How Can You Model Car Oscillations Over a Speed Bump Using Excel?

[_Bd_]

Homework Statement

You have a car going over a speed bump, you need to model the oscillation of the car using excel only (no MATLAB or other software)


Simple Schematics:

|Mass of car|
| |
Spring Damper
| |
\ /
| Wheel | _____________/ Speed bump \________________


Car speed: 88 inches/s
Car mass: 700 lb (need to divide over G=386 in/s^2
Bump Max height (Hb) = 5
Bump length (Lb) =36
Bump Equation (y_wheel)= (Hb/2)*(1-COS(2*PI*X/Lb))
This is the vertical displacement of the wheel as it goes over the bump, the derivative (velocity) is then:
y'_wheel= (Hb/2)*(2PI*V'/Lb)SIN(2*PI*X/Lb)
damping frequency (w_d) = We can assume and play with this
damping ratio (zeta) = We can assume and play with this, values between 0 and .9

Homework Equations

Using force analysis ( on each element) and some solving here and there

Using Laplace transforms and my notes I came up with some equations, I won't list those processes since I verified with the professor that they are correct, but here are the results:


Assuming a zeta of .5 and a frequency of .825 Hz
(natural frequency) w_n=w_d/(1-zeta^2)^.5
(damping frequency) w_d= 5.184 radians
(spring constant) Ks= w_n^2*m
(ave. damping constant) Kd= 2*zeta*(K_s*m)^.5
NOTE: Kd will be divided into K_up and K_down which average to K_d and generally Kup < Kdown

The Attempt at a Solution

I've made my excel table, my problem is that I keep getting a weird wave:

Fs = force of spring
Fd= force of damper
y_w = vertical displacement of wheel
y_c = vertical displacement of car (y' is velocity and y'' is acceleration)

Time | Displacement | Y_wheel | Fs | Fd | K(up or down?) |Y''_c | Y'_c | Y_c

Fs=Ks(y_w - y_c)
Fd=Kd(y'_w - y'_c)
Y''_c= Fs+Fd/m
Y'_ci= Y'_c(i-1)+Y''*dt
Y_ci= Y_c(i-1)+Y'*dt
http://imageshack.us/a/img850/4871/tablegf.png


I get a really funky graph that initially looks good but then doesn't seem to "converge" towards the center like all the graphs in the internet.
I've searched all over and many webpages provide excel codes to do this, (so you can just input your variables and let it solve it by itself) but I want to code it myself (learn how to)

http://imageshack.us/a/img838/6452/graphlj.png

If I extend more time it keeps going down and down and down

and it should look similar to this
http://ese.wustl.edu/ContentFiles/Research/UndergraduateResearch/CompletedProjects/WebPages/2006/as12/identify%20single_files/image017.gif

 

[mfb]

Imperial units :/

What happens if you set the damping to 0?
I think the signs of the damping are wrong in some way - it is dampening in the upwards direction, but increasing the energy in the downwards direction.

 

[Astronuc]

http://imageshack.us/a/img838/6452/graphlj.png

If I extend more time it keeps going down and down and down

and it should look similar to this
http://ese.wustl.edu/ContentFiles/Research/UndergraduateResearch/CompletedProjects/WebPages/2006/as12/identify%20single_files/image017.gif

One appears to have an equation of a form: ( b - ax ) sin (ωx) rather than a damped sine wave.

The cited image shows the behavior obtained from the convolution of a step function and function formed by the sum of a constant and damped sine wave.

I think though one wants a damped sine wave where the amplitude corresponds to the displacement from its equilibrium.

 

[rude man]

I assume that when you wrote Y''_c= Fs+Fd/m you actually implemented
Y''_c= (Fs+Fd)/m.

The equations look right. Of course I can't comment on how you implemented y_w(t) in Laplace.

I don't understand the distinction between ks_up and ks_down. Seems to me the force on the car is always ks(y_w - y_c) where ks is the one and only spring constant. That assumes y = y_c - y_w = 0 before hitting the bump. The fact that the spring is compressed at that point does not change the equation.

Finally, if all else looks right, have you tried different values for dt? Like 10dt or dt/10? Finite-difference equations are strange critters!

You also know I'm sure that there are many levels of transforming a differential equation into a finite-difference one. Some of those turnkey simulations you mention may well have used a higher-order Runge-Kutta or other equation.

 

[rezeew]

It is great that you are using Excel to model the oscillations of a car going over a speed bump. Excel is a powerful tool for data analysis and can also be used for simple mathematical modeling. However, it is important to keep in mind that Excel is not a dedicated mathematical modeling software like MATLAB and may have limitations in terms of accuracy and complexity of the model.

In order to accurately model the oscillations of the car, it is important to consider all the forces acting on the car, such as gravity, normal force, and the forces from the spring and damper. I would suggest using a force analysis approach and applying Newton's laws to determine the equations of motion for the car. This will help you to accurately model the dynamics of the car and its interaction with the speed bump.

Additionally, it is important to carefully choose the values for the damping frequency and damping ratio. These parameters can greatly affect the behavior of the system and may need to be adjusted in order to achieve the desired results. It would be helpful to consult with your professor or do some further research on the effects of these parameters on the oscillation behavior of the car.

Overall, it is great that you are taking the initiative to code the model yourself and learn how to use Excel for mathematical modeling. However, it is important to keep in mind the limitations of Excel and to carefully consider all the forces and parameters involved in the system in order to accurately model the oscillations of the car.