1 MODULE 011 VIBRATION MEASURING DEVICES. 2 Amplification of displacement amplitude (transmissibility of displacement amplitude) DISPLACEMENT TRANSMISSIBILITY.

1

MODULE 011

VIBRATION MEASURING DEVICES

2Amplification of displacement amplitude (transmissibility of displacement amplitude)

DISPLACEMENT TRANSMISSIBILITY

FROM BASE TO MASS IN BASE EXCITATION

Relative / base

Absolute / base

X

Z

Y

r

For larger values of r the relative displacement and the displacement of the base have the same amplitude.

Hence the device can be used to measure base displacement if the frequency of base displacement is at least

three time the device natural frequency

VIBRATION MEASURING DEVICES - SEISMOMETER

relative displacement Zlim 1

base displacementYr

When r → ∞

Seismometer

For large r (r>3) the relative displacement and the displacement of the base have the same magnitude.

Therefore, a device with a low natural frequency can be used to measure base displacement if the frequency of the

base displacement is at least three times higher than the natural frequency of the device. This device is called

seismometer.

Seismometer measures amplitude of displacement

Inman p. 146

ζ = 0.25

ζ = 0.5

Seismometer range

b

n

Natural frequency of device:

Frequency of base excitation:

b

n

3b

n

For seismometer:

VIBRATION MEASURING DEVICES - SEISMOMETER

k = 500N/mc = 39.7 Ns/m

m = 4.38kg

Base excitation:

x =10cos(50*t)

Half amplitude = 10mmCircular excitation frequency 50 rad/sPeriod 0.126s

seismometer.SLDASM

k = 500N/m

VIBRATION MEASURING DEVICES - SEISMOMETER

6

Steady statehalf amplitude: 12mm

Transient response

Steady state response

Results of time response analysis

VIBRATION MEASURING DEVICES - SEISMOMETER

Results of frequency response analysis

(50rad/s, 12.4mm)

VIBRATION MEASURING DEVICES - SEISMOMETER

8Seismometer Measures base displacement

VIBRATION MEASURING DEVICES - SEISMOMETER

Inman p. 146

can be presented as

2 2 20

1lim 1

(1 ) (2 )r r r

therefore is proportional to base acceleration

What happens when r → 0 ?Solution of the relative displacement

VIBRATION MEASURING DEVICES - ACCELEROMETER

10

Accelerometer

For small r (r< 0.5) the relative displacement is approximately proportional to the base acceleration. This proportionality is best for ζ = 0.7

Therefore, a device with a high natural frequency can be used to measure acceleration. This device is called accelerometer.

Accelerometer measures acceleration

ζ = 0.25

ζ = 0.5

ζ = 0.7

0 0.2

0 0.2

b

n

b n

Accelerometer range

b

n

Natural frequency of device

Frequency of base excitation

b

n

VIBRATION MEASURING DEVICES - ACCELEROMETER

Accelerometer measures base acceleration

VIBRATION MEASURING DEVICES - ACCELEROMETER

Displacement amplitude of mass is proportional to base acceleration.

To prove it we subject a SDOF with natural frequency 100rad/s to base acceleration 5m/s2

and then 10m/s2 with frequency range 0-10Hz. This frequency range satisfies the

requirement: 0 0.2 b n

m=10kg

k=100000N/m

ζ=0.7

m=10kg

k=100000N/m

ζ=0.7

5m/s2 10m/s2

accelerometer.SLDASM

2 ( ) ( )nz t y t

VIBRATION MEASURING DEVICES - ACCELEROMETER

Frequency range 0-10 rad/s

Base acceleration 5m/s2

Frequency range 0-10 rad/s

Base acceleration 10m/s2

VIBRATION MEASURING DEVICES - ACCELEROMETER

Frequency range 0-300 rad/s

Base acceleration 5m/s2

Frequency range 0-300 rad/s

Base acceleration 10m/s2

VIBRATION MEASURING DEVICES - ACCELEROMETER