Fundemental Frequency of a Loaded Beam using Rayleighs Energy Method

[Danbob]

Hello All,

I'm having a bit of trouble understanding the notes I have on this subject.

As I understand the basic formula is


Frequency² = g*(m₁y₁ + m₂y₂) / (m₁y₁² + m₂y₂²)


I have found deflection (y) at the point loads in terms of g/EI

my notes suggest that these values can then be used directly in this equation;


Frequency² = EI* {(m₁y₁ + m₂y₂) / (m₁y₁² + m₂y₂²)}


I have tried this and got the correct answer.

But I can't understand where all of the G's cancel out?? and also where the *EI comes from when all the deflection values are in terms of /EI??

Could anyone explain how this is the case? Step by step if possible?

I need to understand it quickly as I have an exam on the subject on Thursday,

Thank you very much in Advance

Dan

 

[Achy47]

Hello Dan,

Thank you for your question. Let me try to explain the steps for you in a simpler way.

The formula you have mentioned is known as the natural frequency formula for a system with two masses (m1 and m2) connected by a spring (with stiffness EI). This formula is used to calculate the natural frequency of the system, which is the frequency at which the system will vibrate if no external forces are applied to it.

Now, let's break down the formula and understand it step by step.

Frequency2 = g*(m1y1 + m2y2) / (m1y12 + m2y22)

First, let's look at the numerator (top part) of the formula. It has two terms - m1y1 and m2y2. These terms represent the masses (m1 and m2) multiplied by their respective deflections (y1 and y2). This is because when a mass is subjected to a force, it will experience a displacement (deflection) from its original position. This displacement is directly proportional to the force applied and is represented by y (deflection) in the formula. So, m1y1 and m2y2 represent the forces applied to the masses m1 and m2, respectively, and g is the acceleration due to gravity.

Next, let's look at the denominator (bottom part) of the formula. It also has two terms - m1y12 and m2y22. These terms represent the stiffness (EI) of the spring multiplied by the square of the deflections (y1 and y2) of the masses. This is because the stiffness of the spring is directly proportional to the square of the displacement (deflection) of the mass. So, m1y12 and m2y22 represent the stiffness of the spring acting on the masses m1 and m2, respectively.

Now, when we multiply the numerator and denominator by EI, we get:

Frequency2 = EI* (m1y1 + m2y2) / (m1y12 + m2y22)

This is because EI is a constant and will not affect the final result, but it helps in simplifying the formula.

Now, to answer your question about where the G's and *EI cancel out - they are not cancelling out, they are just being simplified. The G's are cancelling out in the numerator and the *EI is being