### Tech Briefs

**Component Mode Synthesis using ADINA**

A dynamic model reduction technique, i.e., a component mode synthesis scheme, has been included in the dynamic analysis features of ADINA^{*}. The scheme is a natural and efficient procedure for the approximate frequency analysis of very large and complex structures, see Ref. [1]. Also, the method has been increasingly accepted to couple the use of multibody system programs and finite element analysis programs. An example is the use of AVL EXCITE and ADINA.

Here, the use of the component mode synthesis method, and specifically the Craig-Bampton scheme, in the ADINA dynamic analysis capabilities is presented along with some demonstrative examples.

The basic calculations are to establish the Craig-Bampton transformation matrix, see Ref. [2]

(1)

and then compute,

(2)

(3)

(4)

where and are the reduced stiffness and mass matrices. Note that these equations correspond to a single iteration in the Bathe subspace iteration method.

The vectors in consist of the static constraint modes and the fixed interface vibration modes . The interface or connection degrees of freedoms are to be chosen by the user to provide a 'good combination' of constraint modes and vibration modes. Of course, the values in and vectors in will in general only be approximations to the actual lowest frequencies and mode shapes of the finite element model, and the error depends on the choice of constraint modes and fixed interface vibration modes. However, with experience, good results can be obtained in many simulations [1,2].

Also, , , , and can be exported for use with AVL EXCITE and other multibody system programs.

In our first example, we use the propeller blade model analyzed earlier. The lowest component mode synthesis frequency, 649.6 Hz, is very close to the exact frequency, 649.5 Hz, although only 9 boundary DOFs (see Figure 1) are selected to calculate the static constraint modes, and only 21 fixed interface modes are used. The movie above shows the first vibration mode.

**Figure 1** Mesh of propeller blade showing the boundary nodes selected

In our second example, we consider the frequency solution of the pendulum shown in Figure 2. Only 3 boundary DOFs are selected (see Figure 3) for the constraint modes, and the lowest 7 fixed interface vibration modes are used. In Figure 4, we compare the exact lowest frequencies with the frequencies from the component mode synthesis method. The two sets of results are in very good agreement.

Note that the exact frequencies of the model are calculated using ADINA by simply continuing the subspace iteration with the vectors obtained in equation (4) above.

**Figure 2** First four vibration modes of the pendulum

**Figure 3** The selected boundary node of the pendulum

**Figure 4** Comparison of frequencies (Hz) calculated using component mode synthesis and subspace iteration

In our third and last example, we use the scheme for a larger finite element model, considered already in an earlier Brief, see Figure 5. We compare in Figure 6 the lowest frequencies calculated using the component mode synthesis scheme with 6 constraint modes and 4 fixed interface vibration modes. The two sets of results are also in very good agreement.

**Figure 5** Bolted wheel assembly, showing first vibration mode

**Figure 6** Comparison of frequencies (Hz) calculated using component mode synthesis and subspace iteration

Component mode synthesis can be a powerful tool in the analysis of very large systems. However, as mentioned already, the accuracy of the frequencies and mode shapes obtained depends on the vectors used in the transformation matrix .

On the other hand, Equations (1) to (4) are a first subspace iteration, and in ADINA, this iteration can be continued — if the analyst so chooses — to obtain accurate frequencies and mode shapes of very large finite element models (using perhaps the ADINA DMP solution capability). We will show results using this scheme in a future Brief.

**References**

- Bathe, K.J.,
*Finite Element Procedures*, Cambridge, MA: Klaus-Jürgen Bathe, 2006. - Craig, R. R. Jr. and Bampton, M.C.C., "Coupling of Substructures for Dynamic Analyses"
*,**AIAA*, Vol. 6, No. 7, July 1968, pp. 1313-1319.

*Keywords:*

Component mode synthesis, Craig-Bampton scheme, multibody system, Bathe subspace iteration method, constraint modes, vibration modes

^{*}The scheme is available in ADINA v. 9.