Calculate the resulting uncertainty in omegan al l 10 and l

Question

Problem 1

Consider the cantilever beam shown below. Suppose the uncertainties in the material properties and dimensions are: w_E = 2% w_r =0.0005 in, w_Q = 5 lbm/ft², and w_L = 0.01 in.

The natural frequency of the cantilever beam is given by

ω_n = 11/rL²√(EI/ρΠ)

Calculate the resulting uncertainty in ω_n al L =1.0 and L =4.0 in.

Problem 2
Suppose you are asked to sample the output voltage signal below from an analog sensor using a data acquisition system. Given the signal, complete/solve the following steps:

a) Graphically convert this signal from analog form(shown below) to digital form on the same plot assuming 3 bits of resolution. How many discrete voltage steps are there and what is the voltage step/size resolution?

b) What is the frequency of the signal? In order to properly sample and reproduce the characterstics of the signal in digital form, what should be the minimum sampling frequency and what is the theorm that governs this rule?

Problem 3

A platinum resistance thermometer is placed in a duct to measure the temprature of an airflow stream. The termometer is paced inide a cylindrical shell 6 mm in diameter (d) which has a polished outside surface with ε = 0.08 ± 0.02. The airstream velocity, U is known to be 3 m/s and the pressure is 1.0 atm. The thermometer indicates a temperature of 115^oc. The duct wall temperature is measured at 193^o C.

Calculate the true air temperature and estimate the uncertainty in this temperature. Assume that the uncertainty in the convection heat- transfer coefficient given below is ±15 percent. Also, assume that the uncertainty in the resistance thermometer measurement is ±0.3 C. The Prandtl number, Pr is 0.68 and the kinematic viscosity, v is 27.06 x 10-6 m²/s.