Problem 46 - Concepts and Applications of Finite Element Analysis

Question

Model a simply supported beam by a single element. Let \(L=1.0 \mathrm{~m}, A=\) \(0.0002 \mathrm{~m}^{2}, E I=300.0 \mathrm{~N} \cdot \mathrm{m}^{2}\), and \(\rho=2100.0 \mathrm{~kg} / \mathrm{m}^{3}\). Impose symmetry (and reduce the problem to a single d.o.f.) by setting \(\theta_{2}=-\theta_{1}\). (a) Determine the fundamental frequency \(\omega_{1}\) if there is no axial force. (b) Determine the axial force that makes the frequency \(347 \mathrm{rad} / \mathrm{sec}\). (c) Determine the frequency if the axial force is \(1200 \mathrm{~N}\) in compression.

Step-by-Step Solution

Answer

The fundamental frequency is calculated using the given parameters and the standard formula. The axial force necessary for a specific frequency is derived from the relationship between axial force and natural frequency. Once the axial force is known, it is substituted back into the natural frequency equation to determine the corresponding frequency.

Step 1: Determine the Fundamental Frequency \(\omega\)

The equation for the fundamental frequency of a simply supported beam with no axial force is given by \(\omega=\frac{\pi}{2} \sqrt{\frac{EI}{\rho A L^4}}\). Substitute the given values into the equation to find \(\omega\).

Step 2: Find the Axial Force responsible for Specific frequency

The equation relating axial force to the natural frequency of a simply supported beam is given by \(\omega=\frac{\pi}{2L}\sqrt{\frac{EI}{(\rho A L^2) + (\rho A L^2 \cdot y)}}\), where \(y\) is the ratio of axial force to \(\pi^2EI/L^2\). Rearrange it to solve for \(y\) and use this to find the axial force \(P = y\cdot \pi^2EI/L^2\).

Step 3: Determining the Frequency with known Axial Force

Substitute the known axial force into the equation for \(y\) from Step 2. Then substitute \(y\) into the natural frequency equation to find \(\omega\).

Key Concepts

Natural Frequency

Natural frequency is an intrinsic property of a mechanical system and is hugely important in understanding vibrations. It defines the rate at which an object naturally vibrates when disturbed. In a simply supported beam—like in our problem—it depends on several parameters such as the beam's material properties and design.
The formula for a simply supported beam without any axial force is given as:\[\omega = \frac{\pi}{2} \sqrt{\frac{EI}{\rho A L^4}}\]- **E** represents the modulus of elasticity (a measure of stiffness).- **I** is the moment of inertia which reflects how the beam's cross-section resists bending.- **\(\rho\)** is the density of the material, influencing how the mass distributes.- **A** is the cross-sectional area of the beam.- **L** is the length of the beam.
All these parameters work together to define the beam's tendency to vibrate. If any change occurs, like in material stiffness or beam length, the natural frequency will alter. Understanding this helps us determine how a structure will react to different forces and is crucial in engineering for ensuring safety and functionality.

Axial Force

Axial force is a force applied along the direction of the length of a beam or structure. It can either compress (shorten) or stretch (lengthen) the structure. Think of it like pressing or pulling a rubber band.
In our scenario, we want to find out how large an axial force needs to be while keeping a specific frequency of 347 rad/sec. This connection between axial force and natural frequency is given by:\[\omega = \frac{\pi}{2L} \sqrt{\frac{EI}{(\rho A L^2) + (\rho A L^2 \cdot y)}}\]Where:- **\(y\)** is a dimensionless parameter expressing the ratio of axial force to the critical force (\(\pi^2 EI/L^2\)).
Adjusting the axial force alters the beam's tendency to vibrate. With higher axial compression, the frequency decreases, making it crucial to calculate correctly for secure designs.

Simply Supported Beam

A simply supported beam is a common structural element in engineering. It is supported at both ends but is free to rotate or deflect at the supports. This configuration is simple yet effective for understanding load responses.
Key characteristics:

• **Supports at both ends**: Allows the beam to bear loads across its length without fixed ends.
• **Capabilities to rotate**: The beam can experience bending but will not experience extreme stress at the supports.
• **Subject to various loads**: Supports various types of loads like axial force, making it versatile in structural applications.

Understanding the behavior of simply supported beams helps in predicting how they will react under different load conditions. In our exercise, the beam is modeled with symmetry, simplifying calculations and providing a more focused analysis on how specific forces affect natural frequency. This is essential for designing structures that need safe and reliable vibration performance.