Analysis of dynamic conditions of contact of elements of mechanism with rotational cam and translational bar.

Voloder, Avdo ; Cohodar, Maida ; Karic, Seniha 等

Abstract: Mechanisms with rotational cam and rotational bar have big application as mechanisms for regulation of motion of other mechanisms, which transmit force. In target of regular work of this mechanism, contact between bar and cam must be provided, in other words, force between cam and bar must exist. For satisfaction this conditions, we often use elastic elements (springs). This paper presents analysis of influential parameters on this conditions.

Key words: rotational cam, translational bar, conditions of contact

1. INTRODUCTION

Owing to motion of elements of cam mechanisms, break of contact between cam and bar can be. In some last investigations this problem is analysed (Ozgul & Pasin, 1996, Wu & Angeles, 2001). Figure 1. presents mechanism with rotational cam (1) and translational bar (2), where cam has angular velocity [??] and angular acceleration [??]. Mechanism has spring (3) which is need for contact between cam and bar. In target analysis of conditions of contact between these elements it is need analyse dynamic conditions of motion. The target of this analysis is obtaining of term which shows influence of single parameters on these conditions.

2. ANALYSYS OF PROBLEM

Let we introduce next signs (fig. 1): [phi]-angle of rotation of cam; [tau], n-tangent and normal axis in point of contact of these two elements of mechanism; [??],[??]-perpendicular force and force of friction between cam and bar; [y.sub.0]-beginning displacing between pick of bar and horizontal axis which goes through centre of rotation of cam; s- working displacing of bar; [??]-velocity of pick of bar.

[FIGURE 1 OMITTED]

We put dynamic equation of rotation of bar (Hibbeler, 1995)

[m.sub.B] [d.sup.2]s/[dt.sup.2] = [SIGMA][F.sub.y], (1)

where: [m.sub.B] - mass of bar, t - time.

On other side, we can write

[d.sup.2]s/[dt.sup.2] = [d.sup.2]s/d[[phi].sup.2] + ds/d[phi] [epsilon] = s" x [[omega].sup.2] + s' x [epsilon]. (2)

According terms (1) and (2) we obtain

[m.sub.B] (s" x [[omega].sup.2] + s' x [epsilon]) = Ncos[alpha] - Tsin[alpha] - [F.sub.C], (3)

where: [alpha] - pressure angle of mechanism, [F.sub.C] - force of spring which is need for contact between cam and bar.

Pressure angle is defined as angle between normal on cam in point of tangent between cam and beam and velocity of pick of bar of mechanism. Absolute value of pressure angle of cams must be in definite borders, in target prevention of blockade of such mechanisms. On other side, with increase of absolute value of pressure angle we have decrease level of utility of mechanism. For mechanism with rotational cam and translational bar pressure angle has value (Voloder, et al., 2004)

[alpha] = arctg e+s'/[y.sub.0]+s = arctg e+s'/[square root of ([R.sup.2.sub.0]-[e.sup.2])+s, (4)

where is: e - eccentricity between cam and bar, [R.sub.0] - fundamental radius of cam.

Maximal permissible pressure angle is directly connected with characteristics of friction between bar and cam of mechanism. This angle in technical practice has value until 350 (Norton, 1992).

Force of spring has value

[F.sub.C] = [F.sub.C0] + cs, (5)

where: [F.sub.C0] - beginning force of spring, c - stiffness of spring. 0 C F

When

T = [mu]N (6)

where: [mu] - coefficient of friction between cam and bar, it is following

[m.sub.B](s" x [[omega].sup.2] + s' x [epsilon]) = N x (cos[alpha] - [mu]sin[alpha]) - ([F.sub.C0] + cs). (7)

According last term we obtain perpendicular force N

N = [m.sub.B](s" x [[omega].sup.2] + s' x [epsilon]) + [F.sub.C0] + cs / cos[alpha] - [mu]sin[alpha]. (8)

Condition that contact between cam and bar exists, is: N>0. From term (8) and last condition we obtain

[m.sub.B](s" x [[omega].sup.2] + s' x [epsilon]) + [F.sub.C0] + cs / cos[alpha]-[mu]sin[alpha] >0. (9)

With regard to maximal permissible pressure angle is [[alpha].sub.max] = [35.sup.0], and: [mu] [approximately equal to] 0,1 (for still at still), it is following

cos[alpha] - [mu]sin[alpha]>0, (10)

from (9) next term following

[m.sub.B](s" x [[omega].sup.2] + s' x [epsilon]) + [F.sub.C0] + cs>0. (11)

Last term presents fundamental condition of contact between cam and bar for mechanism with rotational cam and translational bar.

3. EXAMPLE

For mechanism with rotational cam and rotational bar, low of change of displacing of cam is (fig. 2):

in interval: 0 [less than or equal to] [phi] [less than or equal to] [pi]/4, s = 15 x (1 - cos4[phi]),

in interval: [pi]/4 [less than or equal to] [phi] [less than or equal to] [pi]/2, s = 30

in interval: [pi]/2 [less than or equal to] [phi] [less than or equal to] 3[pi]/4, s = 15 x (1 + cos4[phi])

where is: s - mm

It is need calculate beginning force of spring [F.sub.C0], necessary for contact of cam and bar, if next values are known: mass of bar: [m.sub.B] = 0,2 kg; number of revolutions of cam in minute: n = 550 r/min (= const.); stiffness of spring: c = 3000 N/m.

[FIGURE 2 OMITTED]

Solution:

Second derivation of displacing of bar by angle of rotation of cam is

in interval: 0 [less than or equal to] [phi] [less than or equal to] [pi]/4, s" = 240 x cos4[phi],

in interval: [pi]/4 [less than or equal to] [phi] [less than or equal to] [pi]/2, s" = 0,

in interval: [pi]/2 [less than or equal to] [phi] [less than or equal to] 3[pi]/4, s" = -240 x cos4[phi].

Figure 3. presents second derivation of displacing of bar by angle of rotation of cam.

Angular velocity of bar is

[omega] = [pi]n/30 = [pi] x 550/30 = 57,5958 [s.sup.-1].

According term (11) and when [omega] = const. we obtain

[([F.sub.C0]).sub.min] = [(-[m.sub.B]s" x [[omega].sup.2] - cs).sub.max].

We observe function

[xi]([phi]) = -[m.sub.B]s" x [[omega].sup.2] - cs,

which for our example has form

in interval: 0 [less than or equal to] [phi] [less than or equal to] [pi]/4:

[xi]([phi]) = -0,2 x 0,24 x cos4[phi] x [57,5958.sup.2] - 3000 x 0,015(1-cos4[phi])

[FIGURE 3 OMITTED]

[xi]([phi]) = -114,2292 x cos4[phi] - 45 N,

in interval: [pi]/4 [less than or equal to] [phi] [less than or equal to] [pi]/2:

[xi]([phi]) = -90 N,

in interval: [pi]/2 [less than or equal to] [phi] [less than or equal to] 3[pi]/4:

[xi]([phi]) = 0,2 x 0,24 x cos4[phi] x [57,5958.sup.2] - 3000 x 0,015(1 + cos4[phi]), [xi]([phi]) = 114,2292 x cos4[phi] - 45 N.

Maximal value of function [xi]([phi]) is:

[[[xi]([phi])].sub.max] = 69,2292 N.

Consequently, minimal value of beginning force of spring [F.sub.C0], necessary for contact of cam and bar, is

[([F.sub.0C]).sub.min]. = 69,2292 N.

4. CONCLUSION

In target of determination of conditions of contact between cam and bar of mechanism with rotational bar and translational bar, equation which describe these conditions is derived. These conditions are depend of: lows of motion of cam, first and second derivation of displacing of bar by angle rotation of cam and of stiffness of spring which is need for contact between cam and bar. In target illustration this problem, one example is shown.

5. REFERENCES

Hibbeler, R.C. (1995). Engineering Mechanics--Static and Dynamics, Prentice-Hall, Englewood Clifts, ISBN 0-02-354761-8, New Jersey

Norton, R. L. (1992). Design of machinery, McGraw-Hill, Inc. Company, ISBN 0-07-909702-2, New York

Ozgur, K. & Pasin, F. (1996). Separation phenomena in force closed cam mechanisms, Mechanism and Machine Theory Vol. 31. No. 4, pp. 487-499, 1996, Elsevier Science Ltd.

Voloder, A., Kulenovic, M. & Cohodar M. (2005). The Influence of single Parameters of Mechanisms with Rotational Cam and Translational Bar on Pressure Angle, Proceedings of 15 th International DAAAM Symposium, Katalinic, B. (Ed.), pp. 481-482, ISBN 3-901509-42-9, Vienna, November 3-6, 2004. DAAAM International, Vienna

Wu, C.J. & Angeles, J. (2001). The optimum synthesis of an elastic torque-compensating cam mechanism, Mechanism and Machine Theory 36 (2001) pp. 245-259, www.elsevier.com/locate/mechmet