Vertical 2 mass 2 spring problem - IVP

[tusch05]

Homework Statement

mass1 = 10kg connected to ceiling by a spring
mass2 = 2kg connected to mass1 by a spring

2m/s is applied to mass 1 in the downward direction (positive direction) to get the system in motion
At equilibrium Mass 1 is 2m from the ceiling while mass 2 is 4m from the ceiling

10 x1’’ = -100x1 + 120(x2-x1)
2 x2’’ = -120(x2-x1)

find solution over time period 0<=t<=10sec

Need to use Eulers method

Homework Equations

The Attempt at a Solution

Reduce equations to:
x1’’ = 12x2-22x1
x2’’ = 60x1 – 60x2


Let:
x1’ = z
x1’’ = dz/dx
z(0) = 2 m/s ?


Let:
x2’ = v
x2’’ = dv/dt
v(0) = 0 m/s ?


dx1/dx = z = f1(x1,x2,z)
dz/dx = 12x2-22x1 = f2(x1,x2,z)
dx2/dx = v = f3(x1,x2,v)
dv/dx = 60x1 - 60x2 = f4(x1,x2,v)



Not sure if I have these initial conditions set up properly??
Please help, Thanks

 

[KataKoniK]

!



Thank you for your post. It seems like you are on the right track with your solution attempt. However, there are a few things that need to be clarified and corrected in order to find a solution using Euler's method.

Firstly, the initial conditions for the velocities should be given in terms of the initial conditions for the positions. In this case, we are given the initial positions of the masses, but not their initial velocities. Therefore, we need to use the given information to find the initial velocities.

We know that mass 1 is initially at rest (x1' = 0) and it is given an initial velocity of 2 m/s in the downward direction (positive direction). This means that x1'' = -2 m/s^2. We can use this information to find the initial velocity of mass 2, which is connected to mass 1 by a spring. Since the spring is initially at equilibrium, we can assume that the initial velocities of mass 1 and mass 2 are equal, i.e. x2' = z = 2 m/s. This also means that x2'' = dz/dx = -2 m/s^2.

Therefore, the correct initial conditions for the velocities are:
x1' = 0 m/s
x1'' = -2 m/s^2
x2' = z = 2 m/s
x2'' = dz/dx = -2 m/s^2

Secondly, the equations for the accelerations (x1'' and x2'') are incorrect. The correct equations are:
x1'' = -120(x1-x2)/10 = -12(x1-x2)
x2'' = -120(x2-x1)/2 = -60(x2-x1)

Substituting these equations into the Euler's method formula, we get:
x1(t+h) = x1(t) + h*z(t)
z(t+h) = z(t) + h*(-12*(x1-x2))
x2(t+h) = x2(t) + h*v(t)
v(t+h) = v(t) + h*(-60*(x2-x1))

Now, we can use these equations to find the positions and velocities of the masses over the given time period (0<=t<=10 sec). I hope this helps. Let me know if you have any further questions. Good luck with your calculations!

 

[Mene]

I would first commend the student for attempting to solve the problem and using Euler's method to do so. However, I would also suggest that the student double check their initial conditions for x1' and x2' as well as their equations for x1'' and x2''. It may also be helpful to plot the equations and initial conditions to visually understand the system's behavior over time. Additionally, I would recommend checking for any errors or typos in the equations and initial conditions to ensure accurate results. Overall, it seems like the student is on the right track and with some minor adjustments, they should be able to successfully solve the problem.