for future ref) Prof. Steve defined 'd/m=ζ', but typically people define 'd/m=2ζω' (see Harmonic oscillator in wikipedia) which the matlab code follows, and the *_'d' in the code means ζ in that setting._* In the code, that's why there is -2*d*w (-2ζω) instead of -d (-ζ). Also note that we didn't solve the eq. analytically("by hands"), but used numerical analysis in two ways(rough one and fine one).
I consider the use of greek letters to be unnecessary. Regular modern print characters (e.g. x, y) are easier to view, read, and mathematically comprehend.
It's tradition to introduce the Greek alphabet, because there are a lot more than just 26 concepts to name with variables. You might just have 5 variables in a given problem, and could arbitrarily assign any letter you want, but the variable names also serve a purpose to remind you what the term means in the bigger picture of the subject. Sometimes, Greek letters are intentionally used, because there is a relationship between a term represented by a Latin letter, and a similar term represented by its corresponding Greek letter. For instance, Latin a used for acceleration, and Greek alpha used for angular acceleration. The reason omega is commonly used for angular velocity and angular frequency, is that there is no historically corresponding Greek letter to the letter v. So we use the alphabet neighbor to v (i.e. w), and assign the historically corresponding Greek letter for angular velocity, and its related concept of angular frequency.
What was defied as ζ (zeta) in the written section is actually not the damping ratio. The normalized 2nd order homogeneous equation is typically written as xdd + 2ζω*xd + ω^2*x = 0 where ζ in this case *is* the damping ratio, which can be defined as ζ = d/d_crit, and d_crit is the damping value that results in critical damping, which occurs when the discriminate of the characteristic equation is 0. On that note, in the code when bumping up the damping ratio to greater than 1, that's an over-damped case. Critical damping only occurs right on the boundary of pure convergence and oscillatory convergence, which is when the damping ratio is 1 and d=d_crit.
Hi Erik - you seem to have a good grasp of it all - I totally agree with your statement. Can you have a look at my comment - maybe you can help me out? Thank you.
Thank you very much for this presentation (❤❤❤). I can't wait for the next video. Based on this video, if the values of \Zeta and \Omega are available, then the characteristic function is calculated, so x(t) can be obtained by using x0 and xdot0. However, how to calculate the characteristic function in the presence of the response x(t) (Indeed, the inverse problem)?
Haven't you noticed that elastic coefficient carries the same meaning as a damper? Works for force/cross-section of a spring in the elastic range. 19:53 Can you do the lecture left-handedly with a lipstick? It doesn't squeak.
"sometimes you can break the spring, and then this is not true" 😂 You should do a series on the greek alphabet, when you use which letter and why, and maybe what other contexts we may see them in. You may like julia, the programming language. It lets you use greek letters for variable names. Clearly that makes it a better choice than matlab 😄
Definitively, damping doesn't play the same role as spring and mass. Mass and ppring are in total accordance with conservative laws. Not damping. Experimentaters have observed that it is proportional to the velocity and so it has been introduced afterward in the differential equation. Damping is not as pure as spring and mass !
Proportional damping is just an approximation we make, so the problem is practical to solve with the introductory methods in DiffEQ. Friction in the real world could take on numerous different models, such as the stick/slip friction of solid-on-solid contact, or the square of the speed for turbulent drag. There are some examples where proportional damping is a reflection of reality: 1. Dampers that use an extremely viscous fluid, so turbulence is negligible. This is what shock absorbers use, so damping is intentional. 2. Ohmic resistances in an LRC circuit 3. Eddy currents of magnetic brakes. And this is precisely why magnetic brakes cannot be the sole slow-down method, since they can never slow a vehicle down to zero speed. Magnetic brakes can do the bulk of the slow-down, but friction brakes are ultimately needed to bring the speed to a true zero.
What a great video again Steve, a brilliant explanation of a not so easy subject. I have one further question: After solving the characteristic equation for lambda, we have three possibilities: lambda_1 and lambda_2 are both negative, lambda_1 and lambda_2 are complex with negative real part, or lambda_1 is lambda_2 and there is basically just one negative lambda. So we have an over-damped system (no vibration), under-damped system (damped vibration) or critically damped system respectively. This final system will only exists for specific combinations of mass, damping and spring, while the under- and over-damped systems exist for a whole range of choices for mass, damper and spring. Using the initial conditions - x(0) and xdot(0) - we can now solve for the constants C_1 and C_2 in the solution x(t) = C_1 exp(lambda_1 * t) + C_2 exp(lambda_2 * t). This makes sense from a mathematical perspective, it is a linear combination of the two solutions, as well as from a physical perspective. And this works fine for the under- and over-damped systems. But for the critically damped system, you get that C_1 + C_2 = x(0) and (C_1 + C_2) * lambda = xdot(0), which doesn’t allow you to specify C_1 and C_2. The solution is to use x(t) = (C_1 + C_2 * t) exp(lambda * t). I know that. But I don’t understand WHY that would be the right thing to do - from a mathematical perspective as well as from a physical one - it doesn’t comply with my intuition. Is there anyone who can shed some light? Many thanks!
We need to add up two linearly independent terms, in order to form the general solution. Or more generally, for an nth order homogeneous DiffEQ, we need to add up n-terms that are linearly independent. Anytime there is overlap, you need to generate a linearly independent term, by multiplying it by t. This is what we do when we have a forcing function of the same frequency as the natural frequency of an undamped oscillator. The non-homogeneous forcing function adds terms to the solution, but it still is a combination of linearly independent terms. The solution won't just be a linear combination of sin(w*t) and cos(w*t), but rather it will be a linear combination of sin(w*t), cos(w*t) as well as t*sin(w*t) and t*cos(w*t). This is called resonance, since it creates a continuously growing amplitude due to a forcing function of a matching frequency. For a non-matching frequency (call it v) of the forcing function, it would be a linear combination of sin(w*t), cos(w*t) as well as sin(v*t) and cos(v*t). If you graph this for slightly different frequencies, you'll get a pulsing amplitude called beats. In the limit as v approaches w, you get an amplitude that continues to grow linearly with t. For the overdamped case, the two exponential decay rates are our two linearly independent solutions. One of them dominates in the long run, and the other allows for matching the curve to the two initial conditions, and quickly decays away. For the critically damped case, we have to produce another term that is linearly independent of the first. Take the case of y" + 6*y' + 9*y = 0 as a critically damped example. We can show that t*e^(-3*t) satisfies this equation, just like e^(-3*t) does. Start with the premise that y = t*e^(-3*t) y' = (1 - 3*t)*e^(-3*t) y" = (9*t - 6)*e^(-3 t) Apply to original diffEQ y" + 6*y' + 9*y = 0 (9*t - 6)*e^(-3*t) + 6*(1 - 3*t)*e^(-3*t) + 9*t*e^(-3*t) =?=0 Cancel out e^(-3*t), since every term has it, and it can't equal zero except at infinity: (9*t - 6) + 6*(1 - 3*t) + 9*t =?=0 Expand, and group t-terms first: 9*t - 18*t + 9*t - 6 + 6 = ?=0 (18 - 18)*t + (6 - 6) = 0, QED Thus it is satisfied by t*e^(-3*t) as one of the solutions. We can do the same for y=e^(-3*t): y' = -3*e^(-3*t) y" = 9*e^(-3*t) 9*e^(-3*t) + 6*(-3)*e^(-3*t) + 9*e^(-3*t) =?= 0 9 + 6*(-3) + 9 =?= 0 9 - 18 + 9 = 0, QED Thus it is satisfied, that e^(-3*t) is the other solution.
You might play devil's advocate, and wonder why we can't also do the same with y=t^2*e^(-3*t), to the critically damped differential equation y" + 6*y' + 9*y = 0. Give it a try, and you'll see that there is a problem. y = t^2*e^(-3*t) y' = -t*(3 t - 2)*e^(-3*t) y" = (9*t^2 - 12*t + 2)*e^(-3*t) (9*t^2 - 12*t + 2)*e^(-3*t) - 6*t*(3*t - 2)*e^(-3*t) + 9*t^2*e^(-3*t) =?= 0 (9*t^2 - 12*t + 2) - 6*t*(3*t - 2) + 9*t^2 =?= 0 9*t^2 - 12*t + 2 - 18*t^2 + 12*t + 9*t^2 =?= 0 Gather: (9 + 9 - 18)*t^2 + (12 - 12)*t + 2 =?= 0 You can see that the coefficients cancel on t^2 and t, but the standalone constant of 2 doesn't cancel. So it is impossible for this to be the solution. Not without a third order of differentiation, or a resonating non-homogeneous forcing function, to warrant it. Since this ansatz produces a constant we can't cancel, we can rule out this ansatz, and stick with the two ansatzen of e^(-3*t) and t*e^(-3*t)
The characteristic equation of the second derivative include square lambda, so the equation is not linear. But, how can the superposition rule be satisfied?
