**Mark A. Corbo President Stanley B. Malanoski Engineering Consultant No Bull Engineering Guilderland, N.Y.**

Rotating machinery inherently vibrates torsionally at the system’s undamped natural frequencies, producing sinusoidal torque and speed variations in the machine. Rotating components, such as gears, shafts, and couplings generally have slight imperfections which can excite torsional vibrations. Resonance results when component-excitation frequencies match a system’s natural frequency. This can produce large cyclic torques that often lead to broken gear teeth, fatigue in shafts or couplings, and early failure.

Designers avoid vibration problems at the design stage by determining the system’s undamped natural frequencies and charting them on Campbell diagrams (See “Tuning Out Torsional Vibration,” Machine Design, February 6, 1997.) Armed with this information, they can then determine interference points and assess a machine’s ability to withstand them.

**Identifying interference points**

Campbell diagrams help determine the machine’s interference points. Interference points are intersections between natural frequencies and excitation frequency lines and have potentially adverse effects on the system. Steady state and transient are the two types of interference points. Steady-state interference points include all intersecting points within the operating-speed range, ±10 to 15% to provide a design margin.

Transient interference occurs when a machine starts up, and includes all intersecting points below the operating-speed range that are triggered by excitation sources with substantial pulsating torques. For practical purposes, the threshold is 5% of transmitted torque. Accordingly, include all intersecting points driven by synchronous motors or variable-frequency drives. Other sources can generally be ignored.

After locating interference points, filter out those with no impact on the design. This concentrates time-consuming, steady-state damped analysis on points that could severely test the drivetrain. The crux of the inspection is to identify points that could generate large cyclic torques and stresses.

During undamped, free vibration, shaft elements physically change their shape from completely straight to slightly deflected at several points along their length. Interference points excite a specific natural frequency, or mode. Modes have zero vibration at points called nodes and large vibration in areas called active regions. Thus, dismiss interference points with excitations close to nodes because they shouldn’t cause problems and focus on excitations in active regions. If excitation magnitudes for several points are similar (which is often the case) the point in the most active portion is often the only one requiring analysis. These rules do not apply to interference points driven by generic 13 and 23 excitations because their exact locations cannot be pinpointed.

Running speed also helps eliminate interference points. In turbomachines, for example, transmitted torque is proportional to speed squared. Because excitation torques are often a fixed percentage of transmitted torque, if a given mode has interference points at several different speeds, the worst case typically occurs at the highest speed.

Mode number is another important factor. In general, response varies inversely with mode number because lower-number modes contain more energy and are more easily excited. Thus, if two or more modes excite at the same torque and location, only analyze the lower mode.

**Damped-vibration analysis**

If the prior steps reveal no critical interference points, analysis is complete and the machine is satisfactory. It’s more likely, however, that interference points remain which require damped torsional-vibration analysis.

Damped analysis evaluates the machine’s ability to withstand resonance. This involves applying an excitation to the machine and calculating the resulting vibratory torques and stresses for each shaft element. Compare these to allowable values to evaluate the drivetrain’s structural adequacy.

Damped analysis uses the same lumped model as undamped analysis, modified to incorporate damping elements. To convert all inertias, stiffnesses, and damping coefficients in geared systems to equivalent values on the lowest-speed shaft, multiply by the square of the gear ratio.

When using equivalent systems, however, calculated displacements and torques in equivalent elements have no physical meaning. To determine actual torques and displacements in high-speed shafts, reverse the procedure for generating the equivalent system. Convert equivalent displacements and torques at the low-speed shaft through all gear ratios until reaching the desired shaft. Convert displacements by multiplying equivalent values by gear ratio, divide to convert torques.

The first step in performing damped analysis is determining excitation-torque magnitudes. Although drivetrain test data is the best source of this information, the accompanying table lists conservative values for most turbomachinery applications. After determining excitation torques, identify all damping sources in the assembly. External and internal are the two types of damping. Disk elements such as motor rotors or pump impellers are common external dampers. Model these as linear dashpots between the disk and ground with damping proportional to the disk’s absolute velocity as:

*τ _{ext} = c_{ext} × ω*

Viscous-fluid drag on immersed impellers is another external-damping source. This tends to be negligible in pneumatic machines such as turbines and compressors due to the low viscosity of air. Pump impellers immersed in viscous liquids, on the other hand, can provide substantial damping with magnitude typically estimated using conventional pump-windage equations.

Fluid-film journal bearings also produce external damping. Viscous friction in these bearings depends on speed and is, therefore, a theoretical damping source. In most practical applications viscous-friction damping is inconsequential. If lateral motion accompanies torsional vibration, however, it forces fluid in the bearing’s radial clearance to flow circumferentially and generate a squeeze-film effect. Gear meshes exhibit such motion with coupling between lateral and torsional modes. This can cause the bearing to be a significant, even predominant, damping source, producing damping ratios as high as 10% of the critical damping coefficient.

Slip damping is another source of damping. Slip damping occurs at locations such as flanged joints, splines, keyed connections, and shrink fits. It arises from friction between two vibrating surfaces.

Internal damping involves energy dissipation in shaft elements. Shaft-material hysteresis and couplings are examples of internal-damping sources. Model these as dashpots in parallel with corresponding torsional springs (which represent shafts). Internal damping depends on the angular-velocity difference between adjacent disks:

*τ _{int} = c_{int} × (ω_{2} – ω_{1})*

Material hysteresis, a damping mechanism present in all shafts, is energy dissipation when a material undergoes a complete cycle of stress changes within its elastic region. Friction within the shaft causes energy losses released as heat. In elastomeric-damping couplings, this can generate damping as high as 25% of the critical value.

Two common methods quantify component damping. The first, and most straightforward, uses the conventional viscous-damping coefficient from the above equations. Damping is the same for all vibration modes with this method.

A value called the damping ratio can also measure a component’s damping. This is the ratio of damping coefficient to critical-damping coefficient. The latter is the minimum damping value that prevents oscillations when the system is initially displaced, released, and allowed to vibrate freely before coming to rest. For any shaft element the critical-damping coefficient is:

*c _{cr, ij} = 2 × k_{ij} /ω_{n}*

Engineers commonly specify damping with a damping ratio for the entire machine. This is an empirical value, chosen from experience, but commonly 1.0% for ungeared systems. Apply the ratio to every shaft element in the model. The damping coefficient for any shaft element is proportional to its stiffness, therefore absolute damping coefficients for all shaft elements differ from one another and vary with vibration mode.

Because it is difficult to calculate the magnitudes of journal bearing, material hysteresis, and slip damping, also account for these with an empirical damping ratio of 1.0% applied to every shaft element. To account for squeeze-film effects increase the value to 1.5% for geared systems where journal bearings support gearshafts. Calculate all other sources of damping.

One primary source of damping in turbomachinery is the torque versus speed characteristic of the driven load, such as a pump impeller or electric generator. If resisting torque increases with speed, the load is a positive-damping source and damping coefficient is merely the instantaneous value of the torque-speed curve at the operating point.

Note that damping is dissipative only if the slope and corresponding damping coefficient are positive. If slope and damping coefficient are negative, the load enhances vibration and could trigger instabilities. The same holds for system drivers such as motors and turbines. In this case the damping coefficient has equal magnitude but opposite sign to the slope of the torque-speed curve. A driver, accordingly, provides positive damping when slope is negative at the operating point. The sign reverses because the driver is an energy source rather than a dissipater.

When generating torque-speed curves it is critical to hold all relevant parameters constant or misleading damping-coefficient values, such as reversed signs, could result. On impellers, for instance, keep flow rate and all fluid properties constant and hold voltage fixed for motors and generators.

**Checking interference points**

After determining excitation torques and damping coefficients perform damped analysis for each interference point under consideration. Apply only one excitation torque to the model because a specific excitation drives each interference point. If generic 13 or 23 excitations (which could occur anywhere in the system) drive the interference point, apply an excitation torque to the disk with the maximum amplitude in the mode shape. This covers the worst-case scenario.

Run a computerized steady-state damped analysis using a torsional-vibration analysis program. Find twist angle, cyclic torque, and cyclic stress in each shaft element and use these values to evaluate the machine’s acceptability. First check shaft fatigue. Each shaft element is subjected to mean stress from transmitted torque and cyclic stress from torsional vibration. Increase the nominal cyclic stress using stress-concentration factors that reflect material notch sensitivity and the presence of keyways, splines, and fillet radii.

Plot the corrected cyclic stress on a shear Goodman diagram for the shaft material to determine equivalent fully reversing stress. Compare this equivalent stress, in turn, with the material’s shear endurance limit, modified for factors such as size, temperature, and surface finish. Divide endurance limit by equivalent stress to calculate a safety factor. If the resulting safety factor is 2.0 or greater, the shaft is structurally adequate.

The second acceptability criterion compares peak torques with continuous ratings at all couplings and gear meshes. Peak torque is the sum of mean and cyclic torques. If all peak torques are below continuous ratings, the design is acceptable.

The third criterion prohibits torque reversals at all backlash joints, such as gears and splines. Torque reversal occurs when cyclic torque exceeds mean transmitted torque, allowing negative instantaneous torque. When this happens, the drive surfaces periodically separate and pound against each other, leading to tooth pitting, wear, and fracture.

The preceding discussion applies to steady-state damped analysis. There are several situations, however, that require transient damped analysis, such as synchronous-motor startup or a slab entering steel-mill rollers. Most transient torsional analyses use numerical time-marching methods with differential equations integrated numerically and all parameter values defined in terms of their values at the previous time instant. Unfortunately, many of these methods encounter instabilities unless time steps are significantly smaller than the period corresponding to the machine’s highest natural frequency.

This is a problem for machines modeled with many disks because they inherently have many natural frequencies and require infinitesimally small time steps. To avoid this problem, reduce large models to smaller equivalent models with three to five disks. Check this simplified model and alter it to ensure the lowest one or two natural frequencies are unchanged. Then perform the entire transient analysis with the simplified model.

The results of transient analysis consist of dynamic torques and stresses in each shaft element, similar to steady-state analysis. Acceptance criteria, however, are quite different. First, do not compare stress levels to endurance limits because transient cases produce finite numbers of cycles. Instead, from the Goodman diagram, determine equivalent fully reversing stresses for each stress cycle occurring during the transient cycle. Then obtain the allowable number of cycles for each stress condition from an S-N curve for the shaft material. Determine the number of applied cycles for each stress condition by multiplying the number of cycles per transient by the expected number of transients in the drivetrain life. Use the applied and allowable number of cycles in a cumulative-damage algorithm, such as a Miner’s summation, to determine structural adequacy.

Compare peak torques in gears and couplings to the maximum, not continuous, torque ratings, also in contrast to steady-state cases. Finally, there’s no need to check for torque reversals because they usually do not present a problem in transient conditions.

**Eliminating problems**

If damped analysis finds the machine unacceptable, try correcting the problematic natural frequency by altering shaft stiffnesses or disk inertias in active regions of the system. Active regions are defined as regions in the mode shape exhibiting a significant slope (for shaft elements) or amplitude (for disk elements). The designation of such regions is a matter of judgement. Altering elements in inactive regions will have little impact on natural frequencies.

Because the simplest modification is often a coupling change, explore this avenue first. Fortunately, the fundamental mode in many turbomachines (the most common source of problems) twists exclusively in one coupling. Thus, a coupling change often pays dividends. If this alternative is not sufficient, investigate altering other shaft stiffnesses and disk inertias.

Another less-common way to solve the problem is changing the order number of the exciting component by changing a parameter such as the number of impeller blades or gear teeth. This, however, usually hampers performance.

One alternative that is sometimes viable is reducing the excitation magnitude. If the excitation source is a motor, generator, or variable-frequency drive, design changes often reduce torque fluctuations. If the driver is a synchronous motor, switching from a solid pole to a laminated pole design sometimes solves the problem. If a gear mesh produces the excitation, tighter profile tolerances can help.

A last resort is to add damping, such as an elastomeric or hydraulic coupling, to an active region of the system. To maximize the coupling’s effectiveness place it between the excitation source and the vulnerable region. This provides both isolation and damping.

Regardless of the solution, any change creates an entirely new system from a torsional vibration standpoint. Accordingly, repeat the entire procedure with the new system.

For any machine this consists of a continuous oscillation at the excitation frequency superimposed on the steady running speed, N. Consequently, the rotational speeds of the machine’s disks are no longer constant. Instead, their angular velocities sinusoidally vary between N – ω and N + ω. Additionally, in direct contrast to the undamped case, the disks do not vibrate in phase with one another because the system’s damping introduces differing phase shifts to each disk. In the resonant case, which is the primary concern, the response would be infinite if it were undamped. Damping, however, causes response to build up until total energy input to the machine by various excitation sources equals total energy dissipated by the dampers. Steady-state vibration commences at this point. |

The shaft is straight at the node so interference points with excitations near nodes will not cause problems. Vibration hits peak values at one or negative one in the mode shape. |

Additionally, when powering up ac motors, large transient torques occur at line frequency and then quickly diminish. These can be as large as 10.0 pu in induction motors and 8.0 pu for synchronous motors. The transient torque values at right are for reference only and should not be used for design. Consult the supplier for accurate values when dealing with ac motors or variable-frequency drives. |