ME 365 Review 2

ME 365 Exam 1 Review Guide

This study guide reviews common ME 365 Exam 1 topics, including state variables, transfer functions, Laplace transforms, static gain, uncertainty, probability, block diagrams, and equations of motion.


Note: This article is rewritten as an educational review guide. The explanations are organized for learning and exam preparation.

1. State Variables

For a system with two outputs and second-order derivatives, each second-order coordinate requires two state variables.

\[ 2+2=4 \text{ state variables} \]

2. Block Diagram Transfer Function

When finding \(G(s)=Y(s)/U_2(s)\), temporarily set the other input to zero and follow the direct signal path.

\[ G(s)=A(s) \]

3. Free Response

For free response, set the input equal to zero and solve the homogeneous equation.

\[ \dot{y}_{free}+\frac{3}{2}y_{free}=0 \]
\[ y_{free}(t)=e^{-\frac{3}{2}t} \]

4. Bias Error

Bias error is the difference between the average measured value and the true value.

\[ B_H\approx |\bar{H}-H_{true}| \]
\[ B_H\approx |5.04-5.00|=0.04\text{ mm} \]

5. Calibration Bias

From the calibration curve, the output appears shifted downward by about one unit.

\[ b\approx -1 \]

6. Damper Tension

A damper force depends on relative velocity. Pay attention to the sign convention of the chosen coordinates.

\[ T_d=-c(\dot{x}+\dot{y}) \]

7. Transfer Function from a Differential Equation

To find a transfer function, use zero initial conditions.

\[ \ddot{y}+4\dot{y}+6y=5u \]
\[ (s^2+4s+6)Y(s)=5U(s) \]
\[ G(s)=\frac{Y(s)}{U(s)}=\frac{5}{s^2+4s+6} \]

8. Static Gain

Static gain is found by evaluating the transfer function at \(s=0\).

\[ K_0=G(0) \]
\[ K_0=2\frac{0+3}{0+4}=\frac{3}{2} \]

9. Two’s Complement Binary

The first bit is 0, so the number is positive.

\[ (00110000)_2=2^5+2^4 \]
\[ (00110000)_2=(48)_{10} \]

10. Normal Distribution Probability

Convert the value to a z-score.

\[ z=\frac{x-\mu}{\sigma} \]
\[ z=\frac{0.7-1.0}{0.2}=-1.5 \]
\[ P(X<0.7)=P(Z<-1.5)\approx 6.7\% \]

11. Sensor Resolution and Uncertainty

The ADC quantization interval is:

\[ Q=\frac{10-(-10)}{2^8}=0.078125\text{ V} \]

Convert voltage resolution to force resolution using the sensor sensitivity.

\[ \text{input resolution}=\frac{Q}{1.30}\approx 0.06\text{ N} \]

The uncertainty can be estimated by combining nonlinearity and quantization effects.

\[ U\approx 0.08\text{ N to }0.10\text{ N} \]

12. Laplace Response with Impulse Input

Apply the Laplace transform and include initial conditions.

\[ \ddot{y}+6\dot{y}+25y=10\delta(t) \]
\[ Y(s)=\frac{2s+22}{s^2+6s+25} \]

Complete the square:

\[ s^2+6s+25=(s+3)^2+4^2 \]
\[ y(t)=2e^{-3t}\cos(4t)+4e^{-3t}\sin(4t) \]

13. Block Diagram Reduction

Write equations for each block and summing junction, then substitute until only \(Y(s)\) and \(U(s)\) remain.

\[ G(s)=\frac{Y(s)}{U(s)}=\frac{s+1}{s^2+4s+11} \]

14. Uncertainty in Mass Moment of Inertia

For a slender rod:

\[ I_{rod}=\frac{1}{12}\rho AL^3 \]

Only length uncertainty is considered.

\[ u_I=\left|\frac{\partial I}{\partial L}\right|u_L \]
\[ u_I=\frac{1}{4}\rho AL^2u_L \]
\[ u_I\approx 1.20\text{ kg}\cdot\text{m}^2 \]

15. Equation of Motion for Rack and Pinion System

Use rotational dynamics, translational dynamics, and the kinematic constraint between rack displacement and gear rotation.

\[ R_1\theta_1=x_2 \]
\[ (m_2R_1^2+I_1)\ddot{x}_2+k_Tx_2=-R_1^2f(t) \]

Key Study Tips

  • For transfer functions, set initial conditions to zero.
  • For free response, set the input to zero.
  • For static gain, evaluate \(G(0)\).
  • For uncertainty propagation, use partial derivatives.
  • For equations of motion, define coordinates and sign conventions first.
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