Science topic

# Harmonic Analysis - Science topic

Ths group is devoted to Harmonic Analysis (Fourier Analysis, Time Frequency A., Gabor Analysis...)
Questions related to Harmonic Analysis
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Using Python, I would like to convert PSD (G2/Hz vs Frequency) diagrams to Acceleration vs Time diagrams. Would someone be able to provide some insight into this matter, because I would like to know first whether or not this can be accomplished and if so, how?
DearRavi Patel:
At the first, and as you know we can say the following information:
Vibration Research software uses Welch’s method for power spectral density (PSD) estimation. This method applies the fast Fourier transform (FFT) algorithm to the estimation of power spectra.
In regards to his method, Peter D. Welch said, “[the] principal advantages of this method are a reduction in the number of computations and in required core storage, and convenient application in non-stationarity tests.”
The process begins with Gaussian-distributed time-domain input data—i.e., a time history file.
Relying on the above، you can benefit from this valuable article about your topic:
"Calculating PSD from a Time History File"
I hope it will be helpful ...
Best regards ...
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Dear all,
I am trying to perform harmonic analysis with the forcing term which is not harmonic but periodic. I converted the forcing term to harmonic series of sine and cosine. Now the problem arises when I have to incorporate this force in ANSYS analysis.
Is there any way to input non harmonic forcing term, or the another way to input series of sine and cosine term ??
I agree with Hiba Kaoaz . Harmonic response analysis simulates how a structure will respond to sinusoidally repeating dynamic loading. You can define load as a specific sin and cos function with domain and frequency to your model and harmonic analysis, but you should be done with modal analysis before that. Still, if you have a specific function for your forcing vibration, you can use it in the APDL program of your model.
regards;
Ehtisham
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Hello, I am doing a harmonic analysis in 2D on Ansys APDL. I have a disc (mass element + plane182 elements linked to the mass element with rigid constraints) confined in a fluid29. I want to impose a circumferential mean flow effect to my fluid, using BF,,VMEN,,U. But once I run the simulation no frequency appear in the Fourier spectrum, as if no force was applied. However it works when I apply a force with the F command, so I think the problem is with the BF command which is not taken into account. can someone help me ?
follow
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How to get the frequency response in terms of voltage from the piezoelectric patch using the ANSYS workbench? Like in this paper,
follow
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Hello everyone, I am analyzing the harmonic response of a beam in a frequency range [0-50 Hz] but I noticed that the results for the 0 Hz frequency are not provided by ANSYS workbench. would like to know is there a possibility to have the amplitude of the displacement of the beam for the frequency 0Hz?
That's the least I could do.
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hi,
Similarly to the problem addressed in this topic :
I would like to reduce the size of the .rst file as much as possible in order to minimize the time requires for the simulations. The OUTRES function allowed me to significantly reduce the size of the files with the following commands :
OUTRES,erase
OUTRES,ALL,NONE,
OUTRES,NSOL,ALL,
However, I would like to reduce the size even further. A possibility would be to save solution associated to a restricted number of nodes only. The solutions related to the other nodes may be not saved since they are not used. So I tried the following commands (where 'nxtrnl' is the name of the component grouping the nodes of interest):
OUTRES,erase
OUTRES,all,none
OUTRES,nsol,all,nxtrnl
But the results are very different from what I usually get (magnitude 1e6 larger).
So, would you know if is it possible to save into the .rst file only the solution for a restricted number of nodes ?
Thanks a lot !
Simon
Hi Simon, reducing the scope of saved results will only result in decreasing the size of the RST file, but will hardly have an effect on solution time. You are doing it the right way. If results are not consistent, then it could only be a bug although it would be a huge surprise for me (have been using this fonctionnality for years in and out). Lastly, specifying an array (keytimes) instead of "all" might be an option, or storing data every other step. But again, this will be a 50% reduction, not one order of magnitude.
Normally, nodal results are very compact, so RST file size should never be a problem. How many nodes and time steps are we talking here?
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This is an openzmeter capturing the voltage of my house (Almería - Spain). It should be 230V - 50Hz with a more or less sinusoidal wave, but instead, I have a lot of PQ events like in the photo attached. What is happening in the grid??
Mystery solved. The problem was caused by the meter itself. A new update on a processing thread was responsible for the malfunctioning. Thanks anyway for your valuable help and ideas.
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Hi all, I am using PZT to excite the beam structure and measure the impedance using dynamic signal analyzer. And I also build the finite element model in ANSYS to do harmonic analysis and get the impedance data. My current work is to update the simulation model based on experimental measurements. I read some papers, but all use trial-and-error method. I am wondering if there is an efficient method to conduct model updating.
If make the whole FE model in an optimization loop, the harmonic analysis takes really long time for each iteration. On the other hand, there are too many parameters, and most of them have influence on impedance curve. So the parameters to be updates are also undetermined.
Any suggestions or references are appreciated, thanks.
Dear Yang Zhang ,
I think you can reduce the parameters of the system if you used a physically based model for the response to the acoustic waves. There is an electric circuit equivalent model that mimics the acoustic system.
In this model the mass, the elasticity constant, and t he dimensions of the objects are the parameters of the model.
This is just a proposal of an electronic engineer to solve such problem.
Best wishes
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Hello,
I have a question regarding the zero-sequence voltage injection in three-phase SSTs with bypassed modules in a failure case. I investigated some literature and it is clear that only the fundamental harmonic can shift power between the 3 arms (for power balance).
A 3rd harmonic is often used to reduce the peak arm voltages (-13.4%). However, I investigated asymmetrical fault cases and a 5th and 7th harmonic can help tofurther reduce the arm voltage by a small percentage like 5% (so less redundant modules are needed):
My question is: Do you know other methods to handle asymmetrical fault conditions in cascaded H-bridges or SSTs? Can I handle grid-imbalances like module faults, since the modules in the phase with an over-voltage have to transmit more power (if no zero-voltage injection takes place)?
I thank you for your suggestions and ideas.
Or do you have experience in three-phase systems with a floating star point?
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my project is power quality and harmonic analysis ubder balancee load conditions under 0-2 Khz.
i was asked to measure each harmonic up to 2 kHz at different power levels and THD. i made a multidrive system upto 20 drives using Dipde rectifier and nonlinear loads
Change the grid impedance (1%, 5% and 15% base impedance) and see the differences.
so what frequency should i take my source and grid impedance values?
What does upto 2khz mean??
Do i keep my source at 2 khz ?
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The monochord page in Wikipedia implies that the diatonic scale is determined by the monochord instrument to be ratios defined by the position of a movable fulcrum.
However, on musical strings found on instruments such as the guitar or violin, the string is not a monochord because the detainment of the string allows the fundamental mode to resonate between only two points, and not three.
It follow then that the fundamental of the monochord has a wavelength 2L while the musical string has a fundamental which is 1.
Significantly the musical string does not depend in any way on the length, tension, mass, or composition (within a bound range useful for harmonic oscillations) because the only requirement is the 12th fret is placed at the string midpoint and then one half the string is divided into 12 equal frequency units using the 12th root of 2.  This is the geometric equivalent of the construction of the square root of 2 by the unit square.
My question is whether the monochord and the musical string are in fact the same because it seems the monochord overtones are multiples of 2L and the musical string does not depend on L at all because it is normalized for length.
The string is 1.  It is one thing, always the same, a constant. It is not 1L. Its just 1. It always has the same shape, which is detained in a concatenation.  So the idea that the string is the sum of all the possible modes of vibration is wrong.  There is only one mode and that is the fundamental.  This means the natural overtones are not the defined mutliples of the fundamental.  The multiples are the octave, and any subset of the octave is also a multiple.  You have a ruler and then 12 equal subunits.
The monochord investigates the effect of changing the wavelength and frequency as continiuous variables, but the music string (which is formed by the union of the pitch value and string position sets on the fundamental) are constants that cannot be continuous because the fundamental is a standing wave.
If multiples of the fundamental are defined by the frets that detain the fundamental, then the definition of the overtones is mathematically distinct from the monochord overtones which are degenerate (that is not nondegenerate) forms based on a lower mode of vibration than actually exists.
Doesn't this mean the string under the square root of 2 is always tempered and the problem tempering the piano results because the strings on piano have different lengths?
Terence B Allen, you should not compare pears with apples.
You write: " We have the natural overtone series as the set N = F0, F1, F2, F3, … and the tempered series is the set T = A1, A#1, B1, C1, C#1, …, A2, …n. "
You should realize that while the first series is a series of frequencies (which would better be expressed as N = F, 2F, 3F, 4F, ..., with F being the fundamental frequency), the second is a series of logarithms of frequency ratios.
If units of the tempered series are 21/12, then the logarithmic series that you call the "tempered series" becomes T = 1, 2, 3, 4, etc.
If these two series ressemble each other, it merely is to the extent that a series of whole numbers may ressemble a series of logarithms. However, if you want to express the series of whole numbers in terms of the logarithmic series with basis 21/2, then it becomes, to the approximation dictated by the logarithmic base, log(N) = 0, 12, 19, 24, ... – or, in numeric values of the logarithms, 1, 2, 2.997 (=~3), 4, etc.
This is not high level mathematics, it merely is low level arithmetics. It has nothing to do with the vibration of strings, the arithmetics would remain exactly the same for notes (or frequencies) produced by any other means.
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Is there a way to input wave speed in a complex form on ANSYS? The wave speed I need to assign a material is 325(1+0.11i) however I have been unable to do this thus far.
Failing that, is there any way around this?
Thank you,
Callum.
see the attached file.
Dr. Salahuddin
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Hi all, I have conducted modal and harmonic analysis for the PZT transducer in ANSYS or ABAQUS with Open and Short circuit respectively. The following is the key process (ANSYS):
——Solid45 and Solid 5 elements, and Circu94 for circuit
——Coupling surface nodes of PZT to two coupling nodes as electrodes ( shown in the figure1 )
——For Open circuit, I connect a resistor between the two electrodes of PZT with 10e-20 ohm
——For Short circuit, I connect a resistor between the two electrodes of PZT with 10e20 ohm
——The whole analysis process, the bottom electrode is set with a zero voltage as ground
I, respectively, conduct modal and harmonic analysis. For modal analysis, I get the first natural frequency is 30.732 Hz for OC, and 30.721 Hz for SC.
For harmonic analysis, the sweeping force is applied at the tip of the cantilever beam and measure the voltage at the top electrode. The responses are plotted as in the attached figure 2.
My purpose is to calculate the electromechanical coupling coefficient:
ksi = sqrt (1- wSC2/wOC2).
So I can only get the ksi with value of 0.03. The structure and parameters are used from the literature [Active-passive hybrid piezoelectric networks for vibration control: comparisons and improvement]; In this paper, the coupling coefficient is 0.1167. There is a huge difference. I read many papers about related analysis, but I can't figure out where my analysis steps are wrong.
In one conference paper I read, it indeed has a large coupling effects. It uses ANSYS to conduct the same analysis, and its results are really good, as shown in figure 3. I nearly use the same analysis steps, so I know I am now approaching to the correct answer now but there is still some details that I may not set correctly. So I am writing to hope someone familiar with the related analysis could help to figure out my issues. Thanks.
Hi Meng, I am new on using ANSYS since I only use ABAQUS before, so my modeling process is 'semi-parametric' in ANSYS (GUI+APDL). I think there is no difficulty in GUI for you. I can share the APDL for the related processes for you.
！ coupling process
NSEL,ALL
NSEL,S,LOC,Z,3.175e-3
NSEL,R,LOC,X,0.012,0.08244
CP,1,VOLT,ALL
*GET,n_ground,node,0,num,min
NSEL,ALL
NSEL,S,LOC,Z,3.442e-3
NSEL,R,LOC,X,0.012,0.08244
CP,2,VOLT,ALL
*GET,n_top,node,0,num,min
allsel,all
ET,10,CIRCU94,0
R,1,10
TYPE,10
E,n_top,n_ground
allsel
FINISH
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Good day,
I design the rail-wheel geometry and the harmonic frequency ranges between 20-10kHz with 500Hz solution intervals and the results makes sense. Most of the time on the related literatures they use 10 or 20 Hz solution intervals. I want to know how to correctly make a choice of number of solution intervals when performing harmonic analysis?
Hello Dr. Durrani,
I agree with you,
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I have modelled a unit-cell of an anechoic structure using ANSYS. The layer is silicone and has been assigned as an acoustics physics region, with an air cavity inside (the air has been modelled and also assigned as an acoustics physics region). There is a steel backplate as well that is structural. In order to simulate the water closure on the silicone layer face I have applied an impedance boundary condition, assigned a port to this front surface and used the body of the same layer to assign the inside surface bodies. A planar wave has been applied using a 'port in duct' excitation condition (with 10000 Pa) and the acoustic absorbance is calculated.
I have used the same material parameters that I have seen in many papers, however I am getting very different results and wondered if anybody could please highlight where I have modelled it incorrectly or explain the observed behaviour? I have included an image of the absorption coefficient with regards to frequency in a 0-6000 kHz range.
Thank you for any help.
Hi
A hand sketch or similar of the setup would help.
Impedance BC only apply for normal angle of incidence. They work on boundaries at a distance from the source, say one wavelength out or so and then, still are reflecting.
As per usual - the advice when tweaking models - start with simulating a Kundt's tube experiment first, as you there can use impedance BCs. Move to 3d from there.
/C
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I am a final year university student and for my honours project I am studying the effects of the size of air-filled cavities in an anechoic layer. The layer is made of silicone and has a steel plate backing. It is surrounded by a water enclosure, the surfaces in the x and y planes of the anechoic layer have a frictionless support applied to them to stop displacement in the normal directions. I am however having trouble with the remainder of the boundary conditions, namely the mesh sizing and whether I have to use commands to apply specific nodal values (using FLUID 30?).
Any help would be greatly appreciated. Thank you.
Hi
The rule of thumb is seven nodes per wavelength for free wave propagation and nine nodes per wavelength for reactive fields.
By the sound of it, try to use nine close to the body and less dense away from it.
/C
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Whether the model parameters such as property or geometry could be tuned to obtain desired frequency response by performing harmonic analysis coupled with direct optimization module of ANSYS workbench?
Can you share the experimental transmissibility response in .csv format? Thanks in advance.
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How to prove or where to find the radii of Starlikeness and Convexity of functions in Dunkl and q-Dunkl setting and what are the difficulties to meet in this subject?
In fact, I am interested in this question.
Using the convolution of harmonic functions, we introduce a generalization for a previously defined class of right half-strip harmonic mappings and determine sharp radii of univalence, full convexity and starlikeness for such functions
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Hello.
I am using ANSYS mechanical APDL.
I currently follow the procedure described in
I wish to add (measurement) noise to all degrees of freedom when performing a harmonic analysis of a structure.
How can I do this? Is there a function which can do this, or is it necessary to apply loads to all degrees of freedom?
I will supply any necessary information you may need.
Kind Regards
Christoffer
Dear Laerkesen,
I suggest you try the following link:
Regards,
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I've been trying to simulate the force transmissibility experiment based on the setup shown in the attached diagram where the transmissibility is defined as the ratio of transmitted force (measured by bottom force sensor) to the exerted force (measured by top force sensor).
I've used a fixed support boundary condition (BC) on the lower face of the bottom force distribution plate and a remote displacement/displacement BC to the upper face of the top force distribution plate. I used these BCs as I believed they are the closest to the real experimental conditions used.
to measure the force, I used force reaction probes on these boundary conditions. Then, I manually plotted the transmissibility ratio ( force reaction measured from the bottom to top plates).
However, the graphs obtained are nowhere near the normal transmissibility curves.
I also tried this way by using frequency response for the top and bottom plates and used the generated amplitudes to manually plotted the transmissibility ratio, however, I did not manage to do this as the frequency response from the top plate is zero (shown in attached figures).
I think you should get the sum of the interface nodes' reaction (the nodes which are located at the interface of sample and steel plate) as the whole force, not all the nodes.
Do not forget to define the steel plates with enough tickness and stiffness.
Wishes,
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I am doing dynamic analysis of water tank using ansys workbench. While choosing the fluid elements from engineering data, it  shows error message. So I have to give properties of water as Modulus of elasticity and poisson's ratio. I think it does not represent actual behaviour of water. At the same time, using acoustic extension, I can define water as Acoustic body. But the input properties are not bulk modulus and boundary admittance.
Please anybody help me to know how to input  fluid property from the engineering data in modal/ harmonic analysis.
If you r looking for mechanical properties of water you can use E=2.4 GPa and poison ratio 0.49 with density d=1 Kg/l
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Dear researchers,
I am going to estimate the main four tidal constituents (M2, S2, K1, & O1) as well as shallow water corrective terms (f4 & f6) from hourly sea-level observations in a shallow water station based on the Admiralty method. Do you know any open-source package in this regard?
Most of the available packages e.g. t_tide and Utide works based on Foreman (IOS) method and do not provide the mentioned corrective terms.
Thanks
Can you explain what are shallow water corrective terms (f4 & f6) ? I study on tides for many years, but I never hear about these terms. If these terms mean oscillations with some specific frequencies, I think you can use S_TIDE which is developed by me (https://www.researchgate.net/project/A-non-stationary-tidal-analysis-toolbox-S-TIDE). Using S_TIDE, you can extract oscillations at any frequencies.
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Suggestions invited
@karar yes one out of ur mentions has no limit i.e. EE but rest two have
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Here the problem is, we cannot able to create uniform mesh for both exponential cantilever beam and rectangular piezoelectric patch. Is there any problem for this analysis?
Thanks for your response Chitaranjan Pany sir. I will try that sir.
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Taylor described harmonic motion on the string using Newtonian physics as a smooth manifold. This is absolutely unequivocal in Incremental Methods Direct and Indirect. The string field is uniform, tangent-cotangent bundles are (almost) everywhere perpendicular to the string. The curvature of the string is constant because the string always follows the shortest path.
As the first to describe the equation of harmonic motion, Taylor should get credit for the principle of least action but Euler wrote the formal integral.
But Euler said No, the curve of the string can be any continuous curve. To prove this Euler wrote a series of functions, presumably with the string modes on the monochord (kanon, or measuring rod in Greek) in mind. The use of a transcendental series is similar to Fourier harmonic analysis.
Euler and Bernoulli apparently disagreed on whether the number of terms in the series was infinite. They may have thought the series added up to 1, but Cantor showed the series in not coherent because it does not converge.
Show the question I have here is whether the string manifold is smooth or merely continuous.
First, there is no addition function on the monochord which allow two modes to add. They cannot add because they have different critical points and a point cannot be critical and not critical at the same time.
Second, if the string curve is a combination of waves with different frequency, and therefore different energy levels then those waves on the string that have higher energy will simply minimize on the fundamental.
On Research Gate and Stack-exchange (where I am an outlaw banned for life, like the Jesuits opposed to infinitesimals), I have asked perhaps a hundred questions that have never been answered.
I mean, come on! Of course Taylor was correct. It is easy to see the string manifold is smooth because manifolds cannot exist without smooth functions!
I'd like to hear from John M Lee, Pavel Grinfeld, Liviu Nicolaescu, Marco Marzzucchelli, Giuseppe Buttazzo. People who know a smooth manifold when they see one.
Just as Euler's idea became Fourier (useful but just not in music), Taylor's principle later became the Lagrangian, later Hamiltonian principle.
The string is fundamental to science so if physics and mathematicians do not understand it, what else do they have wrong?
The questions you cannot answer are the best ones.
I have attached Taylor's diagrams showing how he analyzed string motion. Even in Latin the words "cycloid" and "constant curvature" are clear.
Not my field
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I tried to use LDREAD command to take results from a static analysis into the harmonic analysis, but it fails to read the file. I've been using it as LDREAD,REAC,LAST,,,,'structural','rst',' '. Are there other way to take results from different analysis into another?
Thank you
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I was wondering if there are some references available on harmonics analysis (especially for power systems) from a control theory perspective.
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I was using ansys workbench to analyze the cantilever energy harvester via harmonic analysis. I don't know why the vibration pattern is a curve.
We are talking about the picture that you uploaded i.e. the harmonic response at at 1e-003 MHz, correct? Can you upload a picture of the modal analysis with the mode that occurs closest to 1 e-003 MHz?
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I am doing dynamic analysis of PIPE contained with fluid using ansys workbench. While choosing the fluid elements from engineering data, it  shows error message. I think it does not represent actual behaviour of water Please anybody help me to know how to input  fluid property from the engineering data in modal/ harmonic analysis.
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I am doing harmonic analysis over a beam, I need to single and continuous load over the beam that will transverse the beam with certain speed. The load represents gross axle of a car. I need to know any source, reference, example, or feature that can help me simulating this in ANSYS. Also please if I require to write a script let me know about sources that can benefits me
Thanks.
great
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A new paper in "Constructive Mathematical Analysis" by Prof. Michele Campiti.
G. C, Rota wrote his thesis under Nelson Dunford that characterized the extensions of OD operators via the concept of boundary values. It appeared in Communications of Pure and Applied Mathematics, in 1958. Might what to check that out relative to this paper.
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I have done modal analysis in ANSYS APDL. Some warning is coming after finish the coupling part. Getting some sort of error in the value of results summary. But i have done the same model in ANSYS workbench, the frequency range is coming correctly. Due to this problem in ANSYS APDL, i cannot able to do harmonic analysis for the model to get the frequency response.
Kindly tell me your suggestions regarding that.
This error only refers to the relationship between nodes of the mesh and your underlying geometry. But the results of the modal analysis are unaffected by this.
Did you get the same modal frequencies in Workbench and APDL? If so, everything is fine.
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Hi,
iam doing project on Pzt energy harvesting analysis in ANSYS APDL.
I have done MODAL ANALYSIS.
1. I want to do harmonic and iam not getting steps to do .
2. how to insert an resistor & get voltage output.
APDL codes for set up the resistor
ET,3,CIRCU94,0 ! Set up the resistor
R,1,R
TYPE,3
!E,Up1,Ub2 ! Parallel the resistor to the
E,1,2 ! electrodes of PZT
EPLOT
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Hello,
A structure is tested for a Random Vibration (say Z axis, 0.1PSD, 20-3000Hz), where I obtained response acceleration from measurement system. Now I wanted to obtain same output(amplitude of modes) by giving Harmonic(sine) sweep as input(in Frequency domain) to Shaker machine. I read some on-line papers explaining conversion/equivalence from PSD to harmonic but none of them are actually working. Can someone please refer some good source?
Thank you,
The basic assumption of the random vibration is that you can't predict the actual time realisation of the process, only the statistical measures, which assuming a Gaussian process is the standard deviation. So you can only compare a harmonic and random it in an average sense, which is to have the same input power. And that is what the PSD gives you. PSD is usually approximated as the square of the signal's Fourier Transform. So if you know what PSD value your random process has in a fequency bin, then you can work out what amplitude of sine wave would give similar power. But the only way you can relate the two are through the PSD, comparing actual time histories is irrelevant
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I wish to know if in a harmonic analysis in Ansys (or any other FEA software) we can apply two different loads with phase difference between them.
Please look at page 13 of the presentation attached.
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Dear All This is Suman from IIT Kharagpur, India. I have a very silly query which described as follows. Any sort of insight is will be appreciated. Harmonic analysis is a basic mathematical technique used in various branches. In the field of meteorology and atmospheric sciences, we regularly use harmonic analysis for the elimination of annual (1st harmonic) and semi-annual (2nd harmonic) variation from different data. From the earlier literature it is noticed in most of the cases long term series of monthly data is used for that purpose. For example if there is a complete series of 30 (1951-1980) years monthly data (30*12=360 months), then to eliminate seasonal variability of the original data 1st, 2nd and sometime 3rd harmonics will be subtracted considering a Fourier series of 12 harmonics (for 12 months). Please make me correct it I am wrong. In my case, it is something similar but I don't have whole data as I am interested in seasonal study (Indian Summer Monsoon). I have also 30 years of data having only June, July, August and September. It means the data I have June, July, August of 1951, June, July, August of 1952 June, July, August of 1953 and so on. Now if I want to eliminate the seasonal harmonics then what to do? As I don't have data for all months, it is unscientific to use the above procedure for my data. Is the idea is only applicable for complete data (I mean having all months data)? Please help. Suman Maity Ph.D student IIT Kharagpur India.
Dear Suman,
I am not sure to understand very well what you want. If you just want to filter out some frequencies, it may be good to use a spectral filter. I would suggest the Lanczos' one: https://journals.ametsoc.org/doi/10.1175/1520-0450%281979%29018%3C1016%3ALFIOAT%3E2.0.CO%3B2
If you definitely need to use your method, I would say that for this you cannot just use up to three harmonics on your data. Actually, since it is a 4-month period, the first harmonic gives a single cycle over the 4 months ( a 4-month periodicity) , the second harmonic 2 cycles (a 2-month periodicity), the third harmonic 3 cycles (1.3 month periodicity)... Thus considering only the first harmonic should eliminate your "seasonal" cycle.
For more documentation, please go through "Time Series Analysis in Meteorology and Climatology: An Introduction" by Duchon and Hale.
As for software, NCL is the easiest to use in my sense. However, other languages including R and Python can do the job.
Kindest regards
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please suggest me proper methodology to study the nonlinear vibraition behaviour of the system shown in the figure.
The standard solution by using FEM software is using explicit code. These are ABAQUS explicite or LS-DYNA - they can easily handle this problem as a direct simulation. You need just to choose corresponding finite elements (beam, shells) and include contact. REMARK: of course, due to the contact you need a refined mesh for the contact zone. Contact problem is initially mesh-dependent and you have to show convergence of your results by performing several computations with varios meshes.
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At first, model is prepared with appropriate material properties. After that, how to couple the electrical potentials of the two electrodes contacted to the phosphor bronze layer?
To get open global frequency, how to do the bottom electrode of the PEH is grounded and the voltage DOF on the top surface is coupled?
To get short global frequency, how to do bot the top and bottom electrode of the PEH are grounded?
Finally, how to do harmonic analysis for that model to get steady state voltage output.
Dear raja
Why don't you use COMSOL Multyphysics?! It's way more easy for this types of analysis! I recommend using that .
Good luck
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(Like that of the force of jet on a turbine), I want to model the force reaction from the jet on the turbine shaft in Harmonic Response ANSYS.
As i know, in Ansys Workbench you can simple add a force on a edge, point, surface by right mouse clicking in Harmonic Response module, so you can calculate value by himself and add this force.
But if you want calculate this force in Ansys you need do fluid structure interaction anlysis (FSI). To do FSI analysis in Ansys Workbench you need: 1. Make the model which have solid bodies (your shaft) and fluid bodies (fluid domain). In Geometru module you need choose for solid body in Details of body -" Solid" , for fluid body - "Fluid"
2. Couple this model with CFX module where you supress solid body
3. Couple this model with Static Structual analysis where you supress fluid domain body.
4. Couple the solution cell in CFX module with setup cell in Static Structual
5. Do calculation flowpath in CFX module
6.Enter in Static structual module. In tree you can find "imported body load", right click on it and choose "insert pressure", then you can calculate.
After this steps you can couple solution cell in Static Structual module with Modal module and Harmonic Response module. All loads add in calculation by the same way in step 6.
If FSI is that you need you have to serch for FSI keywords.
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I am doing an axisymmetric harmonic analysis using FLUID29 and SHELL63 elements of a cylinder. I want a plot of velocity variation with frequency at one node. In APDL, it seems only pressure variation option is there. While in Workbench velocity can be evaluated at one frequency only, and it has to be repeatedly evaluated for each frequency value of interest. So, how to get velocity variation with frequency?
Interesting question
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I conducted harmonic analysis using ANSYS workbench with memes and piezoelectric ACT extension trying to get an understanding of the behavior of the piezo material as the displacement varies harmonically. My question is about the term voltage frequency plot term mentioned in the results tab and what does it mean, and is the resulting plot represents the frequency spectrum of the output voltage that corresponds to the frequency spectrum of the input harmonic force and the resulting frequency spectrum of the displacement?
need help please to clarify things and any help regarding the piezo material simulation is also welcomed.
As far as I understood, You are trying to create a Piezoresistive model where if you have displacement (as the results of a force or pressure) you will get different voltage. Yes. I think you got it correctly. The frequency spectrum of out put voltage corresponds to the frequency spectrum of input which can be either force or pressure. I suggest that you create an equivalent model in workbench using command snippet rather than usign ACT extension. The element you need are solid226 or solid227 (depends on what mesh you used with the keyopt of 101.
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Hello, I have been having receiving a strange error when performing Harmonic Analysis in ANSYS Mechanical. When I try to perform a Harmonic Response for more than one frequency, I receive an error message saying "forrtl: severe (153): allocatable array or pointer is not allocated."
I have read through the output files and found something that is maybe relevant, but maybe not. "The maximum number of substeps (NSBMX = 1) is less than the initial value for the number of substeps (NSBSTP = 2)." Is it possible that this could cause a data corruption error?
Other than this, I do not have any ideas. I am having no issues performing Static Structural or Modal Analyses at the moment, and just on Friday I was able to perform a similar Harmonic Analysis without issue. I've also tried to rebuild the model from scratch and that resulted in the same outcome.
Has anyone seen this before and if so how did you solve the issue?
Thanks,
Ben
Dear Claudio
You are absolutely true. According to IEEE, ANSI and some other organizations, software most never hang. Input data must be checked to be in the valid range and quantity. If a problem occurs during the calculations, the program must only "throw an exception" and take proper action depending on error severity (warning, error, sever error, fatal error, ...). As a rule of thumb, program must always quit normally. But this is too ideal. There are bugs in Microsoft and Apple products (despite all strict QA procedures).
On the other hand, most programmers fail to trace the dependencies of their programs. Consequently, they publish their packages with inadequate resources.
When you encounter a problem with a software, after all, it is always a good idea to send a feedback and let the developer know about it.
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Hi, I have what I think is a simple question that I cannot find the solution to. I am modelling a piezoelectric actuator with a stack system (PZ disc, electrode, PZ disc, electrode, etc.), but I am unable to figure out how to properly apply the sinusoidal voltage to the system.
I would like to apply a sinusoidal voltage (let's say 100V to -100V) to half of my electrodes and keep the others as my ground at a frequency which I previously identified through modal analysis.
Could someone assist me with this? Thanks as this is my first piezoelectric simulation.
Sorry for the delayed reply. I was able to use a method similar to that recommended by Mr. Zangabad to apply the voltage. Thank you both for your help Sirs.
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Hello Everyone!!
I am working on Non linear vibration analysis of a simply supported beam in ansys apdl.
I am wondering how can possibly i do the non linear harmonic ansys in apdl.
If it is not possible to do Non linear harmonic analysis what other alternative do i have to do this non linear analysis of beam.
Any help/suggestion will be highly appreciated.
Rubber components used in industrial products are subjected to harmonic loads superposed on static pre-loads. The resulting deformation is in general highly nonlinear. Here payne effect( or fletcher-gent effect) is considered in material description for rubber for harmonic analysis and frequency response analysis.
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Hello Everyone!!
I am working on harmonic vibration analysis in ansys apdl.
I am using 1D idealization of 3D structure in ansys apd (Preprocessor > Sections > Beam > Common Section).
I am wondering if it is possible to do the analysis in ansys workbench using 1D element which has 3D idelization.
Any help/suggestion will be highly appreciated.
Hi,
Of course, you can create beam structures in Ansys Workbench.
In the design modeler :
1- Create the line
2- Create the beam section (concept --> Cross sections)
3- Apply the beam section to the line (line body --> select the cross section)
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I want to do Harmonic analysis of a system containing spring and dash pot as stiffness and damping element.The input to the system is base excitation(acceleration or displacement). what method and solver should be used to solve the problem in ANSYS workbench and how should I apply it.Also I want to know that how to perform QR Damped mode superposition harmonic analysis in ANSYS workbench ?
Kindly go through the attached ppt for the answer hope will be useful
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The sound envelope is observed as a change in sound amplitude over time when a musical string is plucked. The sound envelope has a sharp upswing called the attack followed by exponential volume decay. The shape of the sound envelop is characteristic.
The shape of the vibrating string during the decay is fusiform, sort of cigar-like, but when the string is pulled taut it forms a triangle so the up-sweep of the attack phase represents the relaxation of string perturbation during which time the string is returning to the boundary condition 0 1 0 imposed by the fixed string endpoints.
The attack phase seems to have a higher frequency content than the decay phase which seems to have a simple mathematic form.
So my question is if the decay of the sound envelop is the result of dampened harmonic oscillation, the decay is frequency dependent, right? Then isn’t it true that if the string is vibrating in many different modes at the same time, the decay would not be exponential?
Or am I wrong, it is possible the decay phase is the summation of many difference string modes whose decay is none-the-less exponential?
Is there an equation for the sound envelop that can be used to understand the frequency spectrum content of the sound?
My interest here is to show that the musical string topology is in fact not understood in modern literature.
It is clear that there are two competing theories for the topology of the musical string. The standard theory is the string vibrates in many modes. This is absurd because it requires us to believe that points on the string can be fixed and not fixed at the same time. The competing theory is there is only one mode possible at a time but the string boundary can be briefly perturbed by vibrations that are not multiples of the string fundamental (that is, not overtones). It seems to me that there should be a simple experiment which can clarify which theory is correct.
I have attached some theoretical notes which include diagrams and string boundary condition theory in more detail.
These are great answers! The first brings up Hilbert Space and the second uses the topology of real numbers in the frequency domain.
So we agree the most important principle here is that all harmonic values are simple multiples of the fundamental.
I think we also agree the Octave is exactly 2 times the fundamental.
Then it follows the space is Hilbert.
There is nothing more to say, really, but most people don't get it. The string space is topologized by the fundamental and the octave. The rest just follows mathematically.
The important difference between the Stewart and Ray answer is how the operation of multiplication is defined on the fundamental F. Stewart's answer assumes the harmonic operation of multiplication is the same thing as the multiplication of real numbers. The topology of real numbers makes the assumption the frequency is a real valued, continuous function the same as the real number line. But the frequency is a discrete value, not contiuous. But the octave space is continuous because the octave interval is precise to a point.
In fact the harmonic multiplication table that is defined completely by the Octave rule is not real valued. This makes a log 2 space. Every point is a distance 1 from the fundamental. Not intuitive.
The topology of the space is not altered when the octave is further divided to 12 tones because S12 = 1 is a Cuachy series that is also precise to a point.
So it follows directly from the octave rule that the tonal space created by the string and its octave is a special kind of Hilbert space in which every distance is 1 and every angle is 90 degrees. This is a Hilbert Space because a = a2 is true when a can only be 1 or 0. A disconnected space of singletons called the discrete topology.
This space can be directly observed as disconnected spaces in drawings of musical objects. Deawings of musical objects are useful but cannot be drawn accurately in 2 dimensions (Haas Theorem; think logic cuircuits). Look at any graph of pitch values (say a graph like Euler's tone net which plots the pitch value x versus musical key y). These graphs are easily shown to be non-planar graphs.
The pitch value graph seems to be x-y planar, but Euler's graph is topologically the same as a torus in a closed space. The torus is the real number topology but pure nonsense because the key and the pitch values must be multiples of the same fundamental and cannot not be two independent real values.
Euler's torus cannot reduce to a point which is the fundamental. There are only two possible manifolds here, totus and sphere. The sphere is obviously correct and Hilbertian.
So in Euler's tone net graph, if you count the number of steps around the trinagle that is formed, the hyptonuese has the same number of steps as the x- and y-axis. The steps are the multples of the fundamental that make the distance formula for the space. In this space the triangle formed has an equal distance on all 3 sides; so all angles are equal and 90 degrees, too. (For instance, in a graph of an octave the triangle formed is an octave on all sides: a2 + a2 = a2) This triangle must be on the surface of a sphere, because any planar triangle the angles add to 180 degrees but on a sphere there can be an equilateral right triangle where angles add to 270.
The result is a wierd arithmetic of octave rings in which there is one operation that is at once both addition and multiplication and every equation has the form 1 + 1 = 1.
My prespective on the string is based on the guitar string. A am influenced by the string condisered with frets and musical keys. The musical key on the string is projective, not affine. That is, the difference keys on the string cannot be brought into coincidnece by a change in intonation.
So I see the string topology as completely defined by a map between pitch value numbers 0, 1, 2, ... N and fret value numbers 0, 1, 2 ... n. Then there is the musical key defined by any pitch value on the string which is the tone center defining the key. There are no real numbers here, and the numbers in use are far more than counting numbers because they are closed and open sets. The clopen nature is highly aesthetic in the sense of pleasing number systems!
The space created by the string, octave, and frets is clearly a disconnected space of singletons. The pitch values are by definition multiples of the fundamental and the frets map on to the pitch values so the fret numbers and the key (or scale degree of each note within the key) are also multiples of F, as are the intervals between notes and frets.
The space is continuous because between every point there is an interval 1 and between every interval there is exactly 1 point.
What makes ther guitar tuning so remarkable is the string is closed and open. The tuning space is Borel and Barre, both the intersection and the union of the strings! The string is closed to multipication/addition operations by the ocatave. Any string larger than 12 frets can contain every key. But the string is also open to union with another string whose fundamental is a simple multiple of the original fundamental.
(It is still not possible for any system to have two independent fundamentals, which violates our first law.)
The guitar tuning is a collection of intervals that defined the tuning completely, say 0 5 5 5 4 5 is the ever popular EADGBE tuning called standard. The 0 is a place holder for the fundamental which is the lowest note on the lowest string.
The 0 5 5 5 4 5 intervals make a summation vector 0 5 10 15 19 24 which gives the pitch value numbers of the EADGBE expression. So on each string the fret number is the same as the pitch value of the open string plus the fret number. This makes a vector equation "sumation vector + fret value vector = pitch value vector" which is an example of 1 + 1 = 1 equation.
Here's anothe example. To change standard tuning EADGBE to Drop D tuning DADGBE use the vector 2 0 0 0 0 0 that add two frets to each note on the lowest string. To change the musical key from E to D subtract 2 frets from each string using the vector -2 -2 -2 -2 -2 -2. To change standard to Drop D and lower musical key we have the vector equation 2 0 0 0 0 0 + -2 -2 -2 -2 -2 -2 -2 = 0 -2 -2 -2 -2 -2. These 3 vectors are orthogonal and they form a triangle with equal sides. This is not a space of real numbers.
I give these examples because they show the mathematics are not real number theory. They are also not Z/Z12 nor are they based in a system of rational numbers. The topologic space of the string is atonishing because it is very obvious and clear but simply outside the accepted paradgm. But what is clear is the mathematic literature on music shows there is no moderm mathematical theory of music. The theory of the string and its multiple modes of vibration, sometimes called the corp ronds, is an arachaic joke.
I think it is clear that the string thoery is not properly understood in modern literature. Real analysis is so compelling but violates the spectal resolution theorem: You cannot assume that the frequency domain is a continous real values function. The problem I have is that to see why this is important you have to learn tablature arithmetic. Tablature seems like nonsense but it is a consistent system of numbers.
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Eddington [1] was one of the few who tried to answer the above question, for the case of the decomposition of white light by a prism; however, his answer is ambiguous. Burgers [2] reportedly (see Ref.[3]) reportedly also refers to this case, but his interest lies in the phenomenon of turbulence, where Fourier analysis and other methods of decomposition are employed. Also, Hinze [3], just before mentioning the Burgers paper, says: "... Though a harmonic analysis of the velocity fluctuations [in a turbulent fluid] can be carried out, this fact is no proof that, conversely, the turbulent fuctuations are composed of these harmonics. Compare the similar problem in the case of sound, where one may distinguish between noise (turbulent) and note (composed of a number of harmonics)."
This problem could conceivably appesr in every domain of physics. Is there an unambiguous abnswer to Eddington's dilemma: "discovery or manufacture?"
[1] Eddington, A.S.: "The philosophy of physical science". Ann Arbor, Ann Arbor Press, 1978. Chapter VII, section I.
[2] Burgers, J. M.: Proc. Koninkl. Akad. Wetenschap. vol.51, p. 1073 (1948).
[3]Hinze, O.: "Turbulence". Mc Graw-Hill, 1975, Ch. 1, pp. 7-8.
To summarize:
1) Any function can be expressed as a linear combination of the elements which constitute an orthogonal basis.
2) The Fourier basis is composed of an infinite number of complex exponential functions.
3) Fourier analysis performs well if the sum of the first elements of the Fourier basis converges to the analyzed function. This occurs when the studied function is periodical and when the sampling rate respects Nyquist's criterion.
4) Fourier analysis becomes important in acoustics and optics because complex exponentials are the eigenfunctions of the wave equation which governs acoustics and optics phenomena. ie harmonic excitation will produce harmonic solution, which can be represented adequately in the Fourier domain.
5) For transient phenomenon, other basis (wavelets for example) can be used to converge faster towards the solution.
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A three phase source connected with three phase load through a transformer is simulated in Matlab platform. Value of THD for each phase differs from each other having different fundamental voltages.
Inspect the load; probably it is unbalanced!
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In the article "Tight wavelet frames on local fields", the author states that "Theorem 3.1 is an easy consequence of lemmas 3.2-3.4". How?
Indeed, we must prove that : || f ||2= g∈X(Ψ)|<f,g>|2      for all f∈L2(K)
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Harmonic analysis - Is it a good method for identification of aquifer type for all types of terrains? How this analysis is conducted?
Dr. Truman Prevatt - Thanks for information. I will study the materials, you provided.
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I am currently working with modal analysis of a girder assembly. I have obtained mode shapes and natural frequency. I performed harmonic analysis by considering natural frequencies from modal analysis. I have imposed a force of 5000 kg to the structure as a harmonic load with frequency ranging from 0 to 40 Hz. I have post processed equivalent stress results which was around 2.5 Mpa, When I applied the same force in static structural the stress was 79 Mpa. Clarify why the stress difference is high between these two analysis. In addition list the scope of Harmonic analysis and its implication on modal analysis
Thanks Claudio Pedrazzi  for this detailed clarifying.
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Does there exists a function $g \in L^1(R)$ such that it satisfy the following property
(1)$L^1$ norm of g \leq 1
(2) $\hat{g}(s) =1$ for $s\in [-\delta, \delta]$ and
(3) \hat{g} has compact support ?
More precisely
Denote $R=\mathbb{R}$, $S(R)= Schwarz Class function on R$
I know that $C_{c}^{\infty}(R) \subset S(R)\subset L^1(R)$ and Fourier transform is bijection on S(R).  So one can start with a function with property $(2)$ and $(3)$ above and take $g$ as a inverse Fourier transform of the function. But I don't know that norm of g can be bounded by 1 ?
Dear Debabrata De,
The answer is negative by the following reason. The L1 norm of the function g is, by definition, the integral over R of |g|, and the \hat{g}(0) is the integral of g over R. So, the conditions \hat{g}(0) = 1 and ||g||1 \le 1 can happen only if g is real and non-negative almost everywhere. By the same reason, if ||g||1 \le 1 and \hat{g}(t) = 1 for some non-zero t, then g(x)eitx must be real and non-negative for almost all values of x, which is impossible for a real function g.
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Since there has been considerable work done to answer this famous and important question, we expect a clear and definitive proof of the ABC Conjecture within 48 months.
_________________________________________________________________
On simple forms, we shall also consider the  following mathematical statements:
e  *  f = a + b  and e / f = a - b.
Example:  We can consider the  following Fermat's Equations (1, 2, and 3):
1.  xn + yn = zn;     2.  x(n + ε)/2 = zn/2  + yn/2;   and  3.  x(n - ε)/2 = zn/2 - yn/2 where
e = xn/2,  f = xε/2,  a = zn/2, and b = yn/2  with  the  control parameter, ε,  such that
0 <  ε  <  n.
_________________________________________________________________
"The ABC Conjecture:  Given λ > 0 there exists a positive number, β(λ) > 0, which has the following property.   For any nonzero relatively prime integers, a, b,  and c such that
a + b = c,
we have:
max( |a|,  |b|, |c| )  ≤  β(λ)  *  rad(a*b*c)1+λ.
Notes:
We have relatively prime integers, a, b, and c or
gcd( a, b ) = gcd( a, c ) = gcd( b, c ) = 1.
Moreover, we define the radical, rad(),  of some integer, k, such that  | k | > 1 to be
rad(k)  =  ∏  p  such that  p | k  for which rad(k) is the product of all primes, p, dividing k, taken with multiplicity one."
Book Reference:  'UNDERGRADUATE ALGEBRA', 3rd Edition, Serge Lang, 2005, Springer,  ISBN 0-387-22025-9.  Please refer to relevant pages, 171 - 176,  of textbook.
_________________________________________________________________
'Easy as ABC? Not quite!',
'The probabilistic heuristic justification of the ABC conjecture',
'On Fermat's Equations',
‘DECOUPLING, EXPONENTIAL SUMS AND THE RIEMANN ZETA FUNCTION’,
'ABC Conjecture and Riemann Hypothesis', http://rgmia.org/papers/v7n4/abc.pdf
'On a combination between an equivalence to the Riemann's Hypothesis and the abc conjecture',
'The Riemann Hypothesis: Arithmetic and Geometry ',
'From ABC to XYZ, or Addition versus Multiplication',
'SMOOTH SOLUTIONS TO THE abc EQUATION: THE xyz CONJECTURE',
'On the distribution in short intervals of integers having no large prime factor'
by Prof. J. C. L. et al,
'GEOMETRY OF NUMBERS WITH APPLICATIONS TO NUMBER THEORY',
Dear David
You can see Emzari Papava's page on ABC theorem
Sincerely Yuri
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Hello everybody,
I have performed a 'Transient Structural (TS) Analysis' of a simple 2D plate problem. I have the response as a function of time.
I have no material non-linearity, no discontinuity, no contacts and have also turned 'Large Deflection OFF' in Ansys WB. It is a simple homogeneous linear elastic 2D plate fixed at bottom and loaded in the top edge at a constant frequency of 50 Hz for 1 second.
I do a Fourier Transformation (FT) of the result to get the response in frequency domain. Can I compare this result with what I obtain from a 'Harmonic Analysis' of the same model where the frequency sweep range is from (0 - 50 Hz)? The problem is that, magnitude of stress at 50 Hz in the harmonic analysis does not match with the magnitude obtained after FT of results from a transient analysis.
Is it possible to compare the results of a linear analysis with that of a non-linear (with all possible non-linearities excluded)? If not, why? Am I missing something?
Have a nice day.
Regards,
Shree
EDIT : Thanks for all inputs. Solved.. :)
Following may help:
1. Consider a linear system say  e= L*didt +R * i
2. let e= vm*sin(wt)  and  z= sqrt((wL)^2 +R^2)    thz = atan (wL/R)
3. The response i(t)= K*e^-L/R *t   + vm/z *sin(wt -thz) that is natural + steady state response
4. When excited by e we must wait for transient to die down to access the steady state ie at least 5 or 6 times L/R time constant
5 Note if we take time wt0 = thz we obtain ss right away.
6. Now if we take FT of 3 clearly it will not be equal FS (ie steady state ) except in case  5 holds .
For the  case given in question  we may have a more complex circuit of parallel and series configuration of spring mass system (linear assumed), still we have natural and steady state  response as above. Note however we can obtain the linear parameters L and R by an impulse response to a delta function or maybe a step function ,leading to system identification . We can then use this to obtain steady state harmonic response.
This might help explain your result.
Good Luck
Cheers
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Hello everybody,
I am trying to perform a harmonic analysis of a material sample (say a simple 2D plate) with an edge crack in it using 'Ansys R16'. I see that under sinusoidal loading condition, the two crack faces open or close and interpenetrate (after closing which is not possible practically).
1.  I believe, stiffnesses of the crack faces change during interpenetration and so the results are not credible anymore.
2.  I also cannot define non-linear contact behaviour (between the two crack faces) in a linear (harmonic) analysis.
How do I overcome this problem?
Bigger idea is to fit the frequency response result from the simulation with that of the experimentally determined response.
I am more than ready to provide further information if required.
Regards,
Shreeraman S
EDIT : Thanks for all inputs. Solved.. :)
Dear Shreeraman,
the opening and closing of the crack is a non-linear time-dependent process (which for instance typically leads to higher harmonics in the measured signal). Thus, the properties of the system change with time. I believe if you want to include this process in your model you have to perform a transient analysis rather and a harmonic one. Afterwards, you can always transform the results into the frequency domain by means of a Fourier transform, if your measured data is provided as a  frequency spectrum.
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In kalman filter for state variable estimation we take initial zeros but in ensemble especially for harmonic analysis, which values to take ?
Dear Sir
I suggest you to view this references :
Kohler, M. (1997). Using the Kalman filter to track human interactive motion: modelling and initialization of the Kalman filter for translational motion. Dekanat Informatik, Univ..
Chen, G. (1993). Approximate Kalman Filtering (Vol. 2). World scientific.
There is no magic recipe, for the initialization of filters
It is necessary to try all possible cases
Cordially
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"We must know; we shall know."  -- David Hilbert.
Riemann Hypothesis states that the real part of all nontrivial zeros (s = a ± b * i)  of the Riemann zeta function equals one-half ( or  Re(s = a ± b * i) = a = 1/2 ) where the Riemann zeta function is  Σ 1 / ks  (from k = 1 to k = ∞).
We assume that s = a + b * i and s’ = a - b * i are a conjugate pair of nontrivial complex zeros of the Riemann zeta function which is 1/1s + 1/2s + 1/3s + … or
1/1s’ + 1/2s’ + 1/3s’ + …, respectively.
The real part of s or s’ equals a or Re(s) = Re(s’) = a.
And we note, s + s’ = 2 * a.
Proof of Riemann Hypothesis:
To unravel the mystery of the Riemann Hypothesis, we recall four important facts.
FACT I: The real part of all nontrivial complex zeros of the Riemann zeta function is in the closed interval or critical strip, [0, 1], according to a Riemann Theorem.
FACT II: The are infinitely many nontrivial complex zeros of the Riemann zeta function whose real part equals one-half according to a Hardy Theorem.
FACT III: The sum of each pair of conjugate nontrivial complex zeros of the Riemann zeta function equals to one according to the Harmonic Series (the divergent Euler Zeta Function).
FACT IV: For all positive integers, k > 1, there exists a prime number, p, that
divides k such that either p = k or p ≤ sqrt(k) = k1/2.
Therefore, according to facts, I, II, III, and IV, the Riemann Hypothesis is true!
*****
Moreover,  let us solve the following system of two equations derived from
Σ 1 / ks (from k = 1 to k = ∞)  = 0 + 0*i = 0:
1. Σ cos (b * log (k) ) / ka (from k = 2 to k = ∞) = -1
and
2. Σ sin (b * log (k) ) / ka (from k = 2 to k = ∞) = 0
where 0 ≤ a ≤ 1 and
where s = a ± b*i is a complex number, and s is also the nontrivial zero of the Riemann zeta function.
You have two equations, one and two, with two unknowns, a and b.
What is the solution?
Bonne chance!!
Notes:
I. Looking at equations one and two above, we see periodicity associated with the variable b indirectly and not with the variable a. Why?
II. Both variables, a and b, are of degree one.
What does this information tell us about the solution of equations, one and two, in terms of variables, a and b?
Hint:  We expect all simple zeros, and there's more ... What more can we state factually?
****
“If you can’t explain simply, you don’t understand it well enough.” — Albert Einstein.

What does the Riemann Hypothesis (RH) mean?

RH confirms the existence of prime numbers in an optimal way. Or rather, for all positive integers, k > 1, there exists a prime number, p, which divides k such that either p = k or p ≤ sqrt(k) = k1/2 where RH states the exponent of k is 1/2.

Please keep that fundamental fact in mind when discussing the truth of RH.
*****

'The Riemann Hypothesis, Explained'',
' RIEMANN’S PLAN FOR PROVING THE PRIME NUMBER THEOREM',
'What is the most efficient way of predicting prime numbers accurately?',
*****
*****
@David Cole: I am not going to read your long  proof if the statement in the question is already WRONG. Unless its a kind of sophisticated pedagogical joke or thought experiment. There is a series expression for zeta in the critical strip and it is NOT \sum_n 1/n^s. It is an elementary fact that \sum_n 1/n^s diverges for Re(s)<1. And you do not need to be an expert to know that. You just need an undergraduate level in complex analysis.
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Thanks.
You are welcome also this may help good luck on your research..
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It's well known that in high voltage transformers, the contents of harmonics in the applied voltage are induced by the non-linear behavior in the saturated part of the magnetizing curve of the testing transformer core. In addition, non-sinusoidal magnetization current causes a distortion of the secondary voltage. Is there any solution in order to improve the applied voltage signal and decrease its harmonic components ?
Hi Youcef
I suggest to use a low frequency filter (LC circuit) with high voltage inductance and capacitor which could help you to eliminate odd harmonic components.
Good luck
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Hello Experts,
Any idea about how to model the tempearure rise due to the propogation of ultrasonic waves in the crack specimen? The frequency used for excitation of structures is 40 kHz. I have try to use Modal Analysis and Harmonic Response and coupled with Thermal Analysis. Unfortunately, i cannot connect the Modal Analysis and Harmonic Response to the Thermal Analysis in ANSYS. Is there any solution for this? What could be the other methods to model the tempearure rise due to the propogation of ultrasonic waves in the crack specimen?
Hello Chunli,
Thanks for the recommendation. I am learning COMSOL.
Regards,
• asked a question related to Harmonic Analysis
Question
I have made predictive current control for dc to ac two level 3 phase. When doing THD analysis i get 8.39% whereas 5% is acceptable. How can i get the desired THD levels