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I am confused on where to start exactly on this problem. Does it matter how many loadings there are and what does the length below the trailer come in? I guess I am just having a problem vizualizing it all.
* Here you need to treat the trailer as a spring-mass system with a mass of 500 kg.
* The information provided of "During the loading, each 75 kg added to the loading caused the trailer to sag 3 mm..." is to be used to determine the stiffness of the spring.
* The system experiences a "base excitation" (prescribed displacement as a function of time). This is the trickiest part of the problem. The road surface is a sinusoidal curve with a wavelength of 1.2 meters and a peak-to-peak amplitude of 50 mm. First write out the road surface waviness as a function of, say, z where z is the distance traveled along the road: xB = A*sin(lambda*z), where A = 25 mm and lambda = 2*pi/1.2. Since the trailer is traveling at a constant speed of v, you can write z = v*t. Therefore, the "base excitation" is given by: xB = A*sin((2*pi*v/1.2)*t). Can you recognize the "frequency of excitation" from this expression?
I am still confused as to what to do for this problem. I have found the EOM, x(double dot)+(k/m)x=-(k/m)xB where xB is what you explined. I don't know what to do with this equation. I was assuming that I should next solve the equation in terms of xP(t) and xP(double dot)(t). Is this the correct way to go about doing this?
Bashover, for your EOM make sure your signs are correct. With your equation you have that the spring force is k(x+xb) since they have opposite signs on opposite sides of the equation. your Force term on the right should be positive if im not mistaken.
--to L. Bashover-- Yes, you now need to find the particular solution of your differential equation of motion. Here you assume a solution for the form xP(t) = A*cos(omega*t) + B*sin(omega*t). By substituting this back into the EOM, you can solve for A and B.
Courtney's approach for finding k is right on the money. She is saying that the effective stiffness is the added weight divided by the resulting added static deformation.
rjaneshe makes a good point on signs. It looks like you have the positive directions for x and xB in OPPOSITE directions since you end up with a negative sign in front of the effective forcing on the right hand side of your EOM. This is fine if that was your choice of coordinates. Just make clear your coordinates at the start of the problem and stick with those sign conventions throughout.
10 comments:
* Here you need to treat the trailer as a spring-mass system with a mass of 500 kg.
* The information provided of "During the loading, each 75 kg added to the loading caused the trailer to sag 3 mm..." is to be used to determine the stiffness of the spring.
* The system experiences a "base excitation" (prescribed displacement as a function of time). This is the trickiest part of the problem. The road surface is a sinusoidal curve with a wavelength of 1.2 meters and a peak-to-peak amplitude of 50 mm. First write out the road surface waviness as a function of, say, z where z is the distance traveled along the road: xB = A*sin(lambda*z), where A = 25 mm and lambda = 2*pi/1.2. Since the trailer is traveling at a constant speed of v, you can write z = v*t. Therefore, the "base excitation" is given by: xB = A*sin((2*pi*v/1.2)*t). Can you recognize the "frequency of excitation" from this expression?
Oh ok, this really helps! I think I was just making it more complicated. thanks.
I am still confused as to what to do for this problem. I have found the EOM, x(double dot)+(k/m)x=-(k/m)xB where xB is what you explined. I don't know what to do with this equation. I was assuming that I should next solve the equation in terms of xP(t) and xP(double dot)(t). Is this the correct way to go about doing this?
Also, I do not know how to find k of the spring.
Thanks!
I found the k to be k = [(75kg)*(9.81m/s^2)]/(0.003m)
Bashover, for your EOM make sure your signs are correct. With your equation you have that the spring force is k(x+xb) since they have opposite signs on opposite sides of the equation. your Force term on the right should be positive if im not mistaken.
--to L. Bashover--
Yes, you now need to find the particular solution of your differential equation of motion. Here you assume a solution for the form xP(t) = A*cos(omega*t) + B*sin(omega*t). By substituting this back into the EOM, you can solve for A and B.
Courtney's approach for finding k is right on the money. She is saying that the effective stiffness is the added weight divided by the resulting added static deformation.
rjaneshe makes a good point on signs. It looks like you have the positive directions for x and xB in OPPOSITE directions since you end up with a negative sign in front of the effective forcing on the right hand side of your EOM. This is fine if that was your choice of coordinates. Just make clear your coordinates at the start of the problem and stick with those sign conventions throughout.
Thanks! This helps a lot.
As for finding the stiffness of the spring value, is this an equation in the book or notes that I am overlooking? I'm not sure where this comes from.
F(sp) = k * delta
therefore, k = F/delta
where F = 75*9.81 N, and delta = 0.003 m
Thank you guys!
I just understood and solved this problem using your comments...
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