Mechanical Vibrations: Underdamped vs Overdamped vs Critically Damped

In the previous video in the playlist we saw undamped harmonic motion such as in a spring that is moving horizontally on a frictionless surface just due to Hooke's law. In this video we upgrade and consider friction as well. We get a second order differential equation that is a constant coefficient homogeneous and we know how to solve these. However, qualitatively the type of solution you get depends on the constants. In the cast of underdamped, you get oscillations that decay exponentially. In the case of overdamped, that is too much friction, then it is just exponential decay with no oscillations possible. And then there is the very interesting third case of critically damped that gives a repeated root to the characteristic equation.

0:00 Deriving the ODE
1:58 Solving the ODE (three cases)
3:20 Underdamped Case
5:20 Graphing the Underdamped Case
7:33 Overdamped Case
8:50 Critically Damped


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[zepledfan]

I already graduated with an engineering degree. I don't use a lot of this part of the degree at work but I have always wanted to go back and master the stuff we covered in college. Unfortunately, a lot of school in the US is only about passing a tests. Your videos are helping me go back and actually learn! Thank you so much!!

[spaceiscool]

This has genuinely blown my mind because it has linked comparatively simple concepts i learned in A-Level physics with ODEs to describe the world clearly using math's and i understood why. I was always told adding friction into mechanics makes it so much harder, and I'm sure in other places it is, but here its more simple than i expected.

[NUGGet-]

See, I was tracking with you when you started explaining the friction force, like "Okay, I can remember that" but then you explained it in terms of the car or walking, and how it goes against you, and I was like "Okay now THAT makes sense!" I just gotta say, I wish more teachers explained stuff like you do. You make it relatable.

[minhazulabedinakib]

I'm a mechanical engineering student. This really helped. Thank you so much!
Love from Italy.

[SharkyShocker]

Wonderful explanations. Clear, Concise, and Clean.
Thank you for your time, especially in making sure the "Why?" part is conveyed which is often the hardest to wrap my head around. But I totally get it now!

[georgesadler]

Professor T. Bazett, thank you for a strong Introduction to Mechanical Vibrations in Differential Equations. Mechanical Vibrations is a huge part of Mechanical Engineering, which deals with tools/equipment that vibrate upon impact. Many problems in Mechanical Vibrations are Underdamped.

[madisonmobach]

I've been struggling with this concept in my physics class all term and you just cleared everything up in like 10 minutes! Thank you so much, your videos are the most helpful!!

[erikawimmer]

Hi Trefor. Just wanted to tell you that your Videos are great. For me as a 14 year old math and physics nerd from germany you are gold!

[somyaharjani]

when everyone's just teaching to help us get marks, teachers like u help us find answers to all our curiosities
thankyou so much sir!

[mognokhan]

Your videos are indubitably helpful. Keep up the good work, Dr. Trefor!

[diegosuarez]

Thanks boss, got a Intro to Vibrations Test on Monday and you make this so much more understandable. Very grateful, and I hope you get a lot of blessings!

[xoticbeats_]

Your engaging bright attitude makes it easier for me to learn this stuff thank you

[Altekameraden]

For anybody who gets into vibration analysis or reliability engineering, in mechanical system you can typically only vary mass or stiffness. Although it can be expensive to modify equipment in the two aforementioned fashions, it can prevent fractures. Changing mass, stiffness or both shifts the frequency response away from the current resonant frequency which is what kills shafts, bearing, supports, pipe fittings.

[Momo-bbfn]

8:33 you're a genius!! That's such a good image!

[sethtang]

this man is literally carry my differential equations class

[linaalghamdi]

Thank you so much, you can never imagine how you helped in understanding this part

[DrAtomics]

Love your videos, ironically I was already subscribed to you but my Analytical Methods professor recommended your video series for our class to watch as a mandatory thing before class! 😁 Your content is making it to universities 🤙

[lukecaetano]

You're a great teacher, thank you so much

[manrajmann]

( AWESOMENESS + RESPECT + PASSION + KNOWLEDGE ) * (MATHEMATICS ) = Dr .Trefor Bazett !

[AMR-]

First time I ever hear that friction is a function of velocity. I mean it does make a difference if you're moving or not (static vs kinetic friction coefficient), however, once you start moving it becomes a function of the normal force (independent of velocity). Now, an hydraulic/pneumatic dampener does depends on velocity. Anyway, great video as always!!