Theory and Design for Mechanical Measurements
6th Edition
ISBN: 9781119031703
Author: FIGLIOLA
Publisher: WILEY
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Chapter 2, Problem 2.34P
To determine
Plot the input in the time domain and construct an amplitude spectrum plot by using a displacement that vary between 2 and 5 mm at a frequency of 100 Hz.
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If you have a spring mass damper system, given by m*x_double_dot + c*x_dot + kx = 0 where m, c, k (all positive scalars) are the mass, damper coefficient, and spring coefficient, respectively. x ∈ R represents the displacement of the mass.
Let us then discuss the stability of the system by using Lyapunov stability theorem. Consider the system energy as a candidate Lyapunov function shown in the image.
Discuss the positive definiteness of V (x, x_dot).
Derive the Lyapunov rate of this system (i.e., V_dot ), and discuss the stability property of thesystem based on the information we gain from ̇V_dot .
In class, two approaches—Theorems 1 and 2 below—are discussed to prove asymptotic stability of asystem when ̇V = 0.
Show the asymptotic stability of the system given in Eq. (1) by applying Theorem 1.
Show the asymptotic stability of the system given in Eq. (1) by applying Theorem 2.
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Chapter 2 Solutions
Theory and Design for Mechanical Measurements
Ch. 2 - Prob. 2.1PCh. 2 - Prob. 2.2PCh. 2 - Research and describe the importance of...Ch. 2 - Research and describe the importance of data...Ch. 2 - Determine the average and rms values for the func...Ch. 2 - Prob. 2.6PCh. 2 - Prob. 2.9PCh. 2 - Prob. 2.10PCh. 2 - Prob. 2.11PCh. 2 - Prob. 2.12P
Ch. 2 - Express the function y(t) = 4 sin 2xt + 15 cos...Ch. 2 - Prob. 2.14PCh. 2 - The Fourier series that formed the result for...Ch. 2 - The nth partial sum of a Fourier series is defined...Ch. 2 - For the Fourier series given by where t is time in...Ch. 2 - Determine the Fourier series for the function y(t)...Ch. 2 - Show that y(t) = f2(—jc < t < k), y(t + 2k) = y(t)...Ch. 2 - Find the Fourier series of the function shown in...Ch. 2 - Determine the Fourier series for the function y(t)...Ch. 2 - Determine the Fourier series that represents the...Ch. 2 - Consider the triangle wave shown in Figure 2.26 as...Ch. 2 - Prob. 2.24PCh. 2 - A particle executes linear harmonic motion around...Ch. 2 - Define the following characteristics of signals:...Ch. 2 - Construct an amplitude spectrum plot for the...Ch. 2 - Prob. 2.28PCh. 2 - Sketch representative waveforms of the following...Ch. 2 - Represent the function e(t) = 5 sin 31 At + 2 sin...Ch. 2 - Repeat Problem 2.30 using a data set of 256 num...Ch. 2 - A particular strain sensor is mounted to an...Ch. 2 - Prob. 2.33PCh. 2 - Prob. 2.34PCh. 2 - Consider the upward flow of water and air in a...Ch. 2 - Prob. 2.37PCh. 2 - Prob. 2.38PCh. 2 - For the even-functioned triangle wave signal...Ch. 2 - Prob. 2.40P
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- You are asked to design a unit to condense ammonia. The required condensation rate is 0.09kg/s. Saturated ammonia at 30 o C is passed over a vertical plate (10 cm high and 25 cm wide).The properties of ammonia at the saturation temperature of 30°C are hfg = 1144 ́10^3 J/kg andrv = 9.055 kg/m 3 . Use the properties of liquid ammonia at the film temperature of 20°C (Ts =10 o C):Pr = 1.463 rho_l= 610.2 kf/m^3 liquid viscosity= 1.519*10^-4 kg/ ms kinematic viscosity= 2.489*10^-7 m^2/s Cpl= 4745 J/kg C kl=0.4927 W/m CCalculate the surface temperature required to achieve the desired condensation rate of 0.09 kg/s( should be 688 degrees C) a) Show that if you use a bigger vertical plate (2.5 m-wide and 0.8 m-height), the requiredsurface temperature would be now 20 o C. You may use all the properties given as an initialguess. No need to iterate to correct for Tf. b) What if you still want to use small plates because of the space constrains? One way to getaround this problem is to use small…arrow_forwardA differential element on the bracket is subjected to plane strain that has the following components:, Ɛx = 300 × 10-6, Ɛy = 150 × 10-6, Ɛxy = -750 x 10-6. Use the strain-transformation equations and determine the normal strain Ɛx in the X/ direction on an element oriented at an angle of 0 = 40°. Note, a positive angle, 0, is counter clockwise. x Enter your answer in micro strain to a precision of two decimal places. eg. if your answer is 300.15X106, please enter 300.15.arrow_forwardIf the 50 mm diameter shaft is made from brittle material having an ultimate strength of σult=595 MPa for both tension and compression, determine the factor of safety of the shaft against rupture. The applied force, F, is 140 kN. The applied torque T, is 5.0 kN⚫m. Enter your answer to a precision of two decimal places. T Farrow_forwardЗіс 1 mH 10 Ω m 16 cos 2.5 × 104 A Lic 592 10 Ω 1 μFarrow_forwardHomework#5arrow_forwardHomework#5arrow_forwardOxygen (molar mass 32 kg/kmol) expands reversibly in a cylinder behind a piston at a constant pressure of 3 bar. The volume initially is 0.01 m3 and finally is 0.03 m3; the initial temperature is 17°C. Calculate the work input and the heat supplied during the expansion. Assume oxygen to be an ideal gas and take cp = 0.917 kJ/kg K. For 1 bonus mark explain why (using your understanding of thermodynamics) that oxygen is used in this context rather than water vapour.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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