# A transfer matrix method for study of vibrations of plane bars systems.

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 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). Each bar can effect bending vibrations in two perpendicular plans (xOy and yOz) which are containing the longitudinal axis of the element, torsion vibrations and axial vibrations around and respectively in the length of this axis (Voinea et al.,1989). The points from the heads of an element are named nodes.

2.2 Method's principle

It is constructed a transfer matrix which keeps account of these types of vibrations, a saltus matrix for variations multi-stage sections, transit matrices for concentrated mass and for passing from an element to another if the coordinate system is changed. We begin from the left terminal point and we traverse the system from an element to another till the right terminal point. Depending of 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. Transfer matrix

Fore one element of the system which effects vibrations like in the second chapter, choosing the reference system like in figure 1, with Oy longitudinal axis and Ox axis in the drawing plan, it results that the axial and torsion vibrations will are in the length and respectively around of the Oy axis. So, between the state vectors from the heads of the element k, [{q}.sup.r.sub.k-1] and [{q}.sup.l.sub.k], we can write the matric relation (1):

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

where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

--[v.sub.x], [v.sub.y], [v.sub.z] are displacements;

--[[phi].sub.z], [[phi].sub.y], [[phi].sub.x] are rotations (angular displacements);

--[M.sub.z], [M.sub.x] are bending moments;

--[M.sub.y] is torsion moment;

--[F.sub.x] [F.sub.z] are shearing forces;

--[F.sub.y] is axial force;

-- [omega] is the natural angular frequency of n order; [rho]--density of material; A--area of cross section; E--Young's modulus (longitudinal modulus of elasticity); I--second moment of area on Ox axis/Oz axis;

--S, T, U, V--Krilov--Rayleigh functions;

--M, N--symbols for abbreviation: M = [cos[alpha].sub.c]l, N = [sin[alpha].sub.c]l; l is the length of an element;

--for circular section: [[alpha].sub.r] = [omega] [square root of [rho]/G] where, G is modulus of rigidity (transverse modulus of elasticity);

--for rectangular section: [[alpha].sub.r] = [omega] [square root of [rho] [I.sub.p]]/G [I.sub.d]] where, [I.sub.d] = [beta]hb, [beta] = 0.141 for h/b = 1 and [beta] = 0.333 for h/b [right arrow] [infinity]; [I.sub.p]--polar moment of inertia of the cross section (on Oy axis);

--P, R--symbols for abbreviation: P = [cos[alpha].sub.r]l, R = [sin[alpha].sub.r]l;

[FIGURE 1 OMITTED]

It is easily to see that the elements of the state vectors are quantities in proportion to the displacements, rotations, moments and forces and depend of the geometric and mechanical characteristics.

The indexes are specifying from which axis the deformation takes place.

The matrix (12x12), named [[A].sub.k] is the transfer matrix for the element k (Boiangiu & Alecu, 1999).

4. Special matrices

4.1 Transition matrix

When we move from an element to another and the coordinate system is modified we introduce a transition matrix. This keeps account of the change of axis (from system [Ox.sub.1][y.sub.1][z.sub.1] to Oxyz) and of the geometrical and mechanical characteristics of the new element. If in a node k the coordinate system is changed (figure 2), between the mechanical characteristics at the left and right side of the node k, we can write the matric relation:

[{p}.sup.r.sub.k] = [[u].sub.k][{p}.sup.l.sub.k] (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the matrix [u]k has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[FIGURE 2 OMITTED]

We want a relation between the state vectors in the node k. Because the elements of the state vectors are quantities proportional with the displacements, rotations, moments and forces the matrix [[u].sub.k] must be replaced by another matrix [[C].sub.k] similar with [[u].sub.k]. The relation between the state vectors from the left side to the right side in the node k, can be written:

[{q}.sup.r.sub.k] = [[C].sub.k][{q}.sup.l.sub.k] (3)

The square matrix (12x12), named [[C].sub.k], represents transition matrix through k node. It is a rare matrix.

The elements of the matrix [[C].sub.k] are the elements of the matrix [[u].sub.k] times quantities that keep account of the geometrical characteristics of the elements connected in the node k.

A matrix's element [[[C].sub.k], for example [C.sub.6,4], can be written as follows:

[FIGURE 3 OMITTED]

4.2 Saltus matrix

For a bars system with variable multi-stage section, in the node k where appears a saltus of sections (figure 3), between the left state vectors and right state vectors of the node k, [{q}.sup.l.sub.k] and [{q}.sup.r.sub.k], we can write relation:

[{q}.sup.r.sub.k] = [[B].sub.k][{q}.sup.l.sub.k]. (4)

The square matrix (12x12), written [[B].sub.k], from relation (4) is a rare matrix and it is named saltus matrix in section k. For a homogeneous bars system the non-zero elements of this matrix are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Supplementary indexes are specifying from which axis the deformation takes place.

4.3 Transition matrix over a concentrated mass

If in a node k we meet a concentrated mass [m.sub.k] (figure 4), when we pass through the node k, a salt of forces appears. So, between the left state vector, [{q}.sup.l.sub.k] and the right state vector, [{q}.sup.r.sub.k] we can write the matric relation:

[{q}.sup.r.sub.k] = [[D].sub.k] [{q}.sup.l.sub.k]. (5)

where, the square matrix (12x12), written [[D].sub.k], is the transfer matrix over the concentrated mass. It is a rare matrix. The non-zero elements are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[FIGURE 4 OMITTED]

Because the elements of the state vectors depend of the geometric and mechanical characteristics of an element of bars system, the special matrices [C], [B], [D] action also as a filter. These matrices let pass by a node, from an element of bars system to the next, only displacements, rotations, moments and forces.

5. 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, the transfer matrix, [A], and the matrices [C], [B], [D] intervene. We obtain the matric relation between the state vectors from the end of the element n, [{q}.sup.l.sub.n], and the first node (point of start) [{q}.sup.r.sub.0]:

[{q}.sup.l.sub.n] = [[A].sub.n] ... [[D].sub.k] ... [[B].sub.j] ... [[C].sub.i] ... [[A].sub.1] [{q}.sup.r.sub.0] = [Q][{q}.sup.r.sub.0]. (6)

The square matrix (12x12), written [Q], from relation (6) is called transfer matrix for system. The indexes show the elements (for transfer matrix) or the nodes.

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

6. Example

The method presented here is applied in concrete case of a steel frame in a "U" shape with unequal sides and rigid fixed terminal points (figure 5).

The steel frame has been divided in 3 elements. We obtain 4 points, named nodes, which bound 3 elements.

Because the bars system (the frame) is arranged in the same plane (xOy), the bending vibrations in the xOy plane coupled with the longitudinal vibrations are separated of the bending vibrations in the yOz plane coupled with the torsion vibrations. We have separately, for each element, 2 transfer matrices and 2 matrices for each special matrix. We have also 2 state vectors for each state vector from (1).

[FIGURE 5 OMITTED]

So, they result 2 transfer matrices [Q] for the bars system: one for the bending vibrations in the xOy plane coupled with the longitudinal vibrations and another for the bending vibrations in the yOz plane coupled with the torsion vibrations.

The boundary conditions in the nodes 0 and 3 are:

[v.sub.x] = [v.sub.y] = [v.sub.z] = 0) [[phi].sub.x] = [[phi].sub.y] = [[phi].sub.z] = 0.

The equation (6) will be:

--for the bending vibrations in the xOy plane coupled with the longitudinal vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

or as algebraic system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

--for the bending vibrations in the yOz plane coupled with the torsion vibrations

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

or as algebraic system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

From the condition that the system to admit non-zero solution

--for the bending vibrations in the xOy plane coupled with the longitudinal vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

--for the bending vibrations in the yOz plane coupled with the torsion vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

we obtain the natural angular frequencies.

We have effectuated experiments on the same steel frame. The steel frame has been excited in more points. Also, the response of the system has been measured in more points.

[FIGURE 6 OMITTED]

We compared the natural frequencies obtained by theoretical method with those obtained by experiments. As shown in Table 1, the theoretical results approximately correspond to the experimental results.

7. Case of system statically indeterminate

In case of a statically indeterminate bars systems we divide the system in more cycle and we introduce special matrices and equations in ramification points. All the cycles must start from the same node (point), written "0". The start and the end points of the cycles must be in connection points of the system with the outside or in free end points of the bars. We can find two types of nodes: nodes with one input and more output, named ramification nodes, (figure 6a), and nodes with more input and more output, named meeting nodes (figure 6b).

In a node we consider a main cycle, for example the cycle number 1. The other cycles are secondary cycles (Boiangiu & Alecu, 2008).

In a ramification node we keep the bars from the main cycles and we replace the bars from the secondary cycles with the forces and moments from the heads in this node. The forces and moments from the heads of the bars from the secondary cycles in a ramification node are unknowns.

For this reason, in all state vectors we introduce new lines corresponding to the forces and moments from the heads of the bars from the secondary cycles in ramification nodes. The number of these new lines is equal with the total number of the unknowns from the secondary cycles in the ramification nodes. Also in all transfer and transit matrices we introduce new lines and columns. The non-zero elements are equal with 1 and are situated on the principal diagonal. The number of these new lines and columns is equal with the total number of the unknowns from the secondary cycles in the ramification nodes.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

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 7. The node 2 is a ramification node. It has one start node (0) and two end nodes (3 and 5).

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 cycle 1 is the main.

The ramification node introduces 6 unknowns: [F.sub.2x.sup.2], [F.sub.2y.sup.2], [F.sub.2z.sup.2], [M.sub.2x.sup.2], [M.sub.2y.sup.2], [M.sub.2z.sup.2] (figure 8). So we have to introduce in state vectors 6 new lines. The superscript and subscript numbers show the secondary cycle and respectively the ramification node.

The bending vibrations in the xOy plane coupled with the longitudinal vibrations they would be separated of bending vibrations in the yOz plane coupled with the torsion vibrations, because the bars system is arranged in the same plane (xOy). We have separately for each element 2 transfer matrices and 2 matrices for each special matrix. In each matrix we introduce 3 new lines and columns. We have also 2 state vectors for each state vector from the relation (1) and we introduce in all state vectors 3 new lines:

--for the bending vibrations in the xOy plane coupled with the longitudinal vibrations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

--for the bending vibrations in the yOz plane coupled with the torsion vibrations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The transfer matrices will are: for bending vibrations in the xOy plane coupled with longitudinal vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for bending vibrations in the yOz plane coupled with torsion vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For bending vibrations in the xOy plane coupled with the longitudinal vibrations we introduce the ramification rare matrix [[R].sub.2]. The non-zero elements are:

--for cycle 1: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

--for cycle 2: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For bending vibrations in the yOz plane coupled with the torsion vibrations, we introduce also the ramification rare matrix [R]2. The non-zero elements are:

--for cycle 1: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

--for cycle 2: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For cycle 2, in ramification node 2, intervenes also the transition matrix [[C].sup.2.sub.2].

We obtain the natural angular frequencies like in case of the bars system statically determinate. We obtain for cycle 1 the matric relation:

[{q}.sup.l.sub.5] = [[A].sub.5][[C].sub.4][[A].sub.4][[R].sup.1.sub.2] [[A].sub.2][[C].sub.1][[A].sub.1][{q}.sup.r.sub.0]= [[Q].sup.1][{q}.sup.r.sub.0] (13)

where [[Q].sup.1] is the transfer matrix for cycle 1. The superscript number shows the cycle. We obtain for cycle 2 the matric relation:

[{q}.sup.l.sub.3] = [[A].sub.3][[C].sub.2][[R].sup.2.sub.2][[A].sub.2] [[C].sub.1][[A].sub.1] [{q}.sup.r.sub.0] = [[Q].sup.2][{q}.sup.r.sub.0] (14)

where [[Q].sup.2] is the transfer matrix for cycle 2. The superscript number shows the cycle. From the relations (13) and (14) we obtain the algebraic system in matric form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

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

For the example from the figure 7, the boundary conditions in the nodes 0, 3 and 5 are the same: [v.sub.x] = 0; [[phi].sub.z] = 0; [v.sub.y] = 0; [[phi].sub.y] = 0; [v.sub.z] = 0; [[phi].sub.x] = 0.

Keeping account of these conditions, the relation (15) leads, like in chapters 5 and 6, to a linear and homogeneous algebraic system.

From the condition that the system to admit non-zero solution:

--for the bending vibrations in the xOy plane coupled with the longitudinal vibrations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

--for the bending vibrations in the yOz plane coupled with the torsion vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

we obtain the natural angular frequencies.

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 moments from the heads in this node, but only for the bars that leave the node. These forces and moments are unknowns. The bars from the secondary cycles that come in node are replaced with the forces and moments 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.

[FIGURE 9 OMITTED]

For example we consider a plane bars system as in figure 9. It has one start node (0) and only one end node (5). The node 1 is a ramification node and the node 4 is a meeting node. We divide the bars system in 2 cycles. The cycle 1 (0-1-2-3-4-5) and the cycle 2 (0-1-4-5). The cycle 1 is the main cycle in the node 1.

The ramification node 1 introduces 6 unknowns: [F.sub.1x.sup.2], [F.sub.1y.sup.2], [F.sub.1z.sup.2], [M.sub.1x.sup.2], [M.sub.1y.sup.2], [M.sub.1z.sup.2] (figure 10). We have to introduce in the state vectors 6 new lines. The superscript and subscript numbers show the secondary cycle and respectively the ramification node.

The bending vibrations in the xOy plane coupled with the longitudinal vibrations they would be separated of bending vibrations in the yOz plane coupled with the torsion vibrations, as in the previous example. As in the previous example we have separately for each element 2 transfer matrices and 2 matrices for each special matrix. In each matrix we introduce 3 new lines and columns like in the previous example. Also as in the previous example we have 2 state vectors. We introduce also in all state vectors 3 new lines. In the node 1 we introduce the ramification matrix [[R].sub.1], as in the previous example. In node 4 we introduce a special meeting matrix [[[R.sup.F,M]].sub.4] written only for forces and moments. We can write in node 4:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

Keeping account of the element 5 we can write:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

where, the square matrix [[Q].sup.1] is called transfer matrix for system for the main cycle (cycle 1). The superscript number shows the cycle.

The matrix [[[R.sup.F,M]].sup.2.sub.4] can be obtained from the transition matrix [[C].sub.k]:

[[[R.sup.F,M]].sup.2.sub.4] = [[C].sup.2.sub.4] [[1.sup.F,M]], (20)

where the superscript and subscript numbers shows the cycle, respectively the node and [[1.sup.F,M]] is the unit matrix but only for forces and moments. The matrices [[C].sup.2.sub.4] will are:

for bending vibrations in the xOy plane coupled with longitudinal vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for bending vibrations in the yOz plane coupled with torsion vibrations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[FIGURE 10 OMITTED]

The elements of the matrix [[1.sup.F,M]] are equal with zero excepting: [e.sub.3,3] = [e.sub.4,4] = [e.sub.6,6] = 1, for bending vibrations in the xOy plane coupled with longitudinal vibrations, and [e.sub.2,2] = [e.sub.5,5] = [e.sub.6,6] = 1, for bending vibrations in the yOz plane coupled with torsion vibrations.

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

The matrix [[[R.sup.D]].sup.2.sub.4] can be obtained from the transition matrix [[C].sub.k]:

[[[R.sup.D]].sup.2.sub.4] = [[C].sup.2.sub.4][[1.sup.D]], (21)

where the superscript number shows the cycle, the subscript number shows the node and [[1.sup.D]] is the unit matrix but only for displacements.

The matrix [[1.sup.D]] is a rare matrix. The nonzero elements are:

--for bending vibrations in the xOy plane coupled with longitudinal vibrations, where

[e.sub.1,1] = [e.sub.2,2] = [e.sub.5,5] = 1;

--for bending vibrations in the yOz plane coupled with torsion vibrations, where

[e.sub.1,1] = [e.sub.3,3] = [e.sub.4,4] = 1.

Starting from the relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] we can write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

[[[R.sup.D]].sub.4] is the meeting matrix for displacements in the node 4. The superscript number shows the cycle.

From the relations (19) and (22) we obtain the algebraic system in matric form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

Depending of 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.

For the example from the figure 9, the boundary conditions in the nodes 0 and 5 are the same: [v.sub.x] = 0; [[phi].sub.z] = 0; [v.sub.y] = 0; [[phi].sub.y] = 0; [v.sub.z] = 0; [[phi].sub.x] = 0.

Keeping account of these conditions, the relation (23) leads, like in chapters 5 and 6, to a linear and homogeneous algebraic system.

From the condition that the system to admit non-zero solution:

--in case of the bending vibrations in the xOy plane coupled with the longitudinal vibrations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

--in case of the bending vibrations in the yOz plane coupled with the torsion vibrations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)

we obtain the natural angular frequencies.

If the bars system has more cycles and more ramification and meeting nodes, for solving the problem it is necessary to follow the two way exposed in this chapter.

For example, we consider the bars system from the figure 11. It contains two ramification nodes (1 and 3) and one meeting node (4). We divide the bars system in 3 cycles: the cycle 1 (0-1-2-3-4-5) the cycle 2 (0-1-4-5) and the cycle 3 (0-1-2-3-6-7). The cycles 1 and 3 have common way from the starting point till the node 3. In the node 1 the cycle 2 is secondary and the cycle 1 (or 3) is main cycle. In the node 3 the cycle 3 is secondary and the cycle 1 is main cycle.

The ramification nodes 1 and 3 introduce 12 unknowns (6 unknowns for each ramification node): [F.sub.1x.sup.2], [F.sub.1y.sup.2], [F.sub.1z.sup.2], [M.sub.1x.sup.2], [M.sub.1l.sup.2], [M.sub.1z.sup.2] and [F.sub.3x.sup.3], [F.sub.3y.sup.3], [F.sub.3z.sup.3], [M.sub.3x.sup.3], [M.sub.3y.sup.3], [M.sub.3z.sup.3] We have to introduce in the state vectors 12 new lines. The superscript and subscript numbers show the secondary cycle and respectively the ramification node.

Following the theory exposed in the chapters 6 and 7 we obtain the algebraic system in matric form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

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

[FIGURE 11 OMITTED]

8. Conclusion

The proposed method gives good results, is flexible and can be easily applied on different types of plane bars systems or pipes systems (Alecu & Boiangiu, 2008).

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.

The method has been applied for simple systems with 2 and 3 cycles. For complex configurations of the bars systems it is difficult to identify the independent cycles (the main cycle and the secondary cycles) and the types of nodes. This paper doesn't explain how the cycles are identified.

The method for the identification of the cycles and the types of nodes will be the subject of an other paper. For complicated plane and spatial bars systems, in future, this method can be applied associated with the graph theory.

DOI: 10.2507/daaam.scibook.2009.77

9. References

Alecu, A. & Boiangiu, M. (2008). Dynamic characteristics of spatial pipe system by semianalytical method, Proceedings of 19th International DAAAM Symposium "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions", Katalinik, B. (Ed), pp.11-12, ISSN 17269679, Trnava, October 2008, Published by DAAAM International Vienna 2008

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

Boiangiu, M. & Alecu, A. (2008). Study of vibrations of plane bars systems by using the transfer matrix method, Proceedings of 19th International DAAAM Symposium "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions", Katalinik, B. (Ed), pp.135136, ISSN 1726-9679, Trnava, October 2008, Published by DAAAM International Vienna 2008

Ohga, M., Shigematsu, T. & Hara, T. (1993). A Finite Element-Transfer Matrix Method for Dynamic Analysis of Frame Structure. Journal of Sound and Vibration, Vol.167, No.3, Nov. 1993, pp.401-411, ISSN 0022-460x

Voinea, R.; Voiculescu, D.; Simion, F.-P. (1989). Solid-State Mechanics with applications in Engineering, Romanian 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, pp.169-175, ISSN 0022-460x

This Publication has to be referred as: Boiangiu, M[ihail] & Alecu, A[urel] (2009). A transfer matrix method for study of vibrations of plane bars systems, Chapter 77 in DAAAM International Scientific Book 2009, pp. 798-814, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-901509-69-8, ISSN 1726-9687, Vienna, Austria

Author's data: Univ. Lecturer. Dipl.-Eng. Dr.techn. Boiangiu, M[ihail]; Univ. Lecturer. Dipl.-Eng. Dr.techn. Alecu, A[urel], "Politeclmica" University of Bucharest, Department of Mechanics, Splaiul Independentei, no.313, 060042, Bucharest, Romania, mboiangiu@gmail.com, aurel_alecu@yahoo.com
```Tab. 1. The first five natural frequencies

Mode number   Theoretical results (Hz)   Experimental 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
```