Mathcad - CMS (Component Mode Synthesis) Analysis.xmcd Component Mode Synthesis by Julio C. Banks, P.E. The treatment of large structural systems may be simplified by dividing the system into smaller systems called components . The components are related through the displacement, and force conditions at their junction points. Each component is represented by mode shapes (or functions) . The sum of the component mode shape functions allows the satisfaction of the displacement and force conditions at the junctions [1]. α 1 m 1 α L 2 L 1 = L 1 1 L 2 α L 1 Component 1: M 11 1 5 M 12 1 6 M 21 M 12 M 22 1 7 Component 2: M 33 1.0α M 34 1 2 α M 35 1 5 α M 43 M 34 M 44 1 3 α M 45 1 6 α M 53 M 35 M 54 M 45 M 55 1 9 α Component 3: M 66 α Julio C. Banks, PE [email protected]page 1 of 11
The treatment of large structural systems may be simplified by dividing the system into smallersystems called components . The components are related through the displacement, and force conditions at their junction points. Each component is represented by mode shapes (orfunctions) . The sum of the component mode shape functions allows the satisfaction of thedisplacement and force conditions at the junctions [1].
Reference"Theory of Vibration with Applications, 5th Ed.", Thomson, W. T., and Marie Dillon Dahleh.1.Prentice Hall.ISBN 0-13-651068-X, Pp. 341 through 346.Julio C. Banks MSME Thesis - "Component Synthesis Methods for Vibrating Systems". Tufts2.University, Medford Massachusetts, May 1984.
Since the total number of coordinates used are six and there are four constraint equations, the number of generalized coordinates for the system is two (i.e., there are four superfluouscoordinates corresponding to the four constraint equations. We can thus choose any two (the first example uses the nonzero diagonal criterion) of the generalized coordinates, q. Let p1 = q1, and p6 = q6 be the generalized coordinates, and express p1 ..p6 in terms of q1, and q6according to the following steps:
Results CommentaryANSYS model uses 2D Beam Elements, while CAEFEM model utilized 3D Beam Elements (That is,ANSYS will most closely follow the closed-form solution since the latter is 2D. On the other hand,CAEFEM model had to be restrained in the out-of-plane dimension in order to emulate a 2D planeframe (CAEFEM has 3D beam elements only).
In general, the natural frequencies can be expressed as a function of = L2/L1 . The choices ofdependent, and independent coordinates follows those chosen in the reference. The results areidentical.
Appendix BDefine two (2) procedures to unit-normalize the columns of a matrix. The first algorithm determinesthe maximum magnitude in each column. The second algorithm, expands the first algorithm to the unit-normalization phase of the solution
