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The average values of a function can be determined by
Use this relationship to verify the results of Eq. (19.13).
To prove: The following results,
If the average value formula is
Explanation of Solution
Given:
The average value formula is
Formula used:
The integral formula are
The trigonometric identity are,
Proof:
Consider the identity,
Use average value formula and take
As it is known that the angular frequency
Therefore,
Use integration formula,
On simplifying further,
Hence, the result is
Now consider next result,
Use average value formula and take
As it is known that the angular frequency
Therefore,
Use the integral formula,
Hence, the result is
Now, consider next result,
Use average value formula and take
As it is known that the angular frequency
Therefore,
Use the trigonometric identity,
Use the integral formula,
Hence, the result is
Now, consider next result,
Use average value formula and take
As it is known that the angular frequency
Therefore,
Use the trigonometric identity,
Use the integral formula,
On further simplifying,
Hence, the result is
Now, consider next result,
Use average value formula and take
As it is known that the angular frequency
Therefore,
Use the trigonometric identity,
On further simplifying,
Hence, the result is
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Chapter 19 Solutions
Numerical Methods for Engineers
- Air is used as the working fluid in a simple ideal Brayton cycle that has a pressure ratio of 12, a compressor inlet temperature of 300 K, and a turbine inlet temperature of 1000 K. The properties of air at room temperature are cp=1.005 kJ/kg⋅K and k=1.4 . Determine the required mass flow rate of air for a net power output of 66 MW, assuming both the compressor and the turbine have an isentropic efficiency of 85 percent. The required mass flow rate of air is _________ kg/s.arrow_forwardF T = 450 Nm A ☑ 100 mm B 500 mm 1000 mm Şekil, 52 kN eksantrik yük taşıyan bir sonsuz mili göstermektedir. Mil radyal ve eksenel yük taşıyan rulmanlarla yataklanmıştır. Yük şekilde gösterildiği gibi somuna asılmış ve dönmeyi engellemektedir. Sürtünmeyi yenip yükü kaldırmak için uygulanan tork 450 Nm'dir. A ve B noktalarındaki gerilmeleri hesaplayınız. Milin, AISI 4140 540°C'DE temperlenmiş alaşımlı çelik olduğu bilindiğine göre maksimum kayma gerilmesi teorisine göre emniyetli olup olmadığını belirleyiniz. The figure shows a worm shaft carrying a 52 kN eccentric load. The shaft is supported by bearings that carry radial and axial loads. The load is suspended from the nut, as shown, preventing rotation. The torque applied to overcome friction and lift the load is 450 Nm. Calculate the stresses at points A and B. Knowing that the shaft is made of AISI 4140 alloy steel tempered at 540˚C, determine whether it is safe according to the maximum shear stress theory. 400 mmarrow_forwardAnswer this question asap show all of the workarrow_forward
- Answer this question asap show all of the workarrow_forwardAnswer this question asap show all of the workarrow_forwardStress, ksi 220 200 180 160 140 120 100 80 Question P: Data for an extension spring is shown in the table below. Use only this table for this question! Also shown is an abridged version of Table 18-2 and figure 18 Spring Material ASTM A228 Music wire Max Operating Load: Fo= 21 Type of Service = Average Estimated Wahl Factor: K = 1.200 Required Mean Diameter: D = 0.550 Design Stress in Wire: Td 90,000 psi TABLE 18-2 Wire Gages and Diameters for Springs Gage no. U.S. steel wire gage¹ (in) Music wire gage² (in) 0.6 26 0.0181 0.063 27 0.0173 0.067 28 00162 0.071 29 00150 0.075 30 00140 0.080 31 0.0132 0.085 22 0.0128 0.090 33 00118 0.095 34 0.0104 0.100 35 0.0095 0.106 36 0.0090 Wire diameter, mm Compression and extension springs, Music Wire, ASTM A228 O'S 5.4 5.8 6.2 0.112 1515 1380 Light service 1240 1100 Average service 965 Severe service 825 690 Wire diameter, in OLIO 0.190 0120 0.250 550 Stress, MPa FIGURE 18-9 Design shear stresses for ASTM A228 steel wire (music wire) What is the…arrow_forward
- Endurance limit,, (psi) 100 000 80 000 60 000 Ground 40 000 20 000 As-rolled 0 50 60 70 80 90 100 110 120 Polished Machined or cold drawn As-forged 130 140 150 160 17 Tensile strength, s, (ksi) (a) U.S. customary units What is the minimum shaft diameter of D3 in inches? (Type in a three-decimal number). Note: We want to know the diameter D3, of the shaft, not the diameter at the base of a ring groove, profile keyseat or any other geometric feature on the shaft. Answer: x (3.008)arrow_forwardQuestion G: The machined shaft shown in the diagram below has the following components on it: (A) Sheave (B) Bearing (C) Sprocket (D) Bearing (E) Spur Gear Diameter D3 is located underneath Bearing B. Only the sheave at point A, the sprocket at point C and the spur gear at point E are held in place with rings. Diameter Dy is located underneath Bearing B. Only the sheave at point A, the sprocket at point C and the spur gear at point E are held in place with rings. PPENDIX 3 Design Properties of Carbon and Alloy Steels Material designation (SAE number) Condition Tensile strength Yield strength (ksi) (MPa) (MPa) Bearing Bearing 1020 Hot-rolled 55 379 207 V-belt sheave 6.00 in PD DD 1020 Cold-drawn 61 420 352 Spur gear Chain sprocket 10.00 in PD 20 FD 12.00 in PD 1020 Annealed 60 414 296 (a) Side view of shaft 10401 Hot-rolled 72 496 290 Belt drive to conveyor 1040 Cold-drawn 80 552 1040 OQT 1300 88 607 1040 OQT 400 113 779 1050 Hot-rolled 90 620 leput from water turbine Gear E drives Q to…arrow_forward220 200 180 160 140 120 Stress, ksi 100 80 Question O: Data for an extension spring is shown in the table below. Use only this table for this question! Also shown is an abridged version of Table 18-2 and figure 18. Spring Material ASTM A228 Music wire Max Operating Load: F₁ = 57 Type of Service Average Estimated Wahl Factor: K= 1.200 Required Mean Diameter: D = 0.850 Design Stress in Wire: 1 = 115,000 psi TABLE 18-2 Wire Gages and Diameters for Springs 0.0181 27 0.0175 Gage no. U.S. steel wire gage (in) Music wire gage² (in) 0063 0.067 28 0.0162 0.071 29 0.0150 0.075 30 00140 0.080 31 0.0132 0085 32 00128 0.090 33 00118 0096 34 0.0104 0.100 35 0.0095 36 0.0090 1.8 Wire diameter, mm 0.106 0.112 5.4 5.8 6.2 1515 Compression and extension springs, Music Wire, ASTM A228. 1380 Light service 1240 Average service 1100 965 Severe service 825 690 P10100 OSO 0 0.150 0.170 061'0 0.210 0.230 F 0.250 550 Stress, MPa Wire diameter, in FIGURE 18-9 Design shear stresses for ASTM A228 steel wire (music…arrow_forward
- Please see attachment.arrow_forwardPlease see attachment.arrow_forwardP3: A differential band brake shown in the figure below uses a woven lining having a design value of the friction coefficient f=0.20. Dimensions are b=80 mm, r=250 mm, c=700 mm, a = 150 mm, s=35 mm, and 0=240°. Find 1) the brake torque if the maximum lining pressure is 0.5 MPa, 2) the corresponding actuating force F, and 3) the values of dimensions that would cause the brake to be self-locking. (25%) -240° F-250 mm Band width, b-80 mm Rotation Friction coefficient, -0.20 Maximum lining pressure, P-0.5 MPa 3-35 mm la-150 mm e-700 mm-arrow_forward
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