recap of session vii chapter ii: mathematical modeling mathematical modeling of mechanical systems...
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Recap of Session VII
Chapter II: Mathematical Modeling
• Mathematical Modeling of Mechanical systems
•Mathematical Modeling of Electrical systems
• Models of Hydraulic SystemsLiquid Level SystemFluid Power System
Recap of Session VII
Mathematical Modeling: Thermal Systems
qin = heat inflow rate
Tov = Temperature of the oven
Tamb = Ambient Temperature
T = Rise in Temperature = (Tov - Tamb)
Mathematical Modeling: Thermal Systems
Parts
Tov
Oven
Tamb
qout
qin
Example: Heat treatment oven
Mathematical Modeling: Thermal Systems-I
From Law of Conservation Energy
qin = heat inflow rate
qin = qs + qout --- (1)
qout = heat loss through the walls of the oven
qs = Rate at which heat is stored (Rate at which heat is absorbed by the parts)
Mathematical Modeling: Thermal Systems-II
Thermal Resistance: R=outq
T
R
Tqout --- (a)
Thermal Capacitance = C = Q/T
Heat stored =
dt
dTCqs --- (b)
Mathematical Modeling: Thermal Systems-IIISubstitute (a) and (b) in (1)
qin = qs + qout --- (1)
R
T
dt
dTCqin .
Model inRqTdt
dTRC
Chapter III: System ResponseChapter III: System Response
• Prediction of the performance of control systems requires
1. Obtaining the differential equations2. Solutions
System behaviour can be expressed as a function of time
Such a study: System response or system analysis in time domain
System Response in Time DomainSystem Response in Time Domain
System Response: The output obtained corresponding to a given Input.
Total response: Two parts•Transient Response (yt)•Steady state response (yss)
•Total response is the sum of steady state response and transient response
y = yt + yss
Transient Response (yt):Transient Response (yt):
•Initial state of response and has some specific characteristics which are functions of time.
•Continues until the output becomes steady.
•Usually dies out after a short interval of time.
•Tends to zero as time tends to ∞
Steady State Response (yss)Steady State Response (yss)
• Ultimate Response obtained after some interval of time
• Response obtained after all the transients die out
• It is not independent of time
• As time approaches to infinity system response attains a fixed pattern
Transient and Steady-state Response of a spring system
Transient and Steady-state Response of a spring system
Transient SS
• When the weight is added the deflection abruptly increases•System oscillates violently for some time (Transient)•Settles down to a steady value (Steady state)
Steady State ErrorSteady State Error
• Steady State Response may not agree with Input
• Difference is called steady state error
Steady state error = Input – Steady state response
Input or Response
Steady state error
Timet =0
Input
Response
t ∞
Test Input SignalsTest Input Signals
• Systems are subjected to a variety of input signals (working conditions)
• Most cases it is very difficult to predict the type of input signal
• Impossible to express the signals by means of Mathematical Models
• Common Input Signals
- Step Input
- Ramp Input
- Sinusoidal
- Parabolic
- Impulse functions, etc.,
Common Input Signals
• In system analysis one of the standard input signal is applied and the response produced is compared with input
• Performance is evaluated and Performance index is specified
• When a control system is designed based on standard input signals – generally, the performance is found satisfactory
Standard input signals
Common System Input Signalsa) Step Input
i (t)
t = 0 time
K
Common System Input Signals
a) Step Input
Input is zero until t = 0
Then takes on value K which remains constant for t > 0
Signal changes from zero level to K instantaneously
Mathematically
i (t) = K for t > 0
= 0 for t < 0
for t = 0, step function is not defined
When a system is subjected to sudden disturbance step input can be used as a test signal
Common System Input Signalsa) Step Input-I
Common System Input Signalsa) Step Input- Examples
Examples
Angular rotation of the Shaft when it starts from rest
Change in fluid flow in a hydraulic system due to sudden opening of a valve
Voltage applied on an electrical network when it is suddenly connected to a power source
b) Ramp Input
i (t)
t = 0 time
K*t
Input
Signals is linear function of time
Increases with time
Mathematically i (t) = K*t for t > 0= 0 for t < 0
Example: Constant rate heat input in thermal system
Common System Input Signalsb) Ramp Input
imei (t)
Input k Sin t
i (t) = k Sin t
c) Sinusoidal Input
Mathematically
i (t) = k Sin t
System response in frequency domain
Frequency is varied over a range
Example: Voltage, Displacement, Force etc.,
Common System Input Signalsc) Sinusoidal Input
Order of the SystemOrder of the System
The responses of systems of a particular order are Strikingly similar for a given input
Order of the system: It is the order of the highest derivative in the ordinary linear differential equation with constant coefficients, which represents the physical system mathematically.
kxkydt
dyC .
Illustration: First order systemIllustration: First order system
x (t) i/p
y (t) o/p
C
K
Cy + ky = kx.
kxkydt
dyC .
Order: Order of the highest derivative = 1 First order system
Illustration: Second order systemIllustration: Second order system
m
x (t)
y (t)
K
C
kxkydt
cdy
dt
ydm
kxkycymy
2
2
.
.. .
Order: Order of the highest derivative = 2 Second order system
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