Lecture 27: Review with problems

Lecture 27: Review with problems Problem set 7 was a very real problem set in the sense that there is more than one answer for most of them. I’d like to look more at this, with a view to seeing what is common and what is not.

Problem One

Find the nature of the vibration of the system shown l l m m k k We’ll take this slowly There are some unstated assumptions: that there is no motion in the horizontal direction that the angle is small that gravity does not affect the vibrations

We can demonstrate that gravity does not affect the vibrations, so it’s not really an assumption I will leave that to you, and simply say that the role of gravity is limited to the static problem: it determines the initial offset of the springs around which the system vibrates. I will define the positions of the elements of the system with respect to their static equilibrium positions

l l m m k k How many degrees of freedom? Two: it can go up and down and it can rotate (through a small angle)

Define as many coordinates as you like to make energies easy z3 z2 z1 z q center of mass

Let’s use these to look at the energies: easiest in terms of z1, z2, and z3 z3 z2 z1 (we said that gravity will not affect the vibration)

We actually have five coordinates from which to choose two independent coordinates z3 z2 z1 z q some choices: z1 and z2 z1 and z3 z and q . . . center of mass

We can certainly make other choices, but these seem to me to be the most obvious choices Someequations connecting the variables

How do these work in the energies? I’ll need expressions for the three variables in terms of the chosen two. z1 and z2

Form the Euler-Lagrange equations Rearrange them to make it easier to form the state space

Form the state equations Eigenvalues

The eigenvalues are pure imaginaries, so they represent the vibration frequencies The modes of vibration — how the masses move — are determined by the eigenvectors I will defer consideration of the eigenvectors until later Right now I want to see how the other two formulations work

z1 and z3

Form the state equations The matrix is different, but the eigenvalues are the same The eigenvectors will not be the same.

z and q This one is a little more complicated

and we have the differential equations Form the state space picture

and the eigenvalues are once again the same

Three matrices with the same eigenvalues This means that they are linear transformations of each other, but I don’t want to explore that now

In order to get a better picture of what is going on, I will specify some numbers: l = 1 = w2, which I can do without loss of generality by the appropriate scaling The common eigenvalues become ±0.874032j and 2.28825j from which we can pluck the oscillation frequencies. The eigenvectors are complex conjugates of each other: one pair for each frequency I really cannot deal with eigenvectors in power point; we’ll have to go to Mathematica

How does the Laplace transform deal with this problem? We can choose any of the formulations and work an initial value problem I’ll choose the third, so we have with x(0) = x0

Symbolically The Laplace transform is from which we find the transform of the solution The denominator of the inverse is a polynomial whose degree is the same as the dimension of the state The various numerators are all polynomials of small degree, so we will have a solution amenable to partial fraction expansion followed by inversion Look at the structure for our case

Take the Laplace transform and we will need to go to Mathematica to make reasonable progress, so let’s switch this lecture to Mathematic and look at this, and the eigenvectors

Off to Mathematica

Kinetic Energy z center of mass y Tacit assumption: y is fixed

z center of mass z1 z2 y Tacit assumption: q remains small enough that sideways motion of the spring may be neglected

Lecture 27: Review with problems