# E12 Assignment 06

Description:

Problems:
1. Three Flywheels, FBD

Consider the system shown below with three flywheels.  Starting at the left is J1, connected by a flexible shaft, K, to J2.  J2 is connected by a rigid shaft to a massless disk.  Between this massless disk and the third flywheel, J3, is a friction B (this acts as a clutch).

a) Draw free body diagrams

b) Write equations of motion.

Animation

Solution:

a)

b)

2. FBD of human eye

Consider the model of the eye shown below (we will consider rotations in the horizontal plane only).  The elements K3 and B3 represent the restoring forces of tendons and other structures attached to the eye (these are rotational elements, the torque they generate is proportional to angular position and velocity). The eye's moment of inertia is J (all other elements are massless).  To the left and the right of the eye are two muscles, the medial rectus (moves the eye toward the body's centerline) and the lateral rectus (moves the eye away from the centerline).  The muscles (and associated connective tissue) are modeled by a parallel force generator and friction, which are in series with a spring -- these elements are translational.  Rotational elements are in capital letters, and blue; translational elements are in lowercase and green.  The radius of the eye is "R".

Draw appropriate free body diagrams, and write the set of resulting differential equations associated with each free body diagram.

 Note: Here is drawing of the model as originally presented, along with the measurement of a saccadic eye movement (i.e., a step change in desired gaze angle) The response looks like it is slightly underdamped. From: Bahill, Hsu and Stark "Glissadic Overshoots are due to Pulse Width Errors," Archives of Neurology, 35, (1978), 138-42 From: Bahill and Stark, "The Trajectories of Saccadic Eye Movements," Copyright 1979, Scientific American.

Solution:

3. Rotation and Translation, Spring at center of cylinder

The system shown below consists of a flywheel (mass=M, radius=r, moment of intertia=J=½Mr²).   There is an axle through the middle of the flywheel connected to a massless yoke (light blue) . The axle bearing in the middle of the flywheel is modeled by rotational friction, B.  A force, fa(t), is applied to the yoke and the yoke is a wall by a spring, k. The wheel rolls without slipping on the floor.  The image on the right shows the flywheel/yoke  seen from above.

a) Draw appropriate free body diagrams.

b) Write a single differential equation with fa(t) as input and x(t) (the position of the flywheel) as output.

Solution:

a)

b)

4. System with Rotation and Translation, Spring on edge of cylinder

The system shown below consists of a flywheel (mass=M, radius=r, moment of intertia=J=½Mr²).  The edge of the wheel is connected at its radius to the wall by a spring, k, with a cable that wraps around the cylinder.  The bearing in the middle of the flywheel is modeled by rotational friction, B.  The flywheel is connected to a massless yoke (light blue) to which a force, fa(t), is applied.  The wheel rolls without slipping on the ground.  The image on the right shows the flywheel seen from above.  Be careful with this problem - the spring stretches both because of translation (the wheel moves) and because of rotation (the spring wraps around the wheel as it moves).

a) Draw appropriate free body diagrams.

b) Write a single differential equation with fa(t) as input and x(t) (the position of the center of the flywheel (also the position of the yoke)) as output.

Solution:

a)

Note: the displacements associated with the spring are "2·x".  This is because the spring is extended by an amount x due to the translation of the flywheel, but it also wraps around the flywheel by an amount r·θ=x.  So the total stretch in the spring is 2·x.

b)

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5. Circuit Transfer Function and Analog

a) For the circuit shown, find the transfer function Eout(s)/Ein(s).

b) Show it is of the same form as the transfer function (Xout(s)/Xin(s)) of the mechanical system shown.  Take xout(t)=0 at equilibrium (i.e., ignore gravity).  The two transfer functions should have similar polynomials in numerator and denominator, only the constant coefficients will differ.

Solution:

a) Use voltage divider

b) Use FBD

6. E12 Coupled Pendulum

For the system shown (with two identical pendula coupled by a spring):

a) draw free body diagrams and,

b) write equations of motion in terms of M, ℓ, g, r, k and the two θ's.

Assume that the rod is massless and that the angle is small. Note: you can't ignore gravity in this situation because the torque due to gravity varies with angle.

Solution:

Draw FBD's about pivot. Force of gravity perpendicular to arm of each pendulum is Mgsin(θ). Displacement of left side of spring is ℓ·sin(θ1)≈ℓ·θ1, so extension of spring is ℓ·θ1-ℓ·θ2.

Note that in this case gravity acts in the same was as a spring (its torque is proportional to the angle)

7. E12 RLC from Lab

For the circuit shown:

a) Use impedances and the voltage divider equation to find the transfer function H(s)=Eout(s)/Ein(s) in terms of R, L and C. You should have only positive powers of "s" in your polynomials.

b) Write the differential equation.

c) Find the impulse response if R=250Ω, L=0.112H and C=1μF.

d) Repeat part c for the step response.

Solution:

8. e12negFeedback

Which of the following represent positive feedback, and which negative?  Remember with negative feedback, if there is any change, the system acts in a direction that opposes that change; it acts to keep the system output constant.  Positive feedback amplifies any disturbance.

1. An eye for an eye; or Do unto others as you would have them do unto you.
2. One good turn deserves another.
3. The squeaky wheel gets the grease.
4. You learn from your mistakes.
5. The rich get richer – the poor get poorer.
6. Imagine you run a restaurant where drinks are in a cooler and patrons help themselves.  You would like to maintain a good variety.  You implement a system in which you see what drinks are in the cooler and order more in proportion to the ones that are present.
7. Imagine you run a restaurant where drinks are in a cooler and patrons help themselves.  You would like to maintain a good variety.  You implement a system in which you see what drinks are in the cooler and order more in inverse proportion to the ones that are present.
8. The position of a ball that sits on the top of a hill as shown.
9. When you eat and your blood sugar increases, insulin is released which starts the process of metabolizing the glucose.
10. The position of a ball that sits in the low part of a trough.
11. Come up with your own example of positive feedback.
12. Come up with your own example of negative feedback
Solution:
1. +'ve
2. +'ve
3. -'ve
4. -'ve
5. +'ve
6. +'ve
7. -'ve
8. +'ve
9. -'ve
10. -'ve
11. +'ve
12. -'ve