Figure 5: Campbell (Interference) Diagram Example.
As shown in the Campbell Diagram, there is a potential for resonance while the equipment is operating within the normal speed range that which could be excited by the confluence of the torsional natural frequency and 2X RPM excitation sources. Practically speaking, this would indicate that vibrations occurring at 2X motor speed could excite the torsional natural frequency. Potential sources of this type of vibration would include shaft misalignment. Unless properly diagnosed as a torsional resonance issues, corrective actions might be incorrectly aimed at reducing shaft misalignment. In this example, while improved shaft alignment would result in a decreased excitation source, it will be difficult to reduce the amplitude of torsional vibration due to the nature of resonance.
To avoid amplification of torque due to resonance, American Petroleum Institute (API) recommends a minimum separation margin of 10% between natural frequencies and excitation sources. As shown in the Campbell Diagram, the potential excitation sources are shown in bands to indicate the separation margins. According to API 684, 2005, a torsional analysis should always be performed for new designs. However, this is not widely practiced in other industries and unfortunately torsional vibration might not be considered to be a problem until after a failure has occurred. It should be noted that the safety and reliability of a torsional system is determined by stress, not merely by separation margin from excitation sources that may or may not be of any importance.
While most equipment manufacturers will study the torsional dynamics when the drivetrain is designed, the same analysis is often neglected when drivetrain equipment is upgraded or process conditions are changed to increase plant throughput. The measurement and analysis of torsional vibration (either at the design stage or during retrofit) will help ensure process optimization and equipment reliability by mitigating the risks associated with otherwise potentially harmful operating conditions.
A torsional analysis can have varying levels of complexity. Often, if there is no indication that the natural frequencies will be excited then that is the extent of the analysis. However, due to the uncertainties in the calculations and the potential for off-design operating conditions, it is recommended that the torsional analysis include the calculation of the forced vibration response.
A forced response analysis typically includes the creation of a mathematical model of the drivetrain and the calculation of the resultant stresses from external torques generated from the driving and/or driven equipment. The mathematical model is usually a discrete lumped mass-spring system which represents the major inertias and torsional springs in the drivetrain. Because the motion of one inertia in a drivetrain is dependent upon all of the other inertias, a system of equations based on the laws of motion is developed. Theoretical torques can be applied to the model to simulate different conditions like start-up, impacts, and resonance by altering the time-varying torque function.
The equations of motion for each inertia can be determined through the summation of the forces applied to each inertia as shown in Equations 2 a-b for the two-inertia system shown in Figure 4. From Newton’s universal laws of motion, the summation of forces acting on an object is equal to the object mass multiplied by acceleration. In the torsional reference frame this equates to inertia (J) multiplied by angular acceleration. When an external time-varying torque, T(t), is applied to an inertia there will be equal and opposite reaction forces transmitted throughout the drivetrain in accordance with the laws of motion. The equal and opposite forces are related to the damping and stiffness characteristics of the connecting shafts. Equation 2c shows the system of equations which is then utilized to simultaneously solve for the angular displacement (θn) as a function of time for each inertia. The resultant torque at each shaft location can then be calculated because torque is equal to the relative angular displacement across inertias connected by a known spring stiffness (Equation 2d).
THE REPORT | SEP 2024 | ISSUE 109 | 69
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