# About plane plates in rotation motion.

1. INTRODUCTION

In many technical problems, where appear plane plates having rotation motion, the solving of the problem of determination of the reactions is simple if is known the position of the support of resultant vector of d'Alembert's fictitious forces system. The determination of the central axis is some time difficult.

Also, in problems where appear plates and bars having rotation motion, the centrifugal moments are necessary when it is applied the theorem of the angular momentum. The determination of the centrifugal moments can be some time laborious.

In the technical literature, the values of the centrifugal moments are calculated by the "classical" method, by integration (Teodorescu, 2007; Hibbeler, 2009), or by studying the vibrations of the system (Belyakov & Seiranyan, 2008). It don't exist studies for finding out a formula for the calculus of the centrifugal moments.

In this paper are proposed a rule for the determination of the support of the resultant vector of the d'Alembert's fictitious forces for plates having uniform rotation motion and a formula for calculus of the centrifugal moment for plane plates.

2. THE SUPPORT OF THE RESULTANT VECTOR OF THE D'ALEMBERT FICTITIOUS FORCES SYSTEM

Let us consider a homogeneous plane plate (Figure 1a). This plate rotates around of a vertical axis which is identical with a straight leg. The axis is contained in the plane of the plate. We know also OB = [h.sub.1] and OA = [h.sub.2]. We relate the plate to a Cartesian reference system (Figure 1a) so that the center of mass C to be situated on the Ox axis, C([zi], 0,0), and the Oz axis to be rotation axis.

[FIGURE 1 OMITTED]

For a random point P, with the mass mP, the d'Alembert's fictitious force is ([epsilon] = [??]) (Figure 1b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

The angle between the d'Alembert's fictitious force and a line parallel with the Ox axis is given by:

tg[phi] = [absolute value of [[??].sup.in[tau]]/[absolute value of [[??].sup.inv]] = [epsilon]/[[omega].sup.2]. (2)

This value doesn't depend of the position of the point P. It follows that the system of d'Alembert's fictitious forces is a parallel forces system and can be replaced by the resultant vector. The resultant vector of d'Alembert's fictitious forces system is (Figure 1c):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where m is the mass of the plate. The resultant vector is applied in the center of the parallel forces, denoted here T. The nonzero coordinates of the point T are given by the relations (Voinea et al., 1989):

[x.sub.T] = [integral] xd[F.sup.in]/[integral] d[F.sup.in]; [z.sub.T] = [integral] zd[F.sup.in]/[integral] d [F.sup.in]. (4)

Because the plate is entirely situated on the same side of the rotation axis, all the d'Alembert's fictitious forces have the same sign. It follows that the second expression (4) will be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [z.sub.rC] is the coordinate of the center of mass of the rotation body generated by plate by rotation (Figure 1d).

If the plane plate has uniform rotation motion ([epsilon] = 0), the resultant vector of the d'Alembert's fictitious forces system will be parallel with the Ox axis. In this case the rule is: for plane plate in rotation motion around an axis from its plane, the support of the resultant vector of the d'Alembert's fictitious forces system is perpendicular on the rotation axis and passes by the center of mass of the rotation body generated by plate by rotation. So, for the determination of the reactions it is necessary only the coordinate [z.sub.T] = [z.sub.rC].

[FIGURE 2 OMITTED]

For example, for the plane plate from the figure 2, having uniform rotation motion, the determination of the support of the resultant vector of the d'Alembert's fictitious forces system, by applying the relation (5) is very simple: by rotation, the plate generates a semi sphere. The centre of mass is on the rotation axis at the distance 3R/8 from the center of sphere. In this point the support of the resultant vector of the d'Alembert's fictitious forces system crosses the rotation axis.

The "classical" method, by integration, is laborious.

3. CENTRIFUGAL MOMENTS FOR PLATES

Let us consider a homogeneous plane plate from the figure 1. We apply the theorem of the angular momentum with respect to the point 0. We obtain the equations (Voinea et al., 1989):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where:

--[J.sub.xz] is the centrifugal moment with respect to the Ox and Oz axes;

--[J.sub.z] is the moment of inertia with respect to the Oz axis;

--[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the moment of the external forces with respect to the point O.

Let us apply now the d'Alembert's principle. Keeping account of the relation (3) we obtain (Radoi & Deciu, 1993):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Comparing the equations (6) with (7) and keeping account of the relation (5) it results the follow expression for the centrifugal moment:

[J.sub.xz] = m[xi][z.sup.rC]. (8)

So, the centrifugal moment is equal with the product between the mass of the plate, the coordinate of center of mass of the rotation body, generated by plate, and the coordinate of center of mass of the plate on the perpendicular axis on the rotation axis.

The geometric centrifugal moment will be:

[I.sub.xz] = A[xi][z.sub.rC]. (9)

Keeping account of the second Guldin's law, [V.sub.Oz] = 2[pi][xi]A (where [V.sub.Oz] is the volume of the rotation body generated by plate by rotation around the Oz axis), the relation (8) becomes:

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[J.sub.xz] = [rho][z.sub.rC][V.sub.Oz]/2[pi]. (10)

If the rotation axis crosses the plate (Figure 3a), we split the plate in two parts with areas [A.sub.1], [A.sub.2] and coordinates of centers of mass [[xi].sub.1] respectively [[xi].sub.2] (Figure 3b). These two parts generate by rotation two rotation bodies with the volumes [V.sub.1Oz] and, respectively, [V.sub.2Oz]. In this case the centrifugal moment is:

[J.sub.xz] = [J.sub.1xz] + [J.sub.2xz] = [rho]([z.sub.rC1][V.sub.1Oz] - [z.sub.rC.sub.2][V.sub.2Oz])/2[pi]. (11)

For example, for the plate from the figure 4, the centrifugal moment can be quickly obtained using the relation (10). During the rotation, the plate describes a semi sphere whose volume is 2[pi][R.sup.3] /3 (Figure 4). The center of mass of the semi sphere is on the axis of symmetry (Oz), [z.sub.rC] = 3R/8. So, it results for the centrifugal moment:

[J.sub.xz] = [rho] 3/8 R x 2/3 [pi][R.sup.3]/2[pi] [rho] [R.sup.4]/8

The "classical" method is laborious (integration, change of variable etc.).

4. CONCLUSION

In this paper the authors proposed alternative methods for calculus of position of the central axis and the centrifugal moments, based on the coordinates of centers of mass. Keeping account that in technical literature is easily to find the positions of the centre of mass for more bodies, the methods proposed here are accessible because they replace the calculus with integrals, which can be some time laborious.

The studies shown in this paper ware been done for plane plates. Similar studies can be done and similar formulae can be found out also for curve plane bars.

Last but not least, a special study can be done for finding out a calculus formula for the moments of inertia with respect to the axis.

5. REFERENCES

Belyakov, A., O. & Seyranian, A., P. (2008). Determining the moments of inertia of larges bodies from vibrations in elastic suspension. Mechanics of Solids Journal, Vol.43, No.2, (April 2008) pp 205-217, ISSN 0025-6544

Hibbler, R., C. (2009). Engineering Mechanics Dynamics, Prentice Hall, ISBN 9780136077916, Upper Saddle River

Radoi, M. & Deciu, E. (1993) Mecanica, Editura Didactica [section]i Pedagogica, ISBN 973-30-2917-3 Bucharest

Teodorescu, P., P. (2007). Mechanical systems, classical models, Springer, ISBN 978-1-4020-5441-9, London

Voinea, R.; Voiculescu, D. & Simion, F.-P. (1989). Solid-State Mechanics with applications in Engineering, Romanien Academy, ISBN 973-27-0000-9, Bucharest

In many technical problems, where appear plane plates having rotation motion, the solving of the problem of determination of the reactions is simple if is known the position of the support of resultant vector of d'Alembert's fictitious forces system. The determination of the central axis is some time difficult.

Also, in problems where appear plates and bars having rotation motion, the centrifugal moments are necessary when it is applied the theorem of the angular momentum. The determination of the centrifugal moments can be some time laborious.

In the technical literature, the values of the centrifugal moments are calculated by the "classical" method, by integration (Teodorescu, 2007; Hibbeler, 2009), or by studying the vibrations of the system (Belyakov & Seiranyan, 2008). It don't exist studies for finding out a formula for the calculus of the centrifugal moments.

In this paper are proposed a rule for the determination of the support of the resultant vector of the d'Alembert's fictitious forces for plates having uniform rotation motion and a formula for calculus of the centrifugal moment for plane plates.

2. THE SUPPORT OF THE RESULTANT VECTOR OF THE D'ALEMBERT FICTITIOUS FORCES SYSTEM

Let us consider a homogeneous plane plate (Figure 1a). This plate rotates around of a vertical axis which is identical with a straight leg. The axis is contained in the plane of the plate. We know also OB = [h.sub.1] and OA = [h.sub.2]. We relate the plate to a Cartesian reference system (Figure 1a) so that the center of mass C to be situated on the Ox axis, C([zi], 0,0), and the Oz axis to be rotation axis.

[FIGURE 1 OMITTED]

For a random point P, with the mass mP, the d'Alembert's fictitious force is ([epsilon] = [??]) (Figure 1b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

The angle between the d'Alembert's fictitious force and a line parallel with the Ox axis is given by:

tg[phi] = [absolute value of [[??].sup.in[tau]]/[absolute value of [[??].sup.inv]] = [epsilon]/[[omega].sup.2]. (2)

This value doesn't depend of the position of the point P. It follows that the system of d'Alembert's fictitious forces is a parallel forces system and can be replaced by the resultant vector. The resultant vector of d'Alembert's fictitious forces system is (Figure 1c):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where m is the mass of the plate. The resultant vector is applied in the center of the parallel forces, denoted here T. The nonzero coordinates of the point T are given by the relations (Voinea et al., 1989):

[x.sub.T] = [integral] xd[F.sup.in]/[integral] d[F.sup.in]; [z.sub.T] = [integral] zd[F.sup.in]/[integral] d [F.sup.in]. (4)

Because the plate is entirely situated on the same side of the rotation axis, all the d'Alembert's fictitious forces have the same sign. It follows that the second expression (4) will be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [z.sub.rC] is the coordinate of the center of mass of the rotation body generated by plate by rotation (Figure 1d).

If the plane plate has uniform rotation motion ([epsilon] = 0), the resultant vector of the d'Alembert's fictitious forces system will be parallel with the Ox axis. In this case the rule is: for plane plate in rotation motion around an axis from its plane, the support of the resultant vector of the d'Alembert's fictitious forces system is perpendicular on the rotation axis and passes by the center of mass of the rotation body generated by plate by rotation. So, for the determination of the reactions it is necessary only the coordinate [z.sub.T] = [z.sub.rC].

[FIGURE 2 OMITTED]

For example, for the plane plate from the figure 2, having uniform rotation motion, the determination of the support of the resultant vector of the d'Alembert's fictitious forces system, by applying the relation (5) is very simple: by rotation, the plate generates a semi sphere. The centre of mass is on the rotation axis at the distance 3R/8 from the center of sphere. In this point the support of the resultant vector of the d'Alembert's fictitious forces system crosses the rotation axis.

The "classical" method, by integration, is laborious.

3. CENTRIFUGAL MOMENTS FOR PLATES

Let us consider a homogeneous plane plate from the figure 1. We apply the theorem of the angular momentum with respect to the point 0. We obtain the equations (Voinea et al., 1989):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where:

--[J.sub.xz] is the centrifugal moment with respect to the Ox and Oz axes;

--[J.sub.z] is the moment of inertia with respect to the Oz axis;

--[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the moment of the external forces with respect to the point O.

Let us apply now the d'Alembert's principle. Keeping account of the relation (3) we obtain (Radoi & Deciu, 1993):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Comparing the equations (6) with (7) and keeping account of the relation (5) it results the follow expression for the centrifugal moment:

[J.sub.xz] = m[xi][z.sup.rC]. (8)

So, the centrifugal moment is equal with the product between the mass of the plate, the coordinate of center of mass of the rotation body, generated by plate, and the coordinate of center of mass of the plate on the perpendicular axis on the rotation axis.

The geometric centrifugal moment will be:

[I.sub.xz] = A[xi][z.sub.rC]. (9)

Keeping account of the second Guldin's law, [V.sub.Oz] = 2[pi][xi]A (where [V.sub.Oz] is the volume of the rotation body generated by plate by rotation around the Oz axis), the relation (8) becomes:

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[J.sub.xz] = [rho][z.sub.rC][V.sub.Oz]/2[pi]. (10)

If the rotation axis crosses the plate (Figure 3a), we split the plate in two parts with areas [A.sub.1], [A.sub.2] and coordinates of centers of mass [[xi].sub.1] respectively [[xi].sub.2] (Figure 3b). These two parts generate by rotation two rotation bodies with the volumes [V.sub.1Oz] and, respectively, [V.sub.2Oz]. In this case the centrifugal moment is:

[J.sub.xz] = [J.sub.1xz] + [J.sub.2xz] = [rho]([z.sub.rC1][V.sub.1Oz] - [z.sub.rC.sub.2][V.sub.2Oz])/2[pi]. (11)

For example, for the plate from the figure 4, the centrifugal moment can be quickly obtained using the relation (10). During the rotation, the plate describes a semi sphere whose volume is 2[pi][R.sup.3] /3 (Figure 4). The center of mass of the semi sphere is on the axis of symmetry (Oz), [z.sub.rC] = 3R/8. So, it results for the centrifugal moment:

[J.sub.xz] = [rho] 3/8 R x 2/3 [pi][R.sup.3]/2[pi] [rho] [R.sup.4]/8

The "classical" method is laborious (integration, change of variable etc.).

4. CONCLUSION

In this paper the authors proposed alternative methods for calculus of position of the central axis and the centrifugal moments, based on the coordinates of centers of mass. Keeping account that in technical literature is easily to find the positions of the centre of mass for more bodies, the methods proposed here are accessible because they replace the calculus with integrals, which can be some time laborious.

The studies shown in this paper ware been done for plane plates. Similar studies can be done and similar formulae can be found out also for curve plane bars.

Last but not least, a special study can be done for finding out a calculus formula for the moments of inertia with respect to the axis.

5. REFERENCES

Belyakov, A., O. & Seyranian, A., P. (2008). Determining the moments of inertia of larges bodies from vibrations in elastic suspension. Mechanics of Solids Journal, Vol.43, No.2, (April 2008) pp 205-217, ISSN 0025-6544

Hibbler, R., C. (2009). Engineering Mechanics Dynamics, Prentice Hall, ISBN 9780136077916, Upper Saddle River

Radoi, M. & Deciu, E. (1993) Mecanica, Editura Didactica [section]i Pedagogica, ISBN 973-30-2917-3 Bucharest

Teodorescu, P., P. (2007). Mechanical systems, classical models, Springer, ISBN 978-1-4020-5441-9, London

Voinea, R.; Voiculescu, D. & Simion, F.-P. (1989). Solid-State Mechanics with applications in Engineering, Romanien Academy, ISBN 973-27-0000-9, Bucharest

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Author: | Boiangiu, Mihail; Alecu, Aurel |
---|---|

Publication: | Annals of DAAAM & Proceedings |

Article Type: | Report |

Geographic Code: | 4EUAU |

Date: | Jan 1, 2009 |

Words: | 1383 |

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