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Study of vibrations of plane bars systems by using the transfer matrix method.

1. INTRODUCTION

The precision in manufacturing can be influenced of vibrations. The machine tools, robots, equipments can contain bars structures. To control the vibrations it means to know the dynamic behaviour of these structures.

In literature it studies the coupled vibrations by utilization the finite elements method, but this method introduces specific feature. In few papers (Ohga et al., 1993) (Yu & Craggs, 1995), it has been tried to improve the finite elements method by combination with the transfer matrix method.

In this paper we propose a method of study based on transfer matrix method, applied first time at arborescent bars network. The bars system is divided in cycles and nodes and the transfer matrix method is applied for each cycle.

We introduce specific matrices for bars systems and special matrices for ramification and meeting nodes. We apply the purposed method to concrete cases.

2. DESCRIPTION OF METHOD AND SIMPLIFYING HYPOTHESIS

2.1 Hypothesis

We consider that the system is formed from straight elements (bars) of constant section (figure 1) which are effecting bending vibrations in two perpendicular plans which are containing longitudinal axis of a element xOy and xOz), torsion vibrations and axial vibrations around and respectively in length of this axis (Voinea, et al.,1989).

2.2 Transfer matrix for one element

For one element of the system, choosing the reference system like in figure 1, between state vectors of ends of the k element, [{q}.sup.r.sub.k-1] and [{q}.sup.l.sub.k], we can write matrix relation (1):

[{q}.sup.l.sub.k] = [[A].sub.k][{q}.sup.r.sub.k-1] (1)

The matrix (12x12), named [[A].sub.k] is the transfer matrix for the element k. The elements of this matrix are the Krilov--Rayleigh functions. The elements of the state vectors are quantities in proportion to displacements, rotating, forces and momentums (Boiangiu & Alecu 1999).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.3 Special matrices

In some nodes is possible to appear special matrices (Boiangiu & Alecu 2001). These are rare matrices and can be:

--a transition matrix, [[C].sub.k], when we move from an element to another and the coordinate system is modified;

-a saltus matrix, [[B].sub.k], when in a node k appears a saltus of section;

--a transfer matrix over a concentrated mass in node k, [[D].sub.k].

Between the left state vectors and right state vectors of section k, [{q}.sup.l.sub.k] and [{q}.sup.r.sub.k], we can write relation:

[{q}.sup.r.sub.k] = [[S].sub.k][{q}.sup.l.sub.k] (2)

The matrix [[S].sub.k] can be one from the matrices [[C].sub.k], [[B].sub.k] or [[D].sub.k].

2.4 Determination of the natural angular frequencies

We begin from the left terminal point, 0, and we traverse the system from an element to another till the right terminal point, n. Depending of configuration of the bars system, on route, intervene the transfer matrix, [A], and the matrices [C], [B], [D]. We obtain the matrix relation between the state vectors of ends of the n element, [{q}.sup.l.sub.n] and the first (point of start) [{q}.sup.r.sub.0] (Boiangiu & Alecu 2001):

[{q}.sup.l.sub.n] = [[Q].sub.k][{q}.sup.r.sub.0] (3)

The matrix (12x12), [Q], is called transfer matrix for system.

Depending on the type of the leaning we write the boundary conditions and we obtain an algebraic system. We put the condition that the system to admit non-zero solution and we obtain the natural angular frequencies.

3. EXAMPLE

The method presented here is applied in concrete case of a steel frame in a "U" shape with unequal sides and rigid fixed in the terminal points (figure 2).

[FIGURE 3 OMITTED]

We have effectuated experiments on the same steel frame. As shown in Table 1, the theoretical results approximately correspond to the experimental results.

4. CASE OF SYSTEM STATICALLY INDETERMINATE

In case of the bars systems statically indeterminate we divide the system in more cycles and we introduce special matrix and equations in ramification nodes. All the cycles must start from the same node (point), named "0". The start and the end points of the cycles must be in a connection point of the system with the outside or in a free end point of the bars. We can find two types of nodes: nodes with one input and more output, named ramification nodes, (figure 3a), and nods with more input and more output, named meeting nodes, (figure 3b)

In a node we consider a main cycle, for example the cycle number 1. The other cycles are secondary cycles.

In a ramification node we keep the bars from the main cycle and we replace the bars from the secondary cycles with the forces and momentums, unknown, from the heads in this node. From this reason, in all state vectors we introduce new lines corresponding to these forces and momentums. Also in all transfer and special matrix we introduce new lines and columns. The number of the new lines and columns (in state vectors and matrices) is equal with the total number of the unknowns from the secondary cycles in the ramification nodes. In these nodes we introduce a special matrix named ramification matrix and written [[R].sub.2].

For example we consider a plane bars system as in figure 4a. The node 2 is a ramification node.

We divide the bars system in 2 cycles. The cycle 1 (0-1-2-4-5) and the cycle 2 (0-1-2-3). The node 2 is a ramification node. In this node the main cycle is the cycle 1.

The ramification introduces 6 unknowns: [F.sub.x.sup.2], [F.sub.y.sup.2], [F.sub.z.sup.2], [M.sub.x.sup.2], [M.sub.y.sup.2], [M.sub.z.sup.2], that is why should be introduced in the state vectors 6 new lines.

For the cycle 2, in ramification node 2, intervenes also the transition matrix [C]. We obtain for cycle 1 the matrix relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

We obtain for cycle 2 the matrix relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[Q].sup.1] and [[Q].sup.2] are the transfer matrices for cycle 1 respectively 2. The superscript number shows the cycle.

[FIGURE 4 OMITTED]

From the relations (4) and (5) we obtain the system

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

In a meeting node we keep also the bars from the main cycles. We replace the bars from the secondary cycles, with the forces and momentums from the heads in this node, but only for the bars that leave the node. These forces and momentums are unknowns. The bars from the secondary cycles that come in node are replaced with the forces and momentums at the head of the cycle in this node. If the output is common for a main cycle and a secondary cycle, we keep only the output for main cycle. In a meeting node are written special equations.

For example we consider a plane bars system as in figure 4b. The node 1 is a ramification node and the node 4 is a meeting node. In node 1 we introduce the ramification matrix [[R].sub.1] as in the previous example. In node 4 we introduce a special ramification matrix [[R.sup.F,M]].sub.4] written only for forces and momentums. We can write for cycle 1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where, the square matrix [Q] is called transfer matrix for system. The superscript number shows the cycle.

In node 4 displacements on the cycle 1 are equal with the displacements on the cycle 2. In this case we introduce a special ramification matrix [[R.sup.D]].sub.4] written only for displacements (displacements, rotations). We can write for the node 4:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[[[R.sup.D]].sup.1-2.sub.4] is the ramification matrix for displacements in the node 4. The superscript number shows the cycles.

From the relations (7) and (8) we obtain the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

5. CONCLUSIONS

The proposed method gives good results, is flexible and can be easily applied on different types of plane bars systems.

This method is indicated to be used in computational engineering, especially for the complicated bars systems.

The next step will be to apply the method for spatial bars systems.

For complicated plane and spatial bars systems, in future, this method will be applied associated with the graph theory, because is difficult to identify the cycles and the types of nodes.

6. REFERENCES

Boiangiu, M. & Alecu, A. (1999). Coupled vibrations of crankshaft, Printech, ISBN 973-652-021-8, Bucharest

Boiangiu, M. & Alecu, A. (2001). The simultaneous vibrations analysis of bars systems, Proceedings of 8th International Symposium on Theory of Machines and Mechanisms, pp. 77-82, ISBN 973-8143-38-1, Bucharest, Sept 2001

Ohga, M. et al. (1993). A Finite Element-Transfer Matrix Method for Dynamic Analysis of Frame Structure. Journal of Sound and Vibration, Vol.167, No.3, Nov. 1993

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

Yu, J., Y. & Craggs, A. (1995). A Transfer Matrix Method for Finite Element Models of a Chaine-like Structure Under Harmonic Excitations. Journal of Sound and Vibration, Vol.187, No.4, Nov. 1995
Tab. 1. The first five natural angular
frequencies

Mode Theoretical Experimental
number results (Hz) results (Hz)

Mode 1 40,87 39,2
Mode 2 138,66 132
Mode 3 139,14 138
Mode 4 212,21 230
Mode 5 255,77 253
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Author:Boiangiu, Mihail; Alecu, Aurel
Publication:Annals of DAAAM & Proceedings
Date:Jan 1, 2008
Words:1609
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