If we have a machine is running at 6,000 rpm. It is vibrating at 1 mil Pk-Pk. Calculate the velocity & acceleration of the vibration in mm/s & mm/S2. Explain the calculation.

In short, assuming pure sinusoidal vibration at 6.000 cpm (pure unbalance, for example):

velocity: 5,642 mm/s rmsdisplacement: 3545 mm/s2 rms

Rms values are those commonly used in ISO standards, for pk or pk-pk values you would have to multiply by sqrt(2) and 2*sqrt(2) respectively.

Formulas are derived from the fact that velocity is the integral of acceleration and displacement the integral of velocity, and assuming sinusoidal vibration at one single frequency*, I've compiled them, with an online calculator to do these conversions in: http://onlinevib.synthasite.com/Vibr...calculator.php

* In fact, you are also assuming zero mean acceleration, velocity and displacement, i.e. the machine vibrates around an equilibrium point and does not move around (if it moves from its foundation you'd better turn it off :-)

You can use the online calculator at http://cbmapps.com/apps/34 to do this kind of conversions (provided the vibration is sinusoidal, the theory is in http://www.cbmapps.com/docs/28).

That is maximum velocity. The maximum acceleration will be the derivative of the velocity, i.e.

if: v = Asin(Bt)

Then, a = ABcos(Bt)

In other words, your maximum acceleration will be the maximum velocity times the frequency (by way of chain rule).

Conversion Between Displacement, Velocity and Acceleration

Vibration is a form of movement; in consequence, the relations between acceleration,

velocity and displacement are governed by simple kinematics; acceleration is the derivative

of velocity, which in turn is the derivative of displacement:

Conversely, displacement is the integral of velocity, which in turn is the integral of

acceleration:

For an arbitrary vibration signal, the only way to convert one of these measures into another

would be to know the complete time waveform and differentiate or integrate it. Fortunately,

the integral and derivative of a sinusoidal function are also sinusoidal functions, so for

sinusoidal waveforms these relations simplify to (the intermediary math has been omitted):

From displacement to velocity and acceleration:

From acceleration to velocity and displacement:

With frequency in Hz and phase in radians.

It is important to observe that if one of the three variables —acceleration, velocity or

displacement— is sinusoidal, the other two are also sinusoidal at the same frequency; only

amplitude and phase change.

Phase Relations

Phase relations are fairly intuitive and independent of amplitude and frequency. The phase

difference between acceleration and displacement is always 180°, which means that when

the object reaches its maximum displacement from the equilibrium position, the acceleration

is maximum in the opposite direction (see points 1 and 2 in the figure below). Velocity always

lags acceleration by 90° and leads displacement by 90°: it is maximum when both

acceleration and displacement are zero, that is, when passing through the equilibrium

position (points 3 and 4).

Phase difference between acceleration, velocity and displacement

Amplitude Relations

The amplitude of acceleration, velocity and displacement are related by factors that depend

on vibration frequency. For a given velocity amplitude, for example, the corresponding

displacement amplitude is higher at low frequencies by a factor proportional to 1/f and

acceleration is higher at high frequencies, by a factor proportional to f. This relations explain

why low frequency vibration is emphasized by displacement measures and high frequency

vibration by acceleration, as illustrated in the following figure:

Sinusoidal acceleration and displacement amplitude as a function of frequency for a fixed velocity

amplitude of 1 mm/s rms

Units in this figure were chosen because they are commonly used and to make the curves fit

in the plot. If different units are used, the scale of the curves will vary but their general form

remains the same.

Conversion Formulas

The conversion formulas for amplitude only are summarized in the following table:

Amplitude conversion between sinusoidal acceleration, velocity and displacement.

You want You have

A, f[Hz] V, f[Hz] X, f[Hz]

Acceleration, A = — 2πf V

≈ 6.28f V(2πf)2 X

≈ 39.5f2 X

Velocity, V =1/(2πf) A

≈ 1/(6.28f) A—

2πf X ≈ 6.28f X

Displacement, X =1/(2πf)2 A

≈ 1/(39.5f2) A 1/(2πf) V

≈ 1/(6.28f) V—

To take into account the phase, the formulas are (using the notation aplitude@phase):

Amplitude and phase conversion between sinusoidal acceleration, velocity and displacement.

You want You have

A, f[Hz] V, f[Hz] X, f[Hz]

Acceleration, A@φa =

— 2πf V@(φv+90°)

≈ 6.28f V@(φv+90°)

(2πf)2 X@(φx+180°)

≈ 39.5f2

X@(φx+180°)

Velocity, V@φv =

1/(2πf) A@(φa−90°) ≈ 1/(6.28f) A@(φa−90°)

— 2πf X@(φx+90°)

≈ 6.28f X@(φx+90°)

Displacement,

X@φx =

1/(2πf)2 A@(φa−180°)

≈ 1/(39.5f2) A@(φa−180°)

1/(2πf) V@(φv−90°) ≈ 1/(6.28f) V@(φv−90°)

—

Units

The formulas presented do not modify the type of amplitude measurement (pk, pk-pk or

rms). They do not transform the units used, either. When applying these formulas, care has

to be taken to convert the result to the desired units.

Example

If we want to convert a sinusoidal acceleration of 0.1g rms into velocity in in/s pk, and we

don't care about the phase, we can proceed as follows:

A = 0.1g  = 0.1 x 32.17ft/s2  = 3.217ft/s2  ≈ 38.6in/s2 

f = 4500 cpm = (4500/min)x(1min/60s) = 75/s

V = A/(2πf) ≈ (38.6m/s2) / (6.28 x 75/s) = 0.082in/s

As the acceleration amplitude was rms, so is the obtained velocity. We use the formulas in

the Amplitude section to get:

V ≈ 0.11in/s pk

As you seem, calculations can be tricky... These are the formulas used by the sinusoidal

vibration calculator to convert between sinusoidal displacement, velocity and acceleration.