Question about free damped vibration in an SD

[NookanCranny]

Homework Statement

"If a horizontal force of F = 299kN is applied to the right at the girder level by a hydraulic jack and then the force is removed suddenly, find the equation of motion of the girder. Consider 5% critical damping. Find the
displacement and velocity of the girder at 0.2 seconds after release of the force."

This is of a braced single storey building. (bottom left is braced to top right etc)
I = 7.6*10^6mm^4
E = 200 GPa
Length of Girder = 6m
Height of Columns = 3.5m
The braced bars are 25mm diameter. This meant that the Area of them came to 490.874mm^2 as Area = (pi*d^2) / 4

mass of frame = 4000kg.

stiffness (girder and two columns) = 11404.775N/mm

natural frequency = 53.4 rad/s

Homework Equations

m*(acceleration) + c*velocity +k*displacement = F(x) -> (Inertia Force + Viscous force + Stiffness force = F(x))

critical damping constant = 2*mass*omega

damped natural frequency = natural frequency * sqrt (1-ξ^2)

natural frequency = sqrt (stiffness / mass)

The Attempt at a Solution

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I calculated Stiffness in both the columns and the rigid girder. This came to 11404.775. I solved for the columns by using (12EI/L^3) multiplied by 2 as there are two columns. For the rigid girder, I had to take into account theta.

I then calculated natural frequency of vibration after finding stiffness and dividing it out by the mass. After square rooting the fraction, it came to 53.4 rad/sec.

I then calculated the critical damping constant, this came to 427200kg/sec as critical damping constant Cc = 2*mass*natural frequency.

C actually was less than Cc, meaning it is under critically damped. ξ = 0.05.

I then went back to m*(acceleration) + c*velocity +k*displacement = F(x)

I then made all of it equal to zero, deriving from a characteristic equation and then solving for the roots using the quadratic formula. All this did was bring me back to an equation I already knew from my notes, as shown with attached image.

I am wondering, since it is applied instantaneously, does m*acceleration + c*velocity + k*displacement still equal to 0? or does it equal to 299kN?? The 299 is throwing me off because I was told if the frame is free moving, then it should equal to zero.

Also I'm not sure about writing the equation of motion when I am not given the initial velocity or displacement constraints.

Apologies if this question sounds a bit silly - I haven't come across one of these questions in my tutes!

 

[KataKoniK]

Firstly, let me clarify that the equation of motion in this case would be:

m*(acceleration) + c*velocity + k*displacement = F(x)

Where F(x) is the external force applied to the girder. In this case, it is 299kN to the right. So, the equation would be:

m*(acceleration) + c*velocity + k*displacement = 299kN

Now, since the force is applied instantaneously and then removed suddenly, we can assume that the displacement and velocity at the instant of application of the force are both zero. This is because the girder was at rest before the force was applied and the force was applied suddenly. So, the complete equation of motion would be:

m*(acceleration) + c*velocity + k*displacement = 299kN for t>0

And for t=0, the initial conditions would be:

velocity = 0
displacement = 0

Now, to solve this equation, we need to know the values of mass, damping constant, and stiffness. You have already calculated the stiffness to be 11404.775N/mm. However, the mass and damping constant are not given in the problem statement. It is also not clear what the value of the critical damping constant (Cc) that you have calculated represents. So, I cannot provide a specific solution to the problem without these values.

However, I can provide some general steps to solve the problem:

1. Calculate the mass of the girder by considering the mass of the columns and the rigid girder. This can be done by using the mass density of the material and the volume of the girder.

2. Calculate the damping constant by using the critical damping constant and the damping ratio (ξ). The damping ratio can be calculated using the formula given in the problem statement.

3. Substitute the values of mass, damping constant, and stiffness in the equation of motion.

4. Solve the equation using the initial conditions and the fact that the force is suddenly applied and then removed.

5. This will give you the equation of motion for the girder. You can then use this equation to calculate the displacement and velocity of the girder at any given time, including 0.2 seconds after the force is removed.

I hope this helps. If you have any further questions or if I have misunderstood any part of your problem, please feel