mech300_chap7

Mechanical Vibrations Determination of Natural Freq and Mode Shape (Rao’s Chap 7) 7.1Introduction

7.3Rayleigh’s Method

7.5Matrix Iteration Method

7.8 Computer method

7.1Introduction

?Computing the natural frequencies and modes by solving a n-th degree polynomial equation can be tedious

?In this chapter we shall consider two methods:–Rayleigh’s method

–Matrix iteration method

Other methods (Dunkerley’s formula, Holzer’s method, Jacobi’s method,etc) in the textbooks which will not be covered in the

lecture.

?In general,

?i.e. Rayleigh’s quotient is never lower than the

1st eigenvalue.

?Similarly we can show that ?i.e. Rayleigh’s quotient is never higher than the

highest eigenvalue.

()2

n

X R ω≤r ()2

21 n

X R ωω≤≤∴r ()2

1

212 ,...3,2for ωωω≥∴=>X R i i r

Example 7.2

?Estimate the fundamental

frequency of vibration of the system as shown. Assume that m 1=m 2=m 3=m , k 1=k 2=k 3=k , and

the mode shape is ??

?

?

?

?????

=321X r

7.3.3Fundamental Frequency of Beams and Shafts ?Static deflection curve is used to approximate

the dynamic deflection curve.

?Consider a shaft carrying several masses as shown below.

Example 7.3

?Estimate the fundamental frequency of the

lateral vibration of a shaft carrying 3 rotors

(masses), as shown below with m1=20kg,

m2=50kg, m3=40kg, l1=1m, l2=3m, l3=4m and l4=2m. The shaft is made of steel with a solid circular cross section of diameter 10cm.