bending dynamics of beams
Post on 03-Dec-2015
246 Views
Preview:
DESCRIPTION
TRANSCRIPT
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
BENDING DYNAMICS
of UNIFORM BEAMS
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Configuration and FBD
y
x
Free Body Diagram of element dx:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
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.
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
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
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
• Separation of variables
Free vibrations
where
a is a parameter defined by the application of the boundary conditions
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Free vibrations and: or
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Boundary conditions • Pinned (hinged) end at x=0:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Boundary conditions
• Clamped end at x=0:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
• Free end at x=0:
Boundary conditions
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Boundary conditions • Translational elastic constraints at x=0 and x=l:
x=0: x=l:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Boundary conditions • Rotational elastic constraints at x=0 and x=l:
x=0: x=l:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Boundary conditions • Translational inertial constraints at x=0 and x=l:
x=0: x=l:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Boundary conditions • Rotational inertial constraints at x=0 and x=l:
x=0: x=l:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Examples of results of calculations of natural frequencies and mode
shapes
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Simply supported beam
Characteristic equation:
Natural frequencies:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Simply supported beam
Mode shapes:
i=1 i=2
i=3 i=4
Node
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Clamped-free beam
Characteristic equation:
Natural frequencies:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Clamped-free beam
Mode shapes:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Clamped-free beam Mode shapes:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Free-free beam
Characteristic equation:
Natural frequencies:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Free-free beam
Mode shapes:
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Free-free beam Mode shapes (rigid body modes):
Massimo Ruzzene School of Aerospace Engineering massimo.ruzzene@ae.gatech.edu Ph: (404) 894 3078
AE 6230 Structural Dynamics
Free-free beam Mode shapes:
top related