# Help me build a mathematical model

In summary, @phinds was trying to say that there are two ways to approach solving a vibration problem, and he is still trying to figure out which one he should use.
So I was trying to learn how to build a mathematical model of an oscillating system. The system and FBD is shown below. I just got confused why I got a different value of k from both x force and torsional equilibrium equation? Am I missing something?

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Really ... you are going to post your work unfocused and sideways and then expect us to decipher it?

Yes, cause actually, I'm just trying to learn here. It's not a homework, It's a problem example of a free vibration translational system problem. I understand the solution, but I need to ignore the effect of the torsion caused by the weight. I'm trying to use a different approach (torsional), and see if the results are the same, but I stuck at this. Here are the problem example and the solution.

Yes, cause actually, I'm just trying to learn here.
What @phinds was trying to say is that photo you posted is illegible. If you want help, you should make things easier for the helpers. Please post again, in focus, better lighting, and right side up.

Ok, below is the overview

The two methods cannot be treated as independent, a precise vibration analysis would be based upon a combination of both the lateral force on the beam due to the ball mass plus the effect of the moment imposed upon the beam by the rotational inertia of the ball.
This is somewhat aligned with the earthquake analysis of one structure supporting another secondary structure.

So, in analyzing every vibration problem, we need to analyze both linear and torsional approach. I thought we can just use either one, where one method covers the other. Thanks for the enlightment.

So, in analyzing every vibration problem, we need to analyze both linear and torsional approach. I thought we can just use either one, where one method covers the other. Thanks for the enlightment.

In general, it is not a matter of this approach or that. In general systems, a general displacement will involve both translation and rotation, so the mode shape will include both sorts of displacement. There are some system types where we know that there is motion of only one type. For a spring-mass system, guided by a support, there will be no rotation and hence no need to consider angular motion. For several bodies on a well supported shaft, usually there will be only rotational motion and there is no need to consider translation. For a system involving two shafts coupled by a gear pair, there will be both translation and rotation involved.

## What is a mathematical model?

A mathematical model is a simplified representation of a real-world system or phenomenon using mathematical equations and relationships. It helps to understand and predict the behavior of the system or phenomenon.

## Why is it important to build a mathematical model?

Building a mathematical model allows us to gain a deeper understanding of complex systems and make predictions about their behavior. It also helps in decision-making and problem-solving in various fields such as science, engineering, economics, and social sciences.

## What are the steps involved in building a mathematical model?

The steps involved in building a mathematical model include identifying the system or phenomenon to be modeled, defining the variables and parameters, formulating equations and relationships, and validating the model with real-world data. It also involves refining and improving the model based on feedback and testing.

## What are some common types of mathematical models?

Some common types of mathematical models include linear and nonlinear models, differential equations, statistical models, and optimization models. Each type has its own strengths and limitations, and the choice of model depends on the specific problem being addressed.

## What are the challenges in building a mathematical model?

Building a mathematical model can be challenging due to the complexity of real-world systems and the need to simplify and make assumptions. It also requires a strong understanding of mathematics and the ability to translate real-world phenomena into mathematical equations. Additionally, validating and refining the model can be time-consuming and may require a significant amount of data.

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