Vibration Conversion 5u2e6s

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 m (pure unbalance, for example): velocity: 5,642 mm/s rms displacement: 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 ing 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 have

You want A, f[Hz]

V, f[Hz]

X, f[Hz]

—

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

—

Acceleration, A =

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

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

You have

You want A, f[Hz]

V, f[Hz]

X, f[Hz]

—

2πf V@(φv+90°) ≈ 6.28f V@(φv+90°)

(2πf) 2 X@(φx+180°) ≈ 39.5f2 X@(φx+180°)

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

—

2πf X@(φx+90°) ≈ 6.28f X@(φx+90°)

Acceleration, A@φa = Velocity, V@φv =

Displacement, 1/(2πf) 2 A@(φa−180°) 1/(2πf) V@(φv−90°) ≈ 1/(39.5f2) ≈ 1/(6.28f) X@φx = A@(φa−180°) 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 m = (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.