Second order differential equation for spring-mass systems

Let's look at modeling the motion of a spring-mass system (a harmonic oscillator) using a second-order differential equation. From Newton's Second Law, we arrive at mx'' + cx' + kx = 0 (or a forcing function), where x(t) is the position of the spring-mass over time, m is the mass, c is the damping coefficient, and k is the constant from Hooke's Law. We focus on the effects of damping and how to detect what kind of damping a spring-mass system has based on the roots of the characteristic equation.

Four types of spring motion are discussed based on the roots: undamped (no damping force), underdamped (small damping), critically damped (damping force just prevents oscillation), and overdamped (large damping). Different damping leads to different behaviors, which we can illustrate with MATLAB simulations.

#mathematics #math #differentialequations #ordinarydifferentialequations #stemeducation #harmonicoscillator #hookeslaw #physics #matlab #matlabsimulation #iitjammathematics

[maxrybold]

I used this tutorial to brush up my understanding of characteristic equations that describe the behavior of a spring mass damper system to confirm simulation results, via Desmos, of essentially the same system outlined in a SolidWorks tutorial textbook. Great explanations, thanks!

[Ivan-mpff]

so pleasurable to watch, informative and detailed. Pretty in all aspects. Thank you

[NamelessProducts]

Im trying to learn qualitative analysis of nonlinear 2nd order differential equations and all the examples so far have been in springs. This helped a lot. Thank you.

[thomashowe]

Clear and concise. I wish I had access to this when I took this class.

[navanithnavanith]

Very clear and precise explanation, helped me understand the concept very quickly. You saved my semester marks 😀😀😀

[samblake]

Awesome stuff! Super clear and I love the fade outs and ins!!

[nowardchaselenkana]

clear and straight forward... cheers Doc

[ShakilAhmedBhuiyan]

Very precise lecture. Very easy to understand.

[Jacoblikesyoutube]

I'm a little lost on the step at 16:45, the last step of the first example. x(t) = cos(2t) because it's the only value at the initial condition that equals 0? So in another situation if both trig functions provided a non-zero output, we might end up with x(t) = c_1 * cos(2t) + c_2 * sin(2t)?

Is it effectively always x(t) = c_1 * cos(2t) + c_2 * sin(2t) but the result in the first example simplifies to x(t) = cos(2t)?

[Ivan-mpff]

At about 23:23, critical damping, what would be the corresponding units of C1 and C2 in order to be consistent with the dimension of LHS of the equation, i.e.distance? I am trying to do a dimensional analysis on it. Thank you.

[mattbabik]

I get 12/35 and 2/35 for the last problem when I put it into wolfram alpha to solve