Damped Vibration, 1DOF, Worked example, Need help breaking it down.

[Jowin86]

Homework Statement

https://www.physicsforums.com/attachment.php?attachmentid=39373&stc=1&d=1317244721

Homework Equations

In all honesty I am not sure?

The Attempt at a Solution

This question was used as a worked example in my first tutorial for Dynamics. The lecturer didn't break it down into logical steps or explain where he was getting all the formulae he used so I am left with pages of notes that don't make any sense!?

I can't pick out the formulae from the equations or pick out what it is he's done to find k, spring stiffness and C, damper coefficient and then the equation for displacement as a function of time with the few bits of info there are. One thing that puzzles me is the first line is 2Td=4, so Td=2 and I don't know what T or d stand for?

I guess this is asking a lot but if someone could write out the working out step by step so I can go through it a few times and try to learn it it'd be a life saver! Either that or point me in the direction of an example similar to this on the net? At the moment I am trying to find walkthroughs on the net but can't find anything that explains it. Kind of hard to teach yourself something you don't know :(

Thanks for any help!

 

[Spinnor]

See if you can follow this,

http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html

For more see,

http://www.google.com/search?hl=en&...w.,cf.osb&fp=e6558c79d248f203&biw=721&bih=408

 

[Jowin86]

thanks! That helps a bit but I can't find out what omega(d) is, or what Td is. I've found that T=period...but what's d?? :-/

 

[Jowin86]

Nevermind, omega(d) is damped natural frequency. I am going to ask my lecturer if he can explain it to me next monday :(

 

[rwisz]

Dear student,

I can understand your confusion with this problem. Let me try to break it down for you.

Firstly, the problem is about damped vibration, which means that the motion of the system is being slowed down by a damping force. This is represented by the term "Cv" in the equation, where C is the damper coefficient and v is the velocity of the system.

The system in this problem is a 1 degree of freedom (1DOF) system, which means that it has only one independent variable that describes its motion. In this case, it is the displacement of the system, denoted by "x".

Now, let's look at the equations given in the problem. The first equation is the equation of motion for a damped system, which is given by:

m(d^2x/dt^2) + C(dx/dt) + kx = 0

where m is the mass of the system, C is the damper coefficient, and k is the spring stiffness.

The next equation is the solution to this equation of motion, which is given by:

x(t) = Ae^(-ζωn t)cos(ωdt + φ)

where A is the amplitude of the vibration, ζ is the damping ratio, ωn is the natural frequency of the system, ωd is the damped frequency, and φ is the phase angle.

Now, let's look at the given information in the problem. We are given the natural frequency of the system (ωn) as 1 rad/s, the damping ratio (ζ) as 0.2, and the amplitude (A) as 1 m. We are also given the information that the system goes through 4 complete cycles (2π radians) in 2 seconds (T).

Using this information, we can find the damped frequency (ωd) using the formula:

ωd = ωn√(1 - ζ^2)

Plugging in the values, we get ωd = 0.98 rad/s.

Next, we can find the phase angle (φ) using the formula:

tan φ = (2ζωn)/(ωd)

Plugging in the values, we get φ = 0.41 radians.

Now, we can use the given information that the system goes through 4 complete cycles in 2 seconds to find the period of the system (