Question

Consider the free flexural vibrations of a uniform elastic beam of length $L$, bending stiffness $EI$ and mass per unit length $m$ that is clamped at both edges, as shown, and is represented mathematically using Euler-Bernoulli Theory. (a) Establish the modal boundary conditions for the structure. (b) Derive the frequency equation for the beam. (c) Determine the first three natural frequencies and modal functions. Plot the modal functions.

Short answer

#Answer# 1. Write down the Euler-Bernoulli beam equation governing the flexural vibrations. 2. Apply separation of variables and boundary conditions to the governing equation. 3. Find the frequency equation and natural frequencies by solving the resulting ordinary differential equation for different mode numbers. 4. Find the modal functions by substituting the natural frequencies back into the general solution for the displacement function, and plot them using a plotting tool like MATLAB or Python.

Step-by-step solution

Write down the Euler-Bernoulli beam equation governing the flexural vibrations

Using Euler-Bernoulli Theory, we can write the equation for flexural vibrations of the beam as: $\frac{{\partial }^{4}w}{\partial x^4}=\frac{m}{EI}\cdot \frac{{\partial }^{2}w}{\partial t^2}$ where, $w(x,t)$ is the transverse displacement of the beam at any point $x$ and time $t$, $E$ is the modulus of Elasticity, $I$ is the moment of inertia of the beam and $m$ is the mass per unit length.

Apply separation of variables and boundary conditions

Using separation of variables, we can write the function $w(x,t)$ as a product of a function of $x$ and a function of $t$, i.e., $w(x,t)=X(x)T(t)$ Substituting this into the Euler-Bernoulli beam equation and dividing both sides by $X(x)T(t)$ gives: $\frac{T^{″}(t)}{T(t)}=\frac{EI}{m}\cdot \frac{X^{(4)}(x)}{X(x)}=-{\omega }^2$ where $T^{″}(t)$ denotes the second derivative of $T(t)$ with respect to $t$, $X^{(4)}(x)$ denotes the fourth derivative of $X(x)$ with respect to $x$, and ${\omega }^2$ is a separation constant. Now, the boundary conditions for a clamped-clamped beam are: w(0, t) = 0, w(L, t) = 0 w'(0, t) = 0, w'(L, t) = 0 Applying these boundary conditions, we have: X(0) = 0, X(L) = 0 X'(0) = 0, X'(L) = 0

Find the frequency equation and natural frequencies

To find the frequency equation, we need to solve the following ordinary differential equation (ODE) for X(x): $X^{(4)}(x)+k^{4}X(x)=0$ where $k^4=\frac{m{\omega }^2}{EI}$. The general solution to this ODE is: $X(x)=A\sin (kx)+B\cos (kx)+C\sinh (kx)+D\cosh (kx)$ Applying the boundary conditions, we find: $A+C=0$ $B+D=0$ $(A{k}^3+C{k}^3)\sin (kL)+(B{k}^3-D{k}^3)\cos (kL)=0$ $(Ak-Ck)\sin (kL)+(Bk+Dk)\cos (kL)=0$ Solving for A, B, C, and D, we find that the frequency equation is: $\tan (kL)=\tanh (kL)$ To find the first three natural frequencies, we can solve the frequency equation numerically for the first three positive roots of k. We can then use these values of k to calculate the natural frequencies as: ${\omega }_n=\sqrt{\frac{EI}{m}}{k}_n^2$ where n = 1, 2, 3, for the first three natural frequencies.

Find the modal functions and plot them

To find the modal functions, we can substitute the values of $k_n$ back into the general solution for X(x). The modal functions are then given by: $X_n(x)=A_n\sin (k_{n}x)+B_n\cos (k_{n}x)+C_n\sinh (k_{n}x)+D_n\cosh (k_{n}x)$ where $A_n$, $B_n$ , $C_n$ , and $D_n$ can be determined using the boundary conditions. Once the list of the modal functions is formed, you can plot these modal functions using a plotting tool like MATLAB or Python. This will give you a graphical representation of the shape of the modal functions for the first three natural frequencies.

Key concepts

Flexural Vibrations of Beams

When we talk about the flexural vibrations of beams, we're delving into how beams bend and oscillate under certain conditions. Such vibrations occur when a force is applied and removed, causing the beam to move back and forth around its initial position. For a clear understanding, envision a diving board bouncing up and down after a diver jumps off; that's flexural vibration at a basic level.

In engineering, the Euler-Bernoulli beam theory is commonly used to predict these vibrations, assuming that the plane sections of a beam remain plane and perpendicular to its axis after deformation. This simplification allows us to focus on the bending movements and ignore other modes of deformation like shear.

The equation that governs these vibrations is a fourth-order partial differential equation, which shows the relationship between the beam's deflection and the various forces and properties acting on it.

Modal Boundary Conditions

Boundary conditions are essential in defining how a beam is supported at its edges, which in turn affects its vibrational characteristics. For the exercise in question, we're dealing with a beam that is clamped at both ends. Clamped or fixed supports constrain not only the deflection but also the slope at the beam's ends to be zero.

This leads to modal boundary conditions which describe the specific limitations at the boundaries of the beam for our vibration problem. For our clamped-clamped beam, the modal boundary conditions specify zero deflection ($w(0,t)=0$ and $w(L,t)=0$) and zero slope ($w^{\prime }(0,t)=0$ and $w^{\prime }(L,t)=0$) at both ends of the beam. These conditions are critical for accurately predicting the natural frequencies and mode shapes (modal functions) of the beam.

Frequency Equation Derivation

Deriving the frequency equation is an exciting journey into the world of differential equations and boundary conditions. Starting with the Euler-Bernoulli beam equation and applying the separation of variables technique, we distill the equation into one that describes the spatial component of vibration, or mode shape.

To get this elusive frequency equation, we approach the boundary value problem by inputting the clamped-clamped modal boundary conditions into the general solution of the ODE. This produces a characteristic equation that only modal frequencies can satisfy. In our case, this takes the intriguing form of $\tan (kL)=\tanh (kL)$, where $k$ relates to the angular frequency of vibration.By determining the roots of this transcendental equation, we unlock the natural frequencies of the beam, a crucial step towards understanding how it will behave under dynamic load conditions.

Natural Frequencies of a Beam

Natural frequencies are a bit like a beam's musical notes; they are the specific frequencies at which a beam will resonate. These are intrinsic properties, determined by the beam's material makeup and geometric design, as well as how it's supported. For our clamped-clamped beam, we find these frequencies by solving the frequency equation derived earlier.Once we have the values for $k$ by numerically solving $\tan (kL)=\tanh (kL)$, we can calculate the natural frequencies using ${\omega }_n=\sqrt{\frac{EI}{m}}{k}_n^2$. If you were to strike the beam, the vibrations would naturally tend to settle at these frequencies. Knowing them is crucial for avoiding destructive resonance in structural applications.

Beam Modal Functions

Modal functions, or mode shapes, are the physical representation of how a beam deforms at each of its natural frequencies. They show the unique shape that a beam takes on as it vibrates. In the exercise, calculating modal functions involves substituting our $k_n$ values back into our general solution equation.Once we calculate coefficients $A_n$, $B_n$, $C_n$, and $D_n$ with the boundary conditions, the set of equations for $X_n(x)$ represents the modal functions for our clamped-clamped beam. When plotted, these functions give us a curve for each natural frequency, helping to visualize where maximum displacements occur and where there might be nodes of no displacement. These visualizations are not just theoretical; they guide engineers in reinforcing structures and designing for stability and safety.