Steps:
- FBD: shown above for trailer (pay close attention to the magnitude and direction of the spring force on the trailer).
- Newton/Euler: sum F_x = k(y_B - x) - mg = m*x_dot_dot
- Kinematics: The position of the trailer, z, is given by z = v*t since the trailer moves at a constant speed. The vertical position of the wheel B depends on the position z of the wheel along the road; that is y = y(z). It is your task to write out y(z) as a sinusoidal function (use either a cosine or sine) based on the known wavelength (1.2 meters) and amplitude (0.05/2 meters). Substituting z = v*t into this equation for y gives y = y(t). From this you can identify the frequency omega of the excitation.
- Solve: You are asked to find the steady-state response of the trailer x(t) as a function of time. However, in actuality, you only need to know at what frequency omega the response is the greatest (hint: this occurs at resonance). That is, you need to set omega = omega_n.
Note that you are not directly given the stiffness of the spring for the suspension of the trailer. What you are given is the increase in static deformation of the suspension (0.003 meters) under additional loading on the trailer (75*9.806 newtons). From this, you can find k.
Let us know if you have questions on this.
4 comments:
Looking at problem 8/72 which is another one of those problems that have 2 springs in line with eachother, I still get really confused when you do k(x_b - x) or vice-versa. Does anyone have a trick they use that makes this easier to understand?
mark
mark,
I find it's easy if you just pick one distance in your mind (x_b or x) imagine pushing that in the positive direction while the other doesn't move. Which way if the force on the block going in that condition?
Draw the arrow in that direction, then the formula is whichever distance you picked (x_b or x) - the other.
I just finished working out this problem and got the right answer for the speed, but I'm lost about how to find the amplitude X (not asked for in the problem, but given as an answer).
Anyone?
I got that the |X| was equal to |b/(1-r^2)| since damping coefficient is equal to 0
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