FINITE ELEMENT ANALYSIS CONVERSION FACTORS FOR NATURAL VIBRATIONS OF

BEAMS

Austin Cosby and Ernesto Gutierrez-Miravete

Rensselaer at Hartford

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

Euler-Bernoulli Beam VibrationsGoverning Equation

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 D16 and D26)– Separation of variables solution for the deflection

of simply supported plates not possible– Numerical solution methods required

Typical Boundary Conditions and Natural Frequency Solutions

Finite Element Modeling

– Ansys BEAM3 and SOLID45 elements– Isotropic Material Model – Input Data

Finite Element Modeling: Geometric Input

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

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

Finite Element Model Results: Mode 1 Frequency Comparison

Finite Element Model Results: Modes 1-4 Frequency Comparison

Conversion Factors

BC Mode 1

F-F y = -0.6982x4 + 22.732x3 - 281.4x2 + 1597x - 3589.6F-Free y = -3E-05x4 + 0.0011x3 - 0.0141x2 + 0.0828x + 0.8024

SS y = -0.0002x4 + 0.0053x3 - 0.066x2 + 0.3753x + 0.1796

BC Mode 2F-F y = -0.0051x2 + 0.113x + 0.3069

F-Free y = 0.0002x3 - 0.0076x2 + 0.0954x + 0.5672SS y = -0.0085x2 + 0.159x + 0.2374

BC Mode 3F-F y = -0.0033x2 + 0.0869x + 0.3278

F-Free y = -0.002x2 + 0.0569x + 0.5637SS y = -0.0087x2 + 0.1762x + 0.032

BC Mode 4F-F y = -0.0033x2 + 0.0869x + 0.3278

F-Free y = -0.0028x2 + 0.0705x + 0.4555SS y = -0.0067x2 + 0.1529x + 0.006

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.