Question

For the following bar, (a) find the natural frequencies using 2 and then 3 elements of equal length. Use a consistent mass matrix. Let the bar have E=30×10⁶.

Answer 1

To find the natural frequencies of a bar using 2 and 3 elements of equal length, use the formula f = (n/L) * sqrt(E/m), where f is the frequency, n is the mode of vibration, L is the length of the element, E is the modulus of elasticity, and m is the mass of the element.

Step-by-step explanation:

To find the natural frequencies of a bar using 2 and 3 elements of equal length, we first need to determine the length of each element. Let's assume the total length of the bar is L. If we divide the bar into 2 equal elements, each element would have a length of L/2. Similarly, if we divide the bar into 3 equal elements, each element would have a length of L/3.

The natural frequency of an oscillating bar can be found using the formula f = (n/L) * sqrt(E/m), where f is the frequency, n is the mode of vibration, L is the length of the element, E is the modulus of elasticity, and m is the mass of the element. Since we are assuming equal lengths for each element, the frequency will be the same for all elements.

Let's calculate the natural frequencies for both cases.

Case 1: Using 2 elements of equal length (L/2)

f = (1/(L/2)) * sqrt(E/m)

Case 2: Using 3 elements of equal length (L/3)

f = (1/(L/3)) * sqrt(E/m)