# Vibration and Modal Analysis: Adding massless springs to a shaft

• grejuvaa
In summary: What is the difference?Fixed supportsFirst and second bending modes of fixed supportFirst and second bending modes of free free modal analysisfirst bending modesecond bending mode
grejuvaa
Hello,

I am working on vibrations and modal analysis recently. I have a question that I could not find any answer. Can you please help me?

Imagine a shaft. When we run the free free modal analysis lets assume that first bending mode is 600 Hz. Then we add 2 massless springs to the shaft. The first bending mode comes around 400 Hz. That means it decreased. How is it possible? If we add massless springs does not that mean the stiffness will increase? Can you please explain with formulas or send me some articles or books to understand it better.

Best regards.

Is this homework? If so, we can move it to the homework forum.

Look very carefully at the mode shapes. Compare the mode shape in the free-free case to the case with the springs added. I suspect that the answer will be obvious.

A general rule in modal analysis is to ALWAYS look at the mode shapes.

berkeman
this is not a homework i am just trying to improve myself. Both of them are first bending mode. So i think that is not about mode shape.

It would be nice to have specific numbers rather than assumptions. Adding springs to a system will have of different impact on stiffness if they are added in parallel or in series.

You should show the specifics of your problem - and your calculations - such that we are sure we are all talking about the same thing.

jrmichler said:
Look very carefully at the mode shapes. Compare the mode shape in the free-free case to the case with the springs added. I suspect that the answer will be obvious.
I could sketch a beam with 600 Hz free-free natural frequency, then add massless springs to make the natural frequency 400 Hz right now. But I won't because I want the OP to learn something here.

Part of asking for help is to fully communicate the problem. Show us the mode shapes, beam properties, and spring stiffness.

Thank you for the answers. To be more clear I have uploaded some images. I draw a shaft. When I run free free modal analysis first bending mode is 362,2 and second bending mode is 986,45 but when i fix the shaft from 2 sides the first bending mode decreases to 361,3 and the second to 980,9. What is the reason for that?

Fixed supports

First and second bending modes of fixed support

First and second bending modes of free free modal analysis

first bending mode

second bending mode
Thank you

grejuvaa said:
When I run free free modal analysis first bending mode is 362,2 ... but when i fix the shaft from 2 sides the first bending mode decreases to 361,3
Show the mode shapes for those two cases. Compare the exact locations of the nodal points in the free-free condition to the locations of the inflection points in the fixed-fixed analysis.

## What is the purpose of adding massless springs to a shaft in vibration and modal analysis?

Adding massless springs to a shaft in vibration and modal analysis is typically done to model the flexibility and stiffness of the system more accurately. This allows for a more precise calculation of the natural frequencies and mode shapes, which are critical for understanding how the system will respond to various excitations.

## How do massless springs affect the natural frequencies of a shaft?

Massless springs affect the natural frequencies of a shaft by altering the overall stiffness of the system. Depending on the stiffness of the added springs, the natural frequencies can either increase or decrease. Stiffer springs generally increase the natural frequencies, while more flexible springs tend to decrease them.

## Can you explain the concept of mode shapes in the context of a shaft with massless springs?

Mode shapes represent the deformation patterns of the shaft at specific natural frequencies. When massless springs are added to the shaft, they influence these deformation patterns by altering the distribution of stiffness along the shaft. This results in different mode shapes compared to a shaft without the springs, reflecting the new stiffness characteristics introduced by the springs.

## What are the typical boundary conditions considered when analyzing a shaft with massless springs?

Typical boundary conditions for analyzing a shaft with massless springs include fixed, simply supported, free, and clamped conditions. These boundary conditions define how the shaft is constrained and influence the natural frequencies and mode shapes. The selection of boundary conditions depends on the physical setup and the specific application of the shaft.

## How do you incorporate massless springs into the mathematical model of a shaft for modal analysis?

Incorporating massless springs into the mathematical model of a shaft for modal analysis involves adding the spring constants to the system's stiffness matrix. This is done by identifying the points on the shaft where the springs are attached and modifying the stiffness matrix accordingly. The updated stiffness matrix is then used in conjunction with the mass matrix to solve for the natural frequencies and mode shapes of the system.

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