### Tech Briefs

Frequency Analysis of Shell Structures with Contact

Shell constrained to rings

Shell in contact with rings

In the News of July 30, 2008 we demonstrated how ADINA can be used to perform frequency analyses of nonlinear problems involving contact and bolts. Now we focus on the solution of frequencies of shell structures including contact. The response of shell structures is highly sensitive to the boundary conditions, see reference [1]. This sensitivity is seen in the deformations under loading, and also in their natural frequencies and mode shapes. Hence, the capability to perform a frequency analysis at any solution step of a nonlinear analysis is very valuable for shell structures.

As an illustration, we consider a cylindrical shell constrained at both ends by rigid rings and carrying only its own weight (see Figure 1 for details). When the shell is fully constrained to the rings (that is, in essence, the shell is fixed at both ends), the first 6 natural frequencies are as given in Table 1 below. If however, contact conditions are assumed between the shell and the rigid rings, together with friction, the top part of the shell detaches (due to the shell weight). This small change in boundary conditions significantly alters the natural frequencies as shown in the table. The animations above show the 1st and 6th frequencies and mode shapes for both cases. Note how the mode shapes are also significantly different for both problems.

Figure 1  Cylindrical shell constrained at both ends; this constraint involves
supports at the ends that make the effective length approximately 0.38

 TABLE 1  Frequencies of shell constrained to and in contact with rings

 Freq. No. Freq. (Hz) Shell constrained to rings Shell in contact with rings 1 724.8 108.4 2 724.8 231.3 3 741.3 268.3 4 741.3 414.5 5 938.2 433.5 6 938.2 579.0

This example clearly illustrates the value of the ADINA capability to solve for frequency and mode shapes in nonlinear deformed configurations.

Keywords:
Frequency analysis, contact, shell structures, shell elements, nonlinear finite elements

Reference

1. D. Chapelle and K.J. Bathe, The Finite Element Analysis of Shells — Fundamentals, Springer, 2003.