ME 365 Exam 1 Review
Dynamics • Laplace • Transfer Functions • State Space • Uncertainty
This article reorganizes engineering exam-style problems into a modern review guide for studying and concept review.
Multiple Choice Review
1. State Variable Block Diagram
Given the governing equations:
\[
\dot x_1=-4x_1-3x_2
\]
\[
\dot x_2=5x_1-x_2+u
\]
Determine the constant \(c_2\).
\[
sX_2(s)=5X_1(s)-X_2(s)+U(s)
\]
\[
c_2=-5
\]
2. Steady State Response
The total response is:
\[
y(t)=
-8\cos(3t)+6\sin(3t)
+10e^{-t}\cos(2t)
-4e^{-t}\sin(2t)
\]
The exponential terms decay with time, so the steady-state response is:
\[
y_{ss}(t)=
-8\cos(3t)+6\sin(3t)
\]
3. ADC Quantization Uncertainty
A 4-bit ADC with range 0–10 V has:
\[
2^4=16
\text{ levels}
\]
\[
Q=\frac{10-0}{16}
=
\frac{5}{8}V
\]
Measurement uncertainty:
\[
u=\pm\frac{Q}{2}
=
\pm\frac{5}{16}V
\]
4. Number of State Variables
The reduced transfer function becomes:
\[
\frac{Y(s)}{R(s)}
=
\frac{7s^2+39s+50}
{s^3+14s^2+45s+50}
\]
The denominator is third order.
3 state variables
5. State Space Matrix B
\[
B=
\begin{bmatrix}
2 & 1 & 0\\
0 & 0 & 3
\end{bmatrix}
\]
The number of columns equals the number of inputs.
The system has 3 inputs.
6. Spring Tensile Force
The spring deformation is the relative displacement:
\[
x_1-x_2
\]
\[
T_s=k(x_1-x_2)
\]
7. Kinematic Constraint Equation
Using geometry of the crank-slider mechanism:
\[
x_1=l_2\cos(\theta_2)-r_3\cos(\theta_3)
\]
8. Block Diagram Transfer Function
Set:
\[
U_2(s)=0
\]
\[
Y(s)=A(s)(U_1(s)-B(s)U_1(s))
\]
\[
G(s)=A(s)(1-B(s))
\]
9. Confidence Interval Interpretation
A confidence interval means:
If the experiment is repeated many times, approximately 95% of the intervals will contain the true mean.
10. Bias Error
Measured average:
\[
\bar H=9.94mm
\]
True value:
\[
H_{true}=10mm
\]
\[
B_H\approx|9.94-10|
=
0.06mm
\]
Free Response Problems
11. Inverted Pendulum Transfer Function
\[
m\ddot y-\frac12 ml\ddot\theta=f
\]
\[
\ddot y-\frac23 l\ddot\theta+g\theta=0
\]
Apply the Laplace transform:
\[
ms^2Y(s)-\frac12 mls^2\Theta(s)=F(s)
\]
\[
s^2Y(s)-\frac23 ls^2\Theta(s)+g\Theta(s)=0
\]
\[
\frac{\Theta(s)}{F(s)}
=
\frac{1}
{\frac16 mls^2-mg}
\]
12. Candy Jar Confidence Interval
Nominal candy count:
\[
N=\frac{500}{0.50}=1000
\]
95% confidence interval:
\[
1000\pm5
\]
13. Calibration Uncertainty
Best-fit calibration line:
\[
v=2.0084h-0.023
\]
Maximum nonlinearity occurs near:
\[
h=2cm
\]
\[
|max nonlinearity|
=
|2.0084(2)-0.023-3.759|
\]
\[
u_h\approx\pm0.12cm
\]
14. Equation of Motion
Torsional spring:
\[
T_s=k_T\theta
\]
Torsional damper:
\[
T_d=c_T(\dot\theta-\dot\phi)
\]
Moment equation:
\[
-T_d-T_s=I\ddot\theta
\]
\[
I\ddot\theta
+c_T\dot\theta
+k_T\theta
=
c_T\dot\phi
\]
15. Laplace Total Response
\[
\ddot y+6\dot y+13y=26u_s(t)
\]
Apply the Laplace transform:
\[
Y(s)=
\frac{26}
{s(s^2+6s+13)}
\]
Partial fraction expansion:
\[
Y(s)=
\frac2s
-\frac{2(s+3)}
{(s+3)^2+2^2}
-\frac3
{(s+3)^2+2^2}
\]
\[
y(t)=
2-e^{-3t}
(2\cos(2t)+3\sin(2t))
\]
Exam Tips
- Always define coordinates before deriving equations of motion.
- Transfer functions use zero initial conditions.
- Steady-state response excludes transient exponential terms.
- Confidence intervals describe repeated experiments, not individual measurements.
- Uncertainty propagation often uses partial derivatives.
ME365, ME 365, transfer function, Laplace transform, dynamics, uncertainty propagation, state space model, block diagrams, Purdue engineering, engineering review
