Bending Dynamics of beams

Massimo Ruzzene School of Aerospace Engineering [email protected] Ph: (404) 894 3078

AE 6230 Structural Dynamics

BENDING DYNAMICS

of UNIFORM BEAMS



Configuration and FBD

y

x

Free Body Diagram of element dx:



Assumptions

•  Small displacements v(x,t); •  Bending is uncoupled from torsion; •  Beam is initially straight and untwisted; •  Neglect deformation due to shear; •  Neglect rotary inertias of cross section; •  Beam is uniform.



Equation of motion

•  where: –  Bending stiffness; –  linear mass;

•  Governing equation is 4th order PDE in space and 2nd order in time.

•  The solution requires 4 boundary conditions and 2 initial conditions



•  Separation of variables

Free vibrations

where

a is a parameter defined by the application of the boundary conditions



Free vibrations and: or



Boundary conditions •  Pinned (hinged) end at x=0:



Boundary conditions

•  Clamped end at x=0:



•  Free end at x=0:

Boundary conditions



Boundary conditions •  Translational elastic constraints at x=0 and x=l:

x=0: x=l:



Boundary conditions •  Rotational elastic constraints at x=0 and x=l:

x=0: x=l:



Boundary conditions •  Translational inertial constraints at x=0 and x=l:

x=0: x=l:



Boundary conditions •  Rotational inertial constraints at x=0 and x=l:

x=0: x=l:



Examples of results of calculations of natural frequencies and mode

shapes



Simply supported beam

Characteristic equation:

Natural frequencies:



Simply supported beam

Mode shapes:

i=1 i=2

i=3 i=4

Node



Clamped-free beam





Clamped-free beam

Mode shapes:



Clamped-free beam Mode shapes:



Free-free beam





Free-free beam

Mode shapes:



Free-free beam Mode shapes (rigid body modes):



Free-free beam Mode shapes:

Bending Dynamics of beams

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