1. 1 Numerical Study of the Vibration of a Periodically-supported Beam Final Presentation Dec 10, 2009 Brandon Rush

2. 2 Introduction Periodic structures contain repeated “building blocks” which influence their behavior A finite element code was developed in MATLAB to model their behavior –Static and dynamic analysis –Verified against analytic, numerical, and experimental results

3. 3 Methodology – Static Analysis Linear 2-dimensional Timoshenko beam elements 2 displacements and 1 rotation at each node

4. 4 Methodology – Static Analysis (cont) Verified with the deflection of a simple cantilevered beam P l x y

5. 5 Methodology – Dynamic Analysis Combined mass matrix –Mixes the strengths of lumped and consistent approaches Direct solution method –Provides an exact solution at a requested frequency –No superposition of individual modes –Performance penalty – solve the full matrix for every solution

6. 6 Methodology – Dynamic Analysis (cont) Verified with the vibration of a cantilevered beam From Timoshenko: and

7. 7 Results – Static Analysis Successfully matches the expected deflection One elementMulti-element

8. 8 Results – Dynamic Analysis Mode shapes and frequencies match analytic predictions

9. 9 Results – Multi-bay beam Code matches COMSOL, analytic, and experimental data from published work COMSOL Project code 1 st Mode: 2 nd Mode:

10. 10 Conclusion A finite element code for static and dynamic analysis of periodic beam structures was designed, implemented and validated against analytic, numeric, and published experimental data