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about Newton's laws in
Lecture three, but in the
absence of mass. Today we are going to look at what happens
when the mass at a point is non zero. That is we are going to look
at the sum of forces
being equal to mass times acceleration where the mass is non-zero.
Remember that here f are the forces acting on a point, always
defined to be positive if the force would cause a positive acceleration.
And A is the acceleration.
The second time derivative of the position of the point.
to find B positive, if the acceleration is to the right.
The acceleration, A, is also the first time derivative
of the velocity, v sub m of the point.
Lastly remember that v sub m of the mass, is always assumed
to be measured relative to what is called an an inertial reference frame.
In the case of the spring mass damper
system, the inertial reference frame is the wall.
That is, we measure velocities with respect to the wall.
This diagram could represent a lot of things, like the shocks
on a car or bicycle, or even this Slinky, where now
we think of the bottom of the Slinky as having mass
and the rest of the Slinky as having the spring characteristics.
If I hold the
Slinky horizontally, then gravity has no effect on the linear motion.
But if I hold it vertically, then, of course, the Slinky will oscillate
and that means that gravity is acting as an external force on the system.
The Slinky is a good example to keep in mind because the model
we choose to use really depends on what behavior we are trying to model.
If the Slinky only oscillates like this, then probably the model
that I just told you about will do a good job.
If however, we create extra oscillations like this,
then probably it won't do such a good job, because there are too
many oscillations in the system and such a simple model won't capture them all.
Hence, your engineering judgment really matters.
You cannot just blindly write down a model and expect it to be valid.
Let's look at a model of a mass with a spring and a damper.
The sum of the forces are equal to the mass times acceleration.
So we get F external, minus the spring force,
minus the damping force, because of our sign conventions.
And those are equal to mass times acceleration, which we know is equal
to mass times the second time derivative of the position of the spring.
We already know from lecture three that v sub d is equal to x sub s dot,
giving F external, and now we're going to plug
in the constitutive law, and we're going to plug in
the damping part of the constitutive law as well.
Now that is equal to mass times the second derivative.
Now notice that x sub s shows up everywhere in this equation.
Now, if we solve for x sub s double dot, we get, x sub s double dot
is equal to minus k over m, x sub s minus b
over n, x sub s dot, plus F external over m.
This is a linear
constant-coefficient second-order ordinary differential equation
in the position of the spring, x sub s.
We know that we need a first-order
ordinary differential equation if we are going
to use Euler integration from lecture four
and the analytic solutions from lecture five.
So we need to convert this system
to a first order ODE.
To do so it is helpful to note that the equation can be re-written as x sub s
double dot is equal to f some function of x sub s and x sub s dot.
Now, we can rewrite this double time derivative as x sub s, dot.
But taking d, dt of that.
And that is also equal to f of x sub s,
x sub s dot. This, we are going
to relabel, as just V, because it will help keep track of things in a nice way.
This means that V, dot, is equal to f of x sub s, V.
Because we have relabeled X sub S dot and we
know, that, because of the labeling, x sub s dot is equal to V.
This is now a first order differential
equation, where the relabeling allows us to see
that on the left hand side we have the time derivative of two quantities, and
on the right hand side we have a function of those two quantities.
This exercise of converting from a
second-order differential equation to a first-order
differential equation is something you will do a lot as an engineer.
It's worth practicing.
What should you remember from today?
First, Newton's equations allow you to incorporate mass
into a linear constant coefficient ordinary differential equation.
Secondly hopefully the Slinky convinced you that
when you have mass, you should expect
the possibility of oscillation.
And third, that ODE's have to be converted into first order
differential equations which will mean
increasing the number of differential equations.
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