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## Recap of Session VII

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**Recap of Session VII**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**Mathematical Modeling:**Thermal Systems Tov Tamb qin = heat inflow rate Tov = Temperature of the oven Tamb = Ambient Temperature T = Rise in Temperature = (Tov - Tamb) Oven qout Parts qin Example: Heat treatment oven Mathematical Modeling: Thermal Systems**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)**Thermal Resistance: R=**--- (a) Thermal Capacitance = C = Q/T Heat stored = --- (b) Mathematical Modeling: Thermal Systems-II**Substitute (a) and (b) in (1)**qin = qs + qout --- (1) Model Mathematical Modeling: Thermal Systems-III**Chapter III: System Response**• Prediction of the performance of control systems requires • Obtaining the differential equations • Solutions • System behaviour can be expressed as a function of time • Such a study: System response or system analysis in time domain Chapter III: System Response**System 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 System Response in Time Domain**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**When the weight is added the deflection abruptly increases**• System oscillates violently for some time (Transient) • Settles down to a steady value (Steady state) Transient SS Transient and Steady-state Response of a spring system Transient and Steady-state Response of a spring system**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 Input or Response Steady state error Response t ∞ Time t =0 Steady State Error**Test Input Signals**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**• Common Input Signals • - Step Input • - Ramp Input • - Sinusoidal • - Parabolic • - Impulse functions, etc.,**Standard 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**Common System Input Signals**a) Step Input K i (t) t = 0 time • 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 Common System Input Signalsa) Step Input**Common System Input Signalsa) Step Input-I**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- 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**Input i (t) K*t • 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 t = 0 time Common System Input Signalsb) Ramp Input**c) Sinusoidal Input**Input k Sin t • Mathematically • i (t) = k Sin t • System response in frequency domain • Frequency is varied over a range Example: Voltage, Displacement, Force etc., i (t) ime i (t) = k Sin t Common System Input Signalsc) Sinusoidal Input**Order of the System**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**Illustration: First order system . Cy + ky = kx x (t) i/p K y (t) o/p C Order: Order of the highest derivative = 1 First order system**Illustration: Second order system**. .. x (t) K y (t) m C Order: Order of the highest derivative = 2 Second order system Illustration: Second order system**Response of First Order Mechanical Systems to Step Input**Response of First Order Mechanical Systems to Step Input