils which actually deform when the spring is loaded, as opposed to inactive turns at each end which are in contact with the platen and therefore do not deform though they may move bodily with the platen   (see the animation below). The   free length   Lo of a compression spring is the spring's maximum length when lying freely prior to assembly into its operating position and hence prior to loading. The   solid length   Ls of a compression spring is its minimum length when the load is sufficiently large to close all the gaps between the coils.

Table 1 indicates how   na, Lo and Ls depend upon wire diameter, total turns, pitch and end treatment, however the Table's predictions should be viewed with caution - especially if there are less than seven turns - because of variability in the squaring and/or grinding operations.

The springs illustrated here are right handed, but left hand lays are just as common. The lay usually has no bearing on performance, except when springs are nested inside one another in which case the two lays must differ to avoid interference. Springs with closed ends do not become entangled when jumbled in a container, which is sometimes an important consideration in assembly.

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. Sometimes a spring is deliberately yielded or   pre-set during manufacture as will be explained later.

The animation illustrates the spring working between a minimum operational state ( Flo, δlo) and a maximum operational state ( Fhi, δhi)   { nomenclature explanation}. If the total number of cycles is small - say less than 104 - then loading may be treated as static, otherwise fatigue considerations apply.

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 ≤ 3.

As the spring approaches solidity, small pitch differences between coils will lead to progressive coil- to- coil contact rather 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   δs - δhi ≥ 0.1δhi, 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 :

o embraces the known upward load   F applied externally and axially to the end coil of the spring, and

o 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   α 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 cosα), a compressive force   (F sinα), a torque   (1/2FD cosα) and a bending moment   (1/2FD sinα).

Assuming the helix inclination   α to be small for close- coiled springs approaching solidity ( when working loads are critical ) then   sinα ≈ 0, cosα ≈ 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 :

            τ   =   τtorsion + τdirect   =   Tr/J + F/A

                 =   (FD/2) (d/2)/(πd4/32) + F/(πd2/4) = (1 + 0.5d/D) 8 FD/πd3

( 1)     τ   =   K 8FC /πd2

                        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 = Ks   ≡   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

• K = Kh   ≈   ( C + 0.6)/( C - 0.67)     which accounts for direct shear and also the effect of curvature- induced stress concentration on the inside of the coil (similar to that in curved beams). Kh 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   δ of the load   F follows from Castigliano's theorem. Neglecting small direct shear effects in the presence of torsion :

            δ   =   ∂U/∂F   =   ∂/∂F [ ∫length (T2/2GJ) ds ]       where   T = FD/2

                  =   ∫length (T/GJ) (∂T/∂F) ds   =   (T/GJ) (D/2)*(wire length)

                  =   (FD/2GJ) (D/2) naπD       which leads to

( 2)     k   =   F/δ   =   Gd / 8naC3       in which   na is the number of active coils (Table 1).

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 AE-11 for practical advice on tolerances.

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   δcrit which depends upon the slenderness ratio   Lo/D rather like Euler buckling of columns, thus :

( 3a)     c1 δcrit /Lo   =   1 - √[ 1 - ( c2 D/ λLo )2 ]   in which the constants are defined as

            c1   =   ( 1 + 2ν/( 1 + ν   =   1.23 for steel ;       c2   =   π √[ ( 1 + 2ν/( 2 + ν ]   =   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   λ = 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 :

( 3b)     Lo.crit   =   [ 1 + ( c2D/c1λδ )2 ] c1δ /2

EXAMPLE

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

Stress factors from ( 1) are   Ks = 1+0.5/C = 1.08;   Kh = (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' 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   nt ≈ 113/4 turns. Ends are not ground right down to feather edges in practice, as shown by this cadmium plated hydraulic valve return spring.

From Table 1 the active turns are   na = nt -2 = 93/4; the solid length is   Ls = ntd = 47 mm, and the pitch is   p = ( Lo -2d)/na = ( 85 -8)/93/4 = 7.9 mm. The corresponding helix angle is   α = arctan( p/D) = 5.5o - the spring is certainly close coiled.

From ( 2) :   k = Gd /8 naC3 = 79E3 x 4 / 8 x 9.75 x 6.53 = 14.8 N/mm

The solid deflection   δs = Lo -Ls = 38 mm, so the solidification load is   Fs = k δs = 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   Ks applies in ( 1). The corresponding solid shear stress from ( 1) is therefore :

              τs = Ks 8 FsC / π d2 = 1.08 x 8 x 560 x 6.5 / π 42 = 625 MPa.

Assume that at the maximum working state the above- mentioned 10% minimum clash allowance applies, so that   δs - δhi ≥ 0.1 δhi   ie. δhi ≤ δs/1.1 = 38/1.1   say   δhi = 34 mm.

Proceeding as with the solid state,   Fhi = k δhi = 14.8 x 34 = 505 N and so :

              τhi = Ks 8 Fhi C / πd2 = 1.08 x 8 x 505 x 6.5 /π 42 = 563 MPa - static conditions still assumed.

Considering stability at this maximum state and assuming hinged ends (λ = 1), then   λ Lo/c2D = 1 x 85 /2.62 x 26 = 1.25. Since this is greater than unity buckling could occur, so investigate via ( 3a):

              δc = { 1 - √( 1 - 1/1.252 )} x 85/1.23 = 27 mm

                            and since the maximum deflection   δhi exceeds δc 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   λLo/c2D < 1 which automatically guarantees stability

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