The main purpose of the vibration measuring instrument is to give an output signal which represents, as closely as possible, the vibration phenomenon. This phenomenon may be displacement, velocity or acceleration of the vibrating system and accordingly, the instrument’s which reproduce signals proportional to these are called vibrometers.

SEISMIC INSTRUMENT.

Figure 16 shows the schematic of a seismic instrument which is used to measure any of the vibration phenomenon. It consists of a frame (casing) in which the seismic mass (m) is supported by means of spring (k) and dashpot (c ). The frame is mounted on a vibrating body and vibrates along with it. The relative motion between seismic mass and casing is then utilised to release it to the quantity being measured. The system reduces to a spring mass dashpot system having base or support excitation.

EXPRESSION FOR RELATIVE DISPLACEMENT.

Let the base be given a motion Y = Y sin wt and suppose as a result og vibration of base, the mass ‘m’ gets a motion as shown in figure 17.

The relative motion (y – x) is to be obtained. The equation of motion ‘m’ is $m\ddot{y} + k(x – y) + c(\dot{x} – \dot{y}) = 0$ - - - -(23)

If relative motion x – y = z, equation (22) becomes

$m\ddot{z} + c\dot{z} + kz = - m\ddot{x}$ - - - - (24)

If x =X sin wt, equation (23) becomes

$m\ddot{z} + c\dot{z} + kz= mw^2 \ X \ sin \ wt$

$\therefore$ the amplitude of relative motion z is

$z = \frac{mw^2X}{\sqrt{(k – mw^2)^2 + (cw)^2}}$

$\frac{z}{y} = \frac{mw^2}{\sqrt{(k – mw^2)^2 + (cw)^2}}$

OR

$\frac{z}{x} = \frac{(\frac{w}{w_n})^2}{\sqrt{ [1- (\frac{w}{w_n})^2]^2 + [ 2 \zeta \frac{w}{w_n}]^2}}$

OR $\frac{z}{x} = \frac{(r)^2}{\sqrt{ (1 – r^2)^2 + (2 \zeta r)^2}}$

This equation applies to both vibrometer and accelerometer.

VIBROMETER.

Vibrometer is an instrument to measure the displacement of a vibrating machine part. Its natural frequency is low compared to that of vibration to be measured.

From equation (25), if $w/w_n$ is large, z/x approaches unity.

[Phase between Z and X is 180˚ nearly.]

In other words, at high frequency seismic mass remains stationery.

Often it is said that to get a vibrometer, it is necessary to give instrument, at a natural frequency at least twice as slow as the lowest vibration to be recorded.

In short it may be said that if $w/w_n$ is large the relative motion between the mass and casing represents amplitude of vibration. The instruments then constitute a vibrometer.

ACCELEROMETER.

Accelerometer is an instrument which is constructed like vibrometer and is used like vibrometer. It records acceleration of body with which it is in contact. Its natural frequency $w_n$ is high compared to that of vibration to be measured.

As $w/w_n$ is very small is an accelerometer, in the absence of damping we may write.

$Z = \frac{(w/w_n)^2 X}{1 - \frac{w^2}{w_n^2}} = \frac{w^2X}{w_n^2}$

So the value of Z becomes proportional to $w^2X$ which is the amplitude of the acceleration to be measured.

In both vibrometer and accelerometer the relative motion between seismic mass and base is generally converted into an electrical signal which may be exhibited on an oscilloscope or measured remotely by multi voltmeter.

With accelerometer electrical integrating circuit may be used to obtained velocity or displacement.