# Challenge: Metronome and Cart Equations of Motion 6

Posted by **Seth Popinchalk**,

I haven’t

seen much activity on the Metronome Synchronization Challenge, so I want to provide a peak at the basis

for my solution. I took the approach of modeling a pendulum, and then linking

its dynamics to the dynamics of a cart. The challenge is choosing the right

coordinate systems to assign to each of these bodies, even though they move

together.

**The Pendulum**

If the

pendulum was all by itself, we could describe its motion by modeling the angle,

, as it deflects from vertical. The

only force on the pendulum bob is gravity. Because the pendulum is constrained

to rotation about the pivot point, we only need the component of gravity in pendulum

frame of reference.

If we

incorporate the motion of the cart, we find the linking of the acceleration of

the cart to the acceleration of the pendulum bob. If the cart is accelerating,

the reference frame of the pendulum accelerates, and this means that the

pendulum will experience an opposite force/acceleration.

Combining

these two ideas in the frame centered on the pendulum pivot,

we can write the pendulum equation of motion.

**Momentum
in the system**

To

understand the acceleration of the cart, I want to look at the change in the component of the momentum. Here are

some definitions:

Length of

the pendulum:

Mass of the pendulum bob:

Mass of the cart:

The momentum

of the pendulum bob in the direction

The momentum

of the cart is

The friction

of the cart on the surface is the only resistance I want to model in the system,

and we can describe this force as proportional to the velocity, . This represents of the change of

momentum.

Combine

these terms to describe the momentum in the direction.

Up to this

point, I have looked at the problem of a single pendulum on a cart. Adding

another pendulum is just another momentum term, and to keep track of the

pendulums we will introduce and .

Taking the

derivative of this equation gives us the acceleration of the cart:

**System
Equations**

This

equation for the cart, along with the dynamics of the pendulum provides the

equations of motion for the system. Rewriting these as the highest order

derivatives gives us something we can create in Simulink.

**Are you
up to the challenge?**

Can you

modify the model I provided in my last post to implement these equations? Post your solution by October 15^{th} to the File Exchange with the keyword *metronome*.

**Category:**- Challenge,
- Modeling,
- ODEs

## 6 CommentsOldest to Newest

**1**of 6

Hmmm, is a simple pendulum an appropriate mechanical model for a metronome? What you’ve modeled is a vertically hanging, gravity stabilized pendulum rather than a spring stabilized inverted pendulum. Based on your approach you should “flipud” your metronome scope ;-)

**2**of 6

@ABindermann – you are right… I should flip the metronome scope. Is this an appropriate mechanical model for a metronome? I have played around with a version using an inverted representation and a spring to stabilize it, and I didn’t see much difference in the behavior. The other experiment I tried was adding friction to the metronome and an energy reserve to kick the pendulum back. I found all of this made things more complicated without much benefit.

This is the metronome I was thinking about as I approached the problem:

**3**of 6

I have uploaded a file, but it will take a while to appear. I think the synchronisation will not work with a purely linear model, because that will produce several resonance frequencies. I modelled the driving force of the metronome, and that did the trick.

**4**of 6

The System equation for x” isn’t 100% correct. You forgot to multiply it with “-“. However, if I use the correct equation I can’t achieve phase coupling anymore…

**5**of 6

The system equations have an algebraic loop.

This is against physics… is’nt it?

**6**of 6

I would like to ask you, what is the name of the system （the pendulum with the car ）? If there is friction between the car and the substrate, and there is a horizontal sinusoidal acceleration field, then what is the movement of the ball?

## Recent Comments