Mode Participation Factor in Dynamic Analysis

Hello Engineers,

Hope you have enjoyed my post on Mass Stiffness relationship

As a Dynamic / FEA Engineer, you would have performed modal analysis for determining the natural frequency and mode shape pattern of the structure.

Have you ever noticed the solution statistics once after executing the Modal dynamics problem?

We have a Participation factor calculation table for 6 directions, which includes translation and rotation of mass in X,Y & Z directions.

In this post, I will be explaining how the solver computes the mass participation factor and the components present in the above mentioned table.

Before going into the calculation part, It will be highly significant to know what exactly the mode participation is!!!

For describing the calculation, I am using a cantilevered bar (dimension is not much important here) made of steel material which has a mass of 294 g as shown in the figure:

The dynamics problem is executed in ANSYS mechanical. We do not want to worry much about the software tool provided that computations is going to be the same for all solvers.

I have solved the problem by specifying the number of Eigen frequencies to 12.

The reason for specifying this value is that the solver should not miss any of the important modes of our interest.

A basic thumb rule behind is that 👍

The ratio of effective mass to the total mass must be greater than 0.90 

Here is the participation table along Y direction excitation, 

Tap to Zoom the Participation table

In the above table only Natural frequency and Participation factors are calculated by the solver based on mass and stiffness matrices.  In addition we require only the final ratio value (i.e.,) 0.933 to derive all the parameters in the table.

Let us dive into the calculation part 👇

Effective Mass = (Participation factor)²

So Effective Mass for first mode = (13.410)² = 179.8 g

Effective mass for second mode = (-7.4537)² = 55.57 g     

Ratio is calculated based on the participation factor values 

Ratio for first mode = Participation for first mode / dominant mode

                                             = 13.410/13.410 = 1

        Ratio for second mode  = 7.4537/13.410 = 0.5558

Cumulative mass fraction  is calculated based on Cumulative Effective Mass / Effective Mass.

Cumulative effective mass for mode 1 : 179.817 + 0 = 179.817 g

For mode 2 it is :179.817 + 55.5574 = 235.3744 g

For mode 3 it is :235.3744 + 0.431263 *(10-7) = 235.3744 g

As mentioned previously we can derive the mass participating in the Y direction by the below method:

Effective Mass participating in Y direction  = Ratio of Total to Effective mass * Mass of the cantilevered bar considered for simulation

Effective mass = 0.933 * 294 g = 274.302 g

Cumulative mass fraction for first frequency     

= 179.8 / 274.65 = 0.654

Cumulative mass fraction for second frequency 

    = 235.3744/274.657 = 0.855

Cumulative mass fraction for third frequency   

    = 235.3744/274.657 = 0.855

In this way the solver computes the values in the participation table.

I hope that the participation factor table will not be a myth for you in future ☺

Please let me know your comments and suggestions in the below section.

Keep visiting my blog for more concepts!!!

Understanding Mass stiffness relationship in a practical way

Hello Engineers,

I hope you have enjoyed reading my previous post on Mechanical Vibration in a practical sense

I have received a query from few of my readers regarding the relationship between natural frequency and mass of the system.

In this post, I am planning to make you clear how the natural frequency of the structure is affected by mass and stiffness.

We can consider two cases regarding to understand it in a better way.

Let us dive in to the first case!!!!

Case 1: Two blocks of same mass and different stiffness are compared here. It could be notated as shown below:

M₁=M₂ & K₁ < K₂

Two masses considered here are 1 kilogram each and the each mass is hanged with spring of different stiffness as shown in the below schematic representation:

We do not want to worry much about the boundary conditions provided I will make you clear with the concept in this post. I will make one more post in near future enumerating the boundary conditions and theoretical calculations.

Blocks are constrained in such a way that they can move only along one direction constituting the single degree of freedom system.

This dynamic system is solved in ANSYS mechanical.

Since the mass matrix is normalized, the Eigen vector or  deformation value show in  the plot do not represent the real value.

Only two modes are extracted as we are interested in the excitation of first two natural frequencies.

Above images represent the natural frequencies and mode shapes for the first two modes. Image on the left shows that the first mode excited for mass 1 is 1Hz and on the right is the mass 2 excited at 1.5 Hz.

Case 2: Two blocks of different mass and same stiffness are compared here. It could be notated as shown below:

M₁ < M₂ & K₁ = K₂

From the above image, we can observe that heavier mass 2 is excited at lower frequency of 0.49 Hz than compared with second natural frequency of lighter mass 1 which is 1 Hz. 

We can have a quick comparison and inference between these two cases from the below table:

Hope you have got clear with the natural frequency and its relation with mass and stiffness. 

Kindly let me know your queries and views in below section.

Stay visiting for more concepts!!!!!

Vibration in a practical sense

Hello Engineers,
                Hope you enjoyed reading my posts on getting jobs and pattern of questions asked in an interview.

If you missed it here are the links for it:👇

Tips & tricks
How to answer Interview Questions??

In this post, I will be explaining about dynamic analysis with a real life practical example

Hope you will enjoy it☺

📌  Dynamic Analysis:

            It is mainly used to determine the response of the structure when it is subjected to the vibration and shock loads.

            For any structure undergoing mechanical vibrations, there are two important characteristics which determines its behavior they are:

·         Mode shapes

·         Natural frequencies

Mode shapes show the response pattern of the structure in a particular frequency and they are termed as Eigen vectors.

Natural frequency is the frequency at which the structure will resonate resulting in  higher response and they are termed as Eigen values.

Hope you remember these stuffs are calculated with the help of matrices in numerical methods.

Modal analysis is used to determine the above mentioned characteristics. A lot of things has to be understood before performing modal analysis.

A lot of resources are available to explain it but my focus here is to explain intuitively what the vibration is !!!!

Let me explain it a more practical example which I experienced in my life. One of my friends crashed my Apache bike 😲😲 with some crashes in Visor and made a crack in the speedometer housing. Because of this crack the housing raised a bit upwards.

Whenever I took it for a ride, the separated region will vibrate when the speed of my bike varies between 48 to 56 Km/hr. During ride, I experienced the vibration by means a sound🔊. I can hear the sound only when the speed is between that stipulated range. I cannot hear the sound once after I cross that speed range. To recheck it, I just lowered my speed below the stipulated speed and raised it again!!!

I was astonished by seeing this phenomenon persistently during each rides and several times in a single ride.

I do not want to go much theoretical with vibration formula and calculations because I will be dedicating my next blog to mathematical model of vibration.

📌 Let me explain why it has happened?  

All the components in the bike are assembled in such a way that the system has to vibrate as a whole when the bike exceeds a speed of around 100 km/hr.  Due to the crack in the hatched region, it made the housing to vibrate when I attained the speed of around 48 km/hr.

Then another question which raises up is why it vibrates at the particular speed alone???

Each component in bike has its own natural frequency〰, but the natural frequency will not be the same when it is assembled in a system and it will also vary based on how it is assembled or constrained.

When my bike housing got cracked it got separated from the assembly (just a minor separation of hatched region) and it was calm until the speed of 47 km/hr (not much accurate) and started to vibrate (or we can say resonate ♒ in a technical perspective) from 48 to 56 km/hr.

What I can conclude here 

This is the beauty of vibration!!! 👆

Stay visiting my blog for understanding structural simulation in a more realistic way!!!!