Torsional Analysis Of Reciprocating Machinery: Damping And Forced Response

Early analysis and treatment help prevent catastrophic failure

Forced response scope

Many modern reciprocating machines run over wide speed range, and produce several significant orders of excitation energy. These factors combine to make avoiding all possible critical speed excitation scenarios problematic. As a result, the scope of the forced response evaluation plays a very important role in the ultimate usefulness of a torsional analysis, because it examines whether the machine can tolerate the anticipated critical speed behavior.

One very important activity to consider is that all appropriate compressor load steps are evaluated. Loading schedules, which are normally provided by reciprocating compressor manufacturers for their products, specify individual cylinder pocket or unloader settings for the various configurations available. In some cases, many load steps are possible with a given design, and fully evaluating all of these during a torsional analysis could potentially be cost prohibitive. General guidelines are available for determining which load steps to analyze, but at a minimum the analyst should ensure that fully loaded and unloaded operation, along with representative single-acting (head end or crank end deactivation) and double-acting (both ends active) cases, are included. This is important because the single-acting cases tend to elevate odd excitation orders, while the double-acting cases similarly influence even excitation orders.

Figure 4. Torsional stress calculation waterfall plot for a hypothetical speed range.

One might be tempted to consider only the most loaded (highest power) case provided, presuming that this would cause the highest shaft stress. However, the composite stress level in a shaft is a combination of mean stress (which generally increases with the power level) and dynamic stress (which can be significantly influenced by excitation of the torsional critical speeds). Operating conditions resulting in significant load asymmetry (some cylinders un- loaded, and some fully loaded) should also be included in the analysis, as these have the potential to result in the highest dynamic shaft stress. It is not unusual for the highest composite stress level to occur at a load step that produces less power than the full load case.

Determining the acceptance criteria for torsional oscillation is another topic that requires careful consideration. The torsional analysis results should be compared to any provided manufacturer limit for allowable shaft stress, torque, torsional velocity or viscous damper heat dissipation. In addition, two industry-accepted approaches for evaluating shaft stress include Military Standard 167 (Mil Std 167) and ASME B106M.

The Mil Std 167 approach involves simply dividing the ultimate tensile strength (UTS) of the shaft material by 25. This approach, although straightforward, tends to be quite conservative. The less conservative approach outlined in ASME B106M is known as a strength reduction method, and involves multiplying the UTS by a series of factors which reduce the allowable stress. The Mil Std 167 approach results in an allowable stress of 4% of the UTS, where the ASME B106M approach results in an allowable stress of about 5.3% utilizing the representative factors shown in Figure 3, and neglecting the mean stress factor. The allowable stress is very sensitive to the multiplicative effects of the various factors assumed. To illustrate this sensitivity, arbitrarily increasing each of the size, surface, and uncertainty factors in the example below to 0.9 (not recommended for an actual analysis), would cause the allowable stress to increase to about 10.5% of the UTS. Therefore, the strength reduction factors actually utilized during an analysis require careful consideration, guided by experience and reasonable conservatism to manage the risks involved.

Figure 5. Representative single engine cylinder misfire effects.

Many torsional forced response analyses evaluate acceptability at fixed speed steady state conditions. However, similar calculations conducted over a hypothetical speed range can provide valuable insight into the separation margins between the planned operating speeds and critical speeds, as illustrated in Figure 4. In this example, the stress levels might be considered acceptable at the planned running speed, but could potentially be unacceptable at the critical speed located just above running speed. These types of speed sweep calculations can be valuable tools that allow the analyst to produce waterfall plots for direct comparison with those captured in the field, such as the one shown in Figure 2.

The effects of misfire events should be taken into account during a torsional analysis involving engine drivers. In the example shown in Figure 5, the frequency content below the fourth order is null for normal operation. However, the figure also shows that during a single cylinder misfire event, the orders below the lowest primary order can increase substantially, exciting critical speeds that are not typically a factor during normal operation. Another judgment to consider is which cylinders to include while simulating such an event. One recommended practice involves conducting separate single-cylinder misfire calculations for all cylinders, in order to establish which one causes the highest shaft stress levels for a given machine operating at the anticipated conditions.

The cylinder phasing relationship between engine drivers and reciprocating compressors should also be considered, especially when simultaneous engine and compressor excitations are applied, as this parameter can have a significant effect on the predicted stress behavior. In such cases, the analyst should clearly state the assumed phasing relationship (such as engine cylinder one and compressor cylinder one at top dead center simultaneously) so that the machine can be configured in the field during initial installation and maintenance events to minimize the potential for excessive shaft stress. In some cases, it may be necessary for the analyst to consider several potential phasing relationships to ensure acceptability.


Humans and machines can both benefit greatly from experience, applied at an optimal time, to mitigate frequently overlooked problems before they become catastrophic. In the case of reciprocating machinery torsional rotordynamics, attention to detail with regard to damping and forced response evaluations can play a key role in improving long-term reliability.