User`s guide
45
Key Equations
Strain in a Cantilever under Known Load
The strain in a cantilever beam under a known load applied at the free end is given by:
where is the strain, P is the applied load, L is the length of the beam, x is the distance
between the clamped end and the interested location of strain, E is the elastic modulus, b is the
width of the beam, and T is the thickness of the beam.
In this experiment, the applied load is the weight of a known mass. Therefore, we have
where M is the mass and g is gravity.
Fundamental Frequency
The equation below describes how to predict natural frequencies of cantilever beams.
where E is elastic modulus, L is effective length of beam, is the density, T is the
thickness of the beam. The dimensionless wave number = 2/wavelength.
values for
cantilever beams are: β1L = 1.8751=
, β2L = 4.6941=
, β3L = 7.8548=
, β4L =
10.99557=
, β5L = 14.1372=
, β6L = 17.279=
.
In this experiment, the effective length of the beam is close to the distance between the
outer edge of the clamp and the free end of the beam.
Vibration Amplitude, Velocity, and Acceleration
The group of equations below shows the relationship how altitude and frequency
determine position, velocity and acceleration during vibration.