Eigenmodes
In addition to standard calculations of linear structural analysis, software Fin 3D has ability to determine the natural frequencies and eigen shapes of structures. The aim of this module is partly to allow the user to predict the behaviour of structures where the inertial effects of the mass of the structure can not be ignored and partly to attract attention to cases when any of the natural frequencies of the considered structures are located close to the frequency of the excitation.
In terms of the structural mechanics, the finding natural frequencies and eigen shapes is characterized as a general problem of eigenvalues described by the equation
Where is: | K |
|
M |
| |
r |
|
If the order of matrices K and M is n, then the above equation allows software to calculate n natural frequencies ωi and n eigen shapes ri. The equation also shows that the absolute value of the components of the vector r is not decisive for the description of the shape oscillations. Eigenvectors ri are standardized in the program, while the size of the individual components of the displacement is not displayed.
The stiffness matrix is assembled as in the case of linear analysis. The consistent formulation is used to build the mass matrix M. Mass matrix is then diagonal, but generally full. Calculating the weight of the matrix elements is based on the material density of the individual members. Weight, which does not directly relate to a given structure, but has influence on the dynamic behavior of the structure, can be entered using concentrated masses. Entered mass (weight) is assigned to the nodes (joints), it is possible to define also the eccentricity.
As mentioned above, the equation allows to calculate only the amount of natural frequency, that is equal to the number of degrees of freedom. The literature describes a number of methods for finding a complete solution to the problem. In many cases it is impractical, because only the first few natural frequencies and eigenmodes are important from engineering point of view. Additionally, higher frequencies and mode shapes are usually burdened with considerable error resulting from discretization structures on individual finite elements. Program therefore supports only two the most commonly used methods, as only the first few eigen shapes and frequencies are needed.
Subspace iteration method
The most common is the subspace iteration method. This method counts the number of selected shapes and lowest natural frequency, while in order to increase the speed of convergence of the iteration, it is performed on the higher plurality of eigenvectors, than is required. Therefore we recommend to enter the structure to be at least double the number of degrees of freedom than the required number of custom shapes. Unfortunately, this method does not guarantee that the calculated natural frequencies are the lowest ones. Therefore, the program is equipped by Sturm control informing the user about the possible omission of any of the required eigen shapes. This method fails sometimes for structures which are characterized by clusters with more frequencies. On the other hand, this method is easy to handle with eigen shapes belonging to multiple natural frequencies.
Lanczos method
This method is suitable mainly for structures with a large number of degrees of freedom. If the solution does not require usage of the harddisc, this method is significantly faster than the subspace iteration method. Although this is a very reliable method, it is similarly to the method of subspace iteration completed with Sturm control. Also this method calculates only selected number of the lowest natural frequencies. This method unfortunately does not allow the separation of the eigenmodes belonging to multiple natural frequencies. In this case, we recommend to use the method mentioned above.
Analysis accuracy and convergence of the calculation
As we mentioned in the introduction to this chapter, the accuracy of calculation of the individual natural frequencies depends on the selected level of discretization. Theoretically it is possible to find as many custom shapes and frequencies as the number of degrees of freedom in the structure. In reality, however, shapes, whose level is close to the number of physical degrees of freedom, burdened with considerable numerical error due to very coarse division into elements. In this case it is necessary to choose a finer division of members into individual analysis elements. In addition, the rate of convergence to lower natural modes of oscillation for subspace iteration method depends on the number of vectors that are used in the calculation, and the iteration works, if possible, always with a number of eigen shapes slightly higher than the required number.
Insufficient number of physical degrees of freedom is one of the main reasons for the termination of calculation without founding all requested eigen shapes. Another reason may be inadequate maximum number of iterations or the required accuracy for calculating the natural frequencies. While the maximum number of iterations, which allows the calculation is equal to 200. The required tolerance of accuracy should be higher than 10-4.
At this point, we would like to draw the attention to the special importance of the concept of iteration for Lanczos method. From a theoretical point of view, this is a generation-base vector used in a Rayleigh-Ritz method. In practice, this means that the maximum number of iterations can not be higher than the number of physical degrees of freedom. Users should therefore not be surprised that the calculation of the structure with 6 degrees of freedom was stopped after the sixth iteration, although the specified number of iterations was elected 100. The sufficient attention should be given to the selection of the number of iterations in the Lanczos method, because it influences the speed of calculation. The reason is, that the section of internal memory is allocated for each iteration and such space may stay completely unused. The number of specified iterations should be therefore as close as possible to the number of iterations that are required for the convergence to the specified number of natural frequencies. Unfortunately, there is no general rule, how to proceed in such cases, and the user has to rely on his own judgement and experience. The maximum number of iterations is equal to 200 for this method.