# Dynamic Spring Plunger System Help

#### aculaMD

Hi all!

I'm having trouble trying to mathematically model a dynamic system. I tried to draw the system up and write out the algebraic governing equations in the attachment. In short, a compressed spring is pushing on a syringe plunger, which is pushing a liquid through a small radius tube. I want create a dynamic system model, but I'm kind of stuck! Any help would be appreciated!

Thanks!
J

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#### aculaMD

Realized I accidentally posted this in pre-university instead of university. Sorry!

#### studiot

Hi all!

I'm having trouble trying to mathematically model a dynamic system. I tried to draw the system up and write out the algebraic governing equations in the attachment. In short, a compressed spring is pushing on a syringe plunger, which is pushing a liquid through a small radius tube. I want create a dynamic system model, but I'm kind of stuck! Any help would be appreciated!

Thanks!
J
Even a simple mathematical model of this system is not an algebraic equation or a system of algebraic equations.

The simplest will be one differential equation of motion, more complicated models will entail a system of diff equations.

Are you familiar with differential equations?

#### aculaMD

Even a simple mathematical model of this system is not an algebraic equation or a system of algebraic equations.

The simplest will be one differential equation of motion, more complicated models will entail a system of diff equations.

Are you familiar with differential equations?
Hi studiot,

I am familiar with differential equations, and I figured I would need them. I have an engineering degree, but haven't used anything I learned in school in quite a few years. I was trying to help out a friend with this problem, and realized how much I've forgotten! I'd love to brush up, if you could point me in the right direction!

Thanks!
J

#### studiot

The basic equation of motion refers to the plunger, which is controlled by an elastic force due to the spring and opposed by a viscous force due to the fluid. It will also be subject to an inertia force due to its mass. This inertial force may be negligible.

The equation relates the force to the displacement, X of the plunger by balancing the forces involved.

So the net force = plunger mass x its acceleration + spring elastic force + resistance force

Now the mass x acceleration is self evident and as I said may perhaps be ignored.

The stiffness, s, is defined as the force required to produce unit displacement.

The mechanical resistance,r, is defined as the force required to produce unit velocity.

Using these constants and labelling time as t we have the second order differential equation

$$\displaystyle F(t) = m\frac{{{d^2}X}}{{d{t^2}}} + sX + r\frac{{dX}}{{dt}}$$

Are you with me so far?

I'm sorry that's it for tonight from me, I will look again tomorrow for your thoughts.

• 1 person

#### aculaMD

studiot,

Thank you so much for going through this in detail for me. I really appreciate it.

I'm completely with you to this point. I agree that the plunger inertial force is likely negligible, and we can ignore it to make the solution simpler. Ignoring the inertial force, we have a first order linear differential equation - correct?

Thanks!
J

#### studiot

Engineering? That's good, I had wondered if the MD stood for medical doctor.

What you are analysing looks like a medical syringe pump driver.

The variable of interest in these is, of course, the discharge rate, Q.

So it could be appropriate to introduce the discharge form of Torricelli's equation

$$\displaystyle Q = Ca\sqrt {2gh}$$

and work on combining that with what we already have.

Further is the spring a real spring or an idealisation of the driver mechanism?
I ask because it is possible to devise constant force (pressure) spring mechanisms, that operate over an appropriate range of displacement.
This would surely be more desirable than a variable one?

#### aculaMD

Actually an MD with an engineering undergraduate. The set-up is really as simple as the original image I drew. It is just a thought experiment a friend and I were curious about. With a simple spring, the force on the plunger, and thus the flow rate out, decreases with time, and we were just curious what the flow rate and volume out vs. time would look like.

#### aculaMD

So, moving forward, would it be correct to say that s=-k (spring constant) and r=(8*n*l)/(pi*r^4) (resistant to viscous flow through the tube)?

#### studiot

So, moving forward, would it be correct to say that s=-k (spring constant) and r=(8*n*l)/(pi*r^4) (resistant to viscous flow through the tube)?
I have had a rethink about this problem an now think that I made it a bit overcomplicated with my equation of motion (which is correct but not needed)
My apologies.

As I understand the system, the plunger is pushed by the spring, which must therefore be precompressed so that it can push the plungers as it expands towards its equilibrium postion.
As the spring expands it exerts a diminishing force on the plunger, causing a reducing pressure on the fluid in the barrel of the syringe.
The fluid is thus forced out through the nozzle under a reducing pressure regime so exits at a continually reducing rate.

Under these circumstances an appropriate question is not how much fluid is discharged in a given time, but how much time is taken to discharge a given (presumably prescribed) amount of fluid.

Calculating this involves integrating the Torricelli equation I mentioned earlier, over time and under reducing head conditions.

If you are happy with this description I will post a sketch and some mathematics tomorrow.