Question

The clamped-clamped beam shown is subjected to a concentrated load \(P(t)=\) \(P_{0}\) that is suddenly applied to the center of the span. Determine the response of the beam after the load is suddenly removed.

Short answer

Short Answer: The deflection of a clamped-clamped beam subjected to a concentrated load at its center and then removed is given by the equation: $$w(x,t) = \sum_{n=1}^\infty \frac{2P_0}{(\alpha_n^2\beta^2)\pi^2 n^2 } (-1)^{n+1} \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha_n^2\beta^2t}$$ This equation represents the deflection at any point x along the beam and at any time t after the load is removed.

Step-by-step solution

Determine the deflection equation from the load function P(t)

To find the deflection equation for the clamped-clamped beam, we will use the following general equation: $$w(x,t) = \sum_{n=1}^\infty A_n \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha_n^2\beta^2t}$$ where \(w(x,t)\) is the deflection of the beam, \(A_n\) is the amplitude of response for the nth mode, \(x\) is the distance along the beam, \(t\) is the time, \(L\) is the length of the beam, and \(\alpha_n\) is the dimensionless root of the nth mode, \(\beta\) is the wave speed. The equation of the concentrated load when it is suddenly applied is given by: $$P(t) = P_0$$

Compute the initial conditions

The initial conditions will help us find the coefficients A_n for the modal responses. Compute the initial displacement and initial velocity at any given point x: Initial displacement: \(w(x,0) = 0\) Initial velocity: \(\frac{\partial w}{\partial t}(x,0) = 0\)

Calculate the coefficients for the modal responses

Now we need to find the coefficients A_n for the response modes using the initial conditions: $$A_n = \frac{1}{(\alpha_n^2\beta^2)\pi} \int_0^L P_0 \sin\left(\frac{n\pi x}{L}\right) dx$$ Since the load is applied at the center of the span, the equation becomes: $$A_n = \frac{2P_0}{(\alpha_n^2\beta^2)\pi^2 n^2} (-1)^{n+1}$$

Find the time-dependent response after the load is removed

Substitute the coefficients A_n back into the deflection equation and sum up the terms: $$w(x,t) = \sum_{n=1}^\infty \frac{2P_0}{(\alpha_n^2\beta^2)\pi^2 n^2 } (-1)^{n+1} \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha_n^2\beta^2t}$$

Create an equation for the deflection at any point along the beam after the load is removed

The final equation for the deflection at any point along the beam after the load is removed can be written as: $$w(x,t) = \sum_{n=1}^\infty \frac{2P_0}{(\alpha_n^2\beta^2)\pi^2 n^2 } (-1)^{n+1} \sin\left(\frac{n\pi x}{L}\right) e^{-\alpha_n^2\beta^2t}$$ This equation will give us the deflection of the beam at any point x along the beam and at any time t after the load is removed.

Key concepts

Understanding Clamped-Clamped Beam Vibration

In structural engineering, a clamped-clamped beam refers to a beam that is fixed at both ends, restricting all forms of movement. This means that both rotations and translations are not allowed at the ends, making the beam rigidly supported. This setup significantly impacts the beam's vibration characteristics. When a concentrated load, such as a sudden weight or pressure, is applied in the middle of its span, it begins to vibrate.

Clamped beams tend to have higher stiffness, which in turn affects their natural frequencies. The natural frequency is a critical factor since it represents how a structure inherently vibrates when disturbed by an external force. Because of the clamped conditions, these beams tend to have evenly spaced mode shapes that are sine waves characterized by nodes and antinodes. This means higher resistance to bending because a larger force is required to achieve the same deflection compared to other types of support like simply supported beams.

Exploring Modal Analysis for Beam Response

Modal analysis is a technique used to determine the vibration modes of a structure. In the context of beam analysis, it helps us determine how different modes or shapes affect the system's response to external loads. Each mode is characterized by a particular shape and frequency.

When a clamped-clamped beam is analyzed, the response is expressed as a summation of its mode shapes. These modes are unique to the beam's geometry and boundary conditions and are mathematically represented by sinusoidal functions. Modal analysis utilizes these functions to break down complex vibrations into simpler components, allowing us to comprehend how various frequency components react independently. This approach is vital because it helps minimize errors when predicting the deflection and stress distribution along the beam's length.

Deflection Calculation Fundamentals

Deflection in beams refers to the displacement of the beam from its original position due to an external load. In a clamped-clamped beam, calculating deflection involves integrating the effects of its modes.

The deflection at any point along the beam is determined using the series expansion of sine functions, summing infinite modes due to their ability to define the beam’s movement accurately. The primary goal of this calculation is to find how much the beam bends when subjected to a concentrated load. The intrinsic modal amplitudes, determined via initial conditions and the load's characteristics, directly influence the magnitude of deflection encountered. These calculations inform engineering choices concerning the beam's ability to withstand applied loads without excessive bending.

Concentrated Load Response in Beams

A concentrated load is a load applied at a specific point on the beam, causing significant localized stress and deflection. For a clamped-clamped beam, a load applied at the midpoint initiates a specific response pattern.

When the load is suddenly applied, it causes the beam to bend and vibrate, and this response is modeled through equations that take into account the beam's boundary conditions and material properties. Once the load is removed, the beam will continue to oscillate until all its vibrational energy dissipates.

The response of the beam to such a load is critical in designing structures to ensure they can recover from any sudden impacts or applied forces without undergoing permanent deformation. This aspect of structural design ensures safety and functionality over the structure's lifespan.