The vibrations influence on the field of accelerations of the linear-elastic connecting rod of a mechanism connecting rod lug.

Bagnaru, Dan Gheorghe ; Hadar, Anton ; Grigoras, Stefan 等

1. INTRODUCTION

The researches that were done until now by several authors determined only the field of accelerations for undeformable parts. That is why we decided to determine the field of accelerationsao the rod from a rod-lug mechanism. This approach way is original. In our future researches, we will determine the field of accelerations for spatial moving kinematic elements.

2. THEORETICAL RESULTS

With the Hamilton's variational principal we have obtained the mathematical model of a linear elastic bar movement submitted to vibrations under the shape (Bagnaru, 2005).

[L]{u} +[[M.sub.4]]{[a.sub.0]} + {[V.sub.1]} + [[M.sub.7]]{f} + {[V.sub.2]} = {0}, (1)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[??] the linear elastic displacement; [??] the instantaneous angular speed of a bar; [??] the instantaneous angular acceleration; [[??].sub.0] the acceleration of the bar extremity O; [??] (x, t) the external force reported to the lenth unity; [??] (x, t) the external moment reported to the lenth unity; [rho] particular bar mass; A(x) the area of a transversal section bar; E Young modulus;

I = [I.sub.zz] the inaction geometrical moment of the transversal section bar reported to the axis Oz (neutral axis). The matrix [M5], [M6] are done in two variants (see the table 1).

[TABLE 1 OMITTED]

By distributing the coupling terms between the longitudinal and transversal vibrations, as well as the terms wich confers to the mathematical model the quality of a invariant model in time, it results a mathematical model in a first approximation under the shape (Mobley, 1999):

[[L.sub.0]]{u} + [[M.sub.4]] {[a.sub.0]} + {[V.sub.1]} = {0}, (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By applying an iterative method, we obtain the fields of longitudinal respectively transversal displacements, in the first approximation, in the case of OA connecting rod free vibrations of a mechanism R(RRT) in figure 1, under the form (Harrison, 1997):

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

[FIGURE 1 OMITTED]

3. ACCELERATIONS FIELD

The expression of accelerations field (Buculei et al., 1986) becomes in our case the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where u and u are done by the relations (3), and the dynamic parameters by the relations (Johnson, 2002):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

If we stop the iterative process at the third approximation and we situate in the concrete case when the lengths of connecting rod, respectively lug, are L = [L.sub.b] = 1 [m], respectively r = 0.07 [m], we'll obtain the numerical values of accelerations of different points of the connecting rod done in the table 2.

These values are comparable with those obtained experimentally and represented in figure 2.

[FIGURE 2 OMITTED]

The experiments were realised with the help af one equipment composed by an acquisition interface WebDaq/100, from the charge magnifiers Bruel & Kjaer 2635 and Robotron M1 300as well as accelerometers Bruel & Kjaer 4382.

4. EXPERIMENTAL TESTS

The accelerometers used to measure the vibrations were installed on the surface of the installation, on vertical and longitudinally horizontal directions (along the vibration direction on the horizontal).

Tests were effectuated to determine the vibratory answer of the installation, on vertical direction, for different percussion frequencies of irritation.

The sampling frequency, at acquisition, was of 40000Hz.

The numerical integration of answers in acceleration was effectuated, determining the vibration speed of the instalation mass.

The numerical integration of answers in speed was effectuated, determining the vibration displacement of the instalation mass.

By Rapid Fourier Transform of the answer acceleration was determined the frequency spectrum of vibratory answer of the installation using a resolution of 0.1532 Hz.

5. CONCLUSIONS

The amplitude of longitudinal vibrations is much smaller than the amplitude of transversal vibrations and for this reason the longitudinal vibrations can be neglected. The aparition of buckling phenomenon where apear significant longitudinal deformation is done with transversal deformation that have much bigger values. Therefore they determined only bar transversal vibration.

The amplitudes of transversal vibrations have minimum values when the connecting rod is perpendicular on the lug and maximum when the connecting rod and the lug are aligned.

6. REFERENCES

Bagnaru, D. (2005). The vibrations of kinematic elements, SITECH Publishers, ISBN 973-657-854-2, Craiova

Buculei, M., Bagnaru, D., Nanu, Gh. & Marghitu, D. (1986). Calculus method in the analysis of the mechanisms with bars, Scrisul Romanesc Publishers, ISBN 978-606-510-574-4, Craiova

Harrison, H.R. (1997). Advanced Engineering Dynamics, John Wiley & Sons Inc., New York

Johnson, W. (2002). Impact strength of materials, Edward Arnold

Mobley, R.K. (1999). Vibrations Fundamental, Newns, Boston

Tab. 2. Numerical values of accelerations

Mat Revolut Freq. [[omega].sub.0]
 rot/min (Hz) ([s-.sup.1])

Steel 237,3 3,955 24,83
 149,4 2,490 15,63

Mat AccV Acc OT
 (m/[s.sup.2]) (m/[s.sup.2])

Steel 26,183 30,668
 6,98 11,70