Nondimensionlize damped spring

[INeedHelpTY]

Homework Statement

Vertically suspended spring with mass attached. F = ma gives the diff eq: m(d^2y/dt^2) + b(dy/dt) + ky = 0.

y(t) measures the vertical position (upward is positive) of the mass relative to the equilibrium position (how far the mass hangs down if it is not moving).
m = mass of the attached mass
b = damping coefficient of spring (units N*S/m)
k = spring constant (units N/m)

Consider the mass to be released from rest at height A, initial conditions:
y(0) = A and y'(0) = 0

Nondimensionalize the system using t=t_cs, y(t)=y_cz(s). Choose t_c and y_c in a way that permits the investigation of the effect of b. Write the nondimensional system using [tex]B[/tex]=b/sqrt(mk)

The Attempt at a Solution

A

Here's what i got so far:
d/dt = (ds/dt)(d/ds) = (1/t_c *d/ds)
d^2/dt^2 = (1/t_c^2)(d^2/ds^2)

substitute into the system equation on the top gives:
(m/t_c²)(d²/ds²(y_cz(s))) + (b/t_c)(d/ds(y_cz(s))) + ky_cz(s) = 0

(my_c/t_c²)(d²z/ds²) + (by_c/t_c)(dz/ds) + ky_cz(s) = 0

dividing by (my_c/t_c²) gives:
d²z/ds² + (bt_c/m)dz/ds + (t_c²k/m)z = 0

Gives 3 dimensionless Pis:
Pi₁ = (bt_c/m)
Pi₂ = (t_c²k/m)

Unsure about this (am i right?): the corresponding initial conditions are z(0) = A/y_c and dz/ds (0) = 0.

If so, this gives Pi₃ = A/y_c.

Ideas for going further in this problem is that to pick certain Pis and work around to make the system into dimensionless based on dz/ds.

 

[mighty2000]

Then can find the effect of b by looking at Pi1 and Pi2 for different values of B.

I would like to add some input to your solution. Your approach to nondimensionalizing the system is correct. However, there are a few things that can be improved:

1. The units of B should be N/(m*sqrt(N/m)) = sqrt(N/m). This means that B is already dimensionless and does not need to be divided by sqrt(mk).

2. Instead of using A/yc as Pi3, it would be more useful to use A/L, where L is the natural length of the spring (i.e. the length when there is no mass attached). This would make Pi3 dimensionless and give a more meaningful representation of the initial condition.

3. Instead of using Pi1 and Pi2, you can use Pi1 and Pi3 to investigate the effect of b. This is because Pi2 is already a combination of b and k, so using it separately would not give any new information.

4. To investigate the effect of b, you can fix the values of Pi2 and Pi3 and vary Pi1 (which represents b) to see how the system responds. For example, you can fix Pi2 = 1 and Pi3 = 1, and then vary Pi1 from 0 to 10 to see how the system behaves for different values of b.

Overall, your approach is correct and you have identified the important dimensionless parameters in the system. With a few improvements, you can use this nondimensionalized system to investigate the effect of b on the system.