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2.3.3 Determining Vibration Amplitude, Velocity, and Acceleration
Eq.28 shows the relationship between the deflection at the free end of the beam and at
any point on the beam. The distance between the free end and the point is denoted by y.
Eq. 9 and Eq.14 addressed the derivations of strain and deflection of the beam at a point
with distance x from the clamped end.
Therefore, the expression for the deflection can be updated:
Since it is obvious that L=x+y for any point chosen, we have the expression for the peak
altitude in terms of strain, and the location of measured strain, length of the beam, and thickness
of the beam:
And taking a derivation in regards of time gives the peak velocity of the tip:
And a second order derivative of the deflection gives the peak acceleration:
The root mean square (abbreviated RMS), is a statistical measure of the magnitude of a
varying quantity. It is especially useful when variants are positive and negative, e.g., sinusoids,
RMS is used in various fields.
The RMS value of a set of values (or a continuous-time waveform) is the square root of
the arithmetic mean (average) of the squares of the original values (or the square of the function
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