conversion between displacement, velocity and acceleration _ cbmapps

11/27/13 Conversion Between Displacement, Velocity and Acceleration | CBMApps.com

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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 (30) 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 trough the

equilibrium position (points 3 and 4).



Phase difference between acceleration, velocity and displacement

Amplitude Relations

The amplitude (31) 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:

You want You have

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

Acceleration, A = 2f V

6.28f V

(2f)2 X

39.5f2 X

Velocity, V =1/(2f) A

2f X



Amplitude conversion between sinusoidal acceleration, velocity and

displacement.

1/(6.28f) A 6.28f X

Displacement, X =1/(2f)2 A

1/(39.5f2) A

1/(2f) 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@a =

2f V@(v+90)

6.28f V@(v+90)

(2f)2 X@(x+180)

39.5f2 X@(x+180)

Velocity,

V@v =

1/(2f) A@(a90)

1/(6.28f) A@(a90)

2f X@(x+90)

6.28f X@(x+90)

Displacement,

X@x =

1/(2f)2 A@(a180)

1/(39.5f2) A@(a180)

1/(2f) V@(v90)

1/(6.28f) V@(v90)

Units

The formulas presented do not modify the type of amplitude measurement (pk, pk-pk or rms) (31) . 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/(2f) (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 (http://www.cbmapps.com/docs/31) 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

(http://www.cbmapps.com/appshelp/8) to convert between sinusoidal displacement, velocity and acceleration.

conversion between displacement, velocity and acceleration _ cbmapps

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