# MATLAB Spring mass impact system in Matlab - How to correct it?

#### k.udhay

Summary
A mass of constant velocity colliding on spring is modeled in simulink. Result expected was to be of sinusoidal displacement of spring. However result found was different.
I am new to Simulink and I wanted to start practicing using a spring mass damper system. My first tutorial was this:

Later, I wanted to model a spring system where a mass moving at a known velocity hits the spring. The governing equation and a similar modeling method given in the previous youtube link was also prepared:

Here is the Matlab Simulink model:

The result I expected was a sinusoidal displacement of the spring. However the result was totally different:

If someone can tell what exactly is my mistake and the correction, that will be of huge help. Thanks.

Related MATLAB, Maple, Mathematica, LaTeX, Etc News on Phys.org

#### RPinPA

Homework Helper
You're using an energy equation where all the terms are positive. It is not going to tell you the signs of $x$ or of $v$.

To get the equation of motion, you want to use the fact that the force on the mass is related to the displacement $F = -kx = m \frac {d^2x}{dt^2}$ This force will change sign as $x$ changes sign, which will lead to the sinusoidal behavior.

#### RPinPA

Homework Helper
Please update at some point and let me know if my observation actually led to success. It was just something that jumped out at me looking at your equation.

#### k.udhay

Thank you for your inputs, RPinPA. While I understand force based equation will give me sinusoidal displacement output (as mentioned in the youtube link in OP), I have two questions that I need a clarity:

1. Why will the energy based governing equation not give correct displacement values? It involves velocity (which is a derivative of position) too.

2. If I go with force based governing equation (as you have suggested), where do I input the initial velocity of the mass? This is critical as this determines the amount of deformation of spring (= amplitude of sine wave)

#### RPinPA

Homework Helper
Thank you for your inputs, RPinPA. While I understand force based equation will give me sinusoidal displacement output (as mentioned in the youtube link in OP), I have two questions that I need a clarity:

1. Why will the energy based governing equation not give correct displacement values? It involves velocity (which is a derivative of position) too.
Because you can't tell whether to take the velocity as positive or negative at any given time. Both are solutions.

2. If I go with force based governing equation (as you have suggested), where do I input the initial velocity of the mass? This is critical as this determines the amount of deformation of spring (= amplitude of sine wave)
Well if you were solving this analytically, you'd have a functional expression for $x(t)$ with some free parameters and you'd evaluate its derivative at $t = 0$ to use the information on $v(0)$.

I don't know much about Simulink but I assume you are implementing differential equations and the software takes care of evolving a solution numerically from starting conditions. So here's a more convenient form for that.

A common transformation of 2nd order equations is to turn it into a system of 1st order equations.
So we have this:
$$-kx = m\frac {d^2x}{dt^2}$$

Change it to this equivalent system in the two variables $v$ and $x$:
$$\frac {dx}{dt} = v \\ m\frac {dv}{dt} = -kx$$
Implement those two equations in those two unknowns with the desired initial conditions on $x$ and $v$.

#### RPinPA

Homework Helper
I should emphasize that my comments are about the mathematics of the differential equations you're solving, as I have no expertise in Simulink. I have used Matlab for decades but managed to avoid ever using Simulink or even having it on a computer I was using.

#### k.udhay

Thanks a lot. I will try this and update you.

#### k.udhay

@RPinPA - I tried this in Simulink and there is actually a progress:

However, I see some form of dampening effect in all position, velocity and acceleration plots. I am unable to understand the reason for this. This is my simulink model:

The initial condition (40) tells that dx/dt = 40 when t = 0. I am not very sure if this way of inputting is right. I will be so glad if someone of PF helped me out of this issue. Thank you for guiding me very closely.

"Spring mass impact system in Matlab - How to correct it?"

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving