Torsional Analysis of a Four-Stroke Engine

Reciprocating machines fall into many categories. Despite different applications and designs, e.g. pumps or internal combustion engines with a varying number of pistons, a simple approach to determine torsional modes of regardless which crankshaft assembly can be investigated. The resulting natural frequencies are required by ISO 3046 for rotor dynamic analysis.

Internal Combustion Engine with piston and flywheel geometry
Figure 1 Internal Combustion Engine with piston and flywheel geometry, (https://www.quora.com/What-is-a-starter-flywheel)

Below, a common way to express a crankshaft assembly with massless shaft and mass-inertia elements is presented, whereas the reciprocating and revolving mass around the crack can be expressed as follows:

Formula for Torsional Analysis

The mass-inertias caused by given masses depend partly on the crank is:

Crank Angle Formula

whereas I1 I2 and  correspond to the reciprocating and revolving part, respectively. Due to differing inertias of the whole assembly, a mean value is introduced to express the contribution of the inertia elements in a dynamic process:

mean value - Formula

Another simplification is made to describe the torsional stiffness of the crankshaft, which consists mainly of the main shaft and crankpin. If the shaft and crankpin diameter are of approximately same diameter, the stiffness can be expressed as:

Stiffness Formula

with Shear modulus, shaft diameter  and section length . As a consequence, a substitutive model can be created. The AxSTREAM RotorDynamics™ software developed by SoftInWay Inc. provides a 1D and 2D FE-approach to calculate torsional natural frequencies of the shaft system. This incorporates the mass-inertia elements besides the torsional stiffness of the shaft sections, which can be seen in Figure 1 for a four-stroke engine.

Model of a crankshaft and additional mass-inertia element pins representing the piston and flywheel geometry
Figure 2 Model of a crankshaft and additional mass-inertia element pins representing the piston and flywheel geometry

The resulting 3D mode shapes in Figure 2 can be evaluated, whereas the red contour line corresponds to the torsion of the shaft with applied mass-inertia elements. High angular deviations from the undeformed contour indicate critical sections when excited by a torque close to the natural frequency.

3D Mode shape at a corresponding natural frequency
Figure 3 3D Mode shape at a corresponding natural frequency

Additionally, a Campbell diagram can be plotted, with natural frequencies distributed over a chosen speed range. Interactions of natural frequencies with excitation lines or frequencies, caused by e.g. misfiring or coupling with the clutch, may lead to improper operating conditions or failures and need to be investigated.

Campbell diagram
Figure 4 Campbell diagram representing 5 natural torsional frequencies and 3 arbitrary chosen excitation lines over the speed range of the crankshaft and operating speed conditions framed with dashed lines

A transient torque function, to simulate excitation introduced by the clutch (see Figure 5), can be included in AxSTREAM RotorDynamics™ with the following formula:

Formula AxSTREAM

whereas Mi (Ti) is the torque value at a specific time and the engine rotational speed. Besides this particular excitation case, phenomena such as short circuit or switch-on processes can be simulated.

Rotor Dynamics Charts
Figure 5 a.) Influence of an elastic clutch (Gain corrected) on the overall torque acting on the crankshaft (Active Torque Damping for an ICE-based Domestic CHP System with an SPM Machine Drive – Scientific Figure on ResearchGate. Available from: https://www.researchgate.net/Torque-measurement-and-reconstructed-torque-of-ICE-removing-the-oscillating-torque-due-to_fig5_272397886 [accessed 21 Jun, 2018]); b.) Reconstructed Torque function in AxSTREAM RotorDynamics™
For a time-frame of 0.5 seconds, the response characteristics of the crankshaft has been calculated. High response values indicate resonance behavior and need to be taken into account in the design process of the rotor geometry. In this case, no resonance behavior of the response torque or stress can be seen in Figure 6.

Angle and Torsional stress over time evaluated at an arbitrary chosen position at the crankshaft
Figure 6 Angle and Torsional stress over time evaluated at an arbitrary chosen position at the crankshaft

Interested in learning more about rotordynamics? Join us on September 5th-7th for a three day training on Rotor Dynamics and Bearing Analysis. Read more and register here

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