To determine stability and resistance to external loads, a parameter such as spring stiffness is used. It is also called Hooke's coefficient or elasticity coefficient. In fact, the spring stiffness characteristic determines the degree of its reliability and depends on the material used in production.
The following types of springs are subject to measurement of the stiffness coefficient:
- Compression;
- Sprains;
- Bending;
- Torsion.
Manufacturing of springs of any type you.
What is the spring stiffness?
When choosing ready-made springs, for example for a car suspension, you can determine what stiffness it has by the product code or by the markings that are applied with paint. In other cases, stiffness calculations are made exclusively by experimental methods.
The stiffness of a spring in relation to deformation can be variable or constant. Products whose rigidity remains unchanged during deformation are called linear. And those that have a dependence of the stiffness coefficient on changes in the position of the turns are called “progressive”.
In the automotive industry, with regard to suspension, there is the following classification of spring stiffness:
- Increasing (progressive). Characteristic of a more rigid ride of the car.
- Decreasing (regressive) stiffness. On the contrary, it ensures “softness” of the suspension.
Determination of the stiffness value depends on the following initial data:
- Type of raw materials used in production;
- Diameter of metal wire turns (Dw);
- Spring diameter (the average value is taken into account) (Dm);
- Number of spring turns (Na).
How to calculate spring stiffness
To calculate the stiffness coefficient, the formula is used:
k = G * (Dw)^4 / 8 * Na * (Dm)^3,
where G is the shear modulus. This value can not be calculated, since it is given in tables for various materials. For example, for ordinary steel it is 80 GPa, for spring steel it is 78.5 GPa. From the formula it is clear that the remaining three quantities have the greatest influence on the spring stiffness coefficient: the diameter and number of turns, as well as the diameter of the spring itself. To achieve the required rigidity indicators, it is these characteristics that must be changed.
You can calculate the stiffness coefficient experimentally using the simplest tools: the spring itself, a ruler and a load that will act on the prototype.
Determination of tensile stiffness coefficient
To determine the tensile stiffness coefficient, the following calculations are performed.
- The length of the spring in a vertical suspension with one free side of the product is measured - L1;
- The length of the spring with a suspended load is measured - L2. If you take a load weighing 100g, then it will act with a force of 1N (Newton) - value F;
- The difference between the last and first length indicators is calculated - L;
- The elasticity coefficient is calculated using the formula: k = F/L.
The compression stiffness coefficient is determined using the same formula. Only instead of hanging, the load is installed on the top of a vertically mounted spring.
To summarize, we conclude that the spring stiffness indicator is one of the essential characteristics of the product, which indicates the quality of the source material and determines the durability of the final product.
Laboratory work No. 1.
Study of the dependence of body rigidity on its size.
Goal of the work: Using the dependence of the elastic force on the absolute elongation, calculate the stiffness of springs of different lengths.
Equipment: tripod, ruler, spring, weights weighing 100 g.
Theory. Deformation is understood as a change in the volume or shape of a body under the influence of external forces.When the distance between particles of a substance (atoms, molecules, ions) changes, the forces of interaction between them change. As the distance increases, the attractive forces of burning increase, and as the distance decreases, the repulsive forces increase. who strive to return the body to its original state. Therefore, elastic forces are of an electromagnetic nature. The elastic force is always directed towards the equilibrium position and tends to return the body to its original state. The elastic force is directly proportional to the absolute elongation of the body: .
Hooke's Law: The elastic force that arises during deformation of a body is directly proportional to its elongation (compression) and is directed opposite to the movement of body particles during deformation,, x = Δ l - lengthening of the body, k hardness coefficient[k] = N/m. The stiffness coefficient depends on the shape and size of the body, as well as on the material. It is numerically equal to the elastic force when the body is elongated (compressed) by 1 m.
Graph of the projection of the elastic force F x from lengthening the body.
From the graph it is clear that tgα = k. It is by this formula that you will determine the rigidity of the body in this laboratory work.
The order of work.
1.Fix the spring in the tripod to half its length.
2.Measure the original length of the spring with a ruler l 0 .
3.Hang a load weighing 100g.
4.Measure the length of the deformed spring with a ruler l.
5.Calculate the elongation of the spring x 1 = Δ l = l l 0 .
6. A load at rest relative to a spring is acted upon by two
forces compensating for each other: gravity and elasticity
7.Calculate the elastic force using the formula, g = 9.8 m/s 2 - free fall acceleration
8. Hang a load weighing 200g and repeat the experiment according to steps 4-6.
9. Enter the results into the table.
Table.
|
No. |
Initial length, m |
Final length, m |
Absolute elongation |
Elastic force |
Hardness, tgα =k, N/m |
10. Select a coordinate system and constructgraph of the projection of the elastic force F control from spring extension.
11. Using a protractor, measure the angle between the straight line and the abscissa axis.
12.Use the table to find the tangent of the angle.
13.Draw a conclusion about the value of rigidity to 1 and enter the result into the table.
14.Fix the spring in the tripod to its full length and repeat the experiment point by point 4-13.
15.Compare values k 1 and k 2 .
16.Draw a conclusion about the dependence of stiffness on spring parameters.
TO test questions.
1. The figure shows a graph of the dependence of the modulus of elastic force on the elongation of the spring. Using Hooke's law, determine the spring stiffness.
Indicate the physical meaning of the tangent of the angle between the straight line and the abscissa axis, the area of the triangle under the section OA of the graph.
2. A spring with a stiffness of 200 N/m was cut into 2 equal parts. What is the stiffness of each spring.
3.Indicate the points of application of the elastic force of the spring, gravity and the weight of the load.
4.Name the nature of the elastic force of the spring, gravity and the weight of the load.
5. Solve the problem. To stretch the spring by 4 mm, 0.02 J of work must be done. How much work must be done to stretch the spring by 4 cm?
When exposed to external forces, bodies are capable of acquiring acceleration or deformation. Deformation is a change in the size and (or) shape of a body. If, after removing the external load, the body restores its size and shape completely, then such deformation is called elastic.
Let the spring in Fig. 1 be acted upon by a tensile force directed vertically downward.
When exposed to a deforming force ($\overline(F)$), the length of the spring increases. An elastic force ($(\overline(F))_u$) arises in the spring, which balances the deforming force. If the deformation is small and elastic, then the elongation of the spring ($\Delta l$) is proportional to the deforming force:
\[\overline(F)=k\Delta l\left(1\right),\]
where the proportionality coefficient is the spring stiffness $k$. The coefficient $k$ is also called the elasticity coefficient, the stiffness coefficient. Stiffness (as a property) characterizes the elastic properties of a body subjected to deformation - this is the body’s ability to resist external force and maintain its geometric parameters. The stiffness coefficient is the main characteristic of stiffness.
The spring stiffness coefficient depends on the material from which the spring is made and its geometric characteristics. Thus, the stiffness coefficient of a twisted cylindrical spring, which is wound from round wire, subjected to elastic deformation along its axis, is calculated using the formula:
where $G$ is the shear modulus (a value depending on the material); $d$ - wire diameter; $d_p$ - spring coil diameter; $n$ - number of spring turns.
Spring stiffness units
The International System of Units (SI) unit for stiffness is newton divided by meter:
\[\left=\left[\frac(F_(upr\ ))(x)\right]=\frac(\left)(\left)=\frac(N)(m).\]
The stiffness coefficient is equal to the amount of force that must be applied to the spring to change its length per unit distance.
Spring connection stiffness
When connecting $N$ springs in series, the stiffness of the connection is calculated using the formula:
\[\frac(1)(k)=\frac(1)(k_1)+\frac(1)(k_2)+\dots =\sum\limits^N_(\ i=1)(\frac(1) (k_i)\left(2\right).)\]
If the springs are connected in parallel, then the resulting stiffness is:
Examples of problems on spring stiffness
Example 1
Exercise. What is the potential energy ($E_p$) of deformation of a system of two parallel-connected springs (Fig. 2), if their stiffnesses are equal: $k_1=1000\ \frac(N)(m)$; $k_2=4000\ \frac(N)(m)$, and the elongation is $\Delta l=0.01$ m.
Solution. When connecting springs in parallel, we calculate the stiffness of the system as:
We calculate the potential energy of the deformed system using the formula:
Let's calculate the required potential energy:
Answer.$E_p=0,\ 25$ J
Example 2
Exercise. What is the work ($A$) of the force tensing a system of two springs connected in series with stiffnesses $k_1=1000\ \frac(N)(m)\ \ and$ $k_2=2000\ \frac(N)(m)$ , if the elongation of the second spring is $\Delta l_2=0.\ 1\ m$?
Solution. Let's make a drawing.
When springs are connected in series, each of them is subject to the same deforming force ($\overline(F)$), using this fact and Hooke’s law, we will find the elongation of the first spring:
The work done by the elastic force when stretching the first spring is equal to:
Taking into account the elongation of the first spring obtained in (2.1), we have:
Work of the second elastic force:
The work done by the force that stretches the spring system as a whole will be found as:
Substituting the right-hand sides of expressions (2.3) and (2.4) into formula (2.5), we obtain:
Let's calculate the work:
\[A=\frac(2000\cdot (((10)^(-1)))^2)(2\cdot 1000)\left(2000+1000\right)=30\ \left(J\right) .\]
Answer.$A$=30 J
Forceelasticity- this is the power which occurs when the body is deformed and which seeks to restore the previous shape and size of the body.
The elastic force arises as a result of electromagnetic interaction between the molecules and atoms of a substance.
The simplest version of deformation can be considered using the example of compression and extension of a spring.
In this picture (x>0) — tensile deformation; (x< 0) — compression deformation. (Fx) - external force.
In the case when the deformation is the most insignificant, i.e. small, the elastic force is directed in the direction that is opposite to the direction of the moving particles of the body and is proportional to the deformation of the body:
Fx = Fcontrol = - kx
Using this relationship, Hooke's law, which was established experimentally, is expressed. Coefficient k is commonly called body rigidity. The stiffness of a body is measured in newtons per meter (N/m) and depends on the size and shape of the body, as well as on the materials from which the body is composed.
In physics, Hooke's law for determining the compression or tension deformation of a body is written in a completely different form. In this case, the relative deformation is called
Robert Hooke
(18.07.1635 - 03.03.1703)
English naturalist, encyclopedist
attitude ε = x/l . At the same time, stress is the cross-sectional area of a body after relative deformation:
σ = F / S = -Fcontrol / S
In this case, Hooke’s law is formulated as follows: the stress σ is proportional to the relative deformation ε . In this formula the coefficient E called Young's modulus. This module does not depend on the shape of the body and its dimensions, but at the same time, it directly depends on the properties of the materials from which the body consists. For various materials, Young's modulus fluctuates over a fairly wide range. For example, for rubber E ≈ 2·106 N/m2, and for steel E ≈ 2·1011 N/m2 (i.e. five orders of magnitude more).
It is quite possible to generalize Hooke's law in cases where more complex deformations occur. For example, consider bending deformation. Let's consider a rod that rests on two supports and has a significant deflection.
From the side of the support (or suspension), an elastic force acts on this body; this is the support reaction force. The reaction force of the support when the bodies come into contact will be directed strictly perpendicular to the contact surface. This force is usually called the normal pressure force.
Let's consider the second option. The body lies on a stationary horizontal table. Then the reaction of the support balances the force of gravity and it is directed vertically upward. Moreover, body weight is considered the force with which the body acts on the table.
RIGIDITY
RIGIDITY
A measure of the compliance of a body to deformation under a given type of load: the more fluid, the less. In the strength of materials and the theory of elasticity, liquid is characterized by a coefficient (or total internal force) and a characteristic deformation of the elastic solid. bodies. In the case of tension-compression of the rod, it is called. coefficient ES in the ratio e=P/(ES) between the tensile (compressive) force P and relative. elongation k of the rod (5 - cross-sectional area, E - Young's modulus, (see ELASTIC MODULES). When a round rod is torsionally deformed, the value GIр is called, included in the ratio q = M/GIp, where G is the shear modulus, Iр - polar section, M - torque, q - relative angle of twist of the rod. When bending a beam, EI enters into the ratio c = M/E1 between the bending moment M (moment of normal stress in the cross section) and the curvature c of the curved axis of the beam (/ is the axial moment of inertia of the cross section).In the theory of plates and shells, the concept of cylindrical liquid is used: D = Eh3 12(1-v2), where h is the thickness (of the shell), v is the Poisson coefficient, liquid is also determined for some complex structures.
Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .
RIGIDITY
The ability of a body or structure to resist formation deformations. If the material obeys Hooke's law then the characteristics of J. are elastic moduli E - under tension, compression, bending and G- when shifting. ES in relation e= F/ES between tensile (compressive) force F and relates. elongation e of a rod with cross-sectional area S. When a rod of circular cross-section is torsioned, the liquid is characterized by the value GI p(Where Ip- polar moment of inertia of the section) in the ratio q=M/GI p, between the torque M and relates. angle of twist of the rod q. When bending a beam, the value is equal to EI, is included in the ratio ( =M/EI between bending moment M(moment of normal stresses in the cross section) and the curvature of the curved axis of the beam (,(where I- axial moment of inertia of the cross section), and when bending plates and shells, fluid is understood as a value equal to Eh 3 /12(l - n 2), where h is the thickness of the plate (shell), n is the coefficient. Poisson. AND. has creatures. value when calculating structures for stability.
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .
Synonyms:
Antonyms:
See what “HARDNESS” is in other dictionaries:
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radiation hardness- water hardness - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics energy in general Synonyms water hardness EN radiation hardnesshardnessHh ...
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A set of properties determined by the content of Ca2+ and Mg2+ ions in water. The total concentration of Ca2+ ions (calcium fluid) and Mg2+ (magnesium fluid) is called total fluid. There are Zh. v. carbonate and non-carbonate. Carbonate liquid..... ... Great Soviet Encyclopedia
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HARDNESS, rigidity, plural. no, female (book). distracted noun to hard. Rigidity of character. Excessive hardness of water makes it unfit for drinking. Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 … Ushakov's Explanatory Dictionary
