The spring stiffness formula is perhaps the most important point in the topic of these elastic elements. After all, it is rigidity that plays a very important role in why these components are used so widely.

Today, virtually no industry can do without springs; they are used in instrument and machine tool building, agriculture, the production of mining and railway equipment, energy, and other industries. They faithfully serve in the most important and critical places of various units, where their inherent characteristics are required, first of all, the stiffness of the spring, the formula of which in general is very simple and familiar to children from school.

Features of work

Any spring is an elastic product, which is subjected to static, dynamic and cyclic loads during operation. The main feature of this part is that it deforms under external force, and when the impact stops, it restores its original shape and geometric dimensions. During the period of deformation, energy is accumulated, during restoration - its transfer.

It is this property to return to its original form that has brought widespread use of these parts: they are excellent shock absorbers, valve elements that prevent excess pressure, accessories for measuring instruments. In these and other situations, due to the ability to deform elastically, they perform an important job, so high quality and reliability are required from them.

Types of springs

There are many types of these parts, the most common are tension and compression springs.

• The first of them without load have a zero pitch, that is, the coil is in contact with the coil. In the process of deformation, they stretch, their length increases. The termination of the load is accompanied by a return to its original form - again coil to coil.
• The latter, on the contrary, initially wind with a certain step between the turns, and shrink under load. The contact of the turns is a natural limiter for continued exposure.

Initially, it was for the tension spring that the ratio of the mass of the load suspended on it and the change in its geometric size was found, which became the basis for the formula for the stiffness of the spring through mass and length.

What other types of springs are

The dependence of deformation on the applied external force is also valid for other types of elastic parts: torsion, bending, disk-shaped, and others. It does not matter in which plane forces are applied to them: in the one where the axial line is located, or perpendicular to it, the deformation produced is proportional to the force under which it occurred.

Main characteristics

Regardless of the type of springs, the features of their work associated with constant deformation require the following parameters:

• The ability to maintain a constant value of elasticity for a given period.
• plasticity.
• Relaxation resistance, due to which deformations do not become irreversible.
• Strength, that is, the ability to withstand various types of loads: static, dynamic, shock.

Each of these characteristics is important, however, when choosing an elastic component for a particular job, first of all, its stiffness is of interest as an important indicator of whether it is suitable for this case and how long it will work.

What is stiffness

Rigidity is a characteristic of a part that shows how easy or simple it will be to compress it, how much force must be applied to do this. It turns out that the deformation that occurs under load is the greater, the greater the applied force (after all, the elastic force that arises in opposition to it has the same value in modulus). Therefore, it is possible to determine the degree of deformation, knowing the force of elasticity (applied force) and vice versa, knowing the necessary deformation, it is possible to calculate what force is required.

Physical foundations of the concept of rigidity / elasticity

The force acting on the spring changes its shape. For example, tension/compression springs shorten or lengthen under the influence of an external force. According to Hooke's law (this is the name of the formula that allows you to calculate the coefficient of spring stiffness), force and deformation are proportional to each other within the limits of elasticity of a particular substance. In opposition to the load applied from the outside, a force arises that is the same in magnitude and opposite in sign, which is aimed at restoring the original dimensions of the part and its shape.

The nature of this elastic force is electromagnetic, it arises as a result of a special interaction between the structural elements (molecules and atoms) of the material from which this part is made. Thus, the greater the rigidity, that is, the more difficult it is to stretch / compress the elastic part, the greater the coefficient of elasticity. This indicator is used, in particular, when choosing a particular material for the manufacture of springs for use in various situations.

How did the first version of the formula come about

The formula for calculating the stiffness of a spring, which is called Hooke's law, was established experimentally. In the course of experiments with loads of different masses suspended on an elastic element, the magnitude of its stretching was measured. So it turned out that the same test part under different loads undergoes different deformations. Moreover, the suspension of a certain number of weights, identical in mass, showed that each added/removed weight increases/reduces the length of the elastic element by the same amount.

As a result of these experiments, the following formula appeared: kx \u003d mg, where k is a coefficient constant for a given spring, x is the change in the length of the spring, m is its mass, and g is the acceleration of free fall (approximate value is 9.8 m / s²) .

Thus, the stiffness property was discovered, which, like the formula for determining the coefficient of elasticity, finds the widest application in any industry.

Stiffness formula

The formula studied by modern schoolchildren, how to find the coefficient of spring stiffness, is the ratio of force and magnitude, showing the change in the length of the spring depending on the magnitude of this impact (or

equal to it in the modulus of the elastic force). This formula looks like this: F = -kx. From this formula, the stiffness coefficient of the elastic element is equal to the ratio of the elastic force to the change in its length. In the SI international system of units of physical quantities, it is measured in newtons per meter (N/m).

Another way to write the formula: Young's coefficient

Tensile/compressive deformation in physics can also be described by a slightly modified Hooke's law. The formula includes the values ​​of relative strain (the ratio of the change in length to its initial value) and stress (the ratio of the force to the cross-sectional area of ​​the part). Relative deformation and stress according to this formula are proportional, and the coefficient of proportionality is the reciprocal of Young's modulus.

Young's modulus is interesting in that it is determined solely by the properties of the material, and does not depend in any way on either the shape of the part or its dimensions.

For example, Young's modulus for 100

whether it is approximately equal to one with eleven zeros (unit - N / sq. m).

The meaning of the concept of stiffness coefficient

Rigidity coefficient - coefficient of proportionality from Hooke's law. It is also rightfully called the coefficient of elasticity.

In fact, it shows the amount of force that must be applied to the elastic element in order to change its length by one (in the measurement system used).

The value of this parameter depends on several factors that characterize the spring:

• The material used in its manufacture.
• Forms and design features.
• geometric dimensions.

According to this indicator, you can

to conclude how the product is resistant to the effects of loads, that is, what will be its resistance when an external influence is applied.

Features of the calculation of springs

Showing how to find the stiffness of a spring, the formula is probably one of the most used by modern designers. After all, these elastic parts are used almost everywhere, that is, it is required to calculate their behavior and choose those that will ideally cope with their duties.

Hooke's law very simplistically shows the dependence of the deformation of an elastic part on the applied force; engineers use more accurate formulas for calculating the stiffness coefficient, taking into account all the features of the ongoing process.

For example:

• Modern engineering considers a cylindrical twisted spring as a spiral of wire with a circular cross section, and its deformation under the influence of the forces existing in the system is represented by a set of elementary shifts.
• When bending is deformed, the deformation is considered to be the deflection of a rod located with its ends on supports.

Features of calculating the stiffness of spring connections

An important point is the calculation of several elastic elements connected in series or in parallel.

With a parallel arrangement of several parts, the overall stiffness of this system is determined by a simple sum of the coefficients of the individual components. As you can easily see, the rigidity of the system is greater than that of a single part.

With a sequential arrangement, the formula is more complex: the reciprocal of the total stiffness is equal to the sum of the reciprocals of the stiffness of each component. In this variant, the sum is less than the terms.

Using these dependencies, it is easy to determine the correct choice of elastic components for a particular case.

Nature, being a macroscopic manifestation of intermolecular interaction. In the simplest case of stretching/compression of the body, the elastic force is directed opposite to the displacement of the particles of the body, perpendicular to the surface.

The force vector is opposite to the direction of body deformation (displacement of its molecules).

Hooke's law

In the simplest case of one-dimensional small elastic deformations, the formula for the elastic force has the form:

,

where is the rigidity of the body, is the magnitude of the deformation.

In verbal formulation, Hooke's law reads as follows:

The elastic force arising from the deformation of the body is directly proportional to the elongation of the body and is directed opposite to the direction of movement of body particles relative to other particles during deformation.

Nonlinear deformations

With an increase in the magnitude of the deformation, Hooke's law ceases to operate, the elastic force begins to depend in a complex way on the magnitude of tension or compression.

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See what the "Force of Elasticity" is in other dictionaries:

elastic force- elastic energy - Topics oil and gas industry Synonyms elastic energy EN elastic energy ... Technical Translator's Handbook

elastic force- tamprumo jėga statusas T sritis Standartizacija ir metrologija apibrėžtis Vidinės kūno jėgos, veikiančios prieš jį deformuojančias išorines jėgas ir iš dalies ar visiškai atkuriančios deformuotojo kūno (skysčių, dujų) tūrį ir (kietojo kūno) formą … Penkiakalbis aiskinamasis metrologijos terminų žodynas

elastic force- tamprumo jėga statusas T sritis fizika atitikmenys: angl. elastic force vok. elastische Kraft, f rus. elastic force, f; elastic force, fpranc. force elastique, f … Fizikos terminų žodynas

FORCE- vector quantity measure of mechanical impact on the body from other bodies, as well as the intensity of other physical. processes and fields. Forces are different: (1) S. Ampère, the force with which (see) acts on a conductor with current; direction of the force vector ... ... Great Polytechnic Encyclopedia

"strength" redirects here; see also other meanings. Force Dimension LMT−2 SI units ... Wikipedia

"strength" redirects here; see also other meanings. Force Dimension LMT−2 SI units newton ... Wikipedia

Exist., f., use. max. often Morphology: (no) what? strength for what? strength, (see) what? strength than? strength about what? about strength; pl. what? strength, (no) what? strength for what? forces, (see) what? strength than? forces about what? about forces 1. The ability of the living is called strength ... ... Dictionary of Dmitriev

A branch of mechanics, in which the displacements, deformations and stresses arising in resting or moving elastic bodies under the action of a load are studied. U. t. the basis of calculations for strength, deformability and stability in construction, business, aviation and ... ... Physical Encyclopedia

A branch of mechanics, in which the displacements, deformations and stresses arising in resting or moving elastic bodies under the action of a load are studied. W. t. theoretical. the basis of calculations for strength, deformability and stability in construction. deed… … Physical Encyclopedia

A branch of mechanics (See mechanics) that studies the displacements, strains, and stresses that occur in elastic bodies at rest or in motion under the action of a load. W. t. theoretical basis for calculating strength, deformability and ... ... Great Soviet Encyclopedia

Books

• strength and deformation. Applied Theory of Elasticity Volume 2, A. Feppl. FOREWORD BY THE EDITOR OF THE RUSSIAN TRANSLATION TO THE SECOND VOLUME. The publication of the second volume of the book by A. Feppl and L. Feppl was delayed so much that the initial assumptions about the placement of the row ...

All bodies near the Earth are affected by its attraction. Under the influence of gravity, raindrops, snowflakes, leaves torn off the branches fall to the Earth.

But when the same snow lies on the roof, it is still attracted by the Earth, but it does not fall through the roof, but remains at rest. What prevents it from falling? Roof. It acts on snow with a force equal to gravity, but directed in the opposite direction. What is this power?

Figure 34, a shows a board lying on two stands. If a weight is placed in its middle, then under the influence of gravity the weight will begin to move, but after a while, having bent the board, it will stop (Fig. 34, b). In this case, the force of gravity will be balanced by the force acting on the weight from the side of the curved board and directed vertically upwards. This force is called elastic force. The elastic force arises during deformation. Deformation is a change in the shape or size of the body. One type of deformation is bending. The more the support bends, the greater the elastic force acting from this support on the body. Before the body (weight) was placed on the board, this force was absent. As the weight moved, which bent its support more and more, the elastic force also increased. At the moment the weight stops, the elastic force has reached the force of gravity and their resultant has become equal to zero.

If a sufficiently light object is placed on the support, then its deformation may turn out to be so insignificant that we will not notice any change in the shape of the support. But the deformation will still be! And along with it, the elastic force will also act, preventing the fall of the body located on this support. In such cases (when the deformation of the body is imperceptible and the change in the size of the support can be neglected), the elastic force is called support reaction force.

If some kind of suspension (thread, rope, wire, rod, etc.) is used instead of a support, then the object attached to it can also be held at rest. The force of gravity here will also be balanced by the oppositely directed force of elasticity. In this case, the elastic force arises due to the fact that the suspension is stretched under the action of the load attached to it. stretching another kind of distortion.

The elastic force also occurs when compression. It is she who makes the compressed spring straighten and push the body attached to it (see Fig. 27, b).

A great contribution to the study of the force of elasticity was made by the English scientist R. Hooke. In 1660, when he was 25 years old, he established a law that was later named after him. Hooke's law says:

The elastic force that occurs when a body is stretched or compressed is proportional to its elongation.

If the elongation of the body, i.e., the change in its length, is denoted by x, and the elastic force is denoted by F control, then Hooke's law can be given the following mathematical form:

F control \u003d kx,

where k is the proportionality factor, called rigidity body. Each body has its own rigidity. The greater the rigidity of a body (spring, wire, rod, etc.), the less it changes its length under the action of a given force.

The SI unit of stiffness is newton per meter(1 N/m).

Having done a series of experiments that confirmed this law, Hooke refused to publish it. Therefore, for a long time no one knew about his discovery. Even after 16 years, still not trusting his colleagues, Hooke in one of his books gave only an encrypted formulation (anagram) of his law. She looked

After waiting two years for competitors to claim their discoveries, he finally deciphered his law. The anagram was deciphered as follows:

ut tensio, sic vis

(which in Latin means: what is the tension, such is the force). “The strength of any spring,” Hooke wrote, “is proportional to its stretching.”

Hooke studied elastic deformations. This is the name of deformations that disappear after the cessation of external influence. If, for example, a spring is stretched a little and then released, it will return to its original shape. But the same spring can be stretched so much that, after it is released, it will remain stretched. Deformations that do not disappear after the cessation of external influence are called plastic.

Plastic deformations are used in modeling from plasticine and clay, in metal processing - forging, stamping, etc.

For plastic deformations, Hooke's law is not satisfied.

In ancient times, the elastic properties of some materials (in particular, a tree such as yew) allowed our ancestors to invent onion- a hand weapon designed to throw arrows with the help of the elastic force of a stretched bowstring.

Having appeared about 12 thousand years ago, the bow has existed for many centuries as the main weapon of almost all tribes and peoples of the world. Before the invention of firearms, the bow was the most effective combat weapon. English archers could shoot up to 14 arrows per minute, which, with the massive use of bows in battle, created a whole cloud of arrows. For example, the number of arrows fired at the Battle of Agincourt (during the Hundred Years' War) was approximately 6 million!

The widespread use of this formidable weapon in the Middle Ages caused a justified protest from certain circles of society. In 1139, the Lateran (Church) Council, which met in Rome, banned the use of these weapons against Christians. However, the struggle for "bow disarmament" was not successful, and the bow as a military weapon continued to be used by people for another five hundred years.

The improvement of the design of the bow and the creation of crossbows (crossbows) led to the fact that the arrows fired from them began to pierce any armor. But military science did not stand still. And in the XVII century. the bow was supplanted by firearms.

Nowadays, archery is just one of the sports.

1. In what cases does the elastic force arise? 2. What is called deformation? Give examples of deformations. 3. Formulate Hooke's law. 4. What is hardness? 5. How do elastic deformations differ from plastic ones?

In nature, everything is interconnected and continuously interacts with each other. Each of its parts, each of its components and elements is constantly exposed to a whole complex of forces.

Despite the fact that the number is quite large, they can all be divided into four types:

1. Forces of a gravitational nature.

2. Forces of an electromagnetic nature.

3. Forces of a strong type.

In physics there is such a thing as elastic deformation. Elastic deformation is a deformation phenomenon in which it disappears after external forces cease to act. After such a deformation, the body takes its original shape. Thus, the elastic force, the definition of which says that it occurs in the body after elastic deformation, is a potential force. A potential force, or conservative force, is a force in which its work cannot be dependent on its trajectory, but depends only on the initial and final points of application of forces. The work of a conservative or potential force along a closed path will be zero.

We can say that the elastic force has an electromagnetic nature. This force can be assessed as a macroscopic manifestation of the interaction between the molecules of a substance or body. In any case, in which either compression or stretching of the body occurs, an elastic force is manifested. It is directed against the force that produces the deformation, in the direction opposite to the displacement of the particles of the given body, and is perpendicular to the surface of the body undergoing deformation. Also, the vector of this force is directed in the direction opposite to the deformation of the body (displacement of its molecules).

The calculation of the value of the elastic force that occurs in the body during deformation occurs according to it. According to it, the elastic force is equal to the product of the rigidity of the body and the change in the deformation coefficient of this body. According to Hooke's law, the elastic force that occurs at a certain deformation of a body or substance is directly proportional to the elongation of this body, and it is directed in the direction opposite to the direction in which the particles of this body move relative to other particles at the moment of deformation.

The stiffness index of a certain body or proportional coefficient depends on the material used to make the body. Also, the rigidity depends on the geometric proportions and shape of the given body. In relation to the elastic force, there is also such a thing as such a stress is the ratio of the elastic modulus to the unit area at a given point of the section under consideration. If we associate Hooke's law with this type of voltage, then its formulation will sound somewhat different. The stress of a mechanical type that occurs in a body when it is deformed is always proportional to the relative elongation of this body. It must be borne in mind that the effect of Hooke's law is limited to only small deformations. There are strain limits under which this law operates. If they are exceeded, then the elastic force will be calculated using complex formulas, regardless of Hooke's law.

We continue the review of some topics from the "Mechanics" section. Our today's meeting is devoted to the force of elasticity.

It is this force that underlies the operation of mechanical watches, towing ropes and cables of cranes, shock absorbers of cars and trains are exposed to it. It is tested by a ball and a tennis ball, a racket and other sports equipment. How does this force arise, and what laws does it obey?

How is the force of elasticity born?

A meteorite under the influence of gravity falls to the ground and ... freezes. Why? Does the earth's gravity disappear? No. Power cannot just disappear. At the moment of contact with the ground balanced by another force equal to it in magnitude and opposite in direction. And the meteorite, like other bodies on the surface of the earth, remains at rest.

This balancing force is the elastic force.

The same elastic forces appear in the body for all types of deformation:

• stretching;
• compression;
• shear;
• bending;
• torsion.

Forces resulting from deformation are called elastic.

The nature of the elastic force

The mechanism of the emergence of elastic forces was explained only in the 20th century, when the nature of the forces of intermolecular interaction was established. Physicists have called them "giant with short arms." What is the meaning of this witty comparison?

Forces of attraction and repulsion act between molecules and atoms of matter. Such an interaction is due to the smallest particles that are part of them, carrying positive and negative charges. These powers are big enough.(hence the word giant), but appear only at very short distances.(with short arms). At distances equal to three times the diameter of the molecule, these particles are attracted, "joyfully" rushing towards each other.

But, having touched, they begin to actively repel each other.

With tensile deformation, the distance between molecules increases. Intermolecular forces tend to shorten it. When compressed, the molecules approach each other, which causes the molecules to repulse.

And, since all types of deformations can be reduced to compression and tension, the appearance of elastic forces for any deformations can be explained by these considerations.

Hooke's Law

A compatriot and contemporary studied the forces of elasticity and their relationship with other physical quantities. He is considered the founder of experimental physics.

Scientist continued his experiments for about 20 years. He conducted experiments on the deformation of the tension of springs by hanging various loads from them. The suspended load caused the spring to stretch until the elastic force that arose in it balanced the weight of the load.

As a result of numerous experiments, the scientist concludes: the applied external force causes the appearance of an elastic force equal to it in magnitude, acting in the opposite direction.

The law formulated by him (Hooke's law) is as follows:

The elastic force arising from the deformation of the body is directly proportional to the magnitude of the deformation and is directed in the direction opposite to the movement of particles.

The formula for Hooke's law is:

• F is the modulus, i.e. the numerical value of the elastic force;
• x - change in body length;
• k - coefficient of rigidity, depending on the shape, size and material of the body.

The minus sign indicates that the elastic force is directed in the direction opposite to the particle displacement.

Each physical law has its limits of application. The law established by Hooke can only be applied to elastic deformations, when, after the load is removed, the shape and dimensions of the body are completely restored.

In plastic bodies (plasticine, wet clay) such restoration does not occur.

All solids have elasticity to some degree. The first place in elasticity is occupied by rubber, the second -. Even very elastic materials under certain loads can exhibit plastic properties. This is used for the manufacture of wire, cutting out parts of complex shape with special stamps.

If you have a hand-held kitchen scale (steelyard), then the maximum weight for which they are designed is probably written on them. Let's say 2 kg. When hanging a heavier load, the steel spring inside them will never recover its shape.

The work of the elastic force

Like any force, the force of elasticity, able to do the job. And very useful. She is protects the deformable body from destruction. If she does not cope with this, the destruction of the body occurs. For example, a crane cable breaks, a string on a guitar, an elastic band on a slingshot, a spring on a scale. This work always has a minus sign, since the elastic force itself is also negative.

Instead of an afterword

Armed with some information about elastic forces and deformations, we can easily answer some questions. For example, why do large human bones have a tubular structure?

Bend a metal or wooden ruler. Its convex part will experience tensile deformation, and the concave part will experience compression. The middle part of the load does not carry. Nature took advantage of this circumstance, supplying man and animals with tubular bones. In the process of movement, bones, muscles and tendons experience all kinds of deformations. The tubular structure of the bones greatly facilitates their weight, without affecting their strength at all.

The stems of cereal crops have the same structure. Gusts of wind bend them to the ground, and elastic forces help to straighten up. By the way, the bicycle frame is also made of tubes, not rods: the weight is much less and the metal is saved.

The law established by Robert Hooke served as the basis for the creation of the theory of elasticity. Calculations performed according to the formulas of this theory allow ensure the durability of high-rise structures and other structures.

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