Cantilever Beam Vibration Analysis

Understanding the natural frequency of a structure, like a cantilever beam, is crucial in engineering to prevent resonance and ensure stability. Natural frequency refers to the frequency at which a system tends to oscillate in the absence of any driving or damping force. For a cantilever beam, this frequency depends on its physical properties, such as its material, cross-sectional shape, and length.

Beam Properties and Natural Frequency

The natural frequency ($\omega$) of a vibrating system is generally related to its stiffness ($k$) and mass ($m$) by the formula:

$$ \omega \propto \sqrt{\frac{k}{m}} $$

For a cantilever beam, the stiffness ($k$) is significantly influenced by the material's Young's modulus ($E$), the beam's cross-sectional Moment of Inertia ($T$), and its length ($L$). Specifically, the stiffness is inversely proportional to the cube of the length:

$$ k \propto \frac{E \cdot T}{L^3} $$

The mass ($m$) of the beam is proportional to the material's density ($\rho$), the cross-sectional area ($A$), and the length ($L$):

$$ m \propto \rho \cdot A \cdot L $$

Substituting these relationships into the general frequency formula, we find that the natural frequency ($\omega$) is proportional to the square root of the stiffness divided by the mass:

$$ \omega \propto \sqrt{\frac{E \cdot T / L^3}{\rho \cdot A \cdot L}} = \sqrt{\frac{E \cdot T}{\rho \cdot A \cdot L^4}} $$

This simplifies to:

$$ \omega \propto \frac{1}{L^2} \sqrt{\frac{E \cdot T}{\rho \cdot A}} $$

This equation shows that the natural frequency is inversely proportional to the square of the beam's length ($L^2$).

Impact of Halving Beam Length

The problem states that the original cantilever beam has a length $L_1 = L$. When the beam is broken into two halves, we consider the remaining cantilever beam which will have a new length $L_2$. Assuming the break occurs at the midpoint, the new length is:

$$ L_2 = \frac{L}{2} $$

The other properties, such as the cross-sectional area ($A$) and the moment of inertia ($T$) of the cross-section, remain the same for the remaining part of the beam. The material properties ($E$ and $\rho$) also remain unchanged.

Calculating New Natural Frequency ($\omega_2$)

Using the relationship derived earlier, $\omega \propto \frac{1}{L^2}$, we can compare the original natural frequency ($\omega_1$) with the new natural frequency ($\omega_2$):

For the original beam:

$$ \omega_1 \propto \frac{1}{L_1^2} $$

For the new, shorter beam:

$$ \omega_2 \propto \frac{1}{L_2^2} $$

To find the relationship between $\omega_2$ and $\omega_1$, we can look at their ratio:

$$ \frac{\omega_2}{\omega_1} = \frac{1/L_2^2}{1/L_1^2} = \frac{L_1^2}{L_2^2} $$

Now, substitute the value of the new length $L_2 = L_1/2$:

$$ \frac{\omega_2}{\omega_1} = \frac{L_1^2}{(L_1/2)^2} = \frac{L_1^2}{L_1^2 / 4} = 4 $$

This implies that the new natural frequency is four times the original natural frequency:

$$ \omega_2 = 4 \cdot \omega_1 $$

Comparing Frequencies $\omega_1$ and $\omega_2$

Since the natural frequency $\omega_1$ must be a positive value (as it represents a physical property of the beam), multiplying it by 4 results in a larger positive value. Therefore, the new natural frequency $\omega_2$ is greater than the original natural frequency $\omega_1$.

$$ \omega_2 > \omega_1 $$