We begin our inquiry into mechanical systems with a reminder of Newton's Laws. Now the time honored introduction to Newton's Laws are Feynman's lectures, amazing series of lectures, and I refer you to that book. It's now online, it's hard copy, you can get it in many many different shapes and forms. We're remembering that Newton said that forces cause accelerations, and they're proportional to accelerations through the constant of proportionality called mass. If we measure we need a frame of reference. We're going to put a frame of reference on this figure. And once we have a frame of reference we can measure position Ki in this picture. We can measure velocity that are labeled v. Of course the velocity is the time derivative of the position. And the acceleration that I'm labeling alpha Is the time derivative of velocity? You remember this from Calculus, that if you have a constant acceleration, the horizontal line in this figure. Then your velocity gives you a constant slope curve in the position integrates to a parabolic curve. Please review your calculus in case any of this is confusing or unfamiliar to you. Our notation will be when masses or anything else moves, we'll use kai of t. And when we want to think about their variation in time we'll use Chi dot, that is a little dot over the Greek variable chi to denote the time derivative of the motion chi of t. So this equation says v(t), the velocity at any instant of time t is defined to be, that's what this funny symbol means, it's defined to be chi dot of t. Which is the time derivative d by dt of the position chi of t. When masses move, they have velocity. And they have acceleration, and you remember that the acceleration, alpha of t, is the first derivative. A v. V dot, and therefore it's dt squared of chi, or chi double dot at each instant T. The calculus version of Newton's Law tells us that not f equals MA, but f equals m times d squared t squared or m is balance by the force f. Let's introduce to our moving particle, an acceleration that's impeded by the force of friction. Friction is most commonly modeled as a viscous force that's proportional to the velocity and in opposition to the direction of acceleration. This equation m chi double dot equals negative b times chi dot, expresses in one neat form, all of those ideas. It says that Newton's m a or the inertial force of acceleration is exactly balanced by the viscous drag introduced by the friction that's slowing down the motion in proportion to its velocity. Now, if we take off our physicists' hat, and put on our applied mathematics hat, we're going to realize that this Is a differential equation, namely if we recast kai dot as v and recast kai double dot as v dot. We realize that v dot equals lambda v is an equivalent formulation of Newton's law with Viscous Friction where lambda, that we'll come to think of as the time constant, is given by the ratio of the viscous stamping coefficient and the mass. I hope that you've had a little bit of an introduction to ordinary differential equations. And if you have you'll know that when faced with a linear time and variant differential equation, so called LTI. We can immediately deduce that our friend, the exponential function of time, must give a solution as long as we scale time by the time constant, lambda. So let's check, d by dt of e lambda t is d by dt of this infinite series. And if you apply the derivative operator to each term of this series, d by dt of the constant 1 is 0. Let's bring d by dt in under the summation sign for each of these terms. Take d by dt of the k term, and you remember from calculus, k comes down. k minus one is left as the exponent and k factorial is going to be reduced to k minus 1 factorial by cancellation by k. And so, we get lambda times that same infinite series, and we realise that d by dt of e to the lambda t, is just lambda times that function, e to that lambda t. We realise, now, that a solution for this ode, for any initial condition, is given by v at time, t, from initial condition vo, defined to be e to the lambda t times the initial velocity condition, vo. Again, I'm assuming in this module that you've had a course in ordinary differential equations. Perhaps just a linear ordinary differential equations course. And I'm hoping that you will take some time to review these ideas if they seem unfamiliar or foreign to you. Let's move ahead. Let's add to our acceleration balanced by viscous friction an acceleration balanced as well by a compliant force. This would be the spring force. Due to the stretching of a spring, and you'll recall from your physics classes that the spring or the compliance law asserts a force in opposition to the direction of motion in proportion to the amount stretched. That says that the spring term is negative k times chi. And we have the differential equation m chi double dot equals negative b times chi dot minus k times chi. Now what we'd like to do is think about this rather as a first order system in two dimensions. Then as a second order system in one dimension. Let's say that again. Let's look at this new equation, it's a vector definition. I'm going to think about the two dimensional vector, boldface x, as defined to have two entries, x sub one, x sub two. And I'm defining x1 to be the position variable chi from this differential equation. I'm defining x2 to be this position variable, to be this velocity variable, excuse me, v, from the differential equation. I can now look at the time derivative of each entry of the vector x. And I'm going to denote that thing by putting an x over the vector quantity. I'm putting a dot over the vector quantity x. And what that means is, we should be thinking about the vector of derivatives. When we write down the vector of derivatives, we realize. That d by dt of x 1 is just x 2. How come? Become d by dt of kai is just v. When we write down the derivative of x 2 we get back the differential equation. Namely, we have to go back to the differential equation to realize that kai double dot Is the right hand side of the original ODE divided by n and that's what's been written here. So equation one shows that the second order scale differential equation can be replaced by a first order two dimensional, linear time invariant or LTI Dynamics. Now, I'm going to re-write the right-hand side of that first-order two dimensional equation, in matrix vector form. And again, I'm assuming you have a bit of an introduction to linear algebra, so you'll know how to think about this right hand side term as the product of a constant matrix cap A. Where I've written out A on the side here times the original vector x, whose entries are x1 and x2. This is how we're going to be thinking about our dynamical systems. We're going to be thinking about them not as second order, one dimensional, but as first order two dimensional. Or as we begin to add degrees of freedom, first order 2 n where n is the number of degrees of freedom. Don't worry if degrees of freedom doesn't mean anything to you yet. We're going to be reviewing these ideas carefully in weeks two and weeks three.
