Advanced Methods of Structural Analysis

540 14 Dynamics of Elastic Systems

14.4.1 Differential Equation of Transversal Vibration of the Beam

Differential equation of the uniform beam is

EI

d^4 y

dx^4

Dq; (14.12)

whereyis the transverse displacement of a beam,Ethe modulus of elasticity,Ithe
moment of inertia of the cross section about the neutral axis, andqthe transverse
load per unit length of the beam.
In case of free vibration, the load per unit length is

qDA

d^2 y

dt^2

; (14.13)

whereis the mass density andAis the cross-sectional area.
Equations (14.12)and(14.13) lead to following differential equation of the trans-
verse vibration of the uniform Bernoulli-Euler beam

EI

@^4 y

@x^4

CA

@^2 y

@t^2

D0: (14.14)

If a beam is subjected to forced loadf.x; t/, then the mathematical model is

EI

@^4 y

@x^4

CA

@^2 y

@t^2

Df.x;t/: (14.14a)

Thus the transverse displacement of abeam depends on the axial coordinatex
and timet, i.e.,yDy.x; t/.
Boundary and initial conditions: The classical boundary condition takes into
account only the shape of the beam deflection curve at the boundaries. The nonclas-
sical boundary conditions take into account the additional mass, the damper, as well
as the translational and rotational springs at the boundaries. The classical boundary
conditions for the transversal vibration of a beam are presented in Table14.3.

Notation:yandare transversal deflection and slope;MandQare bending mo-
ment and shear force.

Initial conditions present the initial distribution of the displacement and the initial
distribution of the velocities of each point of a beam attD 0

y.x;0/Du.x/I

dy

dt

.x; 0/D



y.x; 0/D.x/: