Spotts, Marks’, or Shigley. Please recall that the developed equations are based upon a circular cross section. If the cross sectional area is not circular, then the equations must be modified based upon the original integrals used to define iner-tia. Often this type of calculation is not practical due to the complexity of the mechanical part. In these cases, the diagnostician must resort to other
tech-Iarea
niques to determine the inertia properties of the machine element.
In the same way that a scale may be used to determine the weight of a rotor, there are experimental techniques that may be applied to determine iner-tia properties. For example, consider the machine part shown in Fig. 3-12. This could be a blank for a bull gear or a flywheel, or any other machine element for which a mass polar moment of inertia may be required. Due to the complexity of the part it may not be feasible to calculate the inertia directly, but it is possible to experimentally determine the inertia.
One technique consists of suspending the part from a horizontal support as shown in Fig. 3-12. Ideally, this support member should be a knife edge, but more realistically, it will probably be a solid circular rod as depicted. If the mass polar moment of inertia through the axial centerline of the element is desired, then the distance between that centerline and the supporting pivot point Xoffset must be accurately measured. In addition, the part should be weighed to determine the total weight W in pounds. The machine element may now be rocked back and forth as a compound pendulum. By measuring the period of the motion, the iner-tia about the pivot point may be determined. This offset ineriner-tia may now be translated back to the centerline of the machine part by applying the parallel axis equations previously developed.
In actual practice, the Inertia of this compound pendulum may be deter-mined from Marks’ Handbook8. Extracting the appropriate equation, and placing it in terms used within this text, the following equation for overall inertia about the pivot point may be stated:
(3-34) Fig. 3–12 Mechanical
Arrangement For Rocking Test To Determine Mass Polar Moment of Inertia
8 Eugene A. Avallone and Theodore Baumeister III, Marks’ Standard Handbook for Mechani-cal Engineers, Tenth Edition, (New York: McGraw-Hill, 1996), p. 3-16.
Element Weight (W)
Rocking Oscillation 40° to 60°
Fixed and Rigid Support X0ffset
Jz
ozo
W×Xoffset×Period2 4×π2
---=
Inertia Considerations and Calculations 83
If equation (3-34) is inserted into the axis translation equation (3-21), the following is obtained:
Factoring out the common terms, the above may be simplified to:
(3-35)
This is a useful expression since all of the variables are easily determined.
Specifically, the weight W of the machine part can be measured on a scale, and the distance between the pivoting point and the geometric center of the element Xoffset is easily measured. The acceleration of gravity G is constant, and the Period of the swinging motion is determined with a stopwatch in seconds. Nor-mally, a series of runs are made to determine the average period of oscillation.
Experimental techniques are always more credible if the validity of the equations, and the experimental procedures can be verified with a controlled test. With an inertia experiment, the object to be tested should be of simple geometry such as the solid cylinder shown in Fig. 3-13. As indicated on the dia-gram, the average cylinder diameter D is equal to 6.312 Inches, and the average length L is 3.735 Inches. A shop scale indicated that the weight W of the
speci-Fig. 3–13 Rocking Test To Determine Mass Polar Moment Of Inertia Of A Solid Brass Cylinder
men was 36.0 ±0.1 Pounds. Since the cylinder was solid brass, the density was found in Appendix B of this text to be 0.308 Pounds/Inch3. Two holes were drilled and tapped into the side of the cylinder, and two screw eyes were inserted to pro-vide the pivot arm. Since the mass polar moment of inertia along the axial cen-terline is desired, the distance between the pivot point and the cylinder center line Xoffset was measured to be 5.07 Inches.
Before starting the experiment, it is desirable to check the physical proper-ties of the brass cylinder. For example, equation (3-7) may be used to compute the weight of the brass cylinder based upon the average dimensions, and the density of the material as follows:
The calculated value shows excellent agreement with the measured weight of the cylinder, and 36.0 pounds may be confidently used for the ensuing inertia measurements. Some individuals might argue that this type of weight check is unnecessary. However, if the material contained inclusions, or if the density was wrong, or if one or more of the dimensions were incorrect — the weights would not match, and the experimental accuracy would be in jeopardy. So a simple cal-culation such as this weight check is desirable to insure that the physical param-eters are in unison. The other calculation that should be made at this stage is the mass polar moment of inertia of the brass test specimen. From the previously developed equation (3-32) the polar moment of inertia of this brass cylinder is computed in the following manner:
At this point the test calibration setup is established, the final answer is known, and the only remaining variable to be defined for equation (3-35) is the period of the oscillatory motion. Several preliminary swings of the brass cylinder revealed that the time for one complete cycle was less than a second. In addition, the friction between the two eye bolts and the support rod caused the oscillating mass to grind to a stop after only a few cycles. This problem was partially reme-died by putting a tight plastic sleeve on the support rod, and then covering this surface with lithium grease. This friction reduction effort was rewarded by a sub-stantial increase in the number of possible oscillatory cycles. However, it was then discovered that the horizontal support rod was not quite level, and the
Wsolid π ×L ×ρ×D2 ---4
=
Wsolid π×3.735 Inches×0.308 Pounds/Inch3×(6.312 Inches)2
---4 35.997 Pounds
= =
Jmass
solid
π ×L ×ρ×D4 32×G
---=
Jmass
solid
π×3.735 Inches×0.308 Pounds/Inch3×(6.312 Inches)4 32×386.1 Inches/Second2
---=
Jmass
solid
π ×3.735 ×0.308×1 573.33, 32×386.1
--- 0.464 Pound-Inch-Second2
= =
Inertia Considerations and Calculations 85
brass cylinder had a tendency to walk down the rod. This problem was corrected by re-leveling the support rod and the cylinder.
Following these test setup modifications, the actual test was conducted. The brass cylinder was displaced about 30° from the vertical centerline and released.
The peak of the motion at one extremity was visually sighted, and a stopwatch was used to measure the time required for multiple back and forth cycles. The final measured test data is summarized as follows:
Rocking Run #1 7.83 Seconds for 10 cycles Rocking Run #2 7.95 Seconds for 10 cycles Rocking Run #3 7.98 Seconds for 10 cycles Rocking Run #4 8.09 Seconds for 10 cycles Rocking Run #5 7.79 Seconds for 10 cycles Rocking Run #6 7.90 Seconds for 10 cycles Rocking Run #7 7.89 Seconds for 10 cycles Rocking Run #8 7.94 Seconds for 10 cycles Rocking Run #9 7.93 Seconds for 10 cycles Rocking Run #10 7.89 Seconds for 10 cycles Total Time = 79.19 Seconds for 100 cycles Average Time for 1 Cycle = 0.7919 Seconds = Period
It is easy to lose track of the cycle count, or miss a timing point, and negate the accuracy of a data set. These types of errors are evident during the data col-lection work, and erroneous times are identified and discarded. For instance, approximately twenty runs were made to collect the data in the above tabular summary. Ten of the timing runs were not used due to obvious errors in the data accumulation. The ten acceptable test runs reveal an average period of 0.7919 Seconds. This is considered to be a consistent value, and the experimental mass polar moment of inertia may now be computed from equation (3-35).
The experimental value for the polar inertia is 0.502 versus the calculated value for this brass cylinder of 0.464 Pound-Inch-Seconds2. The difference of 0.038 represents an 8% error of the experimental versus the computed value. In some instances this level of deviation is perfectly acceptable. For example, if the part under test was a coupling hub that will be mounted on a power turbine with an inertia of 40.0 Pound-Inch-Seconds2, the small differential of 0.038 Pound-Inch-Seconds2 would be insignificant. However, if the part under test was one of
Jzz W×Xoffset Period
Jzz = 182.52 Pound-Inches×(0.01588–0.01313)Second2 = 0.502 Pound-Inch-Second2
eight impellers to be mounted on a slender shaft, the cumulative error may be unacceptable. In order to explain this error, it is necessary to re-examine the component equations used for the inertia test calculation. Specifically, (3-34) for the total inertia about the pivot point may be solved as follows:
Substituting the test inertia from the above calculation back into equation (3-21), and performing the axis translation, the following result is obtained:
As expected, this result is identical to the previous answer obtained by using the composite equation (3-35). The interesting point of the above calcula-tions is that inertia due to the axis translation is equal to 2.397, versus the over-all test inertia of 2.899 Pound-Inch-Seconds2. Hence, the cylinder inertia is only about 20% of the inertia due to the axis translation. This is not a desirable condi-tion since the axis translacondi-tion is the dominant term. The geometrical configura-tion displayed in Fig. 3-12 would not be as error prone since the XOffset distance resides within the body of the element, and the axis translation term would not dominate the test. Hence, the diagnostician should always be concerned about trusting this type of experimental inertia test with a long XOffset distance between the pivot axis and the desired principal polar moment of inertia axis. The other lesson to be learned is that simplified expressions such as (3-35) may not provide full visibility concerning the potential accuracy of the final results. In some cases it is necessary to revert back to the basic equations, and reexamine the entire calculation and/or experimental test procedure.
With respect to the brass cylinder, it is concluded that improvement of the test accuracy will require a reduction or elimination of the axis translation term.
This could be accomplished with a test that consisted of suspending the mass from cables, and then measuring the period as the mass oscillated in a twisting manner (Fig. 3-14). Since the axis of oscillation is the axial geometric centerline of the element, there is no axis translation involved, and test accuracy should be improved. This type of inertia test is ideal for machine parts such as compressor impellers or turbine disks that contain complex geometrical cross sections. For
Jz
ozo
W×Xoffset×Period2 4×π2
---=
Jz
ozo
36.0 Pounds×5.07 Inches×(0.7919 Seconds)2 4×π2
--- 2.899 Pound-Inch-Second2
= =
Jz
ozo = Jzz+M×(Xoffset)2 2.899 Pound-Inch-Second2 Jzz W
---G×(Xoffset)2 +
=
Jzz 2.899 Pound-Inch-Second2 36.0 Pounds 386.1 Inches/Second2
---×(5.07 Inches)2 –
=
Jzz = (2.899–2.397) Pound-Inch-Second2 = 0.504 Pound-Inch-Second2
Inertia Considerations and Calculations 87
instance, the internal star pattern shown in Fig. 3-14 might be difficult to model with an equivalent inner diameter for the machine part.
For this procedure, the machine element is suspended from three thin cables (3 points determine a plane), spaced at 120° apart. The test piece must be leveled as precisely as possible. If it is not level, then any induced twisting oscil-lations will cause the machine element to wobble during the test. This wobble not only negates the test accuracy, it can prove to be dangerous for parts with any appreciable physical size and weight. After leveling, the average suspension cable length Ls-c and the cable radius Rs-c are accurately measured and recorded.
For best results, each of the suspension cable lengths should be equal, and the radius for all three cables should be identical. As before, the machine element to be tested is weighed on an accurate scale in English units of Pounds.
During execution of this test, the machine element is manually displaced in a twisting manner, and released. The machine part will torsionally twist back and forth, and the period of the twisting oscillations will be measured with a stopwatch. Since friction should not be major problem, the part will oscillate back and forth for many cycles. It is not unusual to observe thirty or more cycles resulting from one initial displacement. Based upon these measured parameters the mass polar moment of inertia may be computed as follows:
(3-36)
The general form of (3-36) was extracted from the Shock & Vibration Hand-book9,and it was converted to the nomenclature used in this text. As previously
Fig. 3–14 Mechanical Arrangement For Twisting Test To Determine Mass Polar Moment of Inertia
9 Cyril M. Harris, Shock and Vibration Handbook, Fourth edition, (New York: McGraw-Hill, 1996), p. 38.5.
Element Weight (W) Suspension Cable Length (Ls-c)
Element Axial Centerline Suspension
Cable
Twisting Oscillation 30° to 40°
Radius (Rs-c)
Jzz W×Rs2–c×Period2 4π2×Ls–c
---=
discussed on the rocking inertia test, it is mandatory to validate the test proce-dure with an actual test on a known geometric shape. For comparative purposes, the solid brass cylinder used for the rocking test will be used for this twisting inertia test as shown in Fig. 3-15. From this diagram it is noted that the average suspension cable radius Rs-c was 3.00 Inches, and the average cable length Ls-c was 33.73 Inches. As before, the total cylinder weight was 36.0 Pounds.
The first test configuration used 48 Inch long suspension cables in an effort to increase the period of the oscillation, and improve the time measurement accuracy. Conceptually this was a good idea, but it turned out to be impractical since the long cables had a tendency to wrap around each other. This proved to be an unmanageable situation, and the support cable lengths were reduced to 33.73 Inches. During the acquisition of test data, the cylinder was twisted about 20°
circumstantially from rest, and released. The peak of the motion at one extrem-ity was visually sighted, and a stopwatch was used to measure the time required for complete back and forth cycles. The test data is summarized as follows:
Twisting Run #1 13.83 Seconds for 10 cycles Twisting Run #2 14.23 Seconds for 10 cycles Twisting Run #3 14.15 Seconds for 10 cycles Twisting Run #4 13.94 Seconds for 10 cycles Twisting Run #5 14.02 Seconds for 10 cycles Twisting Run #6 13.89 Seconds for 10 cycles Twisting Run #7 14.01 Seconds for 10 cycles Twisting Run #8 14.11 Seconds for 10 cycles Twisting Run #9 14.05 Seconds for 10 cycles Twisting Run #10 13.95 Seconds for 10 cycles Total Time = 140.18 Seconds for 100 cycles Average Time for 1 Cycle = 1.4018 Seconds = Period Fig. 3–15 Twisting Test To
Determine Mass Polar Moment Of Inertia Of A Solid Brass Cylinder
Brass Cylinder
Weight (W=36.0 Lbs.)
Suspension Cable Length (Ls-c=33.73") Axial Centerline
Suspension Cable Radius
Twisting Oscillation 30° to 40°
(Rs-c=3.00")
Inertia Considerations and Calculations 89
This is a much smoother test than the rocking inertia previously discussed.
The number of miscounts and aborted runs were substantially reduced, and approximately fifteen runs were made to collect the data shown in the above tab-ular summary. Five of the timing runs were not used due to obvious errors in data accumulation. The ten consistent test runs reveal an average period of 1.4018 Seconds. This was considered to be a consistent value for this experimen-tal procedure, and the mass polar moment of inertia may be computed from equation (3-36) as follows:
The experimental polar inertia from this twisting procedure of 0.478 is quite close to the previously calculated value of 0.464 Pound-Inch-Seconds2. The 3% deviation is quite acceptable for most rotor dynamics calculations. This is particularity true for smaller components that are stacked on a shaft to achieve a final rotor assembly. It should be recognized that both the rocking and the twisting inertia tests have their own domain of application that is dependent on the size and geometry of the machine element.
Just as the weights of individual components are summed up to determine a total rotor weight, the inertia of the component pieces may be added to deter-mine the overall rotor polar inertia. The origin of the inertia values may be from calculations of defined geometries, or from experientially determined inertia val-ues. In any case, as long as the engineering units and the inertia axis are com-mon, the numeric inertia values may be summed up to determine the mass polar moment of inertia for the entire rotating assembly.
In some instances, there is minimal opportunity to determine the inertia of rotor components since the unit cannot be disassembled or unstacked. In these situations, the general inertia characteristics may be estimated based upon available dimensions and probable materials of construction. In other cases, the complexity of the rotor may not allow a reasonable segmentation and estimation of inertia properties. This is particularly true for rotors that are constructed of multiple materials, plus rotors that contain complicated geometric configura-tions. In these instances, another experimental technique may be employed to determine the overall mass polar moment of inertia of the rotor.
This technique is based upon the familiar college physics experiment depicted in Fig. 3-16. In this diagram, a cylinder or drum is mounted in rigid bearings that allow rotation of the cylinder, but restrict any lateral or translation movement of the cylinder. A cord is wrapped around the cylinder at a shaft radius of Rs. It is assumed that this cord is of insignificant weight and diameter, and that it will not stretch with the application of axial tension. Next, a known weight (mass M) is attached to the end of the cord, and allowed to free fall. The
Jzz W×Rs2–c×Period2 4π2×Ls–c
---=
Jzz 36.0 Pounds×(3.00 Inches)2×(1.4018 Seconds)2 4π2×33.73 Inches
--- 0.478 Pound-Inch-Second2
= =
time T required to fall a distance D is measured with a stop watch. The experi-ment normally consists of determining the cylinder mass polar moexperi-ment of iner-tia Jmass based on the four known quantities of radius Rs, mass M, fall distance D, and the average fall time T.
This basic physics problem may be solved by constructing free body dia-grams of the cylinder and the falling mass, developing equations of motion, and then solving for the polar inertia term. Another way to achieve the same result is by performing an energy balance on the system shown in Fig. 3-16. Conservation of energy requires that the change in potential energy is equal to the change in kinetic energy. In this case, the change in potential energy is simply the eleva-tion change in the mass (M x G x D). The overall change in kinetic energy is com-posed of a change in translational energy of the falling mass (M x V2/2), plus the change in the rotational kinetic energy of the rotor (Jmass ω2/2). These traditional physical concepts may be represented mathematically in the following manner:
(3-37)