Given a desired response,

we can design a system

with appropriate parameters ME 421 System Modeling Mechanical Electrical Fluid Response Time Response Ramp Impulse Step Differential

Equations Numerical

Methods Laplace

Transforms Assumed

Solution

Method Solution Transfer

Function Partial

Fractions Controls Tracking Error Analytical Spring Proportional Derivative PID Control Types Integral Lag Control Design Tools Laws Experimental Disturbance

Rejection Damper Mass Standard Forms Leading 1 Component Relations Inductance Open-Loop Constant 1 Capacitance Resistance Closed-loop

Feedback Op-Amp On-Off Laws 2 states (on or off)

Simple, Inexepensive

Lots of Switching

Not fine control Proportional to error

Simple

Limited Control

Instability Possible Component Relations Component Relations Increase system type

Eliminates steady state error Predicts

Speeds up system

Noise Tank Cylinder Valve Like PI but not as effective

High pass filter

Reduces steady state error Laminar Turbulent Orfice Laws Lead Like PD but not as accurate

Low pass filter

Improves speed

Less noise Zc <Pc Zc >Pc Numerator Dynamics Limitations T_cl 2nd order or less

Computationally intensive

Trial and error needed to overcome numerator dynamics Strengths Find multiple gains Zeigler-Nichols Constant 1 Identify dynamic quantities

Instability prediction

Error of system

Value of steady state Error Constants Limitations Unity feedback only

Ess only Strengths No G_cl needed Coupled Systems Hydraulic (Fluid-Mech.) Electric Motors (Elec.-Mech.) Fluid-Thermal Equivalent Systems Complementary Solution 1. Input step, ramp, or impulse

2. Find specifications from graph

3. Put transfer function in standard from

4. Find dynamic quantities Particular Solution Xp Limitations Trial and error method Strengths PID gains from experiment

No model needed Final Solution Method Look at closed loop denominator

# of unstable poles = # of sign changes in first column Free Forced Transient Steady State Root Locus Limitations Requires Matlab (SISOTools) Strengths Find gain, poles, zeros for any constants

No G_cl needed Magnitude Criterion

Angle Criterion SISOTool Dynamic Quantities Example: Sand being loaded into the back of a truck over time Example: A hammer striking a nail MatLab Simulink Software that helps create appropriate control systems Graphical programming that works with matlab

Important uses

Create a ODE to see effect on input

Input a ramp or step into system

Filter inputs

Make different control systems (i.e. on-off, PID) Program that uses written code

Important uses

Solve mathematical equations

Solve ODEs

Solve transfer functions

Making plots

Time Response

Pole Zero Maps

Solve complex systems with multiple inputs Example: Turning on a light Common Denominator Method

Find Common Denominator

Set numerator to B(s)

Match coefficients of power of s

Solve for system Example So what? Solution to ODE gives behavior of system

All solutions can be identified in either form Superposition 1: Steady State and Transient Superposition 2: Free and Forced Combines effects of P, I, and D control Thermal Component Relations Laws Energy Storage Conservation of Energy Heat Transfer Conduction Convection Routh Stability Limitations Stability only

Needs G_cl Strengths Find multiple gains Block Reduction Steady State Error, Difference between desired and actual response at steady state Steady State Response, Value at steady state Settling Time, Time for response to reach and stay withing 2% of steady state Peak Overshoot Maximum response above steady state Damping Coefficient, Percentage Overshoot Percent of steady state that is reached at peak Peak Time, tp Time at which Mp occurs Rise Time, tr Time to cross Xss for the first time Delay Time, td Time to 50% of Xss Specifications Frequency Response To Transfer Function Slopes Phase Other Bode Plot Sinusoidal

Excitation Find output of system with sinusoidal excitation.

Design system accordingly.

(i.e. Design bridge so this doesn't happen) Pole-Zero Map Vibrations Absorbers Isolators Prevent transmission of vibration from or to object for some frequency range Eliminate vibrations at a given frequency Pick a small mass that vibrates at

systems natural frequency Steps Filters Types Cut-Off Frequency Bandwidth High Pass Low Pass Bandpass Notch Format Components Combining Components The portion of the response that eventually disappears Dependent on the input to the system only Dependent on initial conditions only The portion of the response that remains forever

### Present Remotely

Send the link below via email or IM

CopyPresent to your audience

Start remote presentation- Invited audience members
**will follow you**as you navigate and present - People invited to a presentation
**do not need a Prezi account** - This link expires
**10 minutes**after you close the presentation - A maximum of
**30 users**can follow your presentation - Learn more about this feature in our knowledge base article

# ME 421 Concept Map

Concept map for Mechanical Systems and Control

by

Tweet