Problems for Practice

PROBLEMS FOR PRACTICE

On Velocity and Acceleration of the Reciprocating Parts in Engines

1. In a slider crank mechanism, the length of the crank and connecting rod are 150 mm and 600 mm respectively. The crank makes an angle of 60° with the IDC and revolves at a uniform speed of 300 rpm. Find, by analytical method: (i) velocity and acceleration of the slider, and (ii) angular velocity and angular acceleration of the connecting rod. 

[Ans. 4.6 m/s; 61.7 m/s2; 4.17 rad/s; 214 rad/s2]

2. In a slider crank mechanism, the crank and the connecting rod are 200 mm and 800 mm respectively. Determine the crank angle at which the velocity of the piston is maximum and the maximum velocity of the piston. Assuming that the crankshaft rotates at a mean speed of 300 rpm. 

[Ans.  77°; 6.806 m/s]

3. The crank and connecting rod of a steam engine are 300 mm and 1500 mm in length. The crank rotates at 180 rpm clockwise. Determine the velocity and acceleration of the piston when the crank is at 40° from IDC position. Also, determine the position of the crank for zero acceleration of the piston. 

[Ans.  4.194 m/s; 85.35 m/s2; 79.3°]

4. A reciprocating engine has connecting rod 200 mm long and crank 50 mm long. By using Klien's construction, determine: (i) velocity and acceleration of the piston, (ii) angular velocity and angular acceleration of the connecting rod; and (iii) velocity and acceleration of a point X on the connecting rod located at its mid-point when the crank has turned through 45° from IDC clockwise and is rotating at 240 rpm.

[Ans. 1.08 m/s; 22.12 m/s2; 4.525 rad/s; 113.75 rad/s2; 1.1 m/s; 26.54 m/s2] 

5. The crank and connecting rod of a reciprocating engine are 150 mm and 600 mm respectively. The crank is rotating in clockwise at a speed of 450 rpm. Using Klien's construction, determine: (i) velocity and acceleration of the piston; (ii) velocity and acceleration of the point D on the connecting rod which is 150 mm from the crank pin; and (iii) angular velocity and angular acceleration of the connecting rod, at the instant when the crank is at 60° from IDC.

 [Ans. 6.834 m/s; 124.4 m/s2; 6.834 m/s; 266.55 m/s2; 6.127 rad/s; 481.27 rad/s2] 

On Forces on the Reciprocating Parts of an Engine, Neglecting the Weight of the Connecting Rod

6. Find the inertia force for the following data of an I.C engine: Stroke = 200 mm; Bore 175 mm; Engine speed = 500 rpm; Length of connecting rod = 400 mm; Crank angle = 60° from IDC, and Mass of the reciprocating parts = 180 kg 

[Ans.  18.78 kN]

7. The length of crank and connecting rod of a steam engine is 300 mm and 1200 mm respectively. The mass of the reciprocating parts of the steam engine is 250 kg. Its cylinder bore is 0.5 m. When the crank has traveled 60° from IDC, the difference between the driving and the back pressure is 0.35 MPa. If the engine runs at 250 rpm, neglecting the effect of the piston rod diameter, determine:

(i) pressure on slide bars,

(ii) thrust in the connecting rod,

(iii) tangential force on the crank-pin, and 

(iv) turning moment on the crankshaft.

[Ans. 10.96 kN; 50.62 kN; 48.28 kN; 14.48 kN•m] 

8. A horizontal steam engine running at 240 rpm has a bore of 200 mm and stroke of 360 mm. The piston rod is 20 mm in diameter and connecting rod length is 900 mm. The mass of the reciprocating parts is 7 kg and the frictional resistance is equivalent to a force of 500 N. Determine the following when the crank is at 120° from the IDC, the mean pressure being 5000 N/m2 on the cover side and 100 N/m2 on crank side: (i) thrust on the connecting rod, (ii) thrust on the cylinder walls, (iii) loads on the bearings, and (iv) turning moment on the crankshaft. 

[Ans. 133.33 N; 23.08 N; 85.65 N; 1839 N•m]

9. A horizontal steam engine has the following particulars: Bore of its cylinder = 200 mm; stroke = 400 mm; length of connecting rod = 600 mm; mass of the reciprocating parts = 5 kg; diameter of piston rod = 40 mm; at crank angle = 40° from IDC; steam pressure on cover side 45 = 104 N/m2; steam pressure on crank side = 5 × 104 N/m2; Mean speed 120 rpm. Determine: (i) turning moment on the crankshaft, (ii) thrust on the bearings, and (iii) flywheel acceleration. Given that its mass is 5 kg and radius of gyration 500 mm and that the power of the engine is 15 kW. 

[Ans. 2427.44 N•m; 7812.76 N; 987 rad/s2] 

10. A double-acting vertical steam engine has a cylinder 300 mm diameter and 450 mm stroke and runs at 200 rpm. The reciprocating parts have a mass of 225 kg and the piston rod is 50 mm diameter. The connecting rod is 1.2 m long. When the crank has turned through 125° from the top dead centre, the steam pressure above the piston is 30 kN/m2 and below the piston is 1.5 kN/m2. Calculate (a) the piston effort, and (b) the effective turning moment on the crank shaft. 

[Ans.  18.397 kN; 3021.6 N•m] 

11. A steam engine 200 mm bore and 300 mm stroke has a connecting rod 625 mm long. The mass of the reciprocating parts is 15 kg and the speed is 250 rpm. When the crank is at 30° to the IDC and moving outwards, the difference is steam pressures is 840 kN/m2. If the crank pin radius is 30 mm, determine: (i) the force on the crank shaft bearing, and (ii) the torque acting on the frame. 

[Ans.  20.04 kN; 2253 N•m] 

12. A vertical petrol engine 100 mm diameter and 120 mm stroke has a connecting rod 250 mm long. The mass of the piston is 1.1 kg. The speed is 2000 rpm. On the expansion stroke with a crank 20° from TDC, the gas pressure is 700 kN/m2. Determine: (i) net force on the piston, (ii) resultant load on the gudgeon pin, (iii) thrust on the cylinder walls, and (iv) speed above which, other things remaining same, the gudgeon pin load would be reversed in direction. 

[Ans. 2256.8 N; 2265 N; 185.5 N; 2606 rpm]

13. In a double-acting vertical steam engine, steam is admitted at a pressure of 1 MPa gauge. The cut-off takes places at 1/4th of the stroke. The steam expands hyperbolically and then is exhausted at å pressure of 80 kPa gauge. The cylinder bore is 180 mm; length of stroke 270 mm and the length of the connecting rod is 405 mm. If the frictional resistance is equivalent to a force of 400 N, determine the turning moment of the crankshaft when the crank is at 60° from the TDC while it rotates 240 rpm. Total mass of the reciprocating parts is 20 kg. 

[Ans. 2811.91 Nm] 

14. A vertical, single cylinder, single-acting diesel engine has a cylinder diameter 300 mm, stroke length 500 mm, and connecting rod length 4.5 times the crank length. The engine runs at 180 rpm. The mass of the reciprocating parts is 280 kg. The compression ratio is 14 and the pressure remains constant during the injection of the oil 1/10th of the stroke. If the compression and expansion follows the law p.V1.35 = constant, determine: (1) crank- pin effort, (ii) thrust on the bearings, and (iii) turning moment on the crankshaft, when the crank displacement is 45° from the IDC position during expansion stroke. The suction pressure may be taken as 0.1 N/mm2 

[Ans. 109.522 kN; 79.456 kN; 27.38 kN•m] 

15. An internal combustion engine working on four-stroke Otto cycle has a bore of 250 mm and a stroke of 500 mm. The length of the connecting rod is 1000 mm and the mass of the reciprocating parts is 80 kg. The crank rotates at 240 rpm. The compression ratio is 5. The gas pressure at the end of the constant volume heat addition is 3 MPa, suction pressure 0.1 × 105 N/m2 and exhaust pressure 1.5 × 104 N/m2. The law of compression may be assumed as pV1.34 = constant. Atmosphere pressure being 1.03 × 105 N/m2. Find the turning moment on the crankshaft when the crank has turned through an angle of 45° from IDC during power stroke. 

[Ans. 12545.49 N.m]

On Dynamically Equivalent Systems

16. A connecting rod 300 mm long between centres is of mass of 15 kg and has moment of inertia of 7000 kg.mm2 about its center of gravity. Center of gravity is located at a distance of 200 mm from its small end center. Determine the dynamically equivalent two-mass system of the connecting rod, if one of the masses is located at the small end center. 

[Ans. 0.17 kg; 14.83 kg]

17. A connecting rod suspended from a point 25 mm above the center of small end, and 650 mm above its center of gravity, its mass being 37.5 kg. When permitted to oscillate, the time period is found to be 1.87 seconds. Find the dynamical equivalent system constituted of two masses, one of which is located at the small end center.

[Ans. 10 kg; 27.5 kg]

On Correction Couple

18. A connecting rod of an I.C engine has a mass of 2 kg and the distance between the center of gudgeon pin and center of crank pin is 250 mm. The C.G falls at a point 100 mm from the gudgeon pin along the line of centres. The radius of gyration about an axis through the C.G perpendicular to the plane of rotation is 110 mm. Find the equivalent dynamical system if only one of the masses is located at gudgeon pin.

If the connecting rod is replaced by two masses, one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23000 rad/s2 clockwise, determine the correction couple applied to the system to reduce it to a dynamically equivalent system. 

[Ans. 1.1 kg; 0.9 kg; 133.4 N•m]

19. A connecting rod of length 375 mm between centers has a mass of 4 kg. The center of gravity is 250 mm from the small end and the radius of gyration about an axis through the centre of grayity perpendicular to the plane of motion 120 mm. Determine dynamically equivalent system having one mass at the center of small end and the other at a point somewhere in between the center of big end and the C.G. Also, find the distance of this point from the C.G. 

[Ans. 0.75 kg; 3.25 kg; 57.6 mm]

20. Solve the preceding problem, if the masses are to be placed at the centers of small and big ends, find the values of the two masses and correction couple which must be applied to this system in order that its effects may be identical with that of the rod, when the acceleration α = 2600 rad/s2 

[Ans. 1.334 kg; 2.66 kg; 17.52 kg.m]

On Inertia Force on the Reciprocating Parts of an Engine, Considering the Weight of the Connecting Rod

21. The crank and the connecting rod of an engine are 125 mm and 500 mm respectively. The mass of the connecting rod is 60 kg and its center of gravity is 275 mm from the cross-head pin center. The radius of gyration about center of gravity being 150 mm. If the engine speed is 600 rpm for a crank position of 45° from the IDC, determine, using Klien's or any other construction:

(i) the acceleration of the piston, and

(ii) the magnitude, position and direction of inertia force due to the mass of the connecting rod. 

[Ans. 355.5 m/s2; 24.4 kN]

22. Petrol engine has the following particulars:

Stroke = 87.5 mm; Mass of the connecting rod = 1.75 kg; Length of the connecting rod = 175 mm; Center of gravity from big end center = 57.5 mm; Radius of gyration about an axis through the center of gravity and perpendicular to the connecting rod = 75 mm; Engine speed →→3000 rpm. Find the inertia force of the connecting rod and its direction when the crank makes an angle of 40° with IDC. 

[Ans. 6.79 kN]

23. The following data relate to a horizontal reciprocating engine:

Mass of reciprocating parts 120 kg; crank length = 90 mm; Engine speed = 600 rpm; Mass of the connecting rod = 90 kg; Length between centers = 450 mm; Distance of mass center from the small end center = 180 mm; Radius of gyration about mass center axis = 150 mm. Find the magnitude and the direction of the inertia torque on the crank shaft when the crank has turned 30° from the IDC. 

[Ans. 3025 N•m, anticlockwise]

24. The following data refer to a steam engine:

Diameter of piston = 240 mm; Stroke = 240 mm; Stroke = 600 mm; Length of connecting rod 1.5 m; Mass of reciprocating parts = 300 kg; Mass of connecting rod = 250 kg; Speed = 125 rpm; Center of gravity of connecting rod from crank pin = 500 mm; Radius of gyration of the connecting rod about an axis through the center of gravity= 650 mm; Determine graphically or analytically, the magnitude and direction of the torque exerted on the crankshaft when the crank has turned through 30° from IDC. 

[Ans. 3850 Nm]

25. The reciprocating parts of a vertical high-speed engine are of 2 kg mass. The connecting rod is of mass of 4 kg and 300 mm long between its centers, having its center of gravity of 100 mm from its big end bearing. The crank shaft is suspended at its small end bearing axis and allowed to vibrate in a vertical plane. Its time period of oscillation has been found to be 1.06 seconds. If the stroke of the piston is 125 mm, what torque is required to be applied at the crank shaft to overcome the inertia of the moving parts at the instan the crank makes 1400 with TDC while the engine runs at a speed of 1600 rpm?