Bill Goodwine's Courses

Reading: At this point you should have read all of Chapters 1 - 4.

Exercises: 2.11, 2.27, 2.32, 4.17, 4.19, 4.26 and 4.27.

For problem 4.26, if we are describing the vertical motion of the beam, should this be an equation in terms of y? The force is a function of x, so I'm a little confused.

amcgloin wrote:For problem 4.26, if we are describing the vertical motion of the beam, should this be an equation in terms of y? The force is a function of x, so I'm a little confused.

The problem has a typo. If x is along the beam, then the variable in the force equation, F, should be something else, like y.

ETA: this is listed in the typos for the book: http://controls.ame.nd.edu/engdiffeq

For problem 4.27, the problem says the eccentricity of the motor is attached to a mass-spring-damper system, but the figure does not show a damper. So is it just a mass-spring system or is the figure missing the damper? I ask because in a mass-spring-damper system, the steady state solution is the particular solution since the homogeneous solution goes to zero as t goes to infinity. But if my work is correct, a mass-spring system would have to consider the homogeneous solution in the steady state solution as well.

John Hollkamp wrote:For problem 4.27, the problem says the eccentricity of the motor is attached to a mass-spring-damper system, but the figure does not show a damper. So is it just a mass-spring system or is the figure missing the damper? I ask because in a mass-spring-damper system, the steady state solution is the particular solution since the homogeneous solution goes to zero as t goes to infinity. But if my work is correct, a mass-spring system would have to consider the homogeneous solution in the steady state solution as well.

There should be no damping in that problem.

For problem 4.19, part 2, we are asked to maximize the magnitude of the shaking. This occurs when, for low damping ratios, the frequency ratio is near 1, but it changes based on the value of zeta. Is there a way to find exactly when the maximum occurs? All I can find in the book and my notes on the matter is a graph such as figure 4.20, from which this frequency ratio could be estimated, but I'm not sure how to go about calculating it exactly.

jnorby wrote:For problem 4.19, part 2, we are asked to maximize the magnitude of the shaking. This occurs when, for low damping ratios, the frequency ratio is near 1, but it changes based on the value of zeta. Is there a way to find exactly when the maximum occurs? All I can find in the book and my notes on the matter is a graph such as figure 4.20, from which this frequency ratio could be estimated, but I'm not sure how to go about calculating it exactly.

You can just read it from the plots if you want. If you want to compute it exactly, then compute the derivative of the magnitude curve, which gives the extrema where it's zero, naturally.

If for problem 4.27 there is no damping, then is there really a steady state solution? Or should we just calculate the particular solution to the equation of motion of the system?

sbrill wrote:If for problem 4.27 there is no damping, then is there really a steady state solution? Or should we just calculate the particular solution to the equation of motion of the system?

Just go with the particular solution. Also factor out an m_e/m to make the plot.