Foundations of Materials Science and Engineering
6th Edition
ISBN: 9781259696558
Author: SMITH
Publisher: MCG
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Chapter 11.13, Problem 127SEP
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While inspecting the ceramic coating of a component, a small crack is identified. Would it be a big concern?
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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 11 Solutions
Foundations of Materials Science and Engineering
Ch. 11.13 - Define a ceramic material.Ch. 11.13 - Prob. 2KCPCh. 11.13 - Prob. 3KCPCh. 11.13 - Prob. 4KCPCh. 11.13 - Prob. 5KCPCh. 11.13 - Prob. 6KCPCh. 11.13 - Prob. 7KCPCh. 11.13 - What fraction of the octahedral interstitial sites...Ch. 11.13 - Prob. 9KCPCh. 11.13 - Describe the perovskite structure. What fraction...
Ch. 11.13 - Prob. 11KCPCh. 11.13 - Prob. 12KCPCh. 11.13 - Prob. 13KCPCh. 11.13 - Prob. 14KCPCh. 11.13 - Describe the feldspar network structure.Ch. 11.13 - Prob. 16KCPCh. 11.13 - Prob. 17KCPCh. 11.13 - Describe two methods for preparing ceramic raw...Ch. 11.13 - Prob. 19KCPCh. 11.13 - Prob. 20KCPCh. 11.13 - Prob. 21KCPCh. 11.13 - Prob. 22KCPCh. 11.13 - Prob. 23KCPCh. 11.13 - Prob. 24KCPCh. 11.13 - Prob. 25KCPCh. 11.13 - Prob. 26KCPCh. 11.13 - What are the purposes of drying ceramic products...Ch. 11.13 - Prob. 28KCPCh. 11.13 - What is the vitrification process? In what type of...Ch. 11.13 - Prob. 30KCPCh. 11.13 - Prob. 31KCPCh. 11.13 - Prob. 32KCPCh. 11.13 - Prob. 33KCPCh. 11.13 - Prob. 34KCPCh. 11.13 - Prob. 35KCPCh. 11.13 - Prob. 36KCPCh. 11.13 - Prob. 37KCPCh. 11.13 - Prob. 38KCPCh. 11.13 - Why do most ceramic materials have low thermal...Ch. 11.13 - Prob. 40KCPCh. 11.13 - Prob. 41KCPCh. 11.13 - Prob. 42KCPCh. 11.13 - Prob. 43KCPCh. 11.13 - Prob. 44KCPCh. 11.13 - Prob. 45KCPCh. 11.13 - Prob. 46KCPCh. 11.13 - How is a glass distinguished from other ceramic...Ch. 11.13 - Prob. 48KCPCh. 11.13 - Prob. 49KCPCh. 11.13 - Prob. 50KCPCh. 11.13 - Prob. 51KCPCh. 11.13 - Prob. 52KCPCh. 11.13 - Prob. 53KCPCh. 11.13 - Prob. 54KCPCh. 11.13 - Prob. 55KCPCh. 11.13 - Prob. 56KCPCh. 11.13 - Prob. 57KCPCh. 11.13 - Prob. 58KCPCh. 11.13 - Prob. 59KCPCh. 11.13 - Prob. 60KCPCh. 11.13 - Prob. 61KCPCh. 11.13 - Prob. 62KCPCh. 11.13 - Prob. 63AAPCh. 11.13 - Prob. 64AAPCh. 11.13 - Prob. 65AAPCh. 11.13 - Prob. 66AAPCh. 11.13 - Prob. 67AAPCh. 11.13 - Prob. 70AAPCh. 11.13 - Calculate the ionic packing factor for (a) MnO and...Ch. 11.13 - Prob. 72AAPCh. 11.13 - Prob. 73AAPCh. 11.13 - Prob. 74AAPCh. 11.13 - Prob. 75AAPCh. 11.13 - Prob. 77AAPCh. 11.13 - Prob. 78AAPCh. 11.13 - Prob. 79AAPCh. 11.13 - Prob. 80AAPCh. 11.13 - Prob. 81AAPCh. 11.13 - Why are triaxial porcelains not satisfactory for...Ch. 11.13 - Prob. 83AAPCh. 11.13 - Prob. 84AAPCh. 11.13 - Prob. 85AAPCh. 11.13 - What causes the lack of plasticity in crystalline...Ch. 11.13 - Prob. 87AAPCh. 11.13 - Prob. 88AAPCh. 11.13 - Prob. 89AAPCh. 11.13 - A reaction-bonded silicon nitride ceramic has a...Ch. 11.13 - Prob. 91AAPCh. 11.13 - Prob. 92AAPCh. 11.13 - Prob. 93AAPCh. 11.13 - Prob. 94AAPCh. 11.13 - How does the silica network of a simple silica...Ch. 11.13 - Prob. 96AAPCh. 11.13 - Prob. 97AAPCh. 11.13 - Prob. 98AAPCh. 11.13 - Prob. 99AAPCh. 11.13 - Prob. 100AAPCh. 11.13 - Prob. 101AAPCh. 11.13 - Prob. 102AAPCh. 11.13 - Prob. 103AAPCh. 11.13 - Prob. 104AAPCh. 11.13 - Prob. 105AAPCh. 11.13 - Prob. 106AAPCh. 11.13 - Prob. 107AAPCh. 11.13 - Prob. 108SEPCh. 11.13 - Prob. 109SEPCh. 11.13 - Prob. 110SEPCh. 11.13 - Prob. 111SEPCh. 11.13 - Prob. 112SEPCh. 11.13 - Alumina (A12O3) and chromium oxide (Cr2O3) are...Ch. 11.13 - (a) How are the ceramic tiles used in the thermal...Ch. 11.13 - The nose cap and the wing leading edges of the...Ch. 11.13 - Prob. 116SEPCh. 11.13 - Prob. 117SEPCh. 11.13 - Prob. 118SEPCh. 11.13 - Prob. 119SEPCh. 11.13 - Prob. 120SEPCh. 11.13 - Prob. 121SEPCh. 11.13 - Prob. 122SEPCh. 11.13 - Prob. 123SEPCh. 11.13 - Prob. 124SEPCh. 11.13 - Prob. 125SEPCh. 11.13 - Prob. 126SEPCh. 11.13 - Prob. 127SEPCh. 11.13 - Prob. 128SEPCh. 11.13 - Prob. 129SEP
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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. Using linear stability analysis, show that the system is asymptotically stable. Hint: stability of a linear system z_dot = Az is characterized by the eigenvalues of A.arrow_forwardWhat would the electropneumatic diagram of a circuit with the sequence a+b+c+(a-b-c-) look like?arrow_forward### What would the electropneumatic diagram of a circuit with the sequence a+b+c+(a-b-c-) look like, with a counter, in the fluidsim?arrow_forward
- 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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