Case study of a mechanic system composed of four reduced masses and the dynamic absorber placed to one of the extremities of the mechanic system and subjected to four harmonic x.
Key words: the amplitudes, torsion vibrations, reduced masses, dynamic absorber
The present paper deals with the effect of the dynamic absorber on the torsion vibrations executed in the mechanic system composed of reduced masses [m.sub.1], [m.sub.2], [m.sub.3] and [m.sub.4] subjected to four harmonic x. The dynamic absorber is attached to the last reduced mass [m.sub.4] and the harmonic x acts upon the mechanic system is applied to the reduced mass [m.sub.1], [m.sub.2], m3 and m4. All reduced masses of the mechanic system, apart from the reduced masses to which the dynamic absorber is attached execute torsion vibrations.
2. MECHANICAL SYSTEM
We take into consideration the mechanical system formed of four reduced masses presented in the figure 1.
[FIGURE 1 OMITTED]
3. EQUIVALENT SYSTEM
The dynamic absorber having specific dimensions of L,1 and mass M is attached to the reduced mass [m.sub.4].
The mechanical system is subjected to the four harmonic x which acts tangent to the circles of reduced radius r and they are applied on the reduced masses: [m.sub.1], [m.sub.2], [m.sub.3] and [m.sub.4].
We replace the dynamic absorber with a mechanical system with an equivalent system formed of the reduced mass m5 and the reduces section of the shaft with the elastic coefficient c45 who is applied on the reduced mass [m.sub.4] the same momentum as the one applies of the dynamic absorber (Ripianu, 1977).
[FIGURE 2 OMITTED]
We start from a system of five differential equations that governs the vibratory torsion movements of the mechanical system represented in the figure 2 (Ripianu & Craciun, 1985):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
We introduce in the differential system of equations (1) the expressions of linear elongations registered on the circled of reduction of radius r of the torsion vibrations executed by the five reduced masses belonging to the mechanical system presented in the figure 2 (Haddow & Shaw, 2002). They have the expressions:
[a.sub.i] = [A.sub.i] cos([[ohm].sub.x]t-[epsilon]) (i=1~5) (2)
The elongations of the torsion vibrations for the five reduced masses together with them second time derivatives by the simplification of trigonometric function the cos ([[ohm].sub.x]t-[epsilon]) give us a system of five algebraic equations (Ripianu & Craciun, 1999)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
4. SOLUTION OF THE DIFFERENTIAL SYSTEM
We adopt the expressions (2) for the elongations [a.sub.i] (i=1[jenny]5) of the torsion vibrations executed by the five reduced masses (Arghir, 2002).
The linear amplitudes of the torsion vibrations are the solutions of the linear system (3):
[A.sub.i] = [[DELTA].sub.i]/[DELTA] (i=1[jenny]5) (4)
in which determinants A is the algebraic system determinant, [[DELTA].sub.i] (i=1[jenny]5) are the unknown quantities determinants.
Since the amplitude A5 does not refer to the vibrations of the part of a crankshaft and it is without interest for the given system study and the amplitude of this will not be calculated. For the mechanical system formed of mass m5 and the segment of the shaft with the elastic constant of [c.sub.45] to be dynamically equivalent with the dynamic absorber, that is applied on the reduced mass m4 having the same moment as the dynamic absorber (as the same mechanical effect), it is necessary and sufficient for the next relation to be satisfied, in which all the elements noted in figure 1 are represented (Balcau, 2002):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
result the squares of the proper pulsations of the mechanical system formed of mass [m.sub.4] and the section of the shaft with the elastic constant [c.sub.45] respectively from mass m5 and section of the shaft with the elastic constant [c.sup.45] marked by [[omega].sup.2.sub.IV V] respectively .
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
In addition, the harmonic x pulsation is expressed with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
and x represents the order of the harmonic, that is the number of oscillations that the harmonic executes in a time [T.sub.0], in which the shaft executes a complete rotation of 2[pi] radians, and [[omega].sub.0] is angular constant speed of rotation of the reduced crankshaft (Arghir, 1999).
If the dynamic absorber is built in such a way that:
L/1 = [x.sup.2] (8)
The expressions of the determinants [DELTA], [[DELTA].sub.i] (i = 1 ~ 4) become:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[[DELTA].sub.4] = 0 (13)
By replacing the determinants [DELTA], [[DELTA].sub.i] (i=1[jenny]4) in the system (4), we notice that the reduced masses [m.sub.1], [m.sub.2], [m.sub.3] execute torsion vibrations and the reduced mass m4, to which the dynamic absorber is attached and upon which the harmonic [P.sub.x]cos([[ohm].sub.X]t-[epsilon]) does not execute torsion vibration.
[A.sub.1] [not equal to] 0
[A.sub.3] [not equal to] 0
[A.sub.2] [not equal to] 0
[A.sub.4] = 0 (14)
We notice that the property from which the mechanical system subjected to several harmonics x, the only mass that does not execute torsion vibration, it is the one mass that has the dynamic absorber attached.
We notice that, as opposed from the case in which upon the mechanical system we have four harmonic x acting and it is applied on the reduced mass places to one of the extremities of the mechanical system, to which we attach the dynamic absorber, in which case, only one of the reduced masse does not execute torsion vibrations, in this case, in which more harmonic x acts upon the mechanical system, the reduced masses m1, m2, m3 execute torsion vibrations.
Ripianu, A.& Craciun, I. (1999). The dynamic and strength calculus of straight and crank shafts, Transilvania Press Publishing House Cluj
Ripianu, A.& Craciun, I. (1985). Calculul dinamic si de rezistenta al arborilor drepti si cotiti, Editura Dacia, Cluj-Napoca, 1985, pp 122-126
Ripianu, A. (1977). Vibratii mecanice, Atelierul de multiplicare al Institutului Politehnic Cluj-Napoca, pag. 20-22
Arghir, M. (1999). Mechanics II, Statics & Material Point Kinematics, U. T. PRES, Cluj-Napoca 1999, ISBN 9739471-16-1
Arghir, M. (2002). Mechanics II, Rigid Body Kinematics & Dynamics, U. T. PRES, Cluj-Napoca 2002, ISBN 973-83352-20-5
Haddow, A., G.& Shaw, S., W. (2002). Centrifugal pendulum vibration absorbers: an experimental and theoretical investigation. Nonlinear Dynamics 34: 293-307, 2003. 2004 Kluwer Academic Publishers. Printed in the Netherlands
Balcau, M. (2002). Referat nr.1, Studiul actual al cercetarilor efectuate in domeniul absorbitorilor dinamici, Cluj-Napoca, pp 24-29
Balcau, M. (2002). Referat nr.2, Studiul sectorului liber in care fenomenul de rezonanja este eliminat, Cluj-Napoca, pp 9-15
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|Author:||Balcau, Monica Carmen; Arghir, Mariana|
|Publication:||Annals of DAAAM & Proceedings|
|Article Type:||Case study|
|Date:||Jan 1, 2009|
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