Mechanical oscillation task

[Ploppen]

Hi everybody!
I have a big question for you, I have been staring me blind on this problem I got.
And wonder if someof you could help mee with it?

the guestion is this:

"Find the time dependent oscillation u(t) for time t>0 for a damped system with one degree of freedome where c=50 kg/s, k=20 000 N/m, m=2 kg and which suddenly affected by a step function force p(t)=p0 H(t). (that is, the force in zero when t<0 and constant p0=1000 N when t>0.) At time t=0 the mass is in the equilibrium position with zero velocity. (easiest to not use Laplace, even though it is one possible way)."

I whould be soo nice if somebody could help me with this task!

thanks!

best regards
//Tobias

 

[eljose79]

Hello Tobias,

Thank you for sharing your question with us. I am happy to help you with this problem. Let's break down the question and see if we can find a solution.

First, we have a damped system with one degree of freedom. This means that the system is a single mass-spring-damper system with the mass m, spring constant k, and damping coefficient c. The system is also affected by a step function force p(t) which is zero when t<0 and constant p0=1000 N when t>0.

To find the time-dependent oscillation u(t), we can use the equation of motion for a damped system:

m*u''(t) + c*u'(t) + k*u(t) = p(t)

where u''(t) is the second derivative of u(t) with respect to time, u'(t) is the first derivative of u(t) with respect to time, and u(t) is the displacement of the mass from its equilibrium position.

Since we are given the values of m, c, and k, we can plug them into the equation and get:

2*u''(t) + 50*u'(t) + 20,000*u(t) = p(t)

To solve this equation, we need to consider two cases: t<0 and t>0.

Case 1: t<0
In this case, the force p(t) is zero, and the equation becomes:

2*u''(t) + 50*u'(t) + 20,000*u(t) = 0

This is a standard second-order differential equation, and we can solve it using the characteristic equation method. The solution is:

u(t) = A*e^(-25*t) + B*e^(-10,000*t)

where A and B are constants to be determined by the initial conditions.

Case 2: t>0
In this case, the force p(t) is a constant p0=1000 N, and the equation becomes:

2*u''(t) + 50*u'(t) + 20,000*u(t) = 1000

To solve this equation, we can use the method of undetermined coefficients. The particular solution is:

u(t) = 0.05*t + 0.02

The general solution for t>0 is the sum of the particular solution and the

 

[charng]

Hello Tobias,

Thank you for reaching out for help with your mechanical oscillation task. It seems like you have a very interesting problem to solve. I would approach this problem by first setting up the equations of motion for the damped system. From there, I would use the initial conditions (zero position and zero velocity at t=0) to solve for the constants in the general solution for the displacement u(t).

Next, I would use the step function force p(t) to set up the equations for the forced oscillation, and then use the principle of superposition to combine the solutions for the free and forced oscillations. This will give you the time-dependent oscillation u(t) for t>0.

I understand that you prefer not to use Laplace, but it may be helpful to keep in mind that the Laplace transform can be a useful tool for solving differential equations in the time domain. However, if you choose not to use it, there are other methods such as the method of undetermined coefficients or variation of parameters that can also be used to solve the equations of motion.

I hope this helps and I wish you the best of luck in solving your task!

Best regards,