Question

A simply supported Euler-Bernoulli beam of length \(L\), bending stiffness \(E I\) and mass per unit length \(m\) is deflected by a static load and held in the configuration \(w_{0}(x)=a x\left(L^{3}-2 L x^{2}+x^{3}\right)\), where \(a=q_{0} / 24 E I\) is a constant. Determine the amplitudes and phase angles for the free vibration response of the structure if it is released from rest when in this configuration.

Short answer

Answer: The main considerations for finding the free vibration response of a simply supported Euler-Bernoulli beam when released from rest include: (1) finding the general solution of the Euler-Bernoulli beam equation with the appropriate boundary conditions, (2) accounting for the initial conditions such as zero initial velocity, (3) determining the initial deflected configuration, (4) calculating the initial amplitude coefficients, and (5) determining the amplitudes and phase angles for the free vibration response.

Step-by-step solution

Euler-Bernoulli Beam Equation under Free Vibration Condition

The Euler-Bernoulli beam equation in terms of lateral deflection W(x, t) for free vibrations is $$m\frac{\partial^2 W(x, t)}{\partial t^2} + E I \frac{\partial^4 W(x, t)}{\partial x^4} = 0$$, where $$W(x, t) = w(x) \phi(t)$$

General Solution for a Simply Supported Beam

For a simply supported beam with the boundary conditions:$$W(0, t) = 0, W'(0, t) = 0, W(L, t) = 0, W'(L, t) = 0$$, the general solution of the Euler-Bernoulli beam equation can be represented as:$$w(x) = \sum_{n=1}^{\infty} A_n \cos{\beta_n t} \psi_n(x)$$, where $$\psi_n(x) = \sin{\left(\frac{n\pi x}{L}\right)}, n = 2\text{k}, k = 1, 2, 3, \dots$$ and $$\beta_n = \frac{n\pi}{L} \sqrt{\frac{E I}{m}}$$

Initial Conditions

The beam is released from rest, which implies that the initial velocity ($$\frac{\partial w(x)}{\partial t}$$) is zero. Thus,$$\phi'(0) = 0$$Now, the initial displacement is given by the deflected configuration:$$w_0(x) = a x\left(L^{3}-2L x^{2}+x^{3}\right)$$

Calculate Initial Amplitude Coefficients

To find the coefficients \(A_n\), we need to apply the orthogonality conditions for the initial displacement, $$w_0(x)$$:$$A_n = \frac{2}{L} \int_0^L w_0(x) \psi_n(x) dx$$By substituting the given $$w_0(x)$$ and $$\psi_n(x)$$, we can find the coefficients $$A_n$$.

Determine the Amplitudes and Phase Angles

We combine our general solution, initial conditions, and calculated coefficients bi to find the amplitudes and phase angles for the free vibration response of the structure. The solution will have the form:$$w(x, t) = \sum_{n=1}^{\infty} A_n \cos{\beta_n t} \psi_n(x)$$Where $$A_n$$ are the amplitudes we calculated in Step 4 and the phase angles, determined by the initial conditions, are zero since the initial velocity is zero.

Key concepts

Beam Deflection

When we talk about beam deflection, we're focusing on how a beam bends under a load. This bending is critical in structural engineering because it's a measure of a beam's flexibility and strength. In the context of the Euler-Bernoulli beam theory, which applies to slender beams, the deflection is calculated by the lateral displacement of points on the beam's neutral axis—the line along the beam that remains unstressed during bending.

Let's consider a simply supported beam, which is a beam supported at both ends without any support in the middle. When a static load is applied, the beam will bend into a certain shape, which is described by the deflection curve. In the provided exercise, the initial deflection curve is given by the function w_0(x) = a x(L^3 - 2 L x^2 + x^3), representing the shape of the beam before it is released and begins to vibrate freely.

Understanding this initial deflection configuration is the starting point to determine the beam's vibration response. The beam's natural frequencies and mode shapes—patterns of deflection during vibration—are determined based on the beam's physical properties, such as length, bending stiffness, and mass distribution. Knowing the deflection of a beam is not just about figuring out how much it bends; it's about anticipating how it will behave when put into use.

Vibration Response Amplitude

The vibration response amplitude represents the maximum extent a structure displaces from its rest position during vibration. The amplitude is a crucial factor in assessing the vibrational behavior of a structure, as it reflects the intensity of the oscillations. In structural dynamics, designers analyze these amplitudes to ensure the structure can withstand repeated stress over time without failing.

In the exercise's scenario, the beam experiences free vibration once it is released from its deflected position. Since the beam was not in motion initially—implying it was released from rest—we only need to consider the initial deflection, described by the static load configuration, to determine the amplitudes. By calculating the coefficients A_n using the orthogonality conditions and integration, we find the maximum displacement of each vibrational mode, represented by the different values of n. These values of A_n define the specific-height peaks at which the beam's oscillation will reach as it vibrates, giving a complete picture of the response's amplitude over time.

Structural Dynamics Phase Angle

The structural dynamics phase angle is a measure that indicates the initial position of the structure in its vibration cycle at the start of observation—in other words, where in its oscillatory motion the structure begins. The phase angle is significant as it affects how different parts of a structure or different structures vibrate in relation to each other. This can be crucial for avoiding resonant frequencies that may lead to structural damage or failure.

In our exercise, since the beam is released from rest, the initial kinetic energy is zero, meaning there's no initial motion to consider. This simplifies the analysis, as the phase angle for all modes of vibration starts at zero. This zero phase angle implies that the vibration starts at the peak displacement (defined by w_0(x) and the A_n coefficients) and moves symmetrically about the rest position. If the beam had an initial velocity or if there had been an external dynamic load, the phase angles would not necessarily be zero, and the vibration response would need to be analyzed accordingly to account for that initial kinetic influence.