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MECH 372 / ENGR 372
Space Systems Design and Engineering II
 
Assignment #9 - Mechanisms Solutions
  
 
Objective: The purpose of this assignment is to ensure that you have grasped key qualitative (and some very minor quantitative) aspects of the lecture topics involving the introduction to machine elements and motion concepts.  
 
NOTE: There is no solution summary sheet for this assignment.

  

1. Using the Basic Kinematic Equation generates complicated expressions for different types of accelerations when considering motion from the point of view of a rotating frame.  What motivates us to use this complex form of the expression for acceleration?

The 2nd derivative of position with respect to a rotating frame can't be used with Newton's Law, since that Law is only applicable to non-rotating "inertial" frames.  However, we often prefer to think of vectors with respect to a rotating frame since it is more convenient for other reasons.  So - if we want to use derivatives with respect to rotating frames, we use the BKE to generate the other acceleration terms that must be accounted for in order to have an expression that is equivalent to the inertial acceleration.

 

2. The precise form of Newton's 2nd Law is F=dp/dt, where p=mv and when differentiation is done with respect to an inertial frame.  For a constant mass system, the Law can be simplified to F=ma.  What is the simplified form of the Law for a system with varying mass and constant velocity? 

F=(dm/dt)*v

 

3.  Classify each of the following with one of the choices provided [note that the apostrophe, ', denote a derivative wrt time]:

   a.  a = dv/dt    [a kinematic relationship]   vs   [a dynamic relationship]

   b.  F = ma       [a kinematic relationship]   vs   [a dynamic relationship]

   c.  T = Iw'      [a kinematic relationship]   vs   [a dynamic relationship

   d.  x = r*cos(theta)       [a kinematic relationship]   vs   [a dynamic relationship]

   e.  mx'' + bx' + kx = F(t)     [linear equation]   vs   [nonlinear equation]

   f.  Ri+Li'=V(t)     [time invariant equation]   vs    [time-varying equation]

   g.  A 2nd order response with overshoot    [overdamped response]   vs   [underdamped response]

   h.  A 1st order system response   [sinusoidal]   vs   [exponential]

in the above equations, a is acceleration, v is velocity, t is time, F is force, m is mass, T is torque, I is inertia, w is angular velocity, b is damping coefficient, k is spring constant, x is position, R is resistance, i is current, L is inductance, r is magnitude of a position vector, theta is the angle of a position vector, and V is voltage.

 

4. What is mechanical advantage, and (in honor of the upcoming holiday season) explain how a nutcracker exploits this concept.

MA is force multiplication generated through the exploitation of the configuration of simple, passive mechanical devices.   

The nutcracker is a lever - you apply a small force (and large swing in motion) to the end of the lever, and the fulcrum (the lever's axis of rotation) converts this into a large force (and small swing in motion) at the mouth.  This cracks the nut.

5. A lever has a fulcrum placed 2/3 of the way down its length.  What is the mechanical advantage if a load at the short end is lifted by an effort at the long end?

The long portion is 2x the length of the short portion.  So, IMA = 2.

 

6. What are the two primary motivations for using closed loop (feedback) control rather than simple open loop control?

Robustness to disturbances, imperfect knowledge of the system

 

7. What does the acronym PID stand for? Explain the control strategy implied by each term in this acronym.  Also explain why each strategy is potentially a good strategy and when they may be most appropriate.

PID = Proportional - Integral - Derivative

Proportional control is making the output force proportional to the error.  Simply said, the bigger the error, the harder you push.  

Integral control is making the output force proportional to the integral of the error.  This is particularly good for reducing steady state errors (errors where there is a small error that persists over time - so, the integral of the error builds up in order to force the correction)

Derivative control is making the output force proportional to the derivative of the error. This is particularly good for reducing the overshoot of a P controller by backing off on the amount you push if you are already converging quickly on the target destination.

 

8.  Consider 3 types of spacecraft mechanisms: deployment mechanisms, communication antenna pointing mechanisms, and telescope mirror pointing devices.  Rank these three in terms of typical required accuracy (e.g., list them from most accurate to least accurate).

1. mirror pointing.  2. communication antenna pointing mechanism.  3. deployment mechanism

  

9. A dc motor operating at 12V has a max torque of 6 units of torque and a max angular velocity of 1200 rpm.  The desired operating point is 500 rpm and 2 units of torque.  

a. determine Kt/R for the motor given the max numbers

Tmax = Kt*V/R, so Kt/R = 6/12 V = 0.5 torque-units/volts

b. determine Kemf for the motor given the max numbers

Wmax = V/Kemf, so Kemf = 12/1200 = 0.01 V/rpm

c. determine the voltage to be used for the desired operating point

T = Tmax (1 - w/wmax) = 0.5V(1 - w/(V/.01)) = .5V(1 - 0.01w/V) = 0.5(V-0.01w)

2=.5(V-5), so, V=9 volts

check: T=4.5(1-0.0011111w), so for w=500, T=4.5(1-0.55556)=2volts

d. when operating at the standard 12V level, if a 3:1 gearbox is used in order to increase torque, what is the max torque coming out of the gearbox, and what is the max shaft speed coming out of the gearbox?

max gearbox torque is 3 x max motor torque = 18 units of torque

max gearbox speed is 1/3 x max motor speed = 400 rpm

 

10.  Define accuracy, resolution and repeatability given the descriptions provided by Dr. Clark in his mechanisms lecture. (as defined in Dr. Clark's lecture).

Accuracy - The difference between a value and the real value (e.g., truth)
Resolution - The smallest amount of variation that is quantified in a value
Repeatability - The difference between a value and the value achieved from a previous attempt to do the same thing