Engine Excitation Mechanisms


This article was written by Paul Hollingworth for the Transmission and Engine Noise Workshop, Rover Group 1996.

Single Cylinder Engine

Inertia Force - The displacement of the piston with respect to crank angle can be derived from simple trigonometry. This can then be differentiated to yield velocity and acceleration of the piston. The expressions obtained tend to be very complicated and can be simplified into the expression containing only first order (once per revolution), second order (twice per revolution), and a negligible fourth order.
Fi = Inertia force [N]
MREC = Reciprocating mass (piston mass plus approximately 1/4 conrod mass)
θ = Crank angle (zero at top dead centre)
R = Crankshaft radius [m]
L = Conrod length [m]
N = Rotational speed [rpm]

Note: if R/L<0.3 it is accurate enough to use just the first two terms.
Inertia force is obtained by multiplying the piston acceleration by the reciprocating mass and acts only in the line of the cylinders.

Gas Forcing - The rate of rise and peak cylinder pressure of the diesel (13MPa at approximately 20 after TDC) are approximately twice that of the petrol with the angle the peak occurs at typically 5 earlier.

Diesel and petrol combustion is random even at full load, worse at part load and particularly poor at idle. Therefore, it is normal to talk about the average peak cylinder pressure (Pmax mean) and standard deviation of Pmax. This variability both cycle to cycle and cylinder to cylinder is one source of half order excitation.

Equilibrium of Forces - The gas force that acts on the piston also acts on the cylinder head. The force on the piston splits into two components, one acting down the rod and one acting sideways on the cylinder wall. The forces are reacted at the main bearing but a couple exists between the horizontal reaction at the bearing and the piston side force. This couple is equal to the crankshaft output torque, so the crankshaft torque is reacted by forces on the engine structure.

The gas force components of the vertical force at the bearing is equal and opposite to the force acting on the cylinder head, but of course the inertia component is unbalanced.

Torsional Excitation of Crankshaft and Engine Structure - The total torque acting on the crankshaft of the single cylinder engine results from the effect of the gas and inertia forces on the crank slider mechanism.
The torque resulting from piston motion is often called the INERTIA torque and is represented by the equation:
ti = Inertia torque [Nm]

Torque resulting from piston motion alone for a single cylinder engine.

The torque resulting from gas pressure alone is represented by the equation:
tg = Gas torque [Nm]
Pg = Gas pressure [Nm-2]
A = Area of top of piston [m-2]

Torque resulting from gas pressure alone for a single cylinder engine.

The total torque is found by summing these two components. Note that the torque from gas pressure dominates (for the engine firing case).
Total torque for a single cylinder engine.

The sum of the inertia and the gas torques is present at the flywheel and has to be reacted by the engine structure.

If the harmonics of this torque are extracted by Fourier analysis, substantial amplitudes are present for all harmonics including half orders down to order 8 and are still present down to 18. (A half order is a component which does not complete an even number of cycles in 360).

If the INERTIA torque is considered alone, then only orders 1, 2, 3 and 4 are present and can be represented by a formula.

The GAS torque, like the total torque, contains all orders because they are present in the pressure crank angle time history. The torsional excitation of the engine structure is the major source of high and half orders.


EJ Nestorides, "A Handbook on Torsional Vibration", BICERA Research Laboratory

William T Thomson, "Theory of Vibration with Applications"

William T Thomson, "Fundamentals of Automotive Engine Balance"

W Ker Wilson, "The Fundamentals of Engine Balancing", Papers 1 - 5, Gas Oil and Power - July 1955.

See also: Engine Configurations, Engine Orders, Engine Radiated Noise, Flywheel, Internal Combustion Engine.

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Subjects: Automotive Engines Noise & Vibration