Advanced Methods of Structural Analysis

544 14 Dynamics of Elastic Systems

S(x)

V(x)

U(x)

T(x)

Fig. 14.14 Krylov-Duncan functions circular permutations

Table 14.4 Krylov-Duncan functions and their derivatives
Function First derivative Second derivative Third derivative Fourth derivative
S.x/ kV .x/ k^2 U.x/ k^3 T.x/ k^4 S.x/
T.x/ kS.x/ k^2 V.x/ k^3 U.x/ k^4 T.x/
U.x/ kT .x/ k^2 S.x/ k^3 V.x/ k^4 U.x/
V.x/ kU.x/ k^2 T.x/ k^3 S.x/ k^4 V.x/

To obtain frequency equation using Krylov–Duncan functions, the following al-
gorithm is recommended:

Step 1.Represent the mode shape in the form that satisfies boundary conditions at

xD 0. This expression will have only two Krylov–Duncan functions and,

respectively, two constants. The decision of what Krylov–Duncan functions

to use is based on (14.24) and the boundary condition atxD 0.

Step 2.Determine constants using the boundary condition atxDland Table14.4.

Thus, the system of two homogeneous algebraic equations is obtained.

Step 3.The nontrivial solution of this system represents a frequency equation.

Example 14.6.The beam has lengthl, mass per unit lengthm, modulus of elastic-
ityE, and moment of inertia of cross-sectional areaI(Fig.14.15). Calculate the
frequency of vibration and find the mode of vibration.

l

m, EI

x

i=1

i=2

w 1

w 2

Fig. 14.15 Design diagram for simply supported beam and mode shapes vibration foriD 1 and
iD 2

Solution.At the left end.xD0/deflection and the bending moment are zero:

1: X .0/D0;

2: X^00 .0/D0: