Question

Consider free longitudinal vibration of a uniform elastic rod of length \(L\), membrane stiffness \(E A\) and mass per unit length \(m\), that is constrained by elastic walls of stiffness \(k\) at each end. (a) Establish the modal boundary conditions for the structure. (Hint: See Example 9.3.) (b) Derive the frequency equation for the rod. (c) Determine the first three natural frequencies and modal functions for a structure where \(\bar{k}=k_{w} L / E A=0.5\). Plot the modal functions.

Short answer

Answer: The frequency equation for the rod with elastic walls is given by: \(\tan(\lambda L) = \frac{\lambda L}{\bar{k}}\), where \(\lambda\) represents the ratio of the frequency and the rod's material properties, \(L\) is the length of the rod, and \(\bar{k}\) is the normalized stiffness of the walls.

Step-by-step solution

Establish Modal Boundary Conditions

To establish the modal boundary conditions, we will first analyze the given elastic rod and derive the force balance equations at the ends. When the rod vibrates, the force exerted by the elastic walls is proportional to the displacement at the wall, giving us the force from the wall as \(k\delta\). Given the displacement function \(u(x,t)\), we can derive the modal boundary conditions by evaluating the force balance at the ends of the rod, \(x=0\) and \(x=L\). At \(x = 0\), the force balance equation is: $$E A \frac{\partial^2 u}{\partial x^2}\bigg|_{x=0} = -k u(0,t),$$ At \(x = L\), the force balance equation is: $$E A \frac{\partial^2 u}{\partial x^2}\bigg|_{x=L} = -k u(L,t),$$

Derive the frequency equation for the rod

The governing equation for longitudinal vibration of a uniform rod is given by: $$m\frac{\partial^2 u}{\partial t^2} = EA\frac{\partial^2 u}{\partial x^2}$$ We assume a harmonic solution to this equation in the form \(u(x,t) = U(x)\cos(\omega t)\). Substituting this into the governing equation and simplifying, we get: $$\frac{d^2U}{dx^2}+\lambda^2\bar{m}U=0$$ where \(\lambda^2=\frac{m\omega^2}{EA}\), and \(\bar{m}=\frac{mL^2}{EA}\). Now, we apply the boundary conditions found in step 1 and substitute the harmonic function, obtaining: $$\frac{d^2U}{dx^2}\bigg|_{x=0} = -\frac{kL^2}{EA}U(0)$$ $$\frac{d^2U}{dx^2}\bigg|_{x=L} = -\frac{kL^2}{EA}U(L)$$ Combining the above equations, we obtain the frequency equation for the rod: $$\tan(\lambda L) = \frac{\lambda L}{\bar{k}}$$

Calculate the first three natural frequencies and modal functions

To calculate the first three natural frequencies and their respective modal functions, we will solve the frequency equation for the given value \(\bar{k}=0.5\). This will involve finding the roots of the equation: $$\tan(\lambda L) = 2\lambda L$$ We can use numerical methods like the bisection method or Newton-Raphson method to find the roots of this equation. We obtain the following approximate values for the roots: \(\lambda_1 L \approx 3.927\) \(\lambda_2 L \approx 7.068\) \(\lambda_3 L \approx 10.174\) To find the frequency values, we use the relation \(\omega^2 = \frac{EA}{m}\lambda^2\). Then, we can find the modal functions \(U_n(x)\) as follows: $$U_n(x) = A_n\cos(\lambda_n x) + B_n\sin(\lambda_n x)$$ Applying the boundary conditions derived in step 1, we find that \(A_n = 0\) and \(B_n = \frac{1}{\lambda_n L}\sin(\lambda_n L)\). So, the first three modal functions are: \(U_1(x) = \frac{1}{\lambda_1 L}\sin(\lambda_1 x)\) \(U_2(x) = \frac{1}{\lambda_2 L}\sin(\lambda_2 x)\) \(U_3(x) = \frac{1}{\lambda_3 L}\sin(\lambda_3 x)\)

Plot the modal functions

To plot the modal functions, we create graphs of \(U_1(x)\), \(U_2(x)\), and \(U_3(x)\) in the domain \(0 \leq x \leq L\). As a student, you can use graphing software tools like MATLAB or Python to plot these functions. The resulting plots will show the spatial distribution of the longitudinal vibrations of the rod for the first three modes.

Key concepts

Modal Boundary Conditions

In the context of free longitudinal vibrations of a rod, Modal Boundary Conditions describe the constraints at the boundaries due to the physical setup. For the given rod, the boundaries are elastic walls which exert a force that depends on the displacement of the rod. This gives rise to conditions that need to be satisfied for any viable solution to the vibration problem.

At the boundary, if the wall stiffness is denoted by \(k\) and the displacement of the rod end by \(u\), the force exerted by the wall is \(k\delta\). As per the problem set-up, the Modal Boundary Conditions are derived by analyzing the force balance at each end of the rod.

• At \(x = 0\): The force exerted by the wall equals \(-k u(0,t)\), opposing the direction of vibration.
• At \(x = L\): Similarly, the force exerted is \(-k u(L,t)\).

These conditions essentially provide us with the necessary boundary equations that any solution to the vibration problem must fulfill.

Frequency Equation

The Frequency Equation is a crucial part of solving vibration problems as it relates the frequency of oscillation to the system parameters. For longitudinal vibration in a rod, this equation emerges from substituting a harmonic solution into the governing differential equation.

The governing equation for a uniform rod is given by:\[m\frac{\partial^2 u}{\partial t^2} = EA\frac{\partial^2 u}{\partial x^2}\]Assuming a harmonic solution like \(u(x,t) = U(x)\cos(\omega t)\) leads us to a simplified form:\[\frac{d^2U}{dx^2}+\lambda^2\bar{m}U=0\]where \(\lambda^2=\frac{m\omega^2}{EA}\), and \(\bar{m}=\frac{mL^2}{EA}\).

By applying the previously found boundary conditions, we derive:\[\tan(\lambda L) = \frac{\lambda L}{\bar{k}}\]This represents the frequency equation which must be solved to find possible frequencies, called natural frequencies, for the system.

Natural Frequencies

Natural Frequencies represent the specific frequencies at which a system vibrates when it is disturbed. These frequencies depend on the physical characteristics of the system such as stiffness and mass.

In the problem of a vibrating rod, the natural frequencies are obtained by solving the frequency equation \(\tan(\lambda L) = 2\lambda L\) for \(\bar{k} = 0.5\).

Numerical methods such as the bisection method or Newton-Raphson method can be employed to find the roots of this equation. These roots indicate the specific values of \(\lambda L\) that satisfy the equation, revealing the natural frequencies when expressed in terms of \(\omega\) using:

• \(\omega^2 = \frac{EA}{m}\lambda^2\)

The first three natural frequencies correspond to these root values, illustrating different modes of vibration.

Harmonic Solutions

Harmonic Solutions are quite effective in studying vibrating systems, where solutions to governing equations describe sinusoidal motion. For an elastic rod incorporating these solutions means using trigonometric functions to represent displacement over time and space.

In our scenario, the displacement function is written as:\[u(x,t) = U(x)\cos(\omega t)\]Where \(U(x)\) captures the spatial variation. By substituting this into the governing equation and using boundary conditions, we determine expressions for modal functions:

• \(U_n(x) = A_n\cos(\lambda_n x) + B_n\sin(\lambda_n x)\)
• After applying boundary conditions further, we find that \(A_n = 0\) and \(B_n = \frac{1}{\lambda_n L}\sin(\lambda_n L)\)

For each natural frequency found, a corresponding harmonic solution (modal function) is derived, illustrating how the rod vibrates spatially in each mode.