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So today is unforced-- that means zero on the right-hand side, looking for null solutions-- damped-- that means there is a coefficient B in the first derivative. This is really a basic, basic equation. In many applications, A would be the mass. In a spring, for example, A would be a mass.
B is the damping, the friction. And C is the spring constant, the force that pulls the mass back. Or in electronics, B would be the resistance. Just we have to be able to solve it. And we want to look for exponentials. A pure exponential is just right for a constant coefficient equation like that. The derivative brings down an s.
Two derivatives bring down s squared. So I simply have As squared, Bs, and C all together, multiplying e to the st. Well the 0 is not going to come from e to the st, so it has to come from this.
So fundamentally, the whole video and more is about a quadratic equation. As squared plus Bs plus C equals 0. We have to solve it. We have to understand how does the answer depend on these numbers, A and B and C, the constants? So we know the route.
We know the solutions. The quadratic formula tells us that the two solutions-- there are always two, but they could be equal-- have this-- do you recognize this expression? You see the damping coefficient coming in.
You see this all important square root, which tells us, depending on whether B squared is bigger than 4AC, B squared is equal 4AC, B squared is smaller. So smaller B means less damping.
And it would be called underdamping. So here are the possibilities. B could be 0. That puts us back in a previous video when the solution was a pure complex exponential, pure sine cosine. There was no damping. It just oscillated forever.
Now, the new ones with a B are B squared could be smaller. And in that case, what does the solution look like? If B squared is smaller than 4AC, I have to always look back here. So I have something negative.quadratic eigenvalue problem leads to complex eigenvalues and eigenvectors.
The modal de- composition of the equations of motion is usually to be performed in complex space. Linear Algebra Linear algebra relates to points, lines and their planes, and the algebraic analysis of their intersections. A number of new concepts are introduced, such as that of a vector space, which is essentially a collection of vectors that form a defined structure.
And the problem says, the motion is a free undamped motion because there is no external forces free and because we assumed that there is no damping, so it's undamped too, right? DIFFERENTIAL EQUATIONS with Boundary-Value Problemsa Zill Cullen, Exercises for Mathematics.
Free Undamped Motion Spring/Mass Systems: Free Damped Motion Partial differential equations or PDEs are considered in the expanded volume Differential Equations with Boundary-Value Problems, Seventh Edition.
Sep 08, · The problem statement, all variables and given/known data i have uploaded my question.
please check out the initiativeblog.com file. Homework Help: Equation of motion: spring mass system - free undamped vibration Sep 7, #1. So the equation of motion will be exactly the same. Realize that you don't have to know the direction of the.
Note as well that while we example mechanical vibrations in this section a simple change of notation (and corresponding change in what the quantities represent) can move this into almost any other engineering field. This force may or may not be present for any given problem.
Free, Undamped Vibrations. This is the simplest case that we.