# Final review

## 1- Measurement systems
### Goal of experiments
- __1- Engineering/scientific experimentation__
- __2- Operational system__




### Units - Unit conversion - Dimensional analysis
### Errors and Uncertainties


## 2- Dynamic response of measurement systems

\begin{align*}
a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}}x + \dots + a_1 \frac{dy}{dt} + a_0 y = F(t)
\end{align*}

###  0$^{\mathrm{th}}$ order system

\begin{align*}
y(t) = K F(t) = \frac{1}{a_0} F(t) \quad \text{$K=1/a_0$ is called the _static sensitivity_ or _steady-state gain_}
\end{align*}

###  1$^{\mathrm{st}}$ order system

\begin{align*}
a_1 \frac{dy}{dt} + a_0 y = F(t) \quad
\tau \frac{dy}{dt} + y = K F(t)
\end{align*}
with $\tau = a_1/a_0$ the time constant of the system.  

###  2$^{\mathrm{nd}}$ order system

\begin{align*}
a_2 \frac{d^2y}{dt^2} + a_1 \frac{dy}{dt} + a_0 y = F(t) \quad
\frac{1}{\omega^2_n} \frac{d^2y}{dt^2} + \frac{2\zeta}{\omega_n} \frac{dy}{dt} + y = K F(t)
\end{align*}

## 3- Digital data acquisition
### Binary to decimal conversion

### ADC system
__Resolution__
\begin{align*}
\Delta V = \frac{V_{max} - V_{min}}{2^N}
\end{align*}

__Quantization error__
\begin{align*}
Q = \pm \frac{1}{2} \Delta V = \pm \frac{V_{max} - V_{min}}{2^{N+1}}
\end{align*}

__Clipping__
### Discrete Sampling

### Aliasing
Nyquist Shannon sampling rate thereom states:
>_If a function, $y(t)$, contains no frequencies greater than $f_{signal}(MAX)$, it is completely determined if it is sampled with a sampling frequency $f_s$ such that:_
\begin{align*}
f_s > 2 f_{signal}(MAX)
\end{align*}
> if $f_s \ngtr 2f_{signal}(MAX)$
\begin{align*}
f_a =  \left| f - f_s \cdot \mathrm{NINT}\left( \frac{f}{f_s} \right) \right|
\end{align*}

### Fourier Series
\begin{align*}
y(t) = \frac{A_0}{2} + \sum_{n=1}^{\infty} \left( A_n \cos \left(\frac{2\pi nt}{T}\right) + B_n \sin \left(\frac{2\pi nt}{T}\right) \right)
\end{align*}
- __Fundamental harmonic/frequency__: $n=1$, noted $\omega_0=2\pi/T$ or $f_0 = 1/T$.
- $T$: signal period
- Frequencies of $\sin$ and $\cos$ are multiple of the fundamental frequency $\omega$: __$n^{th}$ harmonic__.
- $A_n$, $B_n$: Fourier coefficients
- $A_0/2$: mean of the function

### Fourier Transform, DFT, FFT
\begin{align*}
Y(k \Delta f) = \sum_{n=0}^{N-1} y(n \Delta t) e^{-i (2\pi k \Delta f)(n\Delta t)} \qquad k = 0,\,1,\,2, \dots , \, N-1
\end{align*}

\begin{array}{ll}
N & \text{number of samples taken over sampling interval $T_s$}\\
T_s & T_s = N\Delta t\quad\text{total sampling time}\\
\Delta f & \text{frequency increment/resolution}\\
 & \Delta f = 1/T_s = 1/N\Delta t\\
\Delta t & \text{time increment (or sampling period)}\\
 & \Delta t = T_s/N\\
f_s & \text{sampling rate}\\
 & f_s = 1/\Delta t = N/T_s\\
f_{max} & f_{max} = f_{folding} = f_s/2 = N/2 \cdot \Delta f \quad \text{max resolved frequency}
\end{array}

### Leakage
### Windowing
### Anti-aliasing filter

### Analog signal reconstruction
If an analog signal is sampled properly: reconstruct the analog signal with the _cardinal series_.

## 4- Signal Conditioning

### Basic electronics

### Filters
#### $1^{\mathrm{st}}$ order low-pass filter

\begin{align*}
\omega_{cutoff} & = \frac{1}{RC} \quad  f_{cutoff} =  \frac{\omega_{cutoff}}{2 \pi} = \frac{1}{2\pi RC} \\
G & = \frac{|V_{out}|}{|V_{in}|} =  \frac{1}{\sqrt{1+ \left( \frac{f}{f_{cutoff}}\right)^2 }} \quad
\phi = -\arctan \left( \frac{f}{f_{cutoff}} \right)
\end{align*}
#### Low-pass filter of order $n$
\begin{align*}
G = \frac{1}{\sqrt{1+ \left( \frac{f}{f_{cutoff}}\right)^{2n} }}
\end{align*}
#### $1^{\mathrm{st}}$ order high-pass filter
\begin{align*}
G = \frac{1}{\sqrt{1+ \left( \frac{f_{cutoff}}{f}\right)^2 }} \quad
\phi = \arctan \left( \frac{f_{cutoff}}{f} \right)
\end{align*}

#### High-pass filter of order $n$
\begin{align*}
G = \frac{1}{\sqrt{1+ \left( \frac{f_{cutoff}}{f}\right)^{2n} }} 
\end{align*}

### Op-Amps
ideal op-amps vs real effects (impedance loading, GBP, CMRR)

## 5- Design Stage uncertainty analysis
### Expected uncertainty
RSS: root of the sum of the squares
### Uncertainty of a result
\begin{align}
R = f(x_1, x_2, \dots) \quad u_R = \sqrt{\sum_i \left( u_{x_i} \frac{\partial f}{\partial x_i} \right)^2}
\end{align}
If f is polynomial
\begin{align}
f = x_1^{a_1} \cdot x_2^{a_2} \cdot \dots \quad \frac{u_R}{R} = \sqrt{\sum_i \left( \frac{u_{x_i}}{x_i} a_i \right)^2}
\end{align}
### Combining elemental uncertainties
\begin{align}
u_{sys} = \sqrt{\sum_i u_i^2} \quad \text{with $u_i$ elemental error}
\end{align}


# 6- Design of experiments
## Main concerns of DOE
- Validity of results
- Reliability
- Reproducibility
- Statistical significance
- Define the problem

## Design steps
- Design the experiment
- Construct the experiment
- Acquire data
- Analyze data
- Do confirmatory experiments (if necessary)
- Interpret and report results and methodology. 

## Test matrix design

### Full factorial analysis

$P$: number of parameters, 
$L$: number of levels each parameter is tested at, 
$ N= L^P$ tests.
### Fractional factorial analysis
### Taguchi design array

Optimum Taguchi Design Array for fractional factorial analysis
- Each level of each parameter appears the same number of times in the array.
- Repetitions of parameter-level combinations are minimized as much as possible.




# 7- Strain Gage
\begin{align*}
\sigma_a = \frac{F}{A}, \qquad \epsilon_a = \frac{dL}{L}, \qquad \sigma_a = E \epsilon_a
\end{align*}

\begin{align*}
\frac{dR}{R} = S \epsilon_a
\end{align*}

## Wheatstone bridge

\begin{align*}
\frac{V_o}{V_s} = \frac{1}{4} \left( \frac{dR_1}{R_{ini}} - \frac{dR_2}{R_{ini}} + \frac{dR_3}{R_{ini}} - \frac{dR_4}{R_{ini}} \right)
\end{align*}

### Bridge of order $n$
\begin{align*}
\frac{V_o}{V_s} = \pm \frac{n}{4} \epsilon_a S
\end{align*}
quarter-bridge: $n=1$, half-bridge: $n=2$, full-bridge: $n=4$. 

# 9- Pressure Measurement
\begin{align}
P_{gage} = P_{abs}-P_{atm}
\end{align}



\begin{align}
\left| \frac{P}{P_0} \right| = \frac{1}{\sqrt{\left( 1 - \left( \frac{f}{f_n} \right)^2  \right)^2 + 4h^2 \left( \frac{f}{f_n} \right)^2 }}
\end{align}
with $f_n$ natural frequency, $h$ damping ratio
\begin{align}
f_n = \sqrt{\frac{3 \pi r^2 c^2}{4L V}} \quad h = \frac{2 \mu}{\rho c r^3} \sqrt{3LV}{\pi}
\end{align}
- Mechanical 

Manometer, Bourdon gages

- Electronic

Displacement, deflection, piezo


# 10- Temperature Measurement

## Mechanical
## Thermoresitive 
### Resistance Temperature Device (RTD):
Same than strain gage: $R(T) =R_0 + a (T-T_0)$
### Thermistor

Semi-conductor material: $ R(T) = R_0 \exp \left( \beta \left( \frac{1}{T} - \frac{1}{T_0} \right) \right) $


## Thermoelectric (thermocouple-TC)

Seebeck effet: emf(T) when connecting dissimilar conductors 
- Law of intermediate metals
- Law of intermediate temperatures

## Radiative

# Sample Problems
## Quiz
> What are the major elements of a measurement system?

> When acquiring a repeatable signal of fixed (and unknown) frequency with a digital data acquisition system, how can you determine experimentally if you have aliasing in your data?

> Are there any outliers in the time series data below?  Briefly explain.  On the top graph, the data are in blue, the curve fit in red.   The bottom graph is the residuals with the $95\%$ confidence interval in dashed lines.


> Give two tests to check if your data are randomly distributed.

> What is the benefit of using Taguchi array when designing a matrix of experiments?


### Force Balance
When subjected to a point load on its tip, a rectangular cantilever beam of base (width), $b$, and height (thickness), $h$ has axial strain, $\epsilon$, on the surface of its base:

\begin{align}
\epsilon = \frac{6 W L}{E b  h^2}
\end{align}

where $E$ is the Young's modulus, $W$ the load, and $L$ the distance from the free tip of the beam to the measurement point.



The following parameters are known about the beam:
It has on overall length of 1 m.  $h= 0.00500\, \text{m} \pm 0.4$%, $b = 0.0200\, \text{m}\pm 0.1$%, $E = 200 \,\text{GPa} \pm 0.05$%. Finally, the strain gage is mounted at $L = 0.5\text{ m} \pm 0.005 \text{ m}$.

You wish to measure the strain using a quarter-bride and one strain gage mounted on the surface of the top base at location $L$.  By convention, you would like to have a positive voltage output for a downward deflection of the beam.

>  Draw a Wheatstone bridge and show where the strain gage should be plugged.  Provide an explanation for your choice.

SG top, -> $\epsilon_a > 0$ for $W >0$

SG $R_1$ or $R_3$

> Predict the uncertainty on the force measurement, neglect the uncertainty on the axial strain.

Uncertainty of a result, $W = (L, E, b, h)$

W is polynomial
\begin{align}
\frac{u_W}{W} = \sqrt{\sum_i \left( \frac{u_{x_i}}{x_i} a_i \right)^2}
\end{align}

> Which dimension or parameter contributes the most to your uncertainty?

> How would you increase the resolution of your measurements on the applied force with this system, i.e. which parameter or dimension should you change or refine? 

increase $L$: more sensitivity, smaller $u_L/L$.

> The beam is expected to see a lot of temperature variations and strain gages are sensitive to temperature.  How would you change your measurement system to compensate for temperature effects on your strain measurement?  Explain briefly why.

> Bonus question : 
A two-component force balance is made of two cantilever beams (like the one above), with a beam dedicated to measuring the axial force while the other measures the side force.  In your tests, the axial force is 10 times larger than the side force.  For practical reasons, the beams are made of the same materials and with identical height and length.  How would you size the rest of your system to obtain the most sensitive two-component force balance.

## Signal conditioning and analysis
You are being tasked with designing a load cell to measure oscillations at 260 Hz.  You choose to use a quarter-Wheatstone bridge and size your load cell such that the strain is up to 500 $\mu$-strain.  You are being told that there will be electromagnetic noise in your signal.

You resulting design has the following parameters:
- There is noise on the input: $f_{noise}$ = 60 Hz with amplitude $\pm$ 0.50 mV .
- The strain factor is $S = 2.00$.
- The supply voltage for the Wheatstone bridge is $V_s = 5.00$ V.
- The output is sent to a DAQ board with 12-bit A/D converter and a range of $\pm 10$ V.
- All Wheatstone bridge resistors (and the strain gage) have resistance 350 $\Omega$ including the unloaded strain gage, resulting in a balanced bridge at zero strain.


> Propose a sampling frequency for your data acquisition system that would maximize your frequency resolution.  Briefly justify your choice.

> To make the best use of your digital data acquisition system, what should be the gain of your amplifier stage in your signal conditioning?  In practice would you accomplish this with one or two amplifier stages?  Briefly justify why. 


```python
V_0 = 1/4*2.00*5E-4*5.00 # V
Q = (10--10)/2**(12+1)
print(V_0, Q)
```

    0.00125 0.00244140625



```python
G = 8/0.00125
print(G**(1/3))
```

    18.566355334451114


> Which type of amplifier stages will you use in your design (inverting or non-inverting)?  Explain why. 

> Propose a scheme to filter the noise in your expected data.  In particular select the cutoff frequency(iess) and order of the filter.  Show all your steps.

> Bonus question: Before building your filter, you decide to acquire data to assess the signal and noise in your system.  You obtain the data and spectrum below.  What is your assessment of the actual carrier frequency(ies) of your signal and the nature of the noise?  Propose a new scheme (qualitatively) to treat your data.




```python

```

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Jupyter Notebook
Lectures/13_FinalReview.ipynb
eiriniflorou/GWU-MAE3120_2022
52cd589c4cfcb0dda357c326cc60c2951cedca3b
BSD-3-Clause
2022-01-11T17:38:12.000Z
2022-02-05T05:02:50.000Z
Lectures/13_FinalReview.ipynb
eiriniflorou/GWU-MAE3120_2022
52cd589c4cfcb0dda357c326cc60c2951cedca3b
BSD-3-Clause
Lectures/13_FinalReview.ipynb
eiriniflorou/GWU-MAE3120_2022
52cd589c4cfcb0dda357c326cc60c2951cedca3b
BSD-3-Clause
2022-01-13T17:55:14.000Z
2022-03-24T14:41:03.000Z
Qwen/Qwen-72B
1. YES
2. YES