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PLANE COLLISION WITH FRICTION

Pandrea Marina1, Pandrea Nicolae1,

1 University of Piteşti, Piteşti, ROMANIA

Abstract: In the paper it studies the plane collision with friction of two solids. In this sense it writes the equations that results from general theorems in the common reference system defined by common normal and tangential. It is used the notion of inertances, it writes the relatives velocities and it makes the final calculus for different possible cases.

Keywords: inertances, restitution coefficient, percussion.

1. INTRODUCTION

Plane collision with friction of two free shells was studied by known authors in references [1], [2], [7], [8].

In their papers they consider that the friction percussion acts on the common tangent, it is in opposite direction to the relative tangential velocity and its intensity is proportional with that of the normal percussion, the proportionality factor being a coefficient analog to the coulombian friction coefficient.

We write the equations which result from the general theorems at collisions and by exchanging the frame we rewrite these equations in the reference frame in the contact point, this frame having the axis situated on the common normal, respectively tangent.

In the situation when the tangential speed doesn’t change the sign, the solving of the equation system has no difficulties obtaining the same results both for the Newton model when the restitution coefficient is defined by:

/ (1)

where are the relative initial and final speeds, and for the Poisson model when the restitution coefficient is defined by the relation

/ (2)

where represent the percussions in the compression respectively relaxing phase.

In the case in which the relative tangential speed becomes zero different situations appear name them:

- if by changing the sign of the coefficient the speed changes its sign, then one solves the system of equations and obtains the final results which differs from Newton respectively Poisson model;

- if by changing the sign of the coefficient  the speed doesn’t change its sign, then one considers a tangential “joint” and solves the system of equation obtaining also different results for the two methods.

In the mentioned references are mentioned situations when using Newton model one reaches non-concordances when the kinetic energy’s losing becomes negative and concludes that the Poisson model is the preferred one.

It is even proposed a new model in which the restitution coefficient is defined by the ratio between the variation of the kinetic energy in the relaxing phase and the variation of the kinetic energy in the compression phase.

Mathematical models elaborated in the mentioned papers are laborious with huge calculus relations which can offer results only by numerical calculus.

These aspects can be partially solved [4], [5] introducing the notions of inertances, as we shall proceed in he present paper.

2. REFERENCE SYSTEM. NOTATIONS

Let us consider two plan shells moving in he common plan (fig.1.a), they colliding in point A, the versor of the common normal being , and the normal percussion being P (fig. 1.b).

a) / b)

Figure 1. Mathematical model.

We select the versor of the common tangent in the sense of the initial tangent velocity, and in these conditions the intensities of the initial relative speeds , are positive.

Further on, one considers the notations:

- , the systems of the principal central inertial axes;

- , , coordinates of the point A in the two reference frames;

- , , , , projections of the versors , in the reference system ;

- , , parameters defined by the relations:

;

- , , , the speeds of the points before and after the collision;

- , , , the speeds of the points belonging to the two shells before and after the collision;

- , , , the angular speeds before and after the collision;

- , , , , , projections of the velocities , respective on the axes , ;

- , , , , , projections of the vectors , on the directions of the vectors , ;

- , the intensity of the percussion;

- , , masses of the two bodies;

- , , inertial moments of the two shells with respect to the points ;

- , tangential percussion,

where is a coefficient which is determined experimentally;

- , , , , the magnitudes named inertances defined by the relations:

; ; , / (3)
; ; ; / (4)

- , the relative speeds defined by the relations:

; / (5)

3. GENERAL CALCULUS RELATION

The general theorems for the two shells lead to the equations:

; ; ; / (6)

We multiply the relations (6) respectively by , , and than by , , , keeping into account the equalities:

/ (7)

, ,

one deduces by summing the equalities:

/ (8)
/ (9)

and than the expressions:

/ (10)
/ (11)

Percussion for which is denoted by and it has the expression:

/ (12)

and for this value of the percussion corresponds the normal relative speed:

/ (13)

If the relative tangential speed remains positive then, applying the Newton model one deduces from (10) the final percussion,

/ (14)

In this case the compression percussion is deduced from (10) for i.e:

/ (15)

and the relaxing percussion becomes

/ (16)

and results (in this case) that the Newton model is identical to the Poisson model:

/ (17)

The mathematical conditions to realize this case are either

or (18)

The interesting cases are realized when the relative tangential speeds becomes zero, respectively when:

(19)

4. ANALYSIS OF DIFFERENT CASES

4.1Case when are fulfilled the conditions

: ; / (20)

Corresponding o these conditions, relations (10), (11) become:

/ (21)
/ (22)

and the expressions from table 1 are deduced.

Table 1

Model Newton / Model Poisson
/ / ,


/



4.2Case when are fulfilled the conditions

: ; / (23)

In the mentioned conditions, the relations (22) become:

/ (24)

The second relation (24) proves that for the relative tangential speed doesn’t change its sign and consequently the sign of  must not be changed. In this situation how results hat and therefore the relative tangential speed becomes negative. Avoiding his paradoxical situation can be made only by accepting the solution that for the relative tangential speed is zero, i.e. the joint phenomenon takes place in the tangent direction and consequently the tangential percussion is zero.

As a result, the relations (21), (22) become:

/ (25)
/ (26)

and the equation from the table 2 are deduced.

Table 2

Model Newton / Model Poisson
/ / ,



/



/ / ;



/


4.3Case when are fulfilled the conditions

: ; / (27)

The last two inequalities (27) lead, basing the previous deduction to the conclusion that for the tangential joint takes place and consequently the same relations as in table 2 are produced.

5. NUMERICAL APPLICATION

We consider known:

; ; ; ; ;;

Results successively:

; ;; ;

and in conclusion we obtain the situation from the first line in he table 1.

For the Newton model one obtains the results:

; ; ; ; ; .

If we consider for the model Poisson one obtains the results:

; ; ; ; .

REFERENCES

[1]Brach R. M.: Rigid body collisions. ASME. Journal Applied Mechanics, vol. 56, 1989.

[2]Keller J.B.: Impact with friction. ASME. Journal Applied Mechanics, vol. 93, 1986.

[3]Pandrea N.: Asupra ciocnirii solidelor. St. Cerc. Mec. Apl. Tom 36, nr. 6, 1987.

[4]Pandrea N., Pandrea Marina, Stanescu N.D.: Model mathematic al ciocnirii cu frecare a solidului. Bul. Univ. Petrol-Gaze Ploieşti, vol LV, nr. 4, 2003.

[5]Pandrea N.: About collision of two solids with constraints. Rev. Roum. Sci. Techn. Mec. Appl. Tom 49, nr.1-6, 2004.

[6]Smicală I., Smicală S.C.: Consideraţii privind ciocnirea instantanee a două corpuri ţinând seama de efectul percutant al forţelor de frecare. Tenth Conf. Mech. Vibr. Tom 47, Timişoara, 2002,

[7]Stronge W., J.: Rigid body collisions with friction. Proc. Royal Soc. London, A.431, 1990.

[8]Wang Y., Mason M.T.: Two-dimensional rigid-body collisions with friction. ASME. Jounal Appl. Mech. vol. 59, 1992.

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