Advanced Methods of Structural Analysis

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 543

The mode shape of vibration is

Xi.x/DDsinkixDDsin

i

l

x; iD1; 2; 3; : : :

14.4.3 Krylov–Duncan Method......................................

A general solution of differential equation (14.18) may be presented in the form

X.kx/DC 1 S.kx/CC 2 T.kx/CC 3 U.kx/CC 4 V.kx/; (14.21)

where X.kx/ is the general expression for mode shape; S.kx/; T .kx/;

U.kx/; V .kx/are the Krylov–Duncan functions (Krylov, 1936; Duncan, 1943).

They are present the combination of trigonometric and hyperbolic functions.

S.kx/D

1

2

.coshkxCcoskx/

T.kx/D

1

2

.sinhkxCsinkx/

U.kx/D

1

2

.coshkxcoskx/

V.kx/D

1

2

.sinhkxsinkx/

(14.22)

The constantsCimay be expressed in terms of initial parameters as follows

C 1 DX.0/; C 2 D

1

k

X^0 .0/; C 3 D

1

k^2

X^00 .0/; C 3 D

1

k^3

X^000 .0/ (14.23)

Each combination (14.22) satisfies to equations of the free vibration of a uni-

form Bernoulli-Euler beam. The functions (14.22) have the following important

properties:

1.Krylov–Duncan functions and their derivatives result in the unit matrix atxD 0.

S.0/D1S^0 .0/D0S^00 .0/D0S^000 .0/D 0

T.0/D0T^0 .0/D1T^00 .0/D0T^000 .0/D 0

U.0/D0U^0 .0/D0U^00 .0/D1U^000 .0/D 0

V.0/D0V^0 .0/D0V^00 .0/D0V^000 .0/D 1

(14.24)

2.Krylov–Duncan functions and their derivatives satisfy to circular permutations
(Fig.14.14,Table14.4).
These properties of the functions (14.22) may be effectively used for deriving of
the frequency equation and mode shape of free vibration.