Advanced Methods of Structural Analysis

542 14 Dynamics of Elastic Systems

The general solution of (14.18)is

X.x/DAcoshkxCBsinhkxCCcoskxCDsinkx; (14.19)

whereA, B, C,andDmay be calculated using theboundaryconditions.
The natural frequency!of a beam is defined by equation

!Dk^2

r

EI

m

D

^2

l^2

r

EI

m

; whereDkl: (14.20)

To obtain frequency equation using general solution (14.19), the following algo-
rithm is recommended:

Step 1.Represent the mode shape in the general form (14.19), which contains four

unknown constants.

Step 2.Determine constants using the boundary condition atxD 0 andxDl.

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

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

Example 14.5.Calculate the frequencies of free vibration and find the correspond-
ing mode shapes for pinned-pinned beam. The beam has lengthl, mass per unit
lengthm, modulus of elasticityE, and moment of inertia of cross-sectional areaI.

Solution.The shape of vibration may be presented in form (14.19). For pinned-
pinned beam displacement and bending moment atxD 0 and atxDlequal zero.
Expression for bending moment is

X^00 .x/Dk^2 .AcoshkxCBsinhkxCcoskxDsinkx/ :

ConditionsX.0/D 0 andX^00 .0/D 0 leads to the equations

ACCD 0

ACD 0

ThusADCD 0.

ConditionsX.l/D 0 andX^00 .l /D 0 leads to the equations

BsinhklCDsinklD 0

BsinhklDsinklD 0

ThusBD 0 andDsinklD 0. Non-trivial solution occurs, if sinklD 0 .This
is frequency equation. Solution of this equation iskl D ; 2;:::Thus, the
frequencies of vibration are

!Dk^2

r

EI

m

;! 1 D

3:1416^2

l^2

r

EI

m

;! 2 D

6:2832^2

l^2

r

EI

m