Recap of Session VII Chapter II: Mathematical Modeling Mathematical Modeling of Mechanical systems Mathematical Modeling of Electrical systems Models of Hydraulic Systems Liquid Level System Fluid Power System Recap of Session VII

Mathematical Modeling: Thermal Systems q in = heat inflow rate T ov = Temperature of the oven T amb = Ambient Temperature T = Rise in Temperature = (T ov - T amb ) Mathematical Modeling: Thermal Systems Parts T ov Oven T amb q out q in Example: Heat treatment oven

Mathematical Modeling: Thermal Systems-I From Law of Conservation Energy q in = heat inflow rate q in = q s + q out --- (1) q out = heat loss through the walls of the oven q s = Rate at which heat is stored (Rate at which heat is absorbed by the parts)

Mathematical Modeling: Thermal Systems-II Thermal Resistance: R= --- (a) Thermal Capacitance = C = Q/T Heat stored = --- (b)

Mathematical Modeling: Thermal Systems-III Substitute (a) and (b) in (1) q in = q s + q out --- (1) Model

Chapter III: System Response Prediction of the performance of control systems requires 1.Obtaining the differential equations 2.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 Domain System Response: The output obtained corresponding to a given Input. Total response: Two parts Transient Response (y t ) Steady state response (y ss ) Total response is the sum of steady state response and transient response y = y t + y ss

Transient Response (y t ): 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 (y ss ) 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 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 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 Time t =0 Input Response t ∞

Test 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 Signals a) Step Input i (t) t = 0time 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 Signals a) Step Input-I

Common System Input Signals a) 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 = 0time 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 Signals b) Ramp Input

ime i (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 Signals c) Sinusoidal Input

Order 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.

Illustration: First order system x (t) i/p y (t) o/p C K Cy + ky = kx. Order: Order of the highest derivative = 1 First order system

Illustration: Second order system m x (t) y (t) K C... Order: Order of the highest derivative = 2 Second order system

Response of First Order Mechanical Systems to Step Input