About spatial rigid memory characteristics.

Comanescu, Adriana ; Comanescu, Dinu ; Filipoiu, Ioan Dan 等

1. INTRODUCTION

The rigid memories materialized by planar or spatial cams are the main kinematical elements of various mechanisms applied in industrial equipments (food processing industry, textile industry, manufacturing processes, etc.) and robotics (Antonescu, 2005; Ivanescu, 2007).

The problem of cam planar mechanisms synthesis is comprehensively presented in the literature (Bishop, 2002; Hernandez et al., 2002). This problem is the only approach in some particular cases concerning cylindrical or truncated conic cams (Bloch 2000). This method presume that the spatial cams profile parameters can be determined for various motion laws of a follower which is in contact with the lateral or frontal spatial cam surface (Fig.1). After the calculation of the parameters, by using parametric CAD software, can be design cylindrical, conical and hyperbolical cams. The analytical synthesis of the spatial cam profile, presented in this paper, connected to adequate CAD software is firstly presented in the specialty literature.

2. CYLINDRICAL CAMS SYNTHESIS

In order to determine the line profile parameters of the cylindrical cam it is necessary to choose the following data: T--cam rotation period, r--cam cylinder radius, LV = p - follower displacement amplitude and motion law for the follower.

By developing the cam cylinder a rectangle is obtain, with the length LH = 2[pi]r and the height equal to LV to which a xOy reference system is attached. The current point of the profile line is A([x.sub.Ak], [y.sub.Ak] In this reference system its coordinates depends on the law for the follower, their graphic representations being given in Table 1.

[FIGURE 1 OMITTED]

[TABLE 1 OMITTED]

The spatial curve has the following parameter equations

[x.sub.k] = rcos([[psi].sub.k]) (1)

[y.sub.k] = rsin([[psi].sub.k]) (2)

[z.sub.Ak] = [y.sub.Ak] (3)

where [[psi].sub.k] = [X.sub.Ak] / r.

The cylindrical cams model may be achieved by using adequate design software such as AutoCad, Adams or SolidWorks. The parameters of the spatial curves profile are calculated. By applying a parametrization design, using CAD software, the design of such cams can be obtain, which are presented in Fig.2.

[FIGURE 2 OMITTED]

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3. TRUNCATED CONIC CAMS SYNTHESIS

The follower motion law on the Oz axis is imposed by the technological process. The characteristics to the truncated cone are presented in Fig.3

The truncated cone has the bases radius respectively R and r. The basis circle length is denoted by LH. The follower motion amplitude on Oz is LV. The current point coordinates of the curve placed on the truncated conical surface, A(x,y,[z.sub.A]) may be deduced using the Fig.3 and the following relations:

[k.sub.1] = LH / T (4)

[k.sub.2] = 2 LV / T (5)

x = atan(R-r / LV) (6)

H = R / tan(x) (7)

For example, by adopting the following values for T = 2 s,R = 0,15m , r = 0.08m,LV = 0.8m there are determined the coordinates of the spatial curve current point for different motion laws.

These values were used in the modelling process using CAD software and the models are presented in Fig.4.

4. HYPERBOLOID CAMS SYNTHESIS

In the case of the hyperboloid cams the follower extremity is in contact with the curve described on a hyperboloid.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The hyperboloid surface (Fig.5) may be obtained by rotating a hyperbole around the Oz axis.

The hyperbole equation in the yOz reference system is

[y.sup.2] / [a.sup.2] - [z.sup.2] / [b.sup.2] (8)

where [b.sup.2] = [c.sup.2] - [a.sup.2] and c > a .

The current point of the spatial curve placed on the hyperboloid is A([x.sub.k], [y.sub.k], [z.sub.Ak]). [Z.sub.Ak] is determined in function of the selected motion law. The period of a complete rotation is T. R and r are respectively the maximum radius and the minimum one for the hyperboloid and L.V = p is the displacement amplitude on the Oz. For km time equal intervals can be determine [t.sub.k] = kT / km, where k [member of] [0, km] . At [T.sub.k] time, the rotation angle is [[THETA].sub.k] - [t.sub.k] x [omega] (the cam angular velocity [omega] = 2[pi] / T).

The [[rho].sub.k] it current radius is given by:

[[rho].sub.k] = [square root of [y.sup.2.sub.k]] (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], thus

[x.sub.k] = [[rho].sub.k] cos ([theta].sub.k]) (10)

[y.sub.k] = [[rho].sub.k] cos ([theta].sub.k]) (11)

By using the advanced design techniques were obtained the hyperboloid cams presented in Fig. 6.

5. CONCLUSIONS

For rigid memories defined by spatial cams it is presented an adequate synthesis method.

The spatial parameters of the profile line may be determined by using calculus software and are introduced in the CAD algorithm for modelling. This will make the manufacturing easier because this can be done using CAM.

The future researches include other applications for spatial cams in function of the follower motion laws.

6. REFERENCES

Antonescu, P. (2005). Mechanism and Machine Science, Edit.PRINTECH. ISBN: 9737182916, Bucharest

Bishop, R.H. (2002). The mechatronics handbook, CRC Press. ISBN: 0849300665

Bloch, A.M. (2000). Nonholonomic mechanics and control, Springer. ISBN: 0387955356

Hernandez, A.; Pinto, C.; Petuya, V. & Agirrebeitia, J. ( 2002). Mechanism theory, Serv. Publicaciones de la E. T. S. Ingenieros de Bilbao, ISBN: 8495809117, Bilbao

Ivanescu, M. (2007). From classical to modern mechanical engineering fundamentals, Edit. Acad. Romane, ISBN: 9789732715611, Bucharest