Study about the round springs with horizontal load by Transfer-Matrix Method.

Suciu, Mihaela ; Balc, Gavril ; Teberean, Ioan 等

Abstract: The Transfer-Matrix Method is a numerical method for the calculus in the strengths and stresses domain. We present a study for the calculus of the round spring, challenged with a concentrated horizontal load in its plane, using this method, the Transfer-Matrix Method. With the basic equations of the spring's theory and with the Transfer-Matrix for one spring challenged in its plane, we can write the general expression for the Transfer-Matrix for a round spring, with an application for a round spring challenged with a concentrated horizontal load in its plane, using the Dirac's and Heaviside's distribution functions and operators.

Key words: state vector, Transfer-Matrix Method, round spring, Dirac's function, Heaviside's function.

1. INTRODUCTION

Using the Transfer-Matrix Method, the study of the spring is very easy and interesting. With the basic equations of the spring theory and with Dirac's and Heaviside's functions and operators, we can calculate the six elements for the origin state vector and after, we can calculate, in all spring sections, the state vectors.

2. THE SPRING THEORY AND THE BASIC EQUATIONS OF THE SPRING'S THEORY

We have a round spring with a constant inertia. We have keeped the sign conventions (as well as to the beams) for the internal efforts, for the displacements and for the exterior loads. The displacements owing to the cutter force, is negleted face to the displacements due by the flexion moment (Fig. 1.), (Gery, Calgaro, 1973).

[FIGURE 1 OMITTED]

The basic equations of the spring theory, for an element dx of the spring and an angle e (Fig. 2.), by the relations (1):

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

3. THE SPRING CHALLENGED IN ITS PLANE WITH ITS TRANSFER-MATRIX

For the round spring, at the current point M, the state vector with six elements is [{U}.sub.x] (expression (2)):

[{U}.sub.x] = [{M([theta]), [T.sub.x0]([theta]), [omega]([theta]), v ([theta])}.sup.-1] (2)

For the section 0, we have the state vector [{U}.sub.x]. We can write a matrix relation between this state vector and the state vector (2):

[{U}.sub.x] = [[T].sub.x] [{U}.sub.0] + [{U.sub.e}.x] (3)

where: [[T].sub.x] is the Transfer-Matrix for the passage between the section 0 at the section x and [{U.sub.e}.x] is the state vector for the free term at the section x. More extended, we can write the expression (4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The general expression for the transfer-matrix [T]x of the spring between the origin and the current point M with the parameter [theta] is (5):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The total transfer-matrix for all sections of the spring is obtained if we give at the parameter e the correspondent value for the spring end.

For the round spring, the total Transfer-Matrix for all sections of the spring, is obtained if we give at the parameter e the correspondent value for the spring's end.

4. ROUND SPRING CHALLENGED IN ITS PLANE BY TRANSFER-MATRIX METHOD

We consider a circular spring (Fig. 3.), with a radius R, an angle 2[alpha] and, the origin of the axels system is O, same with the spring origin. The parameter equations are (6):

[FIGURE 3 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

W have the general Transfer-Matrix for the round spring for [theta]=2[alpha]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The vector [{U.sub.e}.x] of the relation (3) and (4) will be calculated function of the exterior density loads.

5. APPLICATION FOR A ROUND SPRING CHALLENGED WITH A CONCENTRATED HORIZONTAL LOAD IN ITS PLANE

We have studied an application for a round spring with a concentrated horizontal load F, challenged in its plane (Fig. 4.), 2[alpha]=1800, [theta]0=900.

[FIGURE 4 OMITTED]

With Dirac's and Heaviside's functions and operators, we can write for the load densities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

We change x by [theta] and we can write:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The others elements of the vector [{U.sub.e}.x] are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

For [theta]=2[alpha] we have the general vector of exterior vertical load F for the spring, [{U.sub.e}.x]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

and the matrix expression for the end O' is (12):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

In this moment, we pose the edge conditions: we have the left edge and the right edge embedded (Fig. 3.).

We pose the edge conditions: we have the left edge and the right edge embedded (Fig. 4.). The conditions at left edge, in the origin, for [theta]=0, are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13) and for [theta]-2[alpha], are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

We have a system with 6 equations and 6 variables. The results give the state vectors for the origin face O and for the end O' of the spring. With the conditions, (13) and (14), the expression (12) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

For [[theta].sub.0]=900 and 2[alpha]=1800, we obtain for (15):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

With the expression (16) we can write one system with 6 equations and 6 variables. After resolution, the results give the state vectors for the origin (face O) and for the end O' of the round spring.

6. CONCLUSIONS

The Transfer-Matrix Method is very easy to applied for the spring calculus. This calculus we can to program it. With this code, we can calculate rapidly, in the origin section and in the end section of the spring, the 12 elements of the two state vectors. With the results, we can calculate all the state vectors in all sections for the complete spring.

7. REFERENCES

Gery, M. & Calgaro, J.-A., (1973), Les Matrices-Transfer dans le calcul des structures, Editions Eyrolles, Paris

Tripa, M., (1967), Strength of Materials, EDP Bucharest