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Republic of the Philippines Nueva Ecija University of Science and Technology Cabanatuan City
Problems and Solutions In ME 414D (Flywheel)
Submitted By: Jefferson Dancel Archie Molina Jun Ceazar Soriano
1. Flywheel turns 550 rev/min (RPM). Determine the magnitude of the normal acceleration of the flywheel point which are at a distance of 15 cm from the rotation axis. Solution: π=
15 550 2 (2π ) = 331.73 πβπ 2 100 60
2. A cast iron flywheel with a mean diameter of 24 inches changes speed from 550 rpm to 530 while it gives up 5800 ft-lb of energy. What is the coefficient of fluctuation. Solution: Cf = V1 β V2 / Vave = N1 β N2 / Nave = 550 β 530 / (550+530/2) = 0.0370 answer
3. A mechanical press is used to punch 6 holes per minute is 25 mm diameter and the plates has an ultimate strength in shear of 420 Mpa. The normal operating speeed 200 rpm. And it slows down to 180 rpm during the process of punching. The flywheel has a mean diameter of one meter and the rim width is 3 times the thickness. Assume that the hub and arm account for 5% of the rim weight concentrated at the mean diameter and the density of cast iron is 7200 kg per cubic meter. Find the power in KW required to drive the press. Solution: E = Β½ Ftp but : Ss = Sus F/A = 420 Mpa F/(πDtp) = 420 F = 420[πDtp] Then: E = Β½(420)(π)(D)(tp)2 = Β½(420)(π)(25)(25)2 = 10 308 351 N-mm = 10 308.351 Nm 1 πππ 60 π
t = 6 βππππ [πππ] = 10 s/holes P = E/t =
10 308 10
= 1031 W power required
answer
4. Find the weight of the flywheel needed by a machine to punch 20.5 mm holes in 15.87 mm thick steel plate. The machine is to make 30 strokes per minute and a hole be punched every stroke, the hole is to be formed during 30 degrees rotation of the puncher crankshaft. A gear train with a ratio of 12 to 1 is to correct the flywheel shaft to
the crankshaft. Let mean diameter of a flywheel rim to the 91.44 cm the minimum flywheel speed is to be 90% of the maximum and assume mechanical efficiency of the machinr to be 80%. Assume an ultimate stress of 49000 psi. Solution: Solving for the energy needed by the process: E = Β½ Ftp Where: F/A = 49000 psi 0.101325 F/(πDtp) = 49000[ ] 14.7 F = 337.75(πDtp) Then: E = Β½(337.75)(π)(D)(tp)2 E = Β½(337.75)(π)(20.5)(15.87)2 E = 2739.195 Nm Solving for the maximum speed of the flywheel: N1 = 30
π π‘πππππ 1 βπππ πππ
[
π π‘ππππ
360
][
30
][
1 πππ£ βπππ
] = 360 rpm
Solving for the minimum speed of the flywheel: N2 = 0.90(N1) = 0.90(360) = 324 rpm Solving for the weight of the flywheel: (based on energy equation) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
ΞKE = -2739.195 N-m Since the flywheel will supply / release this energu, then: -2739.195 = Wf / 2(9.81) [( π x 0.9144 x 324/60)2 β ( π x 0.9144 x 360/60)2] Wf = 952.11 N or 97.05 kgf
answer
5. It is found that the shearing machine requires 205 joules ofenergy to shear a specific gauge of sheet metal. The mean diameter of the flywheel is to be 76.2 cm, the normal operating speed is 200 rpm and slow down to 180 rpm during shearing process. The rim width is 30.48 cm and the weigth of ast iron is 7,196.6 kg/m3. Find the thicknes of the
rim, assuming that the hub and arm account for 10% of the rim weight concentrated on the mean diameter. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-205 = Wf / 2(9.81) [( π x 0.762 x 180/60)2 β ( π x 0.762 x 180/60)2] Wf = 33.89 kg weight of the flywheel Solving for the weight of the rim: Wf = Wrim + WHA 33.89 = 1.10(Wrim)
where: WHA = 0.10 Wrim Wrim = 30.81 kg
Solving for the thickness of the rim, Wrim = Ζ΄(Vrim) = Ζ΄(πDmbt)
where : = 7196.6 kg/m3
30.81 = 7196.6 [π(0.762)(0.3048)(t)] t = 5.867 mm
answer
6. A sheet metal working company purchase a shearing machine from a surplus dealer without a flywheel. It is calculated that the machine will use 2380 joules of energy to shear a 1.2 mm thick sheet metal. The flywheel to be used will have a mean diameter of 91.44 cm with a width of 25.4 cm. The normal operating speed is 180 rpm and slows down to 160 rpm during the shearing process. Assuming that the arms and the hub will account for 12% of the rim weight concentrated at the mean diameter and that the material density is 0.26 lb/cu. In. Compute for the weight of the flywheel. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 V2 = πDmn2 2380 = Wf / 2(9.81) [( π x 0.9144 x 160/60)2 β ( π x 0.9144 x 160/60)2]
Wf = 2996 N = 305 kgf answer
7. Calculate the moment of inertia, mass, for the circular flywheel of which the section. The material of the flywheel is steel, the density of which is 7800 kg.m 3. Solution: We can divide the area into three parts: I1, I2, and I3. I1 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.05 π₯ π
= x (0.1754 β 0.154) 2 = 0.264 kg.m2 I2 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.03 π₯ π
= x (0.154 β 0.1354) 2 = 0.064 kg.m2 I3 =
ππ₯π€π₯π
x (R24 β 0)
2 7800 π₯ 0.015 π₯ π
= x (0.1354 β 0) 2 = 0.061 kg.m2 The total moment of inertia is ITotal = 0.264 + 0.064 + 0.061 = 0.389 kg.m2 8. A solid cast-iron flywheel 12 in. in diameter, rotating at 2500 rpm and having a weight of 50 lbs., is used for a punch press. Calculate the mean torque obtained, if the flywheel's energy is restored during (1/4)th of a revolution. Solution: In a punch press energy is stored in the flywheel for a small portion of the cycle in the form of kinetic energy = (1/2)IΟ2 where I = (1/2)MR2(For a solid circular disk.) = (1/2)(W / g)R2 lbs.-in.sec2. Where; W = weight and R = radius of flywheel. Now, this kinetic energy is restored in the form of work done by the flywheel during the punching operation. β΄Work done = TmΙΈ Where; ΙΈ= Portion of flywheel's rotation during which the work is done = (Ο / 2) radians And, Tm = Mean torque obtained from flywheel rotation. Equating kinetic energy and work done by the flywheel
TmΙΈ = (1/2)IΟ2 Or, Tm = (1/2)[(IΟ2) / ΙΈ]
(1)
Now, I = (1/2)(W/g)R2 g = 386 in./sec.2 = (1/2)[50 / (386)](6)2 = 2.33 lbs.-in.-sec2. And, Ο(angular velocity) = [(2ΟN) / 60][(rads) / (sec)] or, Ο = [{2Ο(2500)} / 60] = 262 rads/sec. Substituting values of I, Ο and ΙΈ in equation (1) Tm = (1/2)[{2.33 Γ (262)2} / (Ο/2)] Tm = 50,910 in.-lb. The same torque is required to apply the brake to stop the flywheel during (1/4)th of a revolution
9. A flywheel used in a punch-press with a total energy requirement of 7000 lbs.-ft. is powered by an electric motor. The torque supplied by the motor is smooth and the torque required by the flywheel is rapidly varying. Both torques are shown in the loadtorque diagram, Figure 1. The motor rpm varies from 1100 to 1400 and the motor supplies (1/4) of its rpm to the flywheel. Calculate the mass-moment of inertia and the outside diameter of the flywheel if the rest of the flywheel's dimensions are in suitable proportions as shown in Figure 2. Assume density of the flywheel material, p = 0.289 lb./in.3 Solution:
MOMENT OF INERTIA: The moment of inertia of the flywheel If can be found by using the relation for change in rotational kinetic energy K.E. ROT, i.e., Ξ΄K.E.ROT = (1/2) If(Οmax2 β Οmin2) From the load-torque diagram, the total energy required for punching = 7000 lbs.-ft.
(1)
Also, energy supplied by the motor for the punching stroke = 1910 Γ (Ο / 4) = 1500 lbs.-ft. Change in K.E. of the flywheel = Energy provided by the flywheel for punching operation = Total energy for punching β Energy provided by the motor = 7000 β 1500 = 5500 lbs.-ft. or Ξ΄K.E. can also be found from the load-torque diagram as Ξ΄K.E. = (1/4) Γ Area of approximate ΞPST = (1/4) Γ 14,000 Γ (Ο / 4) = 5500 lbs.-ft. Also maximum angular velocity of the flywheel Οmax = (1/4) Γ max. angular velocity of the motor = (1/4) Γ [(2ΟN) / 60] = (1/40 Γ [{2Ο(1400)} / 60] = 37 rads./sec. Minimum angular velocity of the flywheel = (1/4) Γ [{2Ο(1100)} / 60] = 29 rads./sec. Substituting all the above values back into eq.(l), 5500 = (1/2) If[(37)2 β (29)2] from which If = 21 lbs.-ft.-sec.2 OUTSIDE DIAMETER OF FLYWHEEL'Doβ: Since polar moment of inertia of the flywheel is Jf = [{Ο(Do4 β Di4)} / 32] [for a hollow disk]
Also, mass moment of inertia of the flywheel If and polar moment of inertia are related together as, If = ebJf = eb[{Ο(Do4 β Di4)} / 32] (2) 3 Since, Ο = 0.289 lbs./in . (given) = 499 lbs./ft?.3 And, b = 0.15Do (from Fig. 2) Di = 0.75Do Further, If = 21 lbs.-ft.-sec.2 Substituting these in eq.(2) gives, 21 = (Ο / 32)(0.15Do)(499)[Do4 β (0.75Do)4] β Do5 = 4.169 or Do = 1.35 ft. 10. A 26 in. mean diameter cast-iron flywheel is used in a shearing machine. After performing the shearing operation at 160 rpm, the flywheel comes back to its normal speed of 190 rpm. The flywheel provides 1200 lbs.-ft. of energy for shearing a sheet
metal. Calculate the thickness of the rim if the width of the rim is 10 in. and arms and hub of the flywheel constitute 12% of the mass of the rim. Solution:
The thickness of the rim tr = (Vr / Ar)
Where,
Vr = Volume of rim Ar = Area of X-section of rim. (Fig. 1) To find Vr, the weight of the rim Wr should be known, Since the arm and the hub have 0.12Wr weight, β΄ Total weight of the flywheel = Wr + 0.12Wr = 1.12Wr Or, 1.12Wr = [{2g(Ξ΄K.E.)} / (VN2 β VS2)] (2) Where, Ξ΄K.E.= Loss of kinetic energy of the flywheel used in shearing operation = 1200 lbs.-ft. (given) VN = Normal working speed VS = Speed during shearing operation, VN = [(ΟDmN) / (12 Γ 60)] = 0.0044 Γ 26 Γ 190 = 22 ft./sec. VS = [(ΟDmN) / (12 Γ 60)] = 0.0044 Γ 26 Γ 160 = 18 ft./sec. Putting these parameters back into eq.(2) 1.12Wr = [(2 Γ 32 Γ 1200) / {(22)2 β (18)2}] β΄ Wr = 428.6 lbs. Specific weight of cast-iron = 0.26 [lb. / (in.3)] β΄ Vr = [Wr / (0.26)] = [(428.6) / (0.26)] = 1648.5 in.3 Also from Figure 1 Ar = w Γ L = w Γ (ΟDm) = 10 Γ (Ο Γ 26)
(1)
= 260Ο in.2 Putting the values of Vr and Ar in eq.(1) tr = [(1648.5) / (260Ο)] = 2 in. 11. A cast iron flywheel with a mean diameter of 75 inches changes speed from 550 rpm to 430 while it gives up 24000 ft-lb of energy. What is the coefficient of fluctuation.
Solution: Cf = V1 β V2 / Vave = N1 β N2 / Nave = 550 β 430 / (550+430/2) = 0.245
answer
12. A 38 in diameter spoked steel flywheel having a 12 in wide x 10 in deep rim rotates at 200 rpm. How long a cut (in inches) can be stamped in one inch thick aluminum plate if ultimate shearing strength of the aluminum is 40,000 lb/in2. During stamping, the force exerted by the stamo varies from a maximum F lb at the point of contact to zero lb when the stamp emerges from the metal. Neglect the weight of the flywheel and spokes and use 0.28 lb/in3 density for flywheel material. Solution: Solving for the mean diameter, Dm = D β t = 48 β 10 = 38 in Solving for the weight of the flywheel: Wf = Wrim + WHA Wf = Ζ΄(Vrim) = Ζ΄(πDmbt)
but : WHA = 0 where : Ζ΄= o.28 lb/in3
Wf =4011.19 lb Solving for the energy needed by the process: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 Assuming that all the energy of the flywheel is released during the process, we have V2 = 0 4011.19 38 200 ΞKE = 2(32.2) [0 β (π π₯ 12 π₯ 60 )2] = -68493.41 ftlb Solving for the force,
E = Β½ Ftp 68493.41 = Β½ F(1/12) F 1 643 842 lb Solving for the length of the aluminum plate Ss = Sus F/A = 40 000 psi F/Lt = 40 000 1 643 842/L(1) = 40 000 L = 41.10 in answer
13. What would be the weight of a flywheel in kg if the weight of the rim is 3 times the sum of the weight of the hub and arms. Given the outside diameter and inside diameter to be 24 in and 18 in respectively and the rim width is 4.5 in. (assume steel flywheel) Solution: where : Ζ΄= 0.284 lb/in3 for steel
Wf = Ζ΄(Vrim) = Ζ΄(πDmbt) Dm = Β½ (D + d) t = Β½ (D β d) then: Wrim = Ζ΄[π(
π·+π 2
)(π)(
= 0.284[π(
24+18 2
π·βπ 2
)]
)(π)(
24β18 2
)]
= 252.94 lb = 114.71 kgf Solving for the weight of the flywheel, Wf = Wrim + WHA
but : Wrim = 3 WHA or WHA = 1/3 (Wrim)
Wf = Wrim + 1/3 (Wrim) = 4/3 (114.71) = 152.95 kgf
answer
14. Find the rim thickness for a cast iron flywheel width of 200 mm, a mean diameter of 1.2 m, a normal operating speed of 300 rpm, a coefficient fluctuation of 0.05 and which is capable of hanging 3000 N-m of kinetic energy. Assume that the hub and arms represent 10% of therim weight and the specific weight of cast iron is 7200 kg/m 3.
Solution: Solve for the minimum speed: Cf = N1 β N2 / Nave = N1 β N2 / (N1 β N2 /2) 0.05 = 300 β N2 / (300 β N2 /2) N2 = 285 rpm The weigth of the flywheel: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
3000 = Wf / 2(9.81) [( π x 1.2 x 285/60)2 β ( π x 1.2 x 300/60)2] Wf = 1740.47 N = 177.42 kgf Weight of the rim: Wf = Wrim + WHA
but : WHA = 0.10(Wrim)
Wf = Wrim + 0.10(Wrim) Wf = 1.10(Wrim) 177.42 = 1.10(Wrim) Wrim = 161.29 kgf Solving for the rim thickness, Wrim = Ζ΄(V) = Ζ΄(πDmbt)
where : Ζ΄= 7200 kg/m3 for cart iron
161.29 = 7200[π(1.2)(0.2)(t)] = 0.02971m t = 29.71 mm
15. It is found that the shearing machine requires 305 joules ofenergy to shear a specific gauge of sheet metal. The mean diameter of the flywheel is to be 86.2 cm, the normal operating speed is 300 rpm and slow down to 280 rpm during shearing process. The rim width is 40.48 cm and the weigth of ast iron is 8,196.6 kg/m3. Find the thicknes of the rim, assuming that the hub and arm account for 10% of the rim weight concentrated on the mean diameter. Solution:
ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-305 = Wf / 2(9.81) [( π x 0.862 x 280/60)2 β ( π x 0.862 x 300/60)2] Wf = 25.81 kg weight of the flywheel Solving for the weight of the rim: Wf = Wrim + WHA 25.81 = 1.10(Wrim)
where: WHA = 0.10 Wrim Wrim = 23.46 kg
Solving for the thickness of the rim, Wrim = Ζ΄(Vrim) = Ζ΄(πDmbt)
where : = 8196.6 kg/m3
23.46 = 8196.6 [π(0.862)(0.4048)(t)] t = 2.611 mm
16. A flywheel has a mean diameter of 4 ft and is required to handle 2200 ft-lb of kinetic energy. The flywheel has a width of 8 in. Normal operating speed is 300 rpm and the coefficient of fluctuation is to be 0.05. find the weight of the rim assuming that the arms and hub are equivalent is 10% of the specific weight. Soluttion: Solving for the weight of the flywheel: (Note : N2 = 285 rpm) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-2200 = Wf / 2(32.2) [( π x 4 x 285/60)2 β ( π x 4 x 300/60)2] Wf = 368.08 lb Solving for the weight of the rim, Wf = Wrim + WHA Wf = 1.10(Wrim) 368.08 = 1.10(Wrim)
= Wrim + 0.10(Wrim) where : WHA = 0.10(Wrim)
Wrim = 334.62 lb
answer
17. The mass of a flywheel is 175 kg and its radius of gyration is 380 mm. Find the torque required to attain a speed of 500 rpm from the rest in 30 seconds. Solution: Solving for the mass moment of inertia: Im = mk2 = 175(0.380)2 = 25.27 kgm2 Solving for the angular acceleration: Ξ± = Wf β Wo / t =
( 2π π₯
500 )β0 60
30
= 1.745 red/s2 Solving for the torque: T = Im Ξ± = 25.27(1.75) = 44.10 N-m
answer
18. A cast iron flywheel is rotated at a speed of 1200 rpm and having a mean rim radius of 1 foot. If the weight of the rim is 30 lbs. What is the centrifugal force? Use factor C=41. Solution: V = 2 πRN 1200 = 2 π(1) 60 = 125.66 fps Solving for the centrifugal force: Fc = WV2/gR (30)(125.66) Fc = 32.2(1) 2 Fc = 14712 lb
19. A flywheel rim which weights 800 pounds has a mean diameter of 48 inches. The speed is to be maintained between 100 and 120 rpm. Considering that the effect of the arms and hub accounts for 12% of the rim weight, determine the capacity of the flywheel. Express your answer in ft-lb. Solution: V1 = (π)DN
=
(π)(48)(120) 12(60)
= 25.13 fps (π)(48)(100) V2 = 12(60) = 20.94 fps ΞKE = 1.12 W [ V22 β V12]/ 2g = 1.12 (800) [ (25.13)2 β (20.94)2]/ 2(32.2) ΞKE = 2,685.68 ft-lbs
20. The moment of inertia of a flywheel is to be calculated for use in a horizontal pressing machine. The machine is operated by a belt driven pulley B of 42 in. diameter, at 300 rpm and a crankshaft S, requiring a work of 7500 lbs.-ft per cycle is driven by a gear G, which in turn is driven by a pinion P with speed reduction ratio of 7 between P and G.The mechanical efficiency of the crankshaft drive is 78% and the flywheel supplies energy for (1/3)rd of crankshaft revolution.
Solution:
The moment of inertia I of the flywheel is found by the equation, I = [{32.2(ΞU)} / (CrΟ2)] Where, ΞU= Change in energy of the flywheel required for pressing-operation Ο = Mean angular speed Cr = Coefficient of speed regulation, = 0.25 (given).
(1)
Now, Where,
ΞU = ES β EB ES = Work required per crankshaft revolution EB = Work done by the belt during operating part of the cycle.
To find EB: Since the speed of pinion P = NP = 300 rpm Speed reduction ratio = r = 7 β΄ Speed of gear G = NG = (NP / 7) = [(300) / 7] = 42.86 rpm. Therefore the crankshaft S has the same rpm, β΄ NS = 42.86 rpm. Now, the horsepower required by the crankshaft to operate the press is given by H.P. = [(TNS) / (33,000)] ; where; T= Torque, lb.-ft. NS= rpm. = [(7500 Γ 42.86) / (33,000)] = 9.74 h.p. β΅ Mechanical friction Ξ·m = 0.78 β΄ Actual horsepower required = [(9.74) / (0.78)] = 12.5 h.p. Now, the net tension (force) in the belt βP' is found by the equation, H.P. = [(P Γ V) / (33,000)] Where, P = Force, lb. V = Speed, ft./min. The belt speed, V = [(ΟDN) / 12] = [(Ο Γ 42 Γ 300) / 12] = 3298.7 ft./min. β΄ P = 33,000 Γ [(H.P.) / V] = 33,000 Γ [(12.5) / (3298.7)] = 125 lbs. Now the distance, the belt traverses around the pulley during the pressing-operation period of the shaft revolution is, L = [{0.33 Γ 3298.7(ft./min.)} / {42.86(rev./min.)}] = 25.4 ft./rev. β΄ EB = P Γ L = 125 Γ 25.4 = 3175 lbs.-ft.
To find ES: The actual work required by crankshaft during pressing- operation is ES = [(7500) / (0.78)] = 9615.4 lbs.-ft. Putting the values of ES and EB in eq.(2)
(2)
ΞU = 9615.4 β 3175 = 6440.4 lbs.-ft. Now, Ο = [(2ΟN) / 60] = [(2 Γ Ο Γ 300) / 60] N = Flywheel's rpm = 31.4 rads./sec. Therefore from equation (1), I = [(32.2 Γ 6440.4) / {0.25 Γ (31.4)2}] = 841.3 lbs.-ft.2
21. A 2.2 kw, 960 rpm motor powers the cam driven ram of a press through a gearing of 6:1 ratio. The rated capacity of the press is 20 kN and has a stroke of 200 mm. Assuming that the cam driven ram is capable of delivering the rated load at a constant velocity during the last 15% of a constant velocity stroke. Design a suitable flywheel that can maintain a coefficient of Speed fluctuation of 0.02. Assume that the maximum diameter of the flywheel is not to exceed 0.6m.
Solution: Work done by the press= U = 20 x 103 x 0.2 x 0.15 = 600 Nm Energy absorbed= work done= 600 Nm Mean torque on the shaft: 2.2 π₯ 103 = 21.88 ππ 960 2 π₯ π π₯ 60
Energy supplied= work don per cycle = 2π x 21.88 x 6 = 825 ππ Thus the mechanical efficiency of the system is = π=
600 825
Therefore the fluctuation in energy is =
= 0.727 = 72%
πΈπ = πΈπππππ¦ πππ πππ β πΈπππππ¦ π π’ππππππ 600 β 825 π₯ 0.075(21.88 π₯ 6 π₯ π π₯ 0.15) 538.125 ππ πΌ=
πΈπ 538.125 = 2 960 πΆπ (πππ£π ) 0.02(2π π₯ 60 )2 = 2.6622 ππ π2 πΌ=
Assuming;
ππ ππ
π π 2 π₯ (π β ππ 2 )π‘ 2 π π
= 0.8 2.6622 =
π 78500 (0.304 β 0.244 )π‘ π₯ 2 9.86 = 59.805 π‘
β΄
π‘=
2.6622 59.805
= 0.0445
ππ = 45 ππ ππ‘ = ππ‘ =
π 2 3+π¦ 1 + 3π¦ 2 π ( )(ππ 2 + ππ 2 β π ) π 8 3+π¦
78500 2 3 + 0.3 1.9 π ( )(0.242 + 0.32 β 0.242 ) 9.81 8 3.3 ππ‘ = 0.543(2π π₯
960 2 ) 60
= 55667 π/π2 = 0.556 πππ Or if ππ‘ = 150 πππ 150π₯106 = 7961.4π2 (0.4125)(0.0376)(0.090)(0.0331) = 0.548π2 π = 16544 πππ/π ππ 2
πππ =
ππ¦ππππ 16544 = π 32π
= 164.65 22. A flywheel used for energy storage. Find (a) the rotational kinetic energy stored in the flywheel, and (b) the energy storage capacity of the flywheel (W-hr/lbm).
Solution: (a) The rotational kinetic energy of the flywheel is, πΌπ2 (πΎπΈ)πππ‘ = 2ππ‘ The mass moment of inertia of a solid circular cylinder rotating about its center, ππ 2 1 27 ππ 2 1 ππ‘ 2 πΌ= = ( )(100 πππ)( ) β( ) = 63.28 ππ‘ 2 β πππ 2 2 2 12 ππ Then,
(πΎπΈ)πππ‘ =
πππ£ 2π πππ πππ (63.28 ππ‘ 2 β πππ)(40,000 πππ)2 β( πππ£ )2 (60 π )2 πππ β ππ‘ 2 (32.174 ) πππ β π 2
= 1.73π₯107 ππ‘ β πππ
Converting to W-hr, (πΎπΈ)πππ‘ = (1.73π₯107 ππ‘ β πππ)β (
1055.0π½ 778.16ππ‘βπππ
)(
βπ 3600π
)(
πβπ π½
) = 6498 π β βπ
(b) The energy storage capacity of the flywheel is the energy stored per unit mass, πππππ‘ =
(πΎπΈ)πππ‘ 6498π β βπ π β βπ = = 6.498 π 100πππ πππ
23. The punch press shown is driven by a motor which rotates the flywheel in the counter clockwise direction at 20 revolutions per minute. During the punch operation, the clutch engages, disconnecting the motor, allowing the flywheel to run the press through its large inertial properties. The total punch operation takes t1 = 1.5 seconds. After the punch operation, the flywheel rotates through a 1/2 revolution with a constant angular acceleration and then the motor engages accelerating the flywheel at 0.1 rad/s2 until it reaches its original angular speed of 20 rpm. The angular acceleration of the punch cycle is shown in the figure provided. Find the total punch cycle time t3.
Solution: Angular speed For any calculations involving angular speed or acceleration, it is always a good idea to convert their units to radians per second or radians per second squared, respectively. π0 = 20
πππ£ 2ππππ πππ 2π πππ = πππ πππ£ 60π 3 π
Using the information provided regarding the angular acceleration, we can determine the angular speed at the end of the punch operation. π =β0 (π‘ β π‘0 ) + π0 For this portion of the cycle, βπ =β0β1, π‘ β π‘π = π‘1, πππ ππ = π0
π1 = β0.2(1.5) +
2π πππ = 1.794 3 π
Angular displacement After the punch operation the flywheel rotates through 0.5 revolutions. We will use this information plus that fact that it rotates at a constant angular acceleration to determine the time and angular speed at end of this stage. 1
π = βπ (π‘ β π‘π )2 + ππ (π‘ β π‘π ) + ππ
π =βπ (π‘ β π‘π ) + ππ
2
For this portion of the cycle, βπ =β1β2, π‘ β π‘π = π‘2 β π‘1, πππ ππ = π1 1
1
π1β2 = 2 β1β2 (π‘2 β π‘1 )2 + π1 (π‘2 β π‘1 ) 0 = π‘22 β 38.88π‘2 + 118.8
π = 2 (β0.1)(π‘2 β 1.5)2 + 1.794(π‘2 β 1.5) π‘2 =
38.88Β±β38.882 β4(118.8) 2
π2 = π1β2 (π‘2 β π‘1 ) + π1 = β0.1(3.35 β 1.5) + 1.794 = 1.61
= 3.35π
πππ π
Punch cycle time To determine the punch cycle time t3, we can again apply the constant acceleration equation. π =βπ (π‘ β π‘π ) + π π For this portion of the cycle, βπ =β2β3, π‘ β π‘π = π‘2 β π‘3, ππ = π2, πππ π = π3 π3 =β2β3 (π‘3 β π‘2 ) + π2
2π 3
= 0.1(π‘3 β 3.35) + 1.61
π‘3 = 8.19π
24. A flywheel rim which weights 800 pounds has a mean diameter of 48 inches. The speed is to be maintained between 100 and 120 rpm. Considering that the effect of the arms and hub accounts for 12% of the rim weight, determine the capacity of the flywheel. Express your answer in ft-lb. Solution: V1 = (π)DN =
(π)(48)(120) 12(60)
= 25.13 fps
V2 =
(π)(48)(100) 12(60)
= 20.94 fps ΞKE = 1.12 W [ V22 β V12]/ 2g = 1.12 (800) [ (25.13)2 β (20.94)2]/ 2(32.2) ΞKE = 2,685.68 ft-lbs
25. A shearing machine requires 2000 ft-lbs of kinetic energy to operate. The mean diameter of its flywheel is 36 in. The rated speed is 66 fps at the mean diameter. The coefficient of fluctuation is 0.20. Compute the weight of the rim is needed if the effects of the flywheel arms and hub are to be neglected. Solution: ΞE = (Wf)(Cf)(V)2/g Wf = (ΞE)(g)/(cf)(v)2 (2000)(32.2) = 0.20(66) 2 = 73.92 lbs
answer
26. A cast iron flywheel with a mean diameter of 36 inches changes speed from 300 rpm to 280 while it gives up 8000 ft-lb of energy. What is the coefficient of fluctuation. Solution: Cf = V1 β V2 / Vave = N1 β N2 / Nave = 300 β 280 / (300+280/2) = 0.069 answer
27. The mass of the flywheel of a shearing machine is 3 tons and its radius of gyration is 0.55 m. At the beginning of its cutting stroke the speed of the flywheel is 110 rev/min. If 12 kJ of work is done in shearing a plate in 3 seconds, calculate the rotation speed of the flywheel at the end cutting stroke in rev/min. Solution: ΞKE = W Β½ [3000(0.552)(11.522 β W22)] = 12000 W2 = 10.31 rad/s = 98.45 rpm answer
28. A flywheel weighing 457 kg has a radius of 375 mm. Compute how much energy, in N-m, does the flywheel loss from 3 rps to 2.8 rps? Solution: V1 = 2π(0.375)(3) = 7.069 m/s V2 = 2π(0.375)(2.8) = 6.597 m/s ΞKE = Β½ [m(V1 β V2)] = 457[(7.069)2 β (6.597)2]/2 = 1473.91 Nm answer
29. A flywheel if manufactured with R2 = 0.2m and R1 = 0.15m, W = 30mm. If it is manufactured from aluminum, calculate its inertia. [density for aluminum is 2720 kg m3]. Solution: π π₯ 2720 π₯ 0.03 If = (0.24 β 0.154) 2 = 0.14 kg.m2 answer
30. Repeat example 17, if the flywheel was: (a) Completely solid. What do you notice? (b) Made of carbon-steel. Solution: (a) Using the same formula, we can substitute a value of zero for R1: π π₯ 2720 π₯ 0.03
If = (0.24 β 04) 2 = 0.205 kg.m2 (b) The density of carbon steel is 7850 kg/m3. Substituting in the formula gives: π π₯ 7850 π₯ 0.03
If = (0.24 β 0.154) 2 = 0.404 kg.m2
31. Calculate the moment of inertia, mass, for the circular flywheel of which the section. The material of the flywheel is steel, the density of which is 7800 kg.m 3.
Solution: We can divide the area into three parts: I1, I2, and I3. I1 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.05 π₯ π
= x (0.1754 β 0.154) 2 = 0.264 kg.m2 I2 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.03 π₯ π
= x (0.154 β 0.1354) 2 = 0.064 kg.m2 I3 =
ππ₯π€π₯π
x (R24 β 0)
2 7800 π₯ 0.015 π₯ π
= x (0.1354 β 0) 2 = 0.061 kg.m2 The total moment of inertia is ITotal = 0.264 + 0.064 + 0.061 = 0.389 kg.m2 answer
32. A cast iron flywheel with a mean diameter of 36 inches changes speed from 300 rpm to 280 while it gives up 8000 ft-lb of energy. What is the coefficient of fluctuation. Solution: Cf = V1 β V2 / Vave = N1 β N2 / Nave = 300 β 280 / (300+280/2) = 0.0513 answer
33. A mechanical press is used to punch 8 holes per minute is 35 mm diameter and the plates has an ultimate strength in shear of 520 Mpa. The normal operating speed 200 rpm. And it slows down to 180 rpm during the process of punching. The flywheel has a mean diameter of one meter and the rim width is 3 times the thickness. Assume that the hub and arm account for 5% of the rim weight concentrated at the mean diameter and the density of cast iron is 7200 kg per cubic meter. Find the power in KW required to drive the press. Solution: E = Β½ Ftp Then:
but : Ss = Sus F/A = 520 Mpa F/(πDtp) = 520 F = 520[πDtp]
E = Β½(520)(π)(D)(tp)2 = Β½(520)(π)(35)(25)2 = 17 867 808 N-mm = 17 867.808 Nm 1 πππ 60 π
t = 8 βππππ [πππ] = 7.5 s/holes 17 868
P = E/t =
7.5
= 2382 W power required
answer
34. Find the weight of the flywheel needed by a machine to punch 30.5 mm holes in 25.87 mm thick steel plate. The machine is to make 40 strokes per minute and a hole be punched every stroke, the hole is to be formed during 30 degrees rotation of the puncher crankshaft. A gear train with a ratio of 12 to 1 is to correct the flywheel shaft to the crankshaft. Let mean diameter of a flywheel rim to the 91.44 cm the minimum flywheel speed is to be 90% of the maximum and assume mechanical efficiency of the machinr to be 80%. Assume an ultimate stress of 59000 psi. Solution: Solving for the energy needed by the process: E = Β½ Ftp Where: F/A = 59000 psi 0.101325 F/(πDtp) = 59000[ 14.7 ] F = 406.68(πDtp) Then: E = Β½(406.68)(π)(D)(tp)2 E = Β½(406.68)(π)(30.5)(25.87)2 E = 1303.963 Nm Solving for the maximum speed of the flywheel: N1 = 40
π π‘πππππ 1 βπππ πππ
[
π π‘ππππ
360
][
30
][
1 πππ£ βπππ
] = 480 rpm
Solving for the minimum speed of the flywheel: N2 = 0.90(N1) = 0.90(480) = 432 rpm Solving for the weight of the flywheel: (based on energy equation) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
ΞKE = -1303.963 N-m Since the flywheel will supply / release this energu, then: -1303.963 = Wf / 2(9.81) [( π x 0.9144 x 432/60)2 β ( π x 0.9144 x 480 /60)2] Wf = 254.95 N answer
35. A shearing machine requires 5000 ft-lbs of kinetic energy to operate. The mean diameter of its flywheel is 38 in. The rated speed is 58 fps at the mean diameter. The coefficient of fluctuation is 0.40. Compute the weight of the rim is needed if the effects of the flywheel arms and hub are to be neglected.
Solution: ΞE = (Wf)(Cf)(V)2/g Wf = (ΞE)(g)/(cf)(v)2 =
(5000)(32.2) 0.30(56)2
= 171.13 lbs
answer
36. It is found that the shearing machine requires 305 joules ofenergy to shear a specific gauge of sheet metal. The mean diameter of the flywheel is to be 86.2 cm, the normal operating speed is 300 rpm and slow down to 280 rpm during shearing process. The rim width is 40.48 cm and the weigth of ast iron is 8,196.6 kg/m3. Find the thicknes of the rim, assuming that the hub and arm account for 10% of the rim weight concentrated on the mean diameter. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-305 = Wf / 2(9.81) [( π x 0.862 x 280/60)2 β ( π x 0.862 x 300/60)2] Wf = 25.81 kg weight of the flywheel Solving for the weight of the rim:
Wf = Wrim + WHA 25.81 = 1.10(Wrim)
where: WHA = 0.10 Wrim Wrim = 23.46 kg
Solving for the thickness of the rim, Wrim = Ζ΄(Vrim) = Ζ΄(πDmbt)
where : = 8196.6 kg/m3
23.46 = 8196.6 [π(0.862)(0.4048)(t)] t = 2.611 mm
answer
37. A sheet metal working company purchase a shearing machine from a surplus dealer without a flywheel. It is calculated that the machine will use 1380 joules of energy to shear a 1.2 mm thick sheet metal. The flywheel to be used will have a mean diameter of 11.44 cm with a width of 25.4 cm. The normal operating speed is 80 rpm and slows down to 60 rpm during the shearing process. Assuming that the arms and the hub will account for 12% of the rim weight concentrated at the mean diameter and that the material density is 0.26 lb/cu. In. Compute for the weight of the flywheel. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 V2 = πDmn2 1380 = Wf / 2(9.81) [( π x 0.1144 x 60/60)2 β ( π x 0.1144 x 80/60)2] Wf = 2695 N
answer
38. A 38 in diameter spoked steel flywheel having a 12 in wide x 10 in deep rim rotates at 200 rpm. How long a cut (in inches) can be stamped in one inch thick aluminum plate if ultimate shearing strength of the aluminum is 20,000 lb/in2. During stamping, the force exerted by the stamo varies from a maximum F lb at the point of contact to zero lb when the stamp emerges from the metal. Neglect the weight of the flywheel and spokes and use 0.28 lb/in3 density for flywheel material. Solution: Solving for the mean diameter, Dm = D β t = 48 β 10 = 38 in Solving for the weight of the flywheel:
Wf = Wrim + WHA Wf = Ζ΄(Vrim) = Ζ΄(πDmbt)
but : WHA = 0 where : Ζ΄= o.28 lb/in3
Wf =4011.19 lb Solving for the energy needed by the process: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 Assuming that all the energy of the flywheel is released during the process, we have V2 = 0 4011.19 38 200 ΞKE = 2(32.2) [0 β (π π₯ 12 π₯ 60 )2] = -68493.41 ftlb Solving for the force, E = Β½ Ftp 68493.41 = Β½ F(1/12) F 1 643 842 lb Solving for the length of the aluminum plate Ss = Sus F/A = 20 000 psi F/Lt = 20 000 1 643 842/L(1) = 20 000 L = 82.192 in answer
39. The flywheel of a machine having weight of 4500 N and radius of gyration of 2 m has cyclic fluctuation of speed from 125 rpm to 120 rpm. Assuming g = 10m/s 2, the maximum fluctuation of energy is? Solution: Mass of flywheel = weight of flywheel/Acceleration due to gravity = 4500/10kg Moment of Inertia = mk2 = 1800 kgm2 Ο1 = 2Ο/60 x 125rad/sec Ο2 = 2Ο/60 x 120rad/sec Emax = 1/2 I(?)2 = 12087.2 N-m = 12100 Nm
40. A circular solid disc of uniform thickness 20 mm, radius 200 mm and mass 20 kg, is used as a flywheel. If it rotates at 600 rpm, the kinetic energy of the flywheel, in Joules is?
Solution: For flywheel K.E = 1/2IΟ2 Ο = 2ΟN/60 = 62.83 rad/s I (for solid circular disk) = 1/2mR2 = 0.4 kg m2 Hence, K.E = 790 Joules. 41. The speed of an engine varies from 210 rad/s to 190 rad/s. During the cycle the change in kinetic energy is found to be 400 Nm. The inertia of the flywheel in kg/m2 is? Solution: Given Ο1 = 210 rad/ sec, Ο2 = 190 rad/ sec, ΞE= 400 Nm As the speed of flywheel changes from Ο1 to Ο2, the maximum fluctuation of energy, ΞE = 1/2I [(Ο1)2 (Ο2)2] I = 0.10 kgm2 42. For a certain engine having an average speed of 1200 rpm, a flywheel approximated as a solid disc, is required for keeping the fluctuation of speed within 2% about the average speed. The fluctuation of kinetic energy per cycle is found to be 2 kJ. What is the least possible mass of the flywheel if its diameter is not to exceed 1 m? Solution: Given N = 1200 rpm, ΞE = 2kJ = 2000 J, D = 1m, Cs = 0.02 Mean angular speed of engine, Ο = 2ΟN/60 = 125.66 rad/ sec Fluctuation of energy of the flywheel is given by, ΞE = IΟ2Cs = 1/2mR2Ο2Cs For solid disc I = mR2/2 m = 51 kg 43. Table 16β6 lists values of the torque used to plot Fig. 16β28. The nominal speed of the engine is to be 250 rad/s. (a) Integrate the torque-displacement function for one cycle and find the energy that can be delivered to a load during the cycle. (b) Determine the mean torque Tm (see Fig. 16β28). (c) The greatest energy fluctuation is approximately between ΞΈ = 15β¦ and ΞΈ = 150β¦ on the torque diagram; see Fig. 16β28 and note that To = βTm. Using a coefficient of speed fluctuation Cs = 0.1, find a suitable value for the flywheel inertia. (d) Find Ο2 and Ο1. Solution: (a) Using n = 48 intervals of βΞΈ = 4Ο/48, numerical integration of the data of Table 16β6 yields E = 3368 in Β· lbf. This is the energy that can be delivered to the load.
(a) Tm =
3368 4π
= 268 lbf Β· in
(b) The largest positive loop on the torque-displacement diagram occurs between ΞΈ = 0β¦ and ΞΈ = 180β¦. We select this loop as yielding the largest speed change. Subtracting 268 lbf Β· in from the values in Table 16β6 for this loop gives, respectively, β268, 2532, 1822, 2162, 1892, 1572, 1322, 942, 798, 535, 264, β84, and β268 lbf Β· in. Numerically integrating T β Tm with respect to ΞΈ yields E2 β E1 = 3531 lbf Β· in. We now solve Eq. (16β64) for I. This gives
I=
E2 β E1 CsΟ2
3531
= 0.1(250)2 = 0.565 lbf Β· s 2 in
(c) Equations (16β62) and (16β63) can be solved simultaneously for Ο2 and Ο1. Substituting appropriate values in these two equations yields Ο2 =
Ο 2
(2 + Cs) =
250 2
(2 + 0.1) = 262.5 rad/s
Ο1 = 2Ο β Ο2 = 2(250) β 262.5 = 237.5 rad/s These two speeds occur at ΞΈ = 180β¦ and ΞΈ = 0β¦, respectively.
44. What would be the weight of a flywheel in kg if the weight of the rim is 6 times the sum of the weight of the hub and arms. Given the outside diameter and inside diameter to be 24 in and 18 in respectively and the rim width is 4.5 in. (assume steel flywheel) Solution:
where : Ζ΄= 0.284 lb/in3 for steel
Wf = Ζ΄(Vrim) = Ζ΄(πDmbt) Dm = Β½ (D + d) t = Β½ (D β d) then: Wrim = Ζ΄[π(
π·+π 2
)(π)(
= 0.284[π(
24+18 2
π·βπ 2
)]
)(π)(
24β18 2
)]
= 252.94 lb = 114.71 kgf Solving for the weight of the flywheel, Wf = Wrim + WHA
but : Wrim = 6 WHA or WHA = 1/6 (Wrim)
Wf = Wrim + 1/6 (Wrim) = 7/6 (114.71) = 133.82 kgf
answer
45. Find the rim thickness for a cast iron flywheel width of 200 mm, a mean diameter of 1.2 m, a normal operating speed of 600 rpm, a coefficient fluctuation of 0.05 and which is capable of hanging 6000 N-m of kinetic energy. Assume that the hub and arms represent 10% of therim weight and the specific weight of cast iron is 5200 kg/m3. Solution: Solve for the minimum speed: Cf = N1 β N2 / Nave = N1 β N2 / (N1 β N2 /2) 0.05 = 300 β N2 / (300 β N2 /2) N2 = 285 rpm The weigth of the flywheel: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
6000 = Wf / 2(9.81) [( π x 1.2 x 285/60)2 β ( π x 1.2 x 600/60)2] Wf = -106.96 kgf
Weight of the rim: Wf = Wrim + WHA
but : WHA = 0.10(Wrim)
Wf = Wrim + 0.10(Wrim) Wf = 1.10(Wrim) 106.96 = 1.10(Wrim) Wrim = -97.23 kgf Solving for the rim thickness, where : Ζ΄= 5200 kg/m3 for cart iron
Wrim = Ζ΄(V) = Ζ΄(πDmbt)
-97.23 = 5200[π(1.2)(0.2)(t)] = 0.02479m t = 24.79 mm
46. A flywheel has a mean diameter of 8 ft and is required to handle 1100 ft-lb of kinetic energy. The flywheel has a width of 8 in. Normal operating speed is 500 rpm and the coefficient of fluctuation is to be 0.05. find the weight of the rim assuming that the arms and hub are equivalent is 10% of the specific weight. Soluttion: Solving for the weight of the flywheel: (Note : N2 = 285 rpm) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-1100 = Wf / 2(32.2) [( π x 8 x 285/60)2 β ( π x 8 x 5 00/60)2] Wf = 2.39 lb Solving for the weight of the rim, Wf = Wrim + WHA
= Wrim + 0.10(Wrim) where : WHA = 0.10(Wrim)
Wf = 1.10(Wrim) 2.39 = 1.10(Wrim) Wrim = 2.17 lb
answer
47. The mass of a flywheel is 115 kg and its radius of gyration is 280 mm. Find the torque required to attain a speed of 450 rpm from the rest in 25 seconds. Solution: Solving for the mass moment of inertia: Im = mk2 = 115(0.280)2 = 90.16 kgm2 Solving for the angular acceleration: Ξ± = Wf β Wo / t =
( 2π π₯
450 )β0 60
25
= 1.885 red/s2 Solving for the torque: T = Im Ξ± = 90.16(1.885) = 169.95 N-m answer
48. A cast iron flywheel is rotated at a speed of 2400 rpm and having a mean rim radius of 3 foot. If the weight of the rim is 55 lbs. What is the centrifugal force? Use factor C=41. Solution: V = 2 πRN 1200 = 2 π(3) 60 = 753.98 fps Solving for the centrifugal force: Fc = WV2/gR (55)(753.98) Fc = 32.2(3) 2 Fc = 323672 lb
49. A flywheel rim which weights 450 pounds has a mean diameter of 14 inches. The speed is to be maintained between 150 and 170 rpm. Considering that the effect of the arms and hub accounts for 12% of the rim weight, determine the capacity of the flywheel. Express your answer in ft-lb. Solution: V1 = (π)DN (π)(14)(170) = 12(60)
= 10.38 fps (π)(14)(150) V2 = 12(60) = 9.163 fps ΞKE = 1.12 W [ V22 β V12]/ 2g = 1.12 (450) [ (10.38)2 β (9.163)2]/ 2(32.2) ΞKE = 186.134 ft-lbs
50. A shearing machine requires 4000 ft-lbs of kinetic energy to operate. The mean diameter of its flywheel is 36 in. The rated speed is 56 fps at the mean diameter. The coefficient of fluctuation is 0.30. Compute the weight of the rim is needed if the effects of the flywheel arms and hub are to be neglected. Solution: ΞE = (Wf)(Cf)(V)2/g Wf = (ΞE)(g)/(cf)(v)2 (4000)(32.2) = 0.30(56) 2 = 136.90 lbs
answer
51. A cast iron flywheel with a mean diameter of 72 inches changes speed from 800 rpm to 680 while it gives up 16000 ft-lb of energy. What is the coefficient of fluctuation. Solution: Cf = V1 β V2 / Vave = N1 β N2 / Nave = 800 β 680 / (800+680/2) = 0.1621 answer
52. The mass of the flywheel of a shearing machine is 3 tons and its radius of gyration is 0.55 m. At the beginning of its cutting stroke the speed of the flywheel is 110 rev/min. If 12 kJ of work is done in shearing a plate in 3 seconds, calculate the rotation speed of the flywheel at the end cutting stroke in rev/min. Solution: ΞKE = W Β½ [3000(0.552)(11.522 β W22)] = 12000 W2 = 10.31 rad/s = 98.45 rpm answer
53. A flywheel weighing 642 kg has a radius of 275 mm. Compute how much energy, in N-m, does the flywheel loss from 5 rps to 4.8 rps? Solution: V1 = 2π(0.275)(5) = 8.64 m/s V2 = 2π(0.275)(4.8) = 8.29 m/s ΞKE = Β½ [m(V1 β V2)] = 642[(8.64)2 β (8.29)2]/2 = 1902.09 Nm answer
54. A flywheel if manufactured with R2 = 0.45 m and R1 = 0.17m, W = 30mm. If it is manufactured from aluminum, calculate its inertia. [density for aluminum is 3420 kg m3]. Solution: π π₯ 3420 π₯ 0.03 If = (0.454 β 0.174) 2 = 6.47 kg.m2 answer
55. Repeat example 17, if the flywheel was: (c) Completely solid. What do you notice? (d) Made of carbon-steel. Solution: (c) Using the same formula, we can substitute a value of zero for R1: π π₯ 3420 π₯ 0.03
If = (0.454 β 04) 2 = 6.609 kg.m2 (d) The density of carbon steel is 7850 kg/m3. Substituting in the formula gives: π π₯ 7850 π₯ 0.03
If = (0.454 β 0.174) 2 = -30896284 kg.m2
56. Calculate the moment of inertia, mass, for the circular flywheel of which the section. The material of the flywheel is steel, the density of which is 7800 kg.m3. Solution:
We can divide the area into three parts: I1, I2, and I3. I1 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.05 π₯ π
= x (0.1754 β 0.154) 2 = 0.264 kg.m2 I2 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.03 π₯ π
= x (0.154 β 0.1354) 2 = 0.064 kg.m2 I3 =
ππ₯π€π₯π
x (R24 β 0)
2 7800 π₯ 0.015 π₯ π
= x (0.1354 β 0) 2 = 0.061 kg.m2 The total moment of inertia is ITotal = 0.264 + 0.064 + 0.061 = 0.389 kg.m2 answer
57. A cast iron flywheel with a mean diameter of 74 inches changes speed from 600 rpm to 480 while it gives up 24000 ft-lb of energy. What is the coefficient of fluctuation. Solution: Cf = V1 β V2 / Vave = N1 β N2 / Nave = 600 β 480 / (600+480/2) = 0.222 answer 58. A mechanical press is used to punch 12 holes per minute is 25 mm diameter and the plates has an ultimate strength in shear of 537 Mpa. The normal operating speeed 200 rpm. And it slows down to 180 rpm during the process of punching. The flywheel has a mean diameter of one meter and the rim width is 3 times the thickness. Assume that the hub and arm account for 5% of the rim weight concentrated at the mean diameter and the density of cast iron is 7200 kg per cubic meter. Find the power in KW required to drive the press. Solution: E = Β½ Ftp but : Ss = Sus F/A = 537 Mpa F/(πDtp) = 537 F = 537[πDtp] Then: E = Β½(537)(π)(D)(tp)2 = Β½(537)(π)(25)(25)2
= 13 179 962 N-mm = 13 179.962 Nm 1 πππ
60 π
t = 12 βππππ [πππ] = 5 s/holes P = E/t =
13 180 5
= 2636 W power required
answer
59. A typical 26-inch bicycle wheel rim has a diameter of 559 mm (22.0") and an outside tire diameter of about 26.2" (665 mm). For our calculation we approximate the radius - r - of the wheel to r = ((665 mm) + (559 mm) / 2) / 2 = 306 mm = 0.306 m The weight of the wheel with the tire is 2.3 kg and the inertial constant is k = 1. The Moment of Inertia for the wheel can be calculated I = (1) (2.3 kg) (0.306 m)2 = 0.22 kg m2 The speed of the bicycle is 25 km/h (6.94 m/s). The wheel circular velocity (rps, revolutions/s) - nrps - can be calculated as nrps = (6.94 m/s) / (2 Ο (0.665 m) / 2) = 3.32 revolutions/s The angular velocity of the wheel can be calculated as Ο = (3.32 revolutions/s) (2 Ο rad/revolution) = 20.9 rad/s The kinetic energy of the rotating bicycle wheel can then be calculated to Ef = 0.5 (0.22 kg m2) (20.9 rad/s)2 = 47.9 J
60. Find the weight of the flywheel needed by a machine to punch 20.5 mm holes in 15.87 mm thick steel plate. The machine is to make 30 strokes per minute and a hole be
punched every stroke, the hole is to be formed during 30 degrees rotation of the puncher crankshaft. A gear train with a ratio of 12 to 1 is to correct the flywheel shaft to the crankshaft. Let mean diameter of a flywheel rim to the 91.44 cm the minimum flywheel speed is to be 90% of the maximum and assume mechanical efficiency of the machinr to be 80%. Assume an ultimate stress of 49000 psi. Solution: Solving for the energy needed by the process: E = Β½ Ftp Where: F/A = 49000 psi 0.101325 F/(πDtp) = 49000[ 14.7 ] F = 337.75(πDtp) Then: E = Β½(337.75)(π)(D)(tp)2 E = Β½(337.75)(π)(20.5)(15.87)2 E = 2739.195 Nm Solving for the maximum speed of the flywheel: N1 = 30
π π‘πππππ 1 βπππ πππ
[
π π‘ππππ
360
][
30
][
1 πππ£ βπππ
] = 360 rpm
Solving for the minimum speed of the flywheel: N2 = 0.90(N1) = 0.90(360) = 324 rpm Solving for the weight of the flywheel: (based on energy equation) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
ΞKE = -2739.195 N-m Since the flywheel will supply / release this energu, then: -2739.195 = Wf / 2(9.81) [( π x 0.9144 x 324/60)2 β ( π x 0.9144 x 360/60)2] Wf = 952.11 N or 97.05 kgf
answer
61. What would be the weight of a flywheel in kg if the weight of the rim is 3 times the sum of the weight of the hub and arms. Given the outside diameter and inside diameter to be 24 in and 18 in respectively and the rim width is 4.5 in. (assume steel flywheel)
Solution: where : Ζ΄= 0.284 lb/in3 for steel
Wf = Ζ΄(Vrim) = Ζ΄(πDmbt) Dm = Β½ (D + d) t = Β½ (D β d) then: Wrim = Ζ΄[π(
π·+π 2
)(π)(
= 0.284[π(
24+18 2
π·βπ 2
)]
)(π)(
24β18 2
)]
= 252.94 lb = 114.71 kgf Solving for the weight of the flywheel, Wf = Wrim + WHA
but : Wrim = 3 WHA or WHA = 1/3 (Wrim)
Wf = Wrim + 1/3 (Wrim) = 4/3 (114.71) = 152.95 kgf
answer
62. Find the rim thickness for a cast iron flywheel width of 500 mm, a mean diameter of 4.2 m, a normal operating speed of 300 rpm, a coefficient fluctuation of 0.05 and which is capable of hanging 3000 N-m of kinetic energy. Assume that the hub and arms represent 10% of therim weight and the specific weight of cast iron is 3800 kg/m3. Solution: Solve for the minimum speed: Cf = N1 β N2 / Nave = N1 β N2 / (N1 β N2 /2) 0.05 = 300 β N2 / (300 β N2 /2) N2 = 285 rpm The weigth of the flywheel: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
3000 = Wf / 2(9.81) [( π x 1.2 x 285/60)2 β ( π x 1.2 x 300/60)2]
Wf = 1740.47 N = 177.42 kgf Weight of the rim: Wf = Wrim + WHA
but : WHA = 0.10(Wrim)
Wf = Wrim + 0.10(Wrim) Wf = 1.10(Wrim) 177.42 = 1.10(Wrim) Wrim = 161.29 kgf Solving for the rim thickness, where : Ζ΄= 3800 kg/m3 for cart iron
Wrim = Ζ΄(V) = Ζ΄(πDmbt)
161.29 = 7200[π(4.2)(0.5)(t)] = 0.006434m t = 6.434 mm
63. It is found that the shearing machine requires 205 joules ofenergy to shear a specific gauge of sheet metal. The mean diameter of the flywheel is to be 76.2 cm, the normal operating speed is 200 rpm and slow down to 180 rpm during shearing process. The rim width is 30.48 cm and the weigth of ast iron is 7,196.6 kg/m3. Find the thicknes of the rim, assuming that the hub and arm account for 10% of the rim weight concentrated on the mean diameter. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-205 = Wf / 2(9.81) [( π x 0.762 x 180/60)2 β ( π x 0.762 x 180/60)2] Wf = 33.89 kg weight of the flywheel Solving for the weight of the rim: Wf = Wrim + WHA 33.89 = 1.10(Wrim)
where: WHA = 0.10 Wrim Wrim = 30.81 kg
Solving for the thickness of the rim, where : = 7196.6 kg/m3
Wrim = Ζ΄(Vrim) = Ζ΄(πDmbt) 30.81 = 7196.6 [π(0.762)(0.3048)(t)] t = 5.867 mm
answer
64. A flywheel has a mean diameter of 4 ft and is required to handle 2200 ft-lb of kinetic energy. The flywheel has a width of 8 in. Normal operating speed is 300 rpm and the coefficient of fluctuation is to be 0.05. find the weight of the rim assuming that the arms and hub are equivalent is 10% of the specific weight. Soluttion: Solving for the weight of the flywheel: (Note : N2 = 285 rpm) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-2200 = Wf / 2(32.2) [( π x 4 x 285/60)2 β ( π x 4 x 300/60)2] Wf = 368.08 lb Solving for the weight of the rim, Wf = Wrim + WHA
= Wrim + 0.10(Wrim) where : WHA = 0.10(Wrim)
Wf = 1.10(Wrim) 368.08 = 1.10(Wrim) Wrim = 334.62 lb
answer
65. It is found that the shearing machine requires 205 joules ofenergy to shear a specific gauge of sheet metal. The mean diameter of the flywheel is to be 76.2 cm, the normal operating speed is 200 rpm and slow down to 180 rpm during shearing process. The rim width is 30.48 cm and the weigth of ast iron is 7,196.6 kg/m3. Find the thicknes of the rim, assuming that the hub and arm account for 10% of the rim weight concentrated on the mean diameter. Solution: ΞKE = Wf / 2G [ V22 β V12]
Where: V1 = πDmn1
V2 = πDmn2
-205 = Wf / 2(9.81) [( π x 0.762 x 180/60)2 β ( π x 0.762 x 180/60)2] Wf = 33.89 kg weight of the flywheel Solving for the weight of the rim: Wf = Wrim + WHA 33.89 = 1.10(Wrim)
where: WHA = 0.10 Wrim Wrim = 30.81 kg
Solving for the thickness of the rim, Wrim = Ζ΄(Vrim) = Ζ΄(πDmbt)
where : = 7196.6 kg/m3
30.81 = 7196.6 [π(0.762)(0.3048)(t)] t = 5.867 mm
answer
66. A sheet metal working company purchase a shearing machine from a surplus dealer without a flywheel. It is calculated that the machine will use 2380 joules of energy to shear a 1.2 mm thick sheet metal. The flywheel to be used will have a mean diameter of 91.44 cm with a width of 25.4 cm. The normal operating speed is 180 rpm and slows down to 160 rpm during the shearing process. Assuming that the arms and the hub will account for 12% of the rim weight concentrated at the mean diameter and that the material density is 0.26 lb/cu. In. Compute for the weight of the flywheel. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 V2 = πDmn2 2380 = Wf / 2(9.81) [( π x 0.9144 x 160/60)2 β ( π x 0.9144 x 160/60)2] Wf = 2996 N = 305 kgf answer
67. A 38 in diameter spoked steel flywheel having a 12 in wide x 10 in deep rim rotates at 200 rpm. How long a cut (in inches) can be stamped in one inch thick aluminum plate if
ultimate shearing strength of the aluminum is 68,000 lb/in2. During stamping, the force exerted by the stamo varies from a maximum F lb at the point of contact to zero lb when the stamp emerges from the metal. Neglect the weight of the flywheel and spokes and use 0.28 lb/in3 density for flywheel material. Solution: Solving for the mean diameter, Dm = D β t = 48 β 10 = 38 in Solving for the weight of the flywheel: Wf = Wrim + WHA
but : WHA = 0
Wf = Ζ΄(Vrim) = Ζ΄(πDmbt)
where : Ζ΄= 0.28 lb/in3
Wf =4011.19 lb Solving for the energy needed by the process: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 Assuming that all the energy of the flywheel is released during the process, we have V2 = 0 4011.19 38 200 ΞKE = 2(32.2) [0 β (π π₯ 12 π₯ 60 )2] = -68493.41 ftlb Solving for the force, E = Β½ Ftp 68493.41 = Β½ F(1/12) F 1 643 842 lb Solving for the length of the aluminum plate Ss = Sus F/A = 68 000 psi F/Lt = 68 000 1 643 842/L(1) = 68 000 L = 24.17 in answer 68. A flywheel weighing 639 kg has a radius of 375 mm. Compute how much energy, in N-m, does the flywheel loss from 3 rps to 2.8 rps? Solution: V1 = 2π(0.375)(3) = 7.069 m/s V2 = 2π(0.375)(2.8) = 6.597 m/s ΞKE = Β½ [m(V1 β V2)] = 639[(7.069)2 β (6.597)2]/2 = 2060.89 Nm answer
69. A flywheel if manufactured with R2 = 0.135m and R1 = 0.148m, W = 30mm. If it is manufactured from aluminum, calculate its inertia. [density for aluminum is 3420 kg m3]. Solution: π π₯ 3420 π₯ 0.03 If = (0.1354 β 0.1484) 2 = -0.0238 kg.m2 answer
70. Repeat example 17, if the flywheel was: (a)Completely solid. What do you notice? (b)Made of carbon-steel. Solution: (a)Using the same formula, we can substitute a value of zero for R1: π π₯ 3420 π₯ 0.03
If = (0.1354 β 04) 2 = 0.535 kg.m2 (b)The density of carbon steel is 7850 kg/m3. Substituting in the formula gives: π π₯ 7850 π₯ 0.03
If = (0.1354 β 0.1484) 2 = -0.0546 kg.m2
71. Calculate the moment of inertia, mass, for the circular flywheel of which the section. The material of the flywheel is steel, the density of which is 3500 kg.m 3. Solution: We can divide the area into three parts: I1, I2, and I3. I1 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.05 π₯ π
= x (0.1754 β 0.154) 2 = 0.119 kg.m2
I2 =
ππ₯π€π₯π
x (R24 β R14)
2 3500 π₯ 0.03 π₯ π
= x (0.154 β 0.1354) 2 = 0.0479 kg.m2 I3 =
ππ₯π€π₯π
x (R24 β 0)
2 3500 π₯ 0.015 π₯ π
= x (0.1354 β 0) 2 = 0.0913 kg.m2 The total moment of inertia is ITotal = 0.119 + 0.0479 + 0.0913 = 0.2582 kg.m2 answer
72. Flywheel turns 450 rev/min (RPM). Determine the magnitude of the normal acceleration of the flywheel point which are at a distance of 10 cm from the rotation axis. Solution: π=
10 450 2 (2π ) = 222.07 πβπ 2 100 60
73. The torque (in N-m) exerted on the crank shaft of a two stroke engine can be described as T = 10000 + 1000 sin 2ΞΈ β 1200 cos 2ΞΈ, where ΞΈ is the crank angle as measured from inner dead center position. Assuming the resisting torque to be constant, the power (in kW) developed by the engine at 100 rpm is? Solution: T=10,000+1000sinΞΈβ1200cos2ΞΈT=10,000+1000sinβ‘ΞΈβ1200cos2ΞΈ its a function of 2ΞΈ2ΞΈ ,so 2ΞΈ2ΞΈ=360, ΞΈΞΈ=180 T=oβ«Ο10,000+1000sinΞΈβ1200cos2ΞΈT=oβ«Ο10,000+1000sinβ‘ΞΈβ1200cos2ΞΈ =[10,000ΞΈβ1000cosΞΈβ12002sin2ΞΈ]Ο0[10,000ΞΈβ1000cosΞΈβ12002sin2ΞΈ]Ο0 =(10,000Ο+1000β0)β(0β1000β0)(10,000Ο+1000β0)β(0β1000β0) =10,000ΟNβ
m10,000ΟNβ
m Now, T=TmeanΓΟTmeanΓΟ so, Tmean=10,000Nβ
mTmean=10,000Nβ
m w=2ΟN/60=2ΟΓ100/60=10.4719rad/sec.w=2ΟN/60=2ΟΓ100/60=10.4719rad/sec.
P=TmeanΓwP=TmeanΓw = 104.719 KW 74. A flywheel is a rotating mechanical device used to store mechanical energy. When attached to a combined electric motor-generator, flywheels are a practical way to store excess electrical energy. Solar farms only generate electricity when it's sunny and wind turbines only generate electricity when it's windy. Combining energy sources like solar and wind with flywheel energy storage devices like a flywheel is one way to create a renewable energy system that is load balanced.
characteristic
value
shape
solid cylinder
material
4340 steel
density
7,850 kg/m3
diameter
100 cm
height
60 cm
max. energy
32 kWh
max. power
8 kW
dc voltage
800 V
Energy storage flywheel
a.
What is the mass of the flywheel?
b.
What is the top angular speed of the flywheel?
c. For how long could a fully charged flywheel deliver maximum power before it needed recharging?
d.
What is the average angular acceleration of the flywheel when it is being discharged? Solution: a. m = Ο(Οr2h) m = (7,850 kg/m3)Ο(0.50 m)2(0.60 m) m = 3,700 kg 2
b. π = π=
π
πΎ
βπ
2 (32,000π)(3,600π ) β 0.50π 3,700ππ
π = 706
c. βπ‘ =
πππ = 112 πππ π
βπ π
32ππβ βπ‘ = 8π βπ‘ = 4β d. β=
βπ βπ‘
πππ 706 π β= 4π₯3,600π πππ β= 0.049 2 π
75. A flywheel has a mean diameter of 4 ft and is required to handle 2200 ft-lb of kinetic energy. The flywheel has a width of 8 in. Normal operating speed is 300 rpm and the coefficient of fluctuation is to be 0.05. find the weight of the rim assuming that the arms and hub are equivalent is 10% of the specific weight. Soluttion: Solving for the weight of the flywheel: (Note : N2 = 285 rpm) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-2200 = Wf / 2(32.2) [( π x 4 x 285/60)2 β ( π x 4 x 300/60)2] Wf = 368.08 lb
Solving for the weight of the rim, Wf = Wrim + WHA
= Wrim + 0.10(Wrim) where : WHA = 0.10(Wrim)
Wf = 1.10(Wrim) 368.08 = 1.10(Wrim) Wrim = 334.62 lb
answer
76. Flywheel turns 600 rev/min (RPM). Determine the magnitude of the normal acceleration of the flywheel point which are at a distance of 17 cm from the rotation axis. Solution: π=
17 600 2 (2π ) = 671.13 πβπ 2 100 60
77. A disk (flywheel) can rotate with maximum angular speed of 6000 revolutions per minute which can provide kinetic energy that allow a truck to move. When there is an average power needed that allow the truck to operate, the truck will be in motion for limited time as the energy eventually will be reduced with time and that will reduce the power (rate of energy consumed per unit time). Solution:
78. The mass of the flywheel of a shearing machine is 3.5 tons and its radius of gyration is 0.60 m. At the beginning of its cutting stroke the speed of the flywheel is 110 rev/min. If 12.5 kJ of work is done in shearing a plate in 3 seconds, calculate the rotation speed of the flywheel at the end cutting stroke in rev/min. Solution: ΞKE = W Β½ [3500(0.602)(11.522 β W22)] = 12500 W2 = 11.52 rad/s = 105.55 rpm
79. The mass of a flywheel is 175 kg and its radius of gyration is 380 mm. Find the tourqe required to attain a speed of 500 rpm from rest in 30 seconds. Solution: Solving for the mass moment of inertia: πΌπ = mπ 2 = 175(0.380)2 = 25.27 kgπ2
Solving for the angular acceleration: a=
ππ β ππ π‘
=
500 ) 60
(2Ο x
30
= 1.745πππ/π 2
Solving for the tourqe: T = πΌπ a = 25.27(1.75) = 44.10 N-m
80. A cast iron flywheel is rotated at a speed of 1200 rpm and having a mean rim radius of 1 foot. If the weight of the rim is 30 lbs. What is the centrifugal force?
Solution: V = 2Ο (1)(1200/60) = 125.66 fps Solving for the centrifugal force: πΉπ = πΉπ
ππ 2 ππ
=
(30)(125.66)2 32.2(1)
=14712 lb
81. The mean coil diameter of a helical-coil spring is 1 inch and a wire diameter of 1/8 inch. Compute the curve correction factor of the spring.
Solution: πΉπ =
πΎπ€ πΎπ
Where, πΎπ€ =
4Cβ1 4Cβ4
D
+
0.615 C
1
ο’where: C = d = 1/8 = 8 = 1.184
πΎπ =
2C+1 2C
= 1.0625
1.184
πΎπ = 1.0625 = 1.1144
82. A flywheel rim which weighs 800 pounds has a mean diameter of 48 inches. The speed is to be maintained between 100 and 120 rpm. Considering that the effect of the arms and hub accounts for 12% of the rim weight, determine the capacity of the flywheel. Express your answer in ft-lb.
Solution: π1 = (Ο)DN =
π2 =
(Ο)(48)(120) 12(60)
(Ο)(48)(100)
ο²KE =
12(60)
= 25.13 fps
= 20.94 fps
1.12W(π1 )2 β (π2)2 2π
=
1.12(800)(25.13)2 β (20.94)2 2 (32.2)
ο²KE = 2,685.68 ft-lbs
83. A cast iron flywheel with a mean diameter of 36 inches changes speed from 300 rpm to 280 while it gives up 8000 ft-lb of energy. What is the coefficient of fluctuation.
Solution:
πΆπ =
π1 β π2 πππ£π
=
π1 β π2 πππ£π
300β280
πΆπ = 300+280/2
πΆπ = 0.069
84. A 38 diameter spoked steel flywheel having a 12 wide x 10 in deep rim rotates at 200 rpm. How long a cut (in inches) can be stamped in one inch thick aluminium plate if ultimate shearing strength of the aluminium is 40,000 lb/in2 . During stamping, the force exerted by the stamp varies from a maximum F lb at the point of contact to zero lb when the stamp emerges from the metal. Neglect the weight of the flywheel and spokes and use 0.28 lb/in3 denstity for flywheel material. Solution: Solving for the mean diameter, π·π = D β t = 48 β 10 = 38 in Solving for the weight of the flywheel: ππ = ππππ + ππ»π΄ ο’but: ππ»π΄ = 0 ππ = y [ππππ ] = y (Οπ·π bt) ο’where: y= 0.28 ππ/ππ3
ππ = 0.28[Ο(38)(12)(10)] = 4011.19 lb Solving for the energy needed by the process: ο²KE =
ππ 2g
[V22 β V12] ο’where: V1 = ΟDmn1
Assuming that all the energy of flywheel is released during the process, we have V2 = 0 ο²KE =
4011.19 2(32.2)
38
[0 β (Ο x12 x
200 2 )] 60
= -68493.41 ftlb
Solving for the force, 1
1
1
E = 2 ftp ο΅οΆ 68493.41 = 2 F (12) F = 1643842 lb
85. What would be the weight of a flywheel in kg if the weight of the rim is 3 times the sum of the weight of the hub and arms. Given the outside diameter and inside diameter to be 24 in and 18 in respectively and the rim width is 4.6 in. (assume steel flywheel). Solution: Solving for the weight of the rim, lb
Wrim = y (V) = y(ΟDmbt) ο’where: y = 0.284 ππ3 for steel 1
Dm = 2 (D + d) 1
T = 2 (D β d) Then: D+d Dβd 24+18 24β18 Wrim = y [Ο ( )(b)( )] = 0.28[Ο ( )(4.5)( )] 2
2
2
2
Wrim = 252.94 lb = 114.71 kgf Solving for the weight of the flywheel, 1
Wf = Wrim + WHA ο’where: Wrim = 3(WHA) or WHA = 3 (Wrim) 1
4
Wf = Wrim + 3 (Wrim) = 3 (114.71) = 152.95kgf
86. Find the rim thickness for a cast iron flywheel with a width of 200 mm, a mean diameter of 1.2 m, a normal operating speed of 300 rpm, a coefficient fluctuation of 0.05 and
which is capable of hanging 3000 N-m of kinetic energy. Assume that the hub and arms represent 10% of the rim weight and the specific weight of cast iron is 7200 kg/m 3 Solution: Solving for the minimum speed: π βπ π βπ Cf = 1π 2 = π11 + π22 ππ£π
0.05 =
300β π2
2
300 + π2 2
N2 = 285rpm The weight of the flywheel: ο²KE =
ππ 2π
[V22 β V12] ο’where: V1 = πDmN1 and V2 = πDmN2
π
π 3000 = 2(9.81) [(πx1.2 x
285 2 ) -(πx1.2 60
x
300 2 )] 60
Wf = 1740.47N = 177.42kgf The weight of the rim: Wf = Wrim + WHA ο’but: WHA = 0.10(Wrim) Wf = Wrim + 0.10 (Wrim) Wf = 1.10(Wrim) 177.42 = 1.10(Wrim) Wrim = 161.29kg, Wrim = y (V) = y (πDmbt) ο’where: y = 7200 kg/m3 for cart iron 161.29 7200 [π(1.2)(0.2)(t)] = 0.02971m
t = 29.71mm
87. A flywheel rim which weights 900 pounds has a mean diameter of 48 inches. The speed is to be maintained between 100 and 120 rpm. Considering that the effect of the arms and hub accounts for 12% of the rim weight, determine the capacity of the flywheel. Express your answer in ft-lb. Solution: V1 = (π)DN
=
(π)(48)(120) 12(60)
= 25.13 fps V2 =
(π)(48)(100) 12(60)
= 20.94 fps ΞKE = 1.12 W [ V22 β V12]/ 2g = 1.12 (900) [ (25.13)2 β (20.94)2]/ 2(32.2) ΞKE = 3,021.40 ft-lbs
88. A cast iron flywheel is rotated at a speed of 3000 rpm and having a mean rim radius of 3.5 foot. If the weight of the rim is 60 lbs. What is the centrifugal force? Use factor C=41. Solution: V = 2 πRN 1500 = 2 π(3.5) 60 = 549.78 fps Solving for the centrifugal force: Fc = WV2/gR Fc =
(60)(549.78) 2 32.2(3.5)
Fc = 85670.74 lb
89. A cast iron flywheel with a mean diameter of 36 inches changes speed from 300 rpm to 260 while it gives up 8000 ft-lb of energy. What is the coefficient of fluctuation. Solution: Cf = V1 β V2 / Vave
= N1 β N2 / Nave = 300 β 280 / (300+260/2) = 0.071 90. The mass of the flywheel of a shearing machine is 3 tons and its radius of gyration is 0.55 m. At the beginning of its cutting stroke the speed of the flywheel is 110 rev/min. If 12 kJ of work is done in shearing a plate in 3 seconds, calculate the rotation speed of the flywheel at the end cutting stroke in rev/min. Solution: ΞKE = W Β½ [3000(0.552)(11.522 β W22)] = 12000 W2 = 10.31 rad/s = 98.45 rpm
91. A flywheel weighing 457 kg has a radius of 375 mm. Compute how much energy, in Nm, does the flywheel loss from 3 rps to 2.8 rps? Solution: V1 = 2Ο(0.375)(3) = 7.069 m/s V2 = 2Ο(0.375)(2.8) = 6.597 m/s ΞKE = Β½ [m(V1 β V2)] = 457[(7.069)2 β (6.597)2]/2 = 1473.91 Nm
92. Calculate the moment of inertia, mass, for the circular flywheel of which the section. The material of the flywheel is steel, the density of which is 7800 kg.m 3. Solution: We can divide the area into three parts: I1, I2, and I3.
I1 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.05 π₯ π
= x (0.1754 β 0.154) 2 = 0.264 kg.m2 I2 =
ππ₯π€π₯π
x (R24 β R14)
2 7800 π₯ 0.03 π₯ π
= x (0.154 β 0.1354) 2 = 0.064 kg.m2 I3 =
ππ₯π€π₯π
x (R24 β 0)
2 7800 π₯ 0.015 π₯ π
= x (0.1354 β 0) 2 = 0.061 kg.m2 The total moment of inertia is ITotal = 0.264 + 0.064 + 0.061 = 0.389 kg.m2
93. A mechanical press is used to punch 8 holes per minute is 35 mm diameter and the plates has an ultimate strength in shear of 520 Mpa. The normal operating speed 200 rpm. And it slows down to 180 rpm during the process of punching. The flywheel has a mean diameter of one meter and the rim width is 3 times the thickness. Assume that the hub and arm account for 5% of the rim weight concentrated at the mean diameter and the density of cast iron is 7200 kg per cubic meter. Find the power in KW required to drive the press. Solution: E = Β½ Ftp Then:
but : Ss = Sus F/A = 520 Mpa F/(πDtp) = 520 F = 520[πDtp]
E = Β½(520)(π)(D)(tp)2 = Β½(520)(π)(35)(25)2 = 17 867 808 N-mm = 17 867.808 Nm 1 πππ 60 π
t = 8 βππππ [πππ] = 7.5 s/holes
P = E/t =
17 868 7.5
= 2382 W power required
94. Find the weight of the flywheel needed by a machine to punch 30.5 mm holes in 25.87 mm thick steel plate. The machine is to make 40 strokes per minute and a hole be punched every stroke, the hole is to be formed during 30 degrees rotation of the puncher crankshaft. A gear train with a ratio of 12 to 1 is to correct the flywheel shaft to the crankshaft. Let mean diameter of a flywheel rim to the 91.44 cm the minimum flywheel speed is to be 90% of the maximum and assume mechanical efficiency of the machinr to be 80%. Assume an ultimate stress of 59000 psi. Solution: Solving for the energy needed by the process: E = Β½ Ftp Where: F/A = 59000 psi 0.101325 F/(πDtp) = 59000[ 14.7 ] F = 406.68(πDtp) Then: E = Β½(406.68)(π)(D)(tp)2 E = Β½(406.68)(π)(30.5)(25.87)2 E = 1303.963 Nm Solving for the maximum speed of the flywheel: N1 = 40
π π‘πππππ 1 βπππ πππ
[
π π‘ππππ
360
][
30
][
1 πππ£ βπππ
] = 480 rpm
Solving for the minimum speed of the flywheel: N2 = 0.90(N1) = 0.90(480) = 432 rpm Solving for the weight of the flywheel: (based on energy equation) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
ΞKE = -1303.963 N-m Since the flywheel will supply / release this energu, then: -1303.963 = Wf / 2(9.81) [( π x 0.9144 x 432/60)2 β ( π x 0.9144 x 480 /60)2]
Wf = 254.95 N
95. It is found that the shearing machine requires 305 joules ofenergy to shear a specific gauge of sheet metal. The mean diameter of the flywheel is to be 86.2 cm, the normal operating speed is 300 rpm and slow down to 280 rpm during shearing process. The rim width is 40.48 cm and the weigth of ast iron is 8,196.6 kg/m3. Find the thicknes of the rim, assuming that the hub and arm account for 10% of the rim weight concentrated on the mean diameter. Solution: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-305 = Wf / 2(9.81) [( π x 0.862 x 280/60)2 β ( π x 0.862 x 300/60)2] Wf = 25.81 kg weight of the flywheel Solving for the weight of the rim: Wf = Wrim + WHA 25.81 = 1.10(Wrim)
where: WHA = 0.10 Wrim Wrim = 23.46 kg
Solving for the thickness of the rim, Wrim = Ζ΄(Vrim) = Ζ΄(πDmbt)
where : = 8196.6 kg/m3
23.46 = 8196.6 [π(0.862)(0.4048)(t)] t = 2.611 mm
96. A 38 in diameter spoked steel flywheel having a 12 in wide x 10 in deep rim rotates at 200 rpm. How long a cut (in inches) can be stamped in one inch thick aluminum plate if ultimate shearing strength of the aluminum is 20,000 lb/in2. During stamping, the force exerted by the stamo varies from a maximum F lb at the point of contact to zero lb when the stamp emerges from the metal. Neglect the weight of the flywheel and spokes and use 0.28 lb/in3 density for flywheel material. Solution: Solving for the mean diameter, Dm = D β t = 48 β 10 = 38 in
Solving for the weight of the flywheel: Wf = Wrim + WHA
but : WHA = 0 where : Ζ΄= o.28 lb/in3
Wf = Ζ΄(Vrim) = Ζ΄(πDmbt) Wf =4011.19 lb
Solving for the energy needed by the process: ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1 Assuming that all the energy of the flywheel is released during the process, we have V2 = 0 4011.19 38 200 ΞKE = 2(32.2) [0 β (π π₯ 12 π₯ 60 )2] = -68493.41 ftlb Solving for the force, E = Β½ Ftp 68493.41 = Β½ F(1/12) F 1 643 842 lb Solving for the length of the aluminum plate Ss = Sus F/A = 20 000 psi F/Lt = 20 000 1 643 842/L(1) = 20 000 L = 82.192 in
97. What would be the weight of a flywheel in kg if the weight of the rim is 6 times the sum of the weight of the hub and arms. Given the outside diameter and inside diameter to be 24 in and 18 in respectively and the rim width is 4.5 in. (assume steel flywheel) Solution: where : Ζ΄= 0.284 lb/in3 for steel
Wf = Ζ΄(Vrim) = Ζ΄(πDmbt) Dm = Β½ (D + d) t = Β½ (D β d) then: Wrim = Ζ΄[π(
π·+π 2
)(π)(
= 0.284[π(
24+18 2
π·βπ 2
)]
)(π)(
24β18 2
)]
= 252.94 lb = 114.71 kgf Solving for the weight of the flywheel, Wf = Wrim + WHA
but : Wrim = 6 WHA or WHA = 1/6 (Wrim)
Wf = Wrim + 1/6 (Wrim) = 7/6 (114.71) = 133.82 kgf
98. A flywheel has a mean diameter of 8 ft and is required to handle 1100 ft-lb of kinetic energy. The flywheel has a width of 8 in. Normal operating speed is 500 rpm and the coefficient of fluctuation is to be 0.05. find the weight of the rim assuming that the arms and hub are equivalent is 10% of the specific weight. Soluttion: Solving for the weight of the flywheel: (Note : N2 = 285 rpm) ΞKE = Wf / 2G [ V22 β V12] Where: V1 = πDmn1
V2 = πDmn2
-1100 = Wf / 2(32.2) [( π x 8 x 285/60)2 β ( π x 8 x 5 00/60)2] Wf = 2.39 lb Solving for the weight of the rim, Wf = Wrim + WHA
= Wrim + 0.10(Wrim) where : WHA = 0.10(Wrim)
Wf = 1.10(Wrim) 2.39 = 1.10(Wrim) Wrim = 2.17 lb
99. A cast iron flywheel is rotated at a speed of 2400 rpm and having a mean rim radius of 3 foot. If the weight of the rim is 55 lbs. What is the centrifugal force? Use factor C=41. Solution: V = 2 πRN
1200
= 2 π(3) 60 = 753.98 fps Solving for the centrifugal force: Fc = WV2/gR Fc =
(55)(753.98) 2 32.2(3)
Fc = 323672 lb 100. A cast iron flywheel with a mean diameter of 36 inches changes speed from 320 rpm to 300 while it gives up 8000 ft-lb of energy. What is the coefficient of fluctuation.
Solution:
πΆπ =
π1 β π2 πππ£π
=
320β300
π1 β π2
πΆπ = 320+300/2
πΆπ = 0.043
πππ£π