Influence of kinematic parameters on the deterministic vibrations of the viscoelastic linear connecting rod: part of a rod lug mechanism.
Stanescu, Marius-Marinel ; Chelu, Angela ; Nanu, Gheorghe 等
1. INTRODUCTION
The researches that were done until now by several authors
determined only the transversal displacements for vibrating beams, but
which do not move. This involves restrictions of the conditions.
We propose an original method, based on an iterative method of
determining the field of transversal displacements of the
linear-viscoelastic rod of a rod lug.mechanism. In our future studies,
for the same problem we analyze now, we will determine the influence of
the aleatoary vibrations.
2. THEORETICAL RESULTS
Let us consider the mathematic model of free vibrations, in a first
approximation (Bagnaru, 1998), in the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
Applying to equation (1) Laplace unilateral transform depending on
time, and replacing module E with [??](s), it results the matricial
equation of the first approximation (in Laplace images), of the
vibrations of the viscoelastic connecting rod OA of mechanism R(RRT)
(Harrison, 1997) in Fig. 1., in the form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The solution of equation (1) is series type and has the following
form:
[u.sup.(1).sub.2] (x,t) = 2/L [[infinity].summation over (n=1)]
[u.sup.(1).sub.2,s] (n,t) x sin ([[alpha].sub.n] x x), (3)
where [u.sup.(1).sub.2,s] (n, t) is the Fourier transforms finite
in sine, of the transversal elastic displacement. The relation (3) shows
the effect of the kinematical parameters of the movement on the
vibration modes.
[FIGURE 1 OMITTED]
3. NUMERICAL APPLICATION
Let us consider the concrete case of the textolite connecting rod
with:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
From relation (3) it results the numerical values of transversal
displacement (in the middle of the connecting rod) presented in tab. 1.
These errors are accepted in techniques, and therefore the
theoretical results are verified by the experimental tests.
4. EXPERIMENTAL TESTS
The experimental tests were conducted on a stand, using the
electronic measurement system Spider 8.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
5. CONCLUSIONS
The displacement fields of some viscoelastic bars, especially the
transversal ones, constitute the support of the afterwords determination
of the stress and deformation states which are usefull in the design of
the mechanisms used in the manufacturing engineering.
We insisted on the transversal displacement field because the
influence of the longitudinal displacements on the stress or
displacements state is insignificant.
The values in table 1 are comparable with those experimentally
obtained in figures 2, 3, 4 and 5, the error being in the first and
second case 10,3 %, and 5% respectively.
These errors are accepted in techniques, so that, in fact, the
theoretical results are verified by the experimental tests.
The vibration energy is mainly contained in the fundamental
harmonic. The rheological materials have the advantage of the existence
of some forces and reduced inertia couples due to the specific mass
which is lower than in the case of metallic materials in comparable
rigidity conditions (Routh, 2003), the displacements being also
comparable with those of the latter.
6. REFERENCES
Bagnaru, D. (1998). About the dynamic response of the viscoelastic
bars of the plane mecanisms PRASIC 98, 5-7 XI 1998, Brasov, pp. 15-18
Bagnaru, D., Marghitu, D.B. (2000). Linear Vibrations of
Viscoelastic Links, 20th Southeastern Conference on Theoretical and
Applied Mechanics (SECTAM-XX), April 16-18, 2000, Callaway Gardens and
Resort, Pine Mountain, Georgia, USA, pp. 1-7
Fu, K.S., Gonzalez, R.C. & Lee, C.S.G. (1997). Robotics,
McGraw-Hill
Harrison, H.R. (1997). Advanced Engineering Dynamics, John Wiley
& Sons Inc., New York
Routh, E.J. (2003). Dynamics of a system of rigid bodies, Part l
& Part 2, Macmillan
Tab. 1. The numerical values of transversal displacement
Transversal Frequency Frequency 1,5381 1,3916
movement on 2,6184 3.2776 (Hz) 1,391 1,8860
the direction (Hz) [Hz] [Hz]
OT [mm] -- -- 0,24 5
V[mm] 15 15,01 -- --
