2103433 Mechanical Vibrations NAV 1

3.2 Applications3.2.1 Rotating Unbalance Masses

Examples:

Unbalance cloth in a

rotating drum of a

washing machine

2103433 Mechanical Vibrations NAV 2

3.2.1 Rotating Unbalance Masses

Consider the system of mass M with a rotating mass m.

The eccentricity is e. The rotational speed is ω.

The system is attached to the fixed foundation via a

spring and a damper shown.

If we only consider the vertical

movement of the system, we can

determine the EOM as follows:

2103433 Mechanical Vibrations NAV 3

3.2.1 Rotating Unbalance Masses

M = total mass

m = rotating mass

E = eccentricity

ω = rotational speed

2103433 Mechanical Vibrations NAV 4

3.2.1 Rotating Unbalance Masses

Model: tmekxxcxM sin2

rir

rH

21)(

2

2

tiemekzzczM 2

)](Im[)( tztx

ti

nn eM

mezzz

222

tiZetz )(

M

me

rir

r

M

me

iZ

nn

212 2

2

22

2

)sin()( tHM

mex

←EOM

2103433 Mechanical Vibrations NAV 5

3.2 Applications

3.2.1 Rotating Unbalance Masses

rir

rH

21)(

2

2

FRF Plot of

Notes:

1. When r is small, |H(ω)|~0;

2. When r is big, |H(ω)|~1;

3. For 0 < ζ < 1/√2 , the maximum

occurs when

and the maximum value is

Increasing ζ222

2

)2()1()(

rr

rH

121

1

2

r

2max12

1)(

H

)sin( tM

mex

0x

2103433 Mechanical Vibrations NAV 6

3.2.1 Rotating Unbalance Masses

Examples: Meirovitch 3.11

A piece of machinery can be regarded as a rigid mass

with a reciprocating rotating unbalanced mass. The

total mass of the system is 12 kg and the unbalanced

mass is 0.5 kg. During normal operation, the rotation of

the masses varies from 0 to 600 rpm. Design the

support system (spring and damper) so that the

maximum vibration amplitude will not exceed 10

percent of the rotating mass’s eccentricity.

2103433 Mechanical Vibrations NAV 7

3.2.1 Rotating Unbalance Masses

Examples: Francis Water Turbine

The schematic diagram of a Francis water turbine is

shown in the figure in which water flows from A into the

blades B and down into the tail race C. The rotor has a

mass of 250 kg and an unbalance (me) of 0.25 kg-m.

The radial clearance between the rotor and the stator

is 5 mm. The turbine operates in the speed range 600

to 3000 rpm. The steel shaft carrying the rotor can be

assumed to be clamped at the bearings. Assume

damping to be negligible.

2103433 Mechanical Vibrations NAV 8

3.2.1 Rotating Unbalance Masses

3

3

l

EIk

Determine

1. Maximum amplitude of x

2. Appropriate value of k

3. Natural frequency

4. Diameter of the shaft so

that the rotor is clear of the

stator

Given that E = 2.07x1011 Pa

and

Examples: Francis Water Turbine (adapted from Rao)

64

4dI

2103433 Mechanical Vibrations NAV 9

3.2.1 Rotating Unbalance Masses

tmekxxM sin2

Examples: Solution

EOM →

Thus,

where the natural frequency is

The value of ω ranges from:

to

Solve for r for Xmax is 5 mm

)1()()( 2

2

22

2

2

2

rM

mer

M

me

Mk

meX

n

rad/s 10060

230003000rpm

rad/s 2060

2600rpm600

M

kn

)1(250

25.0105

)1(

2

23-

2

2

r

r

rM

merX

2103433 Mechanical Vibrations NAV 10

3.2.1 Rotating Unbalance Masses

n

criticalr

rr

9129.0

50250250 22

13.34483.68

16.3149129.083.62

16.31483.62

n

n

Find the ωn

To select the ωn:

1. It has to be greater than 314.16 rad/s to avoid resonance

2. If rad/s Xmax is greater than 5 mm

So pick rad/s

13.34416.314 n

13.344n

2103433 Mechanical Vibrations NAV 11

3.2.1 Rotating Unbalance Masses

Thus, if rad/s

then N/m

The minimum diameter of the shaft is

72 1096.2

13.344

n

n

Mk

cm 7.29

64

331096.2

4

33

7

d

d

l

E

l

EIk