A thermocouple is immersed in a liquid to monitor its temperature fluctuations. Assume the thermocouple acts as a first-order system. The temperature fluctuations (in degrees Celsius) vary in time as T(t) = 50 + 25 cos(4t). The output of the thermocouple transducer system (in V) is linearly proportional to temperature and has a static sensitivity of 2 mV/°C. A step-input calibration of the system reveals that its rise time is 4.6 s. Using MatLAB answer the following: Determine the system's (a) time constant (in seconds), (b) output E(t) (in millivolts), (c) time lag (in seconds) at ω = 0.2 rad/s, (d) solve the response of this first-order system analytically to get temperature distribution equation, (e) solve this first-order system again with MATLAB (ode45) and plot the results along with input forcing function in one figure.

A mechanical system is presented as below. There are four simulation graphs for different values for m, b and c tested for a step response. Identify the graph for m=4 kg, b=0.3 N.s/m and k=1 N/m (Hint: you may need to use matlab simulation to find it). m 1.8 1.6 1.4 b ·f(t) Step Response

A chamber of unknown volume is filled with an unknown amount of ideal gas. A cylinder with an open bottom and calibrated volume is connected to the chamber and is closed with a movable piston. A gas pressure sensor is also mounted on the chamber as shown in the figure. The whole experimental setup is kept at a constant temperature of 300 K. Doris moves the piston from mark 0 to mark 200 mL in steps of 50 mL and records the corresponding pressure in the chamber (see the table below). She moves the piston very slowly and waits several minutes before recording the pressure values.   VCyl(mL) P(105N/m2)) 0 4.35 50 3.85 100 3.50 150 3.15 200 2.90   Determine the volume of the chamber using Doris's data in the table. (Note: This is a problem that requires linearization of data.)   Determine the number of moles of gas. (Note: This is a problem that requires linearization of data.)

d) A thermometer is initially at a temperature of 20 °C and is suddenly placed in a liquid which is maintained at 100 °C. The thermometer indicates 80 °C after a time interval of 4 0, = 0, (1-e) seconds. Knowing that the first order response is given as: estimate the time constant of the system.

Question 1: A pressure gauge which can be modeled as a Linear Time-Invariant (LTI) system has a time response to a unit impulse input h(t) given by (Ge-2t +e-«") u(t). For a certain unknown input, the output y(t) is observed to be (4e7t – 8e2")u(t). For this observed measurement, derive the true pressure input x(t) to the gauge as a function of time. Assume that the input signal is right-sided.

An unmanned submarine is equipped with temperature and depth measuring instruments and hasradio equipment that can transmit the output readings of these instruments back to the surface.The submarine is initially floating on the surface of the sea with the instrument output readingsin steady state. The depth measuring instrument is approximately zero order and the temperaturetransducer first order with a time constant of 50 seconds. The water temperature on the seasurface, T0, is 20C and the temperature Tx at a depth of x metres is given by the relation:Tx = T0- 0.01x(a) If the submarine starts diving at time zero, and thereafter goes down at a velocity of 0.5metres/second, draw a table showing the temperature and depth measurements reported atintervals of 100 seconds over the first 500 seconds of travel. Show also in the table the error ineach temperature reading(b) What temperature does the submarine report at a depth of 1000 metres?

3. The partial differential equation for the small-amplitude vibrations of a beam is given by PA 812 + EI = 0 Ox4 where y(x, t) is the vibration amplitude, p is the material density of the beam, A is the beam cross section, E is the modulus of elasticity, and I is the area moment of inertia of the beam cross section. Using only the parameters given, make this equation dimensionless. Hint: first find a combination of p and/or A and/or E and/or I to make x, y, and t dimensionless. (It is clear that a different combination will be needed to make t dimensionless than that for x and y.)

A balloon carrying a first-order thermometer with 15-sec time constant rises through the atmosphere at 6 m/sec. Assume temperature varies with altitude at 0.15 °C/30m. The balloon radios temperature and altitude readings back to the ground. At 3000 m the balloon says the temperature is 0°C. What is the true altitude at which 0°C occurs?

D6.

Consider the following system: mä(t) + cx(t) + kx(t) = F,8(t) + F28(t – t1) Asumme zero initial conditions. The units are in Newtons. If m = 9.3634 kg, c = 49.8992 Ns/m, k = 90.1813 N/m, F1 = 94.7209 N.s , F2= -85.0916 N.s and t1 = 0.8657 s, the displacement of the system at t2 = 5.4519 s is: %3D %3D 0.0021 mm -0.0218 mm 0.0002 mm -12.3038 mm -0.539 mm