Calculating Cantilever Beam Natural Frequency with Point Load

This question asks for the fundamental natural frequency, measured in radians per second (rad/sec), for a specific mechanical system: a cantilever beam subjected to a point load at its free end.

Understanding Natural Frequency

The natural frequency of a structure is the frequency at which it will oscillate if disturbed from its equilibrium position. For beams, this frequency is related to the beam's stiffness and mass distribution. The fundamental natural frequency is the lowest natural frequency.

Static Deflection of a Cantilever Beam

First, let's determine the static deflection ($\delta$) at the free end of a cantilever beam of length $L$, Young's modulus $E$, and area moment of inertia $I$, when a point load $P$ is applied at the free end. The formula for this static deflection is:

$$ \delta = \frac{PL^3}{3EI} $$

Relating Deflection to Natural Frequency

For a simple single-degree-of-freedom system (which can approximate the fundamental mode of a beam), the undamped natural frequency in radians per second ($\omega$) is related to the static deflection ($\delta$) and the acceleration due to gravity ($g$) by the following formula:

$$ \omega = \sqrt{\frac{g}{\delta}} $$

Deriving the Frequency Formula

Now, we substitute the expression for the static deflection $\delta$ into the natural frequency formula:

$$ \omega = \sqrt{\frac{g}{\left( \frac{PL^3}{3EI} \right)}} $$

Simplifying this expression gives:

$$ \omega = \sqrt{\frac{3EIg}{PL^3}} $$

Matching with Options

Comparing this derived formula with the given options:

• Option 1: \(\sqrt {\frac{{3EIg}}{{P{L^3}}}}\)
• Option 2: \(\sqrt {\frac{{P{L^3}}}{{3EIg}}}\)
• Option 3: \(\frac{1}{{2\pi }}\sqrt {\frac{{3EIg}}{{P{L^3}}}}\) (This represents frequency in Hz, not rad/sec)
• Option 4: \(\frac{1}{{2\pi }}\sqrt {\frac{{P{L^3}}}{{3EIg}}}\) (This represents frequency in Hz, not rad/sec)

The calculated formula matches Option 1.

Therefore, the fundamental natural frequency of the cantilever beam with a point load $P$ acting at the free end, in rad/sec, is \(\sqrt {\frac{{3EIg}}{{P{L^3}}}}\).