EE462-Utilization of Electtrical Energy

class: center, middle

# EE-462 UTILIZATION OF ELECTRICAL ENERGY

# Mechanical Systems

## Ozan Keysan

## [keysan.me](http://keysan.me)

### Office: C-113 <span class="meta">•</span> Tel: 210 7586

---

# Linear Motion
--

### \\(\vec{f} = m \dfrac{d\vec{v}}{dt} \\) 
--

### It is more common to use "u" in mechanical systems

### \\(u = \dfrac{dx}{dt} \\), \\(a = \dfrac{du}{dt} \\)

### \\(f\_{elec} - f\_{mech} = m \dfrac{du}{dt} \\) 
--

### Stored energy: \\(W = \dfrac{1}{2}m u^2\\)

---

# Rotational Motion
--

### \\(\vec{T} = J \dfrac{d\vec{\omega}}{dt} \\)

### \\(\omega= \dfrac{d \theta}{dt} \\)

### \\(T\_{elec} - T\_{mech} = J \dfrac{d \omega }{dt} \\) 
--

### Stored energy: \\(W = \dfrac{1}{2}J \omega^2\\)

---

# Moment of Inertia
--

## What is its unit?
--
: \\(kg m^2\\)

---
# Importance of Inertia in Drive Systems
--

## [Old Diesel Engine](https://www.youtube.com/watch?v=nFQ64a_cGXg)

## [Engine Start](https://www.youtube.com/watch?v=iPyvKOWl7PY)
--

## Flywheel store kinetic energy, reduce speed ripples
--

## Reduces the dynamic performance of the drive 
--

## Large energy to dissipate in dynamic braking 
---

## What is the most common shape in electric motors?
--

## Cylinder
--

### What is the inertia of a solid cylinder?
--

## \\(J = \rho \dfrac{\pi}{2} R^4 L\\)
--

## \\(J = \dfrac{1}{2}M R^2\\)

---
# Rotational Inertia of Different Shapes

---
## Comparison of Different Shapes

<img src="http://i.imgur.com/gsI96Gh.gif" alt="Drawing" style="width:700px;"/>
---
# Analogy between mechanical & electrical systems

<img src="http://www.vias.org/matsch_capmag/img/matsch_caps_magnetics-142.png" alt="Drawing" style="width:700px;"/>
---
# Analogy between mechanical & electrical systems

<img src="http://www.vias.org/matsch_capmag/img/matsch_caps_magnetics-141.png" alt="Drawing" style="width:700px;"/>
---
## Motor with Load Inertia

<img src="./images/ee462/load_inertia.png" alt="Drawing" style="width:700px;"/>
---
## Motor with Shaft: Equivalent Circuit

### [Low-pass Pi Filter](https://en.wikipedia.org/wiki/Electronic_filter)

### [Pi-Section Filters](http://www.kb6nu.com/tag/pi-network/)

### [Modelling Shaft Stiffness in Simulink](https://www.mathworks.com/help/physmod/sps/powersys/ref/mechanicalshaft.html)

### [Stiffness Animation](http://lpsa.swarthmore.edu/Systems/MechRotating/RotMechSysElem.html), [Drag Race](https://www.youtube.com/watch?v=mfJejgODr3E)
---
## Reading Materials

### [Mechanical-electrical analogies](https://en.wikipedia.org/wiki/Mechanical-electrical_analogies)

### [One-Port Elements](http://web.mit.edu/2.14/www/Handouts/OnePorts.pdf)

### [Electrical Analogies of Mechanical Systems](https://www.tutorialspoint.com/control_systems/control_systems_electrical_analogies_mechanical.htm)

### [Mechanical and Electrical Analogies](http://www.vias.org/matsch_capmag/matsch_caps_magnetics_chap1_20.html)

### [First order mechanical systems](https://www.slideshare.net/JARossiter/system-modelling-1st-order-models)

### [Dynamics of a motor](http://www2.ece.ohio-state.edu/~passino/lab2_rotary_dynamics.pdf)

---
# Dynamics of Mechanical Systems: Resonance

### [Transfer function and mathematical modelling](https://www.slideshare.net/vishalgohel12195/transfer-function-and-mathematical-modeling)
--

### [Tacoma Bridge](https://www.youtube.com/watch?v=lXyG68_caV4)

### [Forced vibration-1](https://www.youtube.com/watch?v=OaXSmPgl1os), [Resonant Freq.](https://www.youtube.com/watch?v=LV_UuzEznHs)

### [Torsional Resonance](https://www.youtube.com/watch?v=JLY-yQOpL20)
---
# Resonant Modes

### [Cantilever Vibration](https://www.youtube.com/watch?v=lKT3wBIUFhA)

### [Resonant Modes](https://www.youtube.com/watch?v=uWoiMMLIvco)

### [Modal Shapes](https://www.youtube.com/watch?v=kvG7OrjBirI)
### [Modal Shapes](https://www.youtube.com/watch?v=d3U_m-4XOtg)

---
# Frictional Forces 
--

## Damping
--

---
# Frictional Forces: Coulomb friction

### Constant force (does not change with speed)

---
# Frictional Forces: Viscous friction (Damping)

## Proportional to speed (\\(T= B \omega\\))
---
# Frictional Forces: Static friction

### Extra friction (or stiction) at zero speed

### Usually small and ignored in linear models
---
# Frictional Forces: Coulomb + Viscous

### Extra friction (or stiction) at zero speed

### [How to model in Simulink?](https://www.mathworks.com/help/simulink/slref/coulombandviscousfriction.html)

---
# Windage Torque
--

## Proportional to  \\(\omega^2\\)

## \\(T \propto \omega^2\\)
--

## \\(P \propto \omega^3\\)

### Can be represented as damping if the speed is varying in a narrow range:

## \\(T_B + T_W = B' \omega \\)
---
## Example: Drag Force in a Car

## \\(F = \dfrac{1}{2} \rho A C_D v^2 \\)
---
# Drag Coefficient

<img src="http://www.aquaphoenix.com/lecture/matlab5/images-large/drag_coefficients.jpg" alt="Drawing" style="width:700px;"/>
---
# Drag Coefficient Comparison

<img src="http://insideevs.com/wp-content/uploads/2014/06/ev-CdAs.jpg" alt="Drawing" style="width:800px;"/>
---
## Coupling Mechanisms: Direct Coupling

### Sleeve Coupling

---
## Coupling Mechanisms: Direct Coupling

### Jaw Type Coupling

---
## Coupling Mechanisms

### Direct Coupling

### Spiral Type Coupling

---
## Coupling Mechanisms

## Gearbox

### Gear Ratio = \\( \omega\_{in}:\omega\_{out}\\)

#### [Simulink Gearbox](https://www.mathworks.com/help/physmod/simscape/ref/gearbox.html)
---
## Coupling Mechanisms

## Rack and Pinion

### From rotational to linear motion
---
## Coupling Mechanisms

## Belt Coupling

---

# Coupling Issues

## [Shaft Misalignment](https://www.youtube.com/watch?v=MXAsKFnCqUE)

## [Unbalanced rotor](https://www.youtube.com/watch?v=R2hO--TIjjA)

## [Rotor Critical Speed](https://www.youtube.com/watch?v=dO51IjGKrTM)
---

# Load Types
--

## Centrifugal Loads (Pump, Fan, etc)

## \\(T\_{load} = k \omega^2\\)
## \\(P\_{load} = k \omega^3\\)
---
## Constant Power Loads 
### Paper rolling machine

### Constant tension should be applied at constant linear speed
---
## Constant Torque Loads
### Compressors

## \\(P\_{load} = k \omega\\)
---
## Constant Torque Loads: Conveyors

### Usually there is a high starting torque (to overcome static friction)
---
## Constant Torque Loads: Crane Hoists

### There exist a viscous friction force combined with load mass

### \\(T\_{load} = B \omega + T\_{mech}\\)

### Requires a four-quadrant drive

[Oil-rig Drawworks](https://www.youtube.com/watch?v=yHfnL43H48g), [Drawworks-2](https://www.youtube.com/watch?v=wgUV5ov2jC0)
---
## Constant Torque Loads: Crane Hoists

### Four quadrant operation

---
## You can download this presentation from: [keysan.me/ee462](http://keysan.me/ee462)