The simulation and vizualization of plane truss eigenvibration.

Isic, Safet ; Dolecek, Vlatko ; Karabegovic, Isak 等

Abstract: This paper demonstrates the algorithm for simulation of plane trusses eigenvibration by numerical analysis and visualization of the solutions. It uses the finite element method for numerical analysis. Computer programs may be written to implement this method, or commercial mathematical programs may be used to save analyst time and allow quick graphical representation of obtained results. The MAPLE solution for particular nine bars plane truss is presented.

Key words: Simulation, visualization, finite element method, eigenvibrations, plane truss

1. INTRODUCTION

Eigenvalue of structures free vibration are of the great importance for both mechanical and civil engineers, because it is basic step of analysis of dynamical response. Trusses are the common type of construction which engineers meet practice. If we deal with vibration of such real construction, analytical solution is tedious or even impossible. In this case some numerical method should be introduced. The most popular method for analysis is the finite elements method [Geradin & Rixon, 1998].

Finite elements method deals with large number of linear algebraic equation, and some computer programs must be used. The large commercial FEM packages are expensive and inadequate if we solve some problems temporarily, so it is important to be able to write own programs. Programs may be created using the programming languages (FORTRAN, C++, ...) or mathematical packages like MATLAB, MATHEMATICA, MAPLE, etc.

In this paper we present basic elements of the algorithm for numerical analysis and simulation of free vibration of an arbitrary plane truss. This algorithm may be implemented in any mathematical software or programming language [Isic, 2000].

Eigenvalues of free vibration (vibration modes and corresponding eigenfrequencies) are calculated from the following finite element equation:

([K]+ [[omega].sup.2][M]){U} = 0 (1)

where: [K] is structural stiffness matrix, [M] is mass matrix, U is eigenfrequency and {U}is vibration mode.

Equation (1) presents algebraic eigenvalue problem. Before we are solving it, matrices [K] and [M] must be calculated for the particular truss.

2. INPUT DATA--TRUSS DESCRIPTION

To uniquely describe any plane truss with [N.sub.elem] bars and [N.sub.nod] nodes, e.g. shown on Fig. 1, the following data are required: material properties--modulus of elasticity E and mass density [rho]; coordinates of nodes ([x.sub.i], [y.sub.i]), i = 1, [N.sub.nod]; cross sectional area of members [A.sub.i], i = 1, [N.sub.elem]; indexes of nodes and elements; indexes of the nodes with displacements restricted by supports.

[FIGURE 1 OMITTED]

Nodal coordinates are stored in two dimensional array [X] having two rows and [N.sub.nod] columns

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

From indexes of nodes and elements one two-dimensional array [NOD] is created, having [N.sub.elem] columns. In each column, first coefficient is equal to index of i-th node, and second to j-th node. For the truss on Fig. 1, this array has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Coefficient NOD(1,3) = 3, for example, shows that element 3 has node 3 as its i-th node.

It is necessary to define boundary condition, i.e. which node is restricted to move and size of eigenvalue problem [N.sub.eq]. It is done by "destination array" [ID], which consists of columns equal to the number of truss nodes--[N.sub.nod], and rows equal to number of d.o.f of single node. For example the truss shown on Fig. 1, initially entered [ID] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

E.g. coefficient ID(2,3) = 0 means: displacement of node 3 along axes y is allowed, and e.g. ID(2,1) = 1, means that displacement of node 1 along axes x is restricted.

2.1. Length and orientation of bar elements

On the base of entered data, length of all bars and angles between bars and coordinate axes x are calculated. First it is identified which nodes i and j correspond to k-th element, k = 1, [N.sub.elem]. Using it, corresponding nodal coordinates are selected from [X] and used to calculate the coordinate differences [DELTA][x.sub.k] and [DELTA][y.sub.k] of element nodes. Length [l.sub.k], and the direction (angle [[theta].sub.k]) are calculated from this coordinate difference.

2.2 Check of input data

Input data, consists of large number of coordinates, areas, nodes and element indexes, etc. There is a high possibility for an analyst to make some errors in writing input data, which may produce wrong results. Graphical representation of the input data immediately after they are entered is the best way to exclude any mistake.

Using graphical routines we draw bars in the forms of lines connecting its nodes, with element labels written in the middle of the bar and nodal labels written in nodes. In the case of any error, input data should be revised.

2.3. Removing restricted d.o.f.

Looping through initial [ID] array, column by column, elements corresponding to restricted d.o.f. are putted to zero, and for every element which corresponds to unrestricted d.o.f, value of variable [N.sub.eq] is increased by 1, and this element of [ID] receives this value. For truss on Fig. 1, final [ID] is transformed to the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. STRUCTURAL MATRICES

To calculate constitutive matrix in (1), stiffness and mass matrix of all [N.sub.elem] members will be calculated and expanded to structural size. Stiffness and mass matrices for single plane bar element are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where c = cos([[theta].sub.k]) and s = sin([[theta].sub.k]).

Structural stiffness and mass matrices (i.e., stiffness and mass matrices of the truss) are formed by adding coefficients of element matrices [Cook at al., 1988].

4. EIGENVALUES EXTRACTION

The eigenvalue problem (1) could be transformed in

([[M].sup.-1[K]){U} = [[omega].sup.2] (8)

Solution of (8) may be done by linalg packages if mathematical pacages, e.g. MAPLE, are used. If one is writing own code for solution of (8), method of subspace iterations is recommended [Isiae, 2000; 2002.].

Using any method, both [N.sub.eq] eigenfrequencies [omega] and [N.sub.eq] corresponding eigenvectors {U} will be returned.

To animate vibration, it is necessary to add zero displacement for restricted d.o.f. in eigenvectors {U}. It is done again by using information from [ID] array.

5. VISUALIZATION OF VIBRATION

As the total time of lasting of one full oscillation cycle, time 2[pi] is used. This time interval is divided into [N.sub.timestep] equal subintervals.

Using plotting commands, truss is plotted at the ends of these intervals, with coordinates of nodes ([x.sub.j]([t.sub.i]), [y.sub.j]([t.sub.i])), j = 1, [N.sub.nod], calculated as

[x.sub.j]([t.sub.i]) = [x.sub.j0] + scl [u.sub.j] sin([t.sub.i]), [y.sub.j]([t.sub.i]) = [y.sub.j0] + scl [w.sub.j] sin([t.sub.i]), (9)

where scl is scaling factor, [x.sub.j0] and y are node coordinates in state of the rest stored in [X], and [u.sub.j] and [w.sub.j] are displacement of nodes along x and y axes, recorded in corresponding eigenvectorvector {U}.

Animation of vibration is done using display command in plots package if using MAPLE. If e.g. FORTRAN language is used for writing code, animation is done similarly--plotting truss in positions corresponding to time step [t.sub.i], erasing it, displaying truss in position at step [t.sub.i+1], and so on.

Fig. 2 presents the output of program, based on the presented algorithm, for first eigenmode and frequency of example truss, calculated and vizualized by MAPLE using presented algorithm.

[FIGURE 2 OMITTED]

6. CONCLUSIONS

Algorithm for computer program presented in this paper may be used for quick analysis of plane truss eigenvibration and for educational purposes for learning programming, finite element method and its computer implementation.

Finite elements method requires matrix computation, so implementing this program in some mathematical packages, as MAPLE, allows finite elements analysis of eigenvalues of plane trusses vibration by a user without experience to write a code for matrix computations. Such a program may be also used to help an analyst to develop routines in some higher program languages.

7. REFERENCES

M. Geradin, D.Rixon (1998). "Mechanical Vibrations", John Willey & Sons, 1998.

Cook R., Malkus D.S., Plesha M. (1988). "Concepts and Aplication of Finite Element Analyses", John Willey & Sons, New York, 1988.

S. Isiae (2000). "Analysis of Eigenvibration of Axially Loaded Constructions--Using Finite Elements Method", Master Thessis, Mostar, 2000.

M. L. Abell, J. P. Braselton (1994). "Diferential Equations with Maple V", Academic Press, 1994.

S. Isic (2002). "Exact And Numerical Analysis of Transversal Eigenvibration of Axially Loaded Column", Journal of Mechanical Engineering--Masinstvo, Vol. 6--Br. 3, 2002.