E12 Assignment 07

Description: 

Problems: 
1. E12 Dual/Analog 2a

Write equations for the currents summed at the nodes labeled 1 and 2.  Write the currents in terms of ia, e1 and e2.

Solution: 

2. E12 Dual Analog 2b

a) For the circuit shown, draw over the mechanical system to find the force-current analog.

b) Draw free body diagrams and write equations of motion.

c) Rewrite the equations of motion substituting in the analogous force-current quantities.

d) Verify that the equations from part c are the same as the equations obtained by summing currents at the nodes labeled 1 and 2.

Solution: 

a) Draw over electrical system (background), replacing mechanical elements with their force-current analogs.

Redraw.

b) 

 

c)

d) They are the same.

3. E12 Dual/Analog 2c

a) Find the dual of the circuit shown by drawing over the circuit and substituting the appropriate dual elements.

b) Write the node equations for the original circuit (shown above)circuit and substitute dual quantities.

c) Verify that the equations from part b are appropriate loop equations for part a.

Solution: 

a) Get the dual by drawing over the original system and substituting dual elements.


b)

c)

4. 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)
eye


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: 


 

5. 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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6. Impedance Translation with Transformer

In class (gears on a bicycle) we saw how the effective load changes when gears are being used.  By varying the turns ratio (N1/N2) we can match the apparent load in a circuit (R'load) to the source load (Rs) to maximize power transfer.   See image below - the circuit on the right is equivalent to the one on the left (i.e., currents and voltages are equal).  Show that

Recall that

where e1 and i1 are the voltage across and current through the left coils (and R'load), and likewise for e2 and i2 and the left coils (and Rload).  You can start by writing equations for Rload in terms of e2 and i2, and for R'load in terms of e1 and i1.

      

For the transformer (as for gears and levers) power in is equal to power out.  A lever trades force for velocity (the input can be low force with high velocity, with an output at higher force but lower velocity); a transformer trades current for voltage.

Solution: 

7. Efficiency Improvement with Transformer

If the 8 ohm load is connected directly to the source (shown below) - you may assume Vsource is 1 volt.

  1. Find the power dissipated in Rs (call this PRs),
  2. Find tthe power dissipated in Rload, (call this PRload),
  3. Find tthe total power, Ptot, and
  4. Find tthe efficiency, η, which is the power dissipated in the load divided by the total power.  This is a measure of how well power is transferred from the source to the load.
  5. Determine the turns ratio (N1/N2) needed for a transformer to make the 8 Ohm load appear as 600 Ohms.
  6. Repeat a-d with the 600 Ohm apparent load. Note that although total power supplied by the source decreases by half the efficiency increases enough so that the power dissipated in the load increases by a factor of about 20. 
Solution: