Troubleshooting a transfer matrix program
Comparison of Rotosolve spreadsheet calculated results and analytically-calculated results
I. Simple Beam Cases (calculation option 3)
Beam cases are easy to check since a simple analytical solution is readily available.
The cases in this section are called "simple" because they used the option 3 "simple" for the gyroscopic option, which means that all disk effects are ignored.
The program provides choices for boundary conditions: "hinged", "clamped", "free", which have the same meanings as in normal textbook beam calculations.
The beam was broken into 10 pieces and the lumped-MASS calculation option was set to false (continuos-mass calculation).
In all these cases, the following beam parameters were used:
Length = 1 meter
Diameter = 0.1 meter
Density = 7750 kg/m^3 (more specifically, 7750.37312)
E = 2.31E11 N/m^2 (more specifically, 230974359500)
The thee example simple beam cases were all generated with the inputs shown in the following file: SimpleBeamDemo.xls
A. Free/Free beam,
Geometry:
Free
Free
Results (analytical calculations shown in tab labeled "analytical calculation")
Boundary Conditions: freefree
Frequency
Analytical Calculation
Program Output
1
485.971
485.971
2
1,339.598
1,339.598
3
2,626.148
2,626.148
Mode shapes
Mode shape plot (brg stiffness multiplier =1)
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
1
1.5
x = axial distance (meters)
relative amplitude
f1=485.97hz
f2=1339.6hz
f3=2626.15hz
B. Clamped/Free beam (cantilevered)
Geometry as follows:
Free
Results as follows:
Boundary Conditions: clampedfree
Frequency
Analytical Calculation
Program Output
1
76.372
76.372
2
478.612
478.612
3
1,340.128
1,340.128
Mode shapes as follows:
Mode shape plot (brg stiffness multiplier =1)
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
1
1.5
x = axial distance (meters)
relative amplitude
f1=76.37hz
f2=478.61hz
f3=1340.13hz
C. Hinged-hinged (Simply-supported) beam
Geometry as follows:
Results as follows:
Boundary Conditions: hinged/hinged
Frequency
Analytical Calculation
Program Output
1
214.378
214.377
2
857.512
857.410
3
1,929.403
1,928.105
Mode shapes as follows:
Mode shape plot (brg stiffness multiplier =1)
-1.5
-1
-0.5
0
0.5
1
1.5
00.511.5
x = axial distance (meters)
relative amplitude
f1=214.38hz
f2=857.41hz
f3=1928.11hz
II. Adding concentrated mass along with distributed mass.
M
M
m
The attached file recreates the well-known scenario where a simply-supported beam with distributed beam mass m and concentrated mass M in the center causes a resonant frequency based on an effective stiffness of 48EI/L^3 and an effective mass of M + 0.5*m.
M_PLUS_Halfm.xls
Again the program calculates the result as expected.
III. Introducing bearings
Most real-world rotors will be modeled as free-free boundary conditions. It is a very simple matter to recreate the simply-supported results above by adding an additional 0-length section on the right, adding bearings to the left of the first and last rotor sections, and setting the bearing stiffness values very high, as was done in the following file: SimplySupportedFromFreeFreeWithBearings.xls
K_bearing=infinity
An examination of the critical speed map and the modeshapes in the above file confirms that the high bearing stiffness causes the bearings to act like rigid supports,
The frequency results match the simply-supported results above, as expected:
Results
Frequency
Analytical Calculation
Program Output
1
214.378
213.906
2
857.512
850.231
3
1,929.403
1,894.069
IV. Varying the bearing stiffness.
The model studied is the same as above, except that we have introduced a variable bearing stiffness as follows:
K_bearing=variable
The above system is solved in the attached file: DemoShaftOnBearings2.xls
The same geometry was also solved in "Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis" By Dara Childs, page 123, which can be accessed here:
http://books.google.com/books?id=vKPfBxgQQPoC&pg=PA123&dq=%22These+modes+are+commonly+referred+to+as+stick+modes%22&sig=KKcf-5urzLoMB50PNymoxQlBzv8
A visual comparison of the critical speed maps generated by my spreadsheet with those provided by Childs shows good agreement:
My spreadsheet
Childs
Crit Spd Map Rad/sec
1.E+02
1.E+03
1.E+04
1.E+05
0.010.1110100
Brg Stiffness Multiplier (multiple of Kbrg=48EI/L^3)
radians/sec
first nat freq
second nat freq
third nat freq
Looking toward the left of the critical speed map, we suspect that the first two modes increasing linearly on log-log plot with a slope of 0.5 are rigid-rotor modes. We confirm this with a modeshape plot of these first two frequencies (bearing multiplier 0.01):
Mode shape plot (brg stiffness multiplier =0.01)
-1.5
-1
-0.5
0
0.5
1
1.5
00.511.5
x = axial distance (meters)
relative amplitude
f1=21.15hz
f2=36.77hz
With the rigid rotor simplification, we can analytically verify the first two resonant frequencies at a bearing stiffness multiplier of 0.01.
For the first rigid rotor mode at stiffness multiplier 0.01,
Kbrg = 0.01 * 48*E*I/L^3 = 543338. N/m
M = pi * rho * L * (od^2-id^2)/4 = 60.87 kg
F = sqrt(2*Kbrg/M) / (2*pi) = 21.27hz (matches program-calculated 21.15 very well).
For the second rigid rotor mode at stiffness multiplier of 0.01, we calculate the transverse or diametrical mass moment of inertia of the shaft (assuming all mass concentrated on the centerline... consistent with calculation mode 3) as follows:
)
(
4
1
Id
4
4
inner
outer
R
R
L
-
=
p
r
Id = 5.072363139 kg*m^2
The max kinetic energy is KE = 0.5 *Id2 * ((')2
For small angles, ( ~ sin(() = y / (L/2) (where y is transverse displacement at location of the bearing.
(' = y' / (L/2)
Substitute into KE equation:
KE = 0.5 *Id * (y')2 /(L/2)2
We can preserve kinetic energy by rewriting the above as
KE = 0.5 * Meffective * (y')2 = 0.5 * Meffective * v^2
where Meffective = Id / (L/2)2 = 20.28945255 kg
Thus from an energy standpoint, the rotary inertia Id acts like an effective mass Meffective=Id /(L/2)2 at the location of one of the bearings. Since the fundamental frequency can be calculated from energy considerations (KEmax=PEmax), we can calculate the resonant frequency using this effective mass. (This approach of calculating an effective mass based on energy considerations is described in Thompson's "Mechanical Vibrations" section 2.3 or Rao's "Mechanical Vibrations" example 1.6). The relevant spring stiffness includes both bearings in parallel. The frequency is
f = sqrt(2*Kb/Meffective) /(2*pi) = 36.83283813 hz. This matches the program output 36.77409375 very well (2nd mode for bearing multiplier of 0.01).
Looking toward the right side of the critical speed map toward stiffness multiplier of 100, we see a leveling of the first and second modes. We suspect these reprsent the first and second flexible rotor modes.
For the first mode at bearing stiffness multiplier 100, The modeshape plot appears to confirms a flexible rotor/rigid bearing mode.
Mode shape plot (brg stiffness multiplier =100)
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
00.20.40.60.811.2
x = axial distance (meters)
relative amplitude
f1=213.61hz
We can check the frequency for this mode (first mode at multiplier =100) using a simply-supported beam calculation, which gives a frequency of 214.21 hz. This is reasonably close to the program-calculated resonant frequency of 213.6hz.
For the second mode at bearing stiffness multiplier 100, The modeshape plot appears to confirm a second flexible rotor/rigid bearing mode.
Mode shape for f= 842.86 hz and Kbrg
multiplier = 100
-25
-20
-15
-10
-5
0
5
10
15
20
25
0
0.2
0.4
0.6
0.8
1
1.2
x = axial distance (meters)
y = transverse displacement (no units)
The theoretical modeshape of this second flexible rotor mode is sin(2(*x/L). Examining the area between x=0 and x=L/2, we find it has the same modeshape as the first flexible-rotor/rigid-bearing modeshape of a beam of length L/2. We can use this information to analytically confirm our frequency based on a simply supported beam of length L/2. The analytical solution of the half-length simply-supported beam gives 856.8hz, while the program predicts the second mode at bearing stiffness multiplier of 100 to be 842.4 hz. The small difference can be reconciled by noting that the mode shape does not come completely to 0, so there is some flexibility still present in the bearings (even at bearing stiffness multiplier of 100) which reduces the resonant frequency.
V. Adding tilting disk effects (like bump test scenario) Calculation option 2
If we bump test a rotor with a large disk (especially overhung), the disk inertia causes the natural frequency to lower by virtue of the fact that a moment must be exerted to tilt the disk back and forth. This effect is called "rotary inertia" in beam theory, even though it is not the way we would normally use the word "rotary".
The "simply-supported" beam scenario above (1 meter beam, 0.1 m diameter, etc) was run again using option 2 to add the effects of rotary inertia. The results are shown in this file SimplySupportedWithRotaryInertaVsRaoGood.xls
As shown in the "Analytical check" tab, Rao's "Mechanical Vibrations" provides a formula for calculating the natural frequencies for this geometry (simply-supported uniform beam) including the effects of rotary inertia. The first three resonant frequencies are shown below using simple analytical calc, adding rotary inertia, and comparing to program output:
Analytical, no rotary inertia (simple calculation)
Analytical, with rotary inertia
Program output (with rotary inertia=option 2)
f1
214.3781239
213.7199712
213.7202468
f2
857.5124957
847.1251912
847.1172571
f3
1929.403115
1877.977813
1877.775498
The rotary inertia does not play a very important role in the first mode for this geometry, but increases in importance as the higher order modeshapes introduce more nodes and more tilting. The program matches the analytical calculation very well, even at the higher mode numbers. When I increased the number of elements from 10 to 200 while keeping the total length the same, the program results matched even better (program computed f3=1877.977).
In general, we suspect rotary inertia will play in important role when there are large disks and lots of bending/tilting at the location of the disks.
VI. Gryoscopic Effects calculation option 1
While the disk-tilting effects of option 2 tend to lower critical speed, the gryoscopic effects tend to increase critical speed. Option 1 includes both effects. This results in higher critical speeds than the simple mode (no disk effects) since the increase caused by gyroscopic effects is larger than the decrease caused by the disk tilt effects.
Rigid-rotor gyroscopically-stiffened whirl
"Formulas For Stress, Strain, And Structural Matrices", 2nd ed. by Walter D. Pilkey Table 17-1 gives the following solution for the resonant frequencies of a rigid rotor with a center disk having significant polar and transverse inertia.
This was simulated in the following file RigidRotorGyroDemoWorks1.xls
The mode shapes are as follows:
Mode shape plot (brg stiffness multiplier =1)
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.2
0.4
0.6
0.8
x = axial distance (meters)
relative amplitude
f1=11.77hz
f2=241.85hz
Pilkey's wc1 corresponds to the first mode f1 at 11.77hz where the rigid rotor moves parallel to it's axis with a very simple solution w=sqrt(Ktotal./M).
Pilkey's wc3 corresponds to f2 and represents the forward-rotating gryoscopically-stiffened mode whose 3-d modeshape would resemble two cones with their points meeting at the center of the rotor. Note that for problems involving gryoscopic stiffening, the whirl speed changes as a function of machine speed. Therefore the analytical solution for the critical speed wc3 requires solving an implicit relationship to find the speed where the whirling frequency is equal to the machine speed (as is typical of most problems that include gyroscopic stiffening).
The parameters used for the simulation are defined in the file. The analytical calculations are shown in the analytical tab. The program results match the analytical predications reasonably well:
Analytical
Program
f1
11.7775
11.77269
f2
244.7908
241.8507
VII. Overhung rotor solved/checked for all three calculation modes
This is intended to model the an overhung rotor described in the thread "Gyroscopic effect", at
http://maintenanceforums.com/eve/forums/a/tpc/f/3751089011/m/9291040423/p/1
(except that the smaller disk and shaft stub on the left is omitted for simplicity of the analytical solution).
The rotosolve spreadsheet solution is here: OverhungRotor.xls
The geometry looks as follows:
Rotor Geometry (rotor in horizontal position with only upper half showing)
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.200.20.40.60.81
x = axial distance (meters)
y = profile
od profile
id profile
attached disk od profile
attached disk id profile
Bearing 1
Bearing 2
A complete analytical solution of this geometry (mode 3= simple/point mass, mode 2 = bump test, and mode 1 = critspd) is provided in the following file: FindAlphaR1a.pdf
Critical Speed (like calc option 1)
Bump test frequency (like calc option 2)
My rotosolve spreadsheet was run on the same model, and the results are compared below:
Analytical Solution (hz)
Program output (hz)
Mode 1 = simple / point mass neglects all disk effects
47.2
47.8
Mode 2 = bump test includes disk effects but neglects gyro
~ 35 hz
36.12
Mode 3 = Critical speed includes gryo and disk effects
~59.5
60.44
The mode shapes for the three calculation modes are as follows:
Calc Mode 1 (crit speed)
Calc Mode 2 (bump test)
Calc Mode 3 (simple)
Mode shape plot (brg stiffness multiplier =1)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
00.20.40.60.8
x = axial distance (meters)
relative amplitude
f1=60.44hz
Mode shape plot (brg stiffness multiplier =1)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
00.20.40.60.8
x = axial distance (meters)
relative amplitude
f1=36.12hz
Mode shape plot (brg stiffness multiplier =1)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
00.20.40.60.8
x = axial distance (meters)
relative amplitude
f1=47.82hz
This geometry was checked with the Critspd program, and similar results were obtained.
_1260715176.ppt
K_bearing=variable
_1278084296.xlsChart1
-1-1-1
-0.5371643859-0.2274294260.0519642229
-0.09772682630.39724916670.6428633084
0.27200427410.66201187450.3969312004
0.52024744110.4830290696-0.3278434515
0.60782222940.0000002908-0.7111906071
0.5202476282-0.4830286686-0.3278437995
0.2720046176-0.66201191630.3969308918
-0.0977263783-0.39724967840.6428634009
-0.53716389030.22742863450.0519646904
-0.99999949650.9999991536-0.9999994428
f1=485.97hz
f2=1339.6hz
f3=2626.15hz
x = axial distance (meters)
relative amplitude
Mode shape plot (brg stiffness multiplier =1)
Instructions
Approach
Break your rotor shaft into pieces.
Number the pieces from left (1) to right (n). (these will be entered in rows from top to bottom in RotorSections tab)
For each piece you can specify the length, i.d., o.d. density, young's modulus
For each piece, you can also specify an "attached disk" with associated rho, id, od (same length as shaft piece). It has mass and MOI, but does not contribute to stiffness
If you need a bearing on the extreme right of your rotor, you'll need to add a zero-length piece on the right so you can put that bearing on the left of it.
(see example of simply-supported beam)
How many pieces should your break it into?
For most purposes, breaking the rotor into pieces whose length is the diameter of the shaft is more than enough.
(that algorithm uses distributed stiffness and mass)
(the algorithm uses lumped polar inertia model)
(the modeshape plot examines position of boundary of each piece, not within a piece
Example - a 1-section model will accurately determine all frequencies of a uniform rotor, but will generate a mode-shape of a straight-line plot connecting the ends
Input
Rotor Sections sheet
enter the rotor data (in mks units). Start at top of RotorSections sheet corresponds to left side of rotor.
a button is provided four your convenience to transfer default values from main tab if desired (E and rho for all sections, Kbrg for sections with bearing, and rho_disk for sections with attached disks)
If you want to change parameters like rho or E between sections, you can't use the button
a button is provided to generate a crude plot of your rotor geometry - this helps you double-check that the program interprets your data (bearings, etc) in the same position that you intended
The triangles represent bearings with specified stiffness, not hinged boundary conditions (boundary conditions are not shown on the plot)
You can customize the spreadsheet to add extra columns for your own info if desired
if you want to add into the rotorsections tab a column calculating the the x (axial) coordinate of LHS of each section, type =+SUM(B2:$B$2) in cell M3 and copy it down column M
You may find it handy to add a column off to the right calculating the mass of your shaft sections and attached disks. Of course M=rho*L*pi*()*(OD^2-id^2)/4
* Don't add anything below your data in columns A, B, or H. - this will confuse the formula for number of sections (formula in Main:C2)
Main sheet
Enter the green parameters
Verify the number of sections is right (it should be filled in by formula)
Enter frequency sweep parameters: fstart, fstop, nfsteps. freqtol
fstop=1 (very low) is recommended so program identifies low frequencies. Critical speed map and modeshape labeling (f1=, f2= etc) assume no low frequencies are missed
Identify your option for gyroscopic parameters, 1 2 or 3:
1-Critical Speed calc - include both rotating disk effect and tilting disk effect
2 -Bump test calc - Includes only tilting disk effect
3 -Simple calc - neglects both disk effects. i.e neglects polar and diametral MOI, as if all mass is concentrated at the centerline of the shaft
Identify whether you want to use a bearing stiffness multipliers (true or false)
Enter bearing stiffness multipliers in the specified format, for example '{0.1, 0.2, 0.5, 1, 2, 5, 10}
For each bearing stiffness multiplier, critical speeds will be found and recorded
Select lumped-mass model (true or false)
In general select FALSE. Select true if you get a numerical error
THIS DOES NOT AFFECT HOW YOU INPUT YOUR DATA INTO THE PROGRAM (mass is always entered as continuous vis the rho and rhodisk parameters)
Both models treat gyroscopic properties as lumped (calculated based on input rho, od, length)
Both models treat stiffness as distributed
For rotors without significant gyroscopic effects (no large-diameter disks), the continuous-mass option can give accurate frequency with very few sections
However if you want an accurate mode-shape, you still need to break up into more sections since mode shape is only calculated at boundary betw sections
For some extreme input parameters (for example very high rho combined with low E and I), the continuous model generates numerical error messages due to use of cosh and sinh functions (try cosh(800) in excel
lumped mass model seems to run a little faster
The distributed-mass option will generate an error if you use density=0, the lumped-mass operation will not
Run Critical Speed analysis
Main Sheet - press the grey button labeled "run"
Note that previous results in outsheet tab will be deleted and overwritten. Previous modeshape plots will remain but don't update with the new paramters
View results
outsheet
outsheet will contain a row for each bearing stiffness multiplier, with the critical frequencies listed to the right of the multiplier
Generate Modeshape plots
outsheet tab
You should have a "fresh" set of calculated frequencies (i.e. you you should not have changed input parameters since the frequencies were generated)
Combining old frequencies with new input data will give you garbage
If you have changed input parameters, push the "run" button in "main" tab again to generate fresh set of frequencies
Put your cursor in a cell which contains a frequency of interest.
Press the grey button for modeshape
The program will create a new sheet with a modeshape plot for that critical frequency and bearing stiffness multiplier
Repeat as many as desired. - each one creates a new sheet
Generate Critical Speed Map
outsheet tab
Press the critical speed map tab
New rotor?
If you want to go back and change your rotor definition in "main" and "rotor sections" tabs, be careful - your modeshape plots will not update.
Recommend either: 1 - delete all modeshape sheets or 2 - save old results and start again using a new file
If you get an error message
There are two types
Error Messages from my program - just a message with an "ok" button at the bottom. Read the message
Usually I am telling you there is a problem with your input
Often an excel message will come just after my message. Pay attention to both error messages.
Error Messages from excel - usually has choices "end" or "debug" at the bottom.
"End" will terminate the program
"debug will put you in editor mode and allow you to see what step of the program is causing problems, as well as examine variables in the "locals" window
To terminate the program (necessary before continuing), select "reset" from the "run" menu
To return to excel select "close and return to excel" from the "file" menu
If you got an earlier message from me that you didn't pay attention to, run the program again and read it carefully.
If you are using lumped-mass calc=false, try switching to true
Other misc and minor glitches
When you rerun the program with a new scenario, the old modeshape sheets will be unchanged. Don't be fooled thinking they show the new scenario
Either delete them, rename them so you know what they are, or start a new file
If you change the input model parameters between generating the frequencies and generating modeshapes, modeshape will not be valide/
Modeshape module uses both the frequency and the model as an input - combining a frequency with a model that didn't generate the frequency gives garbage
Blanks on the rotorsections sheet are interpretted as zero's
All inputs are to be mks
* If you press a grey buttons and nothing happens, it's likely that the vba program is hung up in debug mode. This can happen if you get an error message, press debug (vs end), and don't reset
Go to vba window and select "reset" from the run menu. OR - exit and re-enter excel.
Model And Assumpions
Uses the Myklestad / Prohl transfer matrix approach
Two types of matrices: field matrix represents a (distributed) length of shaft/disk, point matrix represents the lumped parameters concentrated at the point between adjacent lengths
The field matrix includes the distributed stiffness and (depending on program option) distributed mass
The distributed mass field matrix is the same as given by Harris' Shock and Vib Handbook Chapter 7
The lumped mass field matrix is similar to Thompson's "Theory of Vibrations" chapter 7
The point matrix includes lumped polar/transverse inertia and also includes effects of bearing force, and (depending on program options) lumped mass
Similar matrices in many references
the MOI properties related to gryoscopic/bump-test/point-mass option are based on "Dynamics Of Rotating Systems" page 146.
Assumed isotropic (i.e. horizontal and veritcal direction stiffnesses are the same on horizontal rotor)
Assume H and V vibrations are uncoupled (except gryoscopic effects which are modeled)
Assumed linearity of all aspects
Neglect damping (undamped analysis)
The support on which the bearing springs are mounted is assumed to be massive and does not move.
The bearing spring may represent series combination of bearing stiffness and support stiffness provided the bearing housing has negligible mass
fluid effects are not modeled (of course)
bearing stiffness assumed constant at the value specified times the bearing stiffness multiplier of interest (in reality stiffness may change with speed)
The model is based on Euler Bernoulli beam model which neglects shear deformation. Shear deformation is negligible for all but the shortest/fattest rotors
The effects of rotary inertia (for example Timoshenko beam) is included in calculation options 1 and 2, but it is lumped (vs distributed)
Any attached disks are assumed to be rigid and ridigly attached to the shaft. Whatever the angle of the shaft, the entire disk takes the same angle
Accuracy
Every trial case that I have checked matches other calculations (based on similar assumptions/models) to within a few percent with one exception:
IF the rotor is very stiff in the middle (including bearing stiffness, high E, high radius), and has a very flexible attachment on the end (smaller E and smaller od)
THEN the program generates many extra frequencies that don't seem to belong.
You can recognize this situation by
Many more resonant frequencies than expected
Mode shapes that show wild wiggles at the ends with very little movement along the length of the rotor
Main
Variable DescriptionSymbolValueUnits
n_10unitlessEnglish Value **UnitsTo match Irvine
DefaultDensity rho *rho_def7750.37312kg/m^30.28lbm/inch^3**0.278
Default Young's Modulus E *e_def230974359500n/m^23.35E+07psi**2.98E+07
Default Bearing Stiffness *kb_def1.78E+07n/m1.00E+05lbf/inch**
fstart1hz
upper bound for frequency sweepfstop5000hz
Number of steps for frequency sweepnfsteps1000unitless
4.999hz
freqtol0.001hz
inclgyro31,2,3(1=critspeed; 2=bumptest,3=simple)
Sweep bearing stiffness using multipliers?sweepbrgTrue or false
brgmultipliertext{0.01,0.02,0.05,0.1,0.2, 0.5, 1, 2, 5, 10,20,50,100}unitlessNot programmed yet:
leftbcfreefree, hinged, or clamped
rightbcfreefree, hinged, or clamped
Use lumped mass calc? (usually false)lumpedmasscalcTrue or false
* - the three "default" parameters identified (rho_def,e_def,kb_def) above accomplish nothing unless you copy them into the rotorsections sheet (using FILL IN DEFAULTS button in RotorSections sheet)
** - the three blue parameters in English units do nothing unless you copy them to the green cells using the button provided in this sheet
output1
radian freqfreqobjective functionrootsobjective fn at root
6.28318530721481239.226694412485.97104058849505398668.21875
37.69282865785.999623160304.7023991339.5978884888-2658840315136
69.102472008410.9987036438895.131642626.1477780151156038325338112
100.51211535915.99731474314213.25424341.15627020264.35904498683085E15THIS SHEET IS EXTRA INFO THAT IS NOT OF MUCH INTEREST (BUT DON'T DELETE IT, THE PROGRAM WRITES TO IT)
131.921758709520.99693313083728.3624905.88461053470.0018826872THIS SHEET HOUSES THE OBJECTIVE FUNCTION VS FREQUENCY FOR THE LAST KMULT CALCULATED
163.331402060125.995218997769149.24351.64672540282.6026291847THE ROOTS FALL WHERE THE OBJECTIVE FUNCTION CROSSES 0
194.741045410730.994441941397923.68782.6760546875-6.06095443234122E30If you want, you can examine the behavior of the objective funciton to decide if your sweep is fine enough
226.150688761335.993802397242990.75888.37347656255.31819040876999E30
257.560332111940.9921347304148654.14110.61683593752.34218255432794E31
288.969975462545.9912130105097536.53114.9094140625-2.099546306628E31
320.379618813150.993210539200593.58120.4507421875-2.45607303794219E30
351.789262163755.9894654407319108.42130.28464843752.02031814411374E31
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