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.
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:
The mass-inertias caused by given masses depend partly on the crank is:
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:
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:
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.
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.
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.
A transient torque function, to simulate excitation introduced by the clutch (see Figure 5), can be included in AxSTREAM RotorDynamics™ with the following formula:
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.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.