# Modal analysis of a pin-jointed frameworks

• jackcolebrook
In summary: However, if you have multiple elements with potential displacements at each node, then the stiffness matrix becomes too large to be manageable.
jackcolebrook
Hi

I'm trying to find the modal frequencies of a pin-jointed frames, to validate an ansys model.

So far I've used direct stiffness method to generate a global stiffness matrix for the framework in Matlab. Is it as simple as mutiplying the stiffness matrix with an inverted mass matrix and finding the eigenvalues?

My gut geeling tells me I'm missing a step. Any help would be great.

When:
Mass = M
Stiffness = K

Then:
vector(modal frequencies) = sqrt(eigenvals(K * M^-1))

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• Eigen.jpg
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Thanks for the help skeleton,

Could you take a look at my workings as see if I am talking cobblers.

If I use direct stiffness method, I can obtain a global stiffness for a framework. This is done by decomposing the frame into sperate trusses. Calulating the stiffness in local co-ordinates, then tranfering into global co-ordinates. Each truss is then arranged into a global stiffness matrix.

This global stiffness matrix now releates forces at each node with a displacements in x and y. Ordinarily I would then use this as a simple method to calulate displacements/stresses in a framework.

Ok, so this is where I get confused.

In my example I have 3 elements with 2 possible displacements at each node. This gives a 6x6 stiffness matrix for the frame.

So calculating f=sqrt(eig(K*inv(m)), gives 6 numbers.

Are these numbers the 1st 6 frequencies of the framework, or the modal frequencies in x and y for each truss? Or have I calulated the wrong stiffness matrix?

Thanks,

#### Attachments

• Modal Frequencies of Pin Jointed Frameworks.doc
30 KB · Views: 259
Your document wrote:
A=10mm dia bar (Area of strut)

Truss members would area, not diameter. So,
A = pi/4*dia^2 = 78 mm2

For structural steel,
Young's modulus: E = 200 GPa,
Shear modulus: G = 77 GPa

You wrote E=74 GPa. Your value is close to G, not E, of steel. Is that what you wanted?

The above two numerical changes would not change the "mechanics" of the equations, only the resulting values.

If you send me your MATLAB file then I'll look at it in answering your primary question.

Hi Skeleton

Apologies, should have been more clear on the document, my writing is terible!

74GPa is the young's modulas of Aluminum and area was based on a 10mm dia bar.

Attached are my MATLAB files. DSM01 set up the problem, truss2d returns a mass and stiffness matrix for a given element in global co-ordinates.

Thanks for the help,

J

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• truss2d.m
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• DSM01.m
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## 1. What is modal analysis?

Modal analysis is a technique used to study the dynamic behavior of a structure or system. It involves determining the natural frequencies and mode shapes of a structure, which are important factors in understanding how the structure will respond to external forces or vibrations.

## 2. What is a pin-jointed framework?

A pin-jointed framework is a type of structure that consists of interconnected members joined together at their ends with pins or hinges. This type of framework is commonly used in trusses, bridges, and other load-bearing structures.

## 3. Why is modal analysis important for pin-jointed frameworks?

Modal analysis is important for pin-jointed frameworks because it helps to identify the natural frequencies and mode shapes of the structure. This information is crucial in determining the structural integrity of the framework and can help engineers make design improvements to optimize its performance.

## 4. How is modal analysis performed on pin-jointed frameworks?

Modal analysis of pin-jointed frameworks is typically performed using computer-aided simulation software. The framework is modeled as a finite element model, and the natural frequencies and mode shapes are calculated by solving the equations of motion for the structure.

## 5. What are some applications of modal analysis for pin-jointed frameworks?

Modal analysis is commonly used in the design and optimization of pin-jointed frameworks for various applications such as bridges, buildings, and aerospace structures. It is also used in the maintenance and monitoring of existing structures to detect any potential structural issues or failures.

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