Slider–Crank Mechanism of an IC Engine

Step-by-Step Solution at One Crank Angle

The standard textbook procedure (two-mass equivalent rod with correction couple), with your numbers substituted at every step.

Kinematics: Velocity & Acceleration

How fast the piston and rod move at every crank angle. These numbers drive all the inertia forces.

Velocity and acceleration diagrams (relative velocity method)

Drawn to scale at the animation angle; they move with the mechanism. Vectors: oc crank-pin velocity ⟂ OC, cp velocity of P relative to C ⟂ CP, op piston velocity along the stroke. In the acceleration diagram, c′x is the radial (centripetal) and xp′ the tangential component of P relative to C.

Fig. 2. Piston displacement, velocity and acceleration over one revolution (solid: exact, dashed: approximate formula).

Fig. 3. Angular velocity and angular acceleration of the connecting rod.

How it is calculated

Cylinder Pressure & Indicator Diagram

The gas pushes the piston; this pressure is the input for the force analysis.

Fig. 4. Cylinder pressure over the four-stroke cycle.

Fig. 5. Indicator () diagram. The enclosed area is the indicated work per cycle; the dot follows the animation.

How it is calculated

First-law energy balance of the closed cylinder, integrated with RK4 (0.25° step). Combustion starts at and lasts . Suction is at , exhaust at , and the crank-case is at . Net heat released at full load is 2.6 kJ per litre of swept volume.

Dynamic Force Analysis

Gas force minus the inertia of the moving parts gives the forces in every link and bearing.

Fig. 6. Gas force, inertia force of the piston and net piston effort.

Fig. 7. Thrust in the connecting rod (+ compression, − tension) and side thrust on the cylinder wall.

Fig. 8. Left: crank effort and radial force . Right: polar load diagram of the crank pin (in the crank frame), used to size the big-end bearing.

How it is calculated

The plots use the exact Newton–Euler solution of piston and rod (rod mass and moment of inertia included): , , . The textbook two-mass method with correction couple gives exactly the same torque; see the Verification tab.

Turning-Moment Diagram & Flywheel

The engine torque swings above and below the steady load torque; the flywheel stores the surplus and gives it back.

Fig. 9. Turning-moment diagram. Red areas: (flywheel speeds up); blue: (flywheel slows down).

Table of areas (textbook method)

Each loop is the area between and between two crossing points. Taking the energy at the first crossing as , the running total gives the energy at every crossing; the largest minus the smallest is .

Fig. 10. Left: fluctuation of energy . Right: crankshaft speed with the designed flywheel, which stays inside the band.

How it is calculated

Balancing of Reciprocating Masses

The moving parts shake the engine frame; a counterweight on the crank can only partly cancel this.

Fig. 11. Left: primary and secondary unbalanced forces over one revolution. Right: shaking force on the frame for , the chosen , and the optimum .

How it is calculated

What Changes When a Parameter Changes?

A bar of +0.8 means a 1 % increase in that parameter raises the chosen result by about 0.8 %; negative bars mean it falls.

Fig. 12. Elasticity of the selected result to each input at the current design (central differences, ±1 %).

Fig. 13. Design space over crank speed and obliquity ratio . Left: flywheel rim mass needed for the chosen . Right: change of peak crank-pin load relative to the current design (★). Click a map to move the design there.

Verification of the Calculations

Every result is cross-checked against an independent method.

Benchmark A: textbook problem. A horizontal engine has  mm,  m,  m,  kg and  rpm, with a net pressure of 0.35 N/mm² at from IDC. It is solved by the hand-calculation chain (approximate acceleration) and by the exact solver.

Benchmark B: air-standard Otto cycle. For the integrated cycle must give .

Live checks on the current design (recomputed on every change):

Fig. 14. Exact turning moment against the common simplified method (two-mass rod, approximate acceleration, no correction couple). The gap is the error of that simplification for this design.

Problem Statement & Notation

Setting. A slider-crank chain . The crank of radius turns counter-clockwise at  rpm about the fixed centre . The connecting rod has length , mass , centre of mass at from the crank pin and radius of gyration . The piston (bore , reciprocating mass ) slides along the line of stroke. A rim-type flywheel of mean diameter and a counterweight balancing the revolving mass plus a fraction of the reciprocating mass are fitted.

Coordinates. is the crank angle from inner dead centre (TDC) and the four-stroke cycle angle (0 TDC suction, 180 BDC, 360 firing TDC, 540 BDC). Forces are positive toward the crank; torque is positive in the direction of rotation.

Data (live).

References.

1. S. S. Rattan, Theory of Machines, McGraw-Hill Education.
2. R. S. Khurmi & J. K. Gupta, Theory of Machines, S. Chand.
3. A. Ghosh & A. K. Mallik, Theory of Mechanisms and Machines, East-West Press.
4. J. J. Uicker, G. R. Pennock & J. E. Shigley, Theory of Machines and Mechanisms, Oxford University Press.
5. J. B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill.