1. FINITE ELEMENT ANALYSIS CONVERSION FACTORS FOR NATURAL VIBRATIONS OF BEAMS Austin Cosby and Ernesto Gutierrez-Miravete Rensselaer at Hartford

2. Euler-Bernoulli Beam Theory The beam has uniform properties The beam is slender (L/h is small) The beam obeys Hooke’s Law There is no axial load Plane sections remain plane during motion The plane of motion is the same as the beam symmetry plane Shear Deformation is Negligible

3. Euler-Bernoulli Beam Vibrations Governing Equation

4. Symmetric Angle Ply Laminates Characteristics – Extensional Stiffness Matrix A is full – Coupling Stiffness Matrix B is empty – Bending Stiffness Matrix D is full (twist coupling stiffnesses D 16 and D 26 ) – Separation of variables solution for the deflection of simply supported plates not possible – Numerical solution methods required

5. Typical Boundary Conditions and Natural Frequency Solutions

6. Finite Element Modeling – Ansys BEAM3 and SOLID45 elements – Isotropic Material Model – Input Data

7. Finite Element Modeling: Geometric Input

8. First Four Mode Shapes For Fixed-Fixed Beam Modeled with Beam Elements

9. First Four Mode Shapes For Fixed-Fixed Beam Modeled with Solid Elements

10. Finite Element Model Results: Mode 1 Frequency Comparison

11. Finite Element Model Results: Modes 1-4 Frequency Comparison

12. Conversion Factors BCMode 1 F-Fy = -0.6982x 4 + 22.732x 3 - 281.4x 2 + 1597x - 3589.6 F-Freey = -3E-05x 4 + 0.0011x 3 - 0.0141x 2 + 0.0828x + 0.8024 SSy = -0.0002x 4 + 0.0053x 3 - 0.066x 2 + 0.3753x + 0.1796 BCMode 2 F-Fy = -0.0051x 2 + 0.113x + 0.3069 F-Freey = 0.0002x 3 - 0.0076x 2 + 0.0954x + 0.5672 SSy = -0.0085x 2 + 0.159x + 0.2374 BCMode 3 F-Fy = -0.0033x 2 + 0.0869x + 0.3278 F-Freey = -0.002x 2 + 0.0569x + 0.5637 SSy = -0.0087x 2 + 0.1762x + 0.032 BCMode 4 F-Fy = -0.0033x 2 + 0.0869x + 0.3278 F-Freey = -0.0028x 2 + 0.0705x + 0.4555 SSy = -0.0067x 2 + 0.1529x + 0.006

13. Conclusions As expected, as the slenderness ratio decreases below 10, the solutions obtained using beam and solid elements diverge. The higher order modes have a larger difference between solutions at slenderness ratios greater than 10 than the lower order modes. The difference between mode 1 frequencies is less than that for the higher order modes. The conversion factors obtained allow accurate modeling and prediction of natural frequencies of vibration of beams of any slenderness ratio by using simple beam elements.