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.
Suppose I want to alter the natural frequency of a object for that I changed the but changing in mass also affects the stiffness as stiffness depends on mass. If I increase mass stiffness of object also increases so how to tackle this scenario? It was a question asked to me in interview. Thank you.
ReplyDeleteHi Vijay,
DeleteThanks for your query!!!
It is obvious that the addition of material will increases its stiffness. If you want to increase the stiffness without adding material then you need to play with section modulus. Distributing the material away from its neutral axis will increase the stiffness without altering the mass.
Consider a cantilever beam of length L, mater density P. K=F/u=3EI/L^3. Beam cross section is rectangular of width W, Depth D. Mass M= P*(W*D*L).
ReplyDeleteNow double the depth 2*D.
Mass increase by 2 times.
Stiffness increase by 2^3 times. Because moment of inertia I in stiffness equation is W*D^3/12.
So for a change in the dimension of object, change in stiffness is more than change in mass, so frequency go up finally.
Also note that if you double width W, frequency will not change.
Thanks for your explanation.
DeleteWhat you explained is correct. In this post my scope is to illustrate the mass and stiffness relationship with the help of spring mass system.
Very nice information sir.
ReplyDeleteI have one question, what is the effect of damping ratio on fatigue failure?
Best info
ReplyDeleteHello All,
ReplyDeleteI would like to ask basic question about vibration. My question is actually about the natural frequency of the structure. I know that the natural frequency is related to stiffness and mass. When I change the mass of the structure, let us say I made an optimization on mass and I have decreased it. What should I expect in the natural frequency of the structure? According to theory, I am expecting that the natural frequency of the structure increase. Am I right? but when I made a modal analysis on basic plate geometry, I see some of the mode increase and some of modes decrease? What is the main reason that I missed in this result? Normally, all modes should increase.
Thanks in advance for your assessments.
Best Regards,
Emre
Hi thanks for query.
DeleteCould you please send the model file to sriram.rajendran12@gmail.com