# Cyclic symmtery

## Main Question or Discussion Point

Hi,

Please can anyone explian what is cyclic symmetry? I'm new to this term and have encountered this in jet engine example

Vishal

Related Other Physics Topics News on Phys.org
AlephZero
Homework Helper
The basic meaning is a pattern that repeats N times around a circle.

A simple example would be the vibration of a circular plate supported at the center. The vibration mode shapes can be described by the number of radial and circumferential nodal lines.

The situation gets more complicated when the structure itself has cyclic symmetry, for example a fan or turbine wheel with N identical blades. This also has vibration modes which look as if they are made up of M "segments", where 0 <= M <= N/2. The motion of each blade may be different if M and N are co-prime numbers, but the motion of the whole wheel can be described by complex numbers representing the motion for one of the N segments. The complex numbers are multiplied by $e^{2 \pi M i /N}$ for each successive segment around the disk. There are some analogies between this and Fourier analysis.

This is an advantage for making finite element models etc, first because you only need to model one sector or 1/N'th part of the complete wheel, not all of it, and second because you can find the mode shapes and frequencies just for one particular value of M, rather than finding literally hundreds of modes for every different value of M all mixed together, and having to search through them to find the ones you are interested in.

Things get even more complicated when the wheel is rotating, because the vibration mode shapes can also rotate, not necessarily at the same angular velocity as the wheel. I'm not going to attempt the math for that from memory, except to say that again the whole motion can be represented by complex numbers which describe the motion of one segment.

Last edited:
Dear sir,

Thank you very much for your reply. Yes, I'm also concerned with vibration analysis.

Yes, I understand when you say:The basic meaning is a pattern that repeats N times around a circle.

Now:

When you say:

A simple example would be the vibration of a circular plate supported at the center. The vibration mode shapes can be described by the number of radial and circumferential nodal lines.

Now, the mode shape denotes vibration pattern of the structure-right? When you say mode shapes may be described by a number of radial and circumferential lines do you mean a vibration pattern wherein the circle vibrates along the circumferential and radial lines only and then the final pattern (thus response) being obtained through superposition of all these radial and circumferential patterns?

What do you mean by the term 'nodal lines' here?

Next, you said:

The situation gets more complicated when the structure itself has cyclic symmetry, for example a fan or turbine wheel with N identical blades. This also has vibration modes which look as if they are made up of M "segments", where 0 <= M <= N/2.

Why do you say M lies between 0 and N/2 (where N is the number of blades)- why not 0 and N/likewise?

IF we assume that motion of all blades is same, how should one proceed?

Vishal

AlephZero
Homework Helper
Now, the mode shape denotes vibration pattern of the structure-right? When you say mode shapes may be described by a number of radial and circumferential lines do you mean a vibration pattern wherein the circle vibrates along the circumferential and radial lines only and then the final pattern (thus response) being obtained through superposition of all these radial and circumferential patterns?

What do you mean by the term 'nodal lines' here?
The "nodal lines" are the points where there is no motion.
See http://online.physics.uiuc.edu/courses/phys193/Lecture_Notes/P193_Lect4_Ch4_Part2.pdf [Broken]

Why do you say M lies between 0 and N/2 (where N is the number of blades)- why not 0 and N/likewise?
There are analogies with "aliasing" in Fourier analysis or the Nyquist sampling theorem here. The patterns where M = m and M = N-m turn out to be identical.

IF we assume that motion of all blades is same, how should one proceed?
Make a model of one sector of the disk and one blade, such that the pattern of the mesh points on the two "cut faces" of the sector are the same. Then impose contraints to make each corresponding pair of nodes have the same displacement, in cylindrical polar coordinates.

When you do the vibration analysis on the sector, you can then plot N copies of the model, rotated round the axis of symmetry, and the displacements will match up across the boundaries.

For this special case (M = 0) you don't need to use complex numbers to make the model.

Last edited by a moderator:
Dear Sir,

Can you throw light on the question before nodal lines (I guess you missed seeing it)

Now, the mode shape denotes vibration pattern of the structure-right? When you say mode shapes may be described by a number of radial and circumferential lines do you mean a vibration pattern wherein the circle vibrates along the circumferential and radial lines only and then the final pattern (thus response) being obtained through superposition of all these radial and circumferential patterns?

Vishal

AlephZero
Homework Helper
The modal lines form a "grid" of radial lines and circles that do not move. The parts of the plate in between those lines vibrate up and down. See Figure 9.5 here: http://weather.org/singer/chapt09.htm

I would have thought there would be some better pictures or videos of this on the web, but for some reason I can't find any.

The response of the structure to some applied loads will usually be a superposition of different mode shapes, exactly the same as any other situation involving dynamics.

Dear Sir,

Thanks again.Some fundamental questions:

1) Can you point a picture wherein all blades are under vibration in any mode ? (flexural/torsional) mode

2) What is meant by an 'edgewise' mode?

3)Often while reading we come across terms 36EO/ 1family (1F) - I know engine order stands for exciting frequency and 1F for first flexural mode- indicating a resonance where exciting frequency matches natural freqiency of first flexural mode. What does 'family' signify here??

Vishal

AlephZero
Homework Helper
1. There are some pictures here that show you the sort of patterns you get:
http://www.softinway.com/news/articles/Vibration-Analysis-of-Large-Steam-Turbines/3.asp [Broken] (click on the links for pages 4 and 5)

2. A "flap" vibration mode is when the blade tip is moving approximately normal to the chord. An "edgewise" mode is when it is moving approximately parallel to the chord. "torsion" is when it is twisting. These are "exact" descriptions for the vibration modes of a flat plate. For real curved blades they are only approximate, but they are still useful.

3. If you do a vibration analysis of a single blade fixed at the root, the first few vibration modes can classified as "flap", "edgewise", or "torsion". If there are several flap modes the lowest frequency one is called 1F, the next one 2F, etc. If you take all the modes in order of their frequences, you will probably get a sequence something like 1F, 1T, 2F, 1E, etc.

For cyclic vibration, the blade modes will look similar, but the vibration frequencies may be a lot different because of the flexibility of the disk. All the cyclic modes where the blade motion is similar to the 1F mode would be called the "first family", the modes similar to the 1T the "second family", the modes similar to 2F the "third family", etc.

Because of the disk flexibility, it is quite common for frequency range of all the modes in one family (with different numbers of nodal diameters) to overlap the frequency range of other families. Describing a mode just by its frequency, or as "the 75th mode of the complete turbine wheel" isn't very useful, but a name like "36th engine order 2rd family" or "36th engine order 1st torsion" is a good description of what the mode looks like.

Last edited by a moderator:
Thanks Sir.

Sir, in that case (An "edgewise" mode is when it is moving approximately parallel to the chord) the edge wise mode is anotehr name for axial mode- that moving along its own axis?

Sir, can you throw some light on effect of damping (mechanical and that due to friction) on vibration of blades? Which one (mechanical or frictional) paly a greter role and why?

Also, to emulate frictional damping in a analysis (FE Analysis) - are there any clever ways? Kind of applying restraints or some other?

Vishal

AlephZero
Homework Helper
Sir, in that case (An "edgewise" mode is when it is moving approximately parallel to the chord) the edge wise mode is anotehr name for axial mode- that moving along its own axis?
Hm... I don't really understand what you mean by "its own axis", and we don't use "axial mode" to describe blade vibration, either for a single blade, or a complete wheel.

Sir, can you throw some light on effect of damping (mechanical and that due to friction) on vibration of blades? Which one (mechanical or frictional) paly a greter role and why?
The main source of damping that comes from from the wheel itself is the aerodynamic damping. That depends on how each individual mode shape interacts with the gas flow, and it can be negative (destabilising) as well as positive.

If by "mechanical damping" you mean the hysteretic damping from the oscillating stresses in the blade and disk material, that will be small unless the design includes some features like viscoelastic materials. That is more likely in a fan than a turbine, because of the operating temperatures involved.

Also, to emulate frictional damping in a analysis (FE Analysis) - are there any clever ways?
THis is one way to do it, either as a transient analyisis or to generate a linearised damper model to include in a steady state dynamic analysis. http://web.itu.edu.tr/sanliturk/Publications/GTP00919.pdf (You may remember that reference from another thread about underplatform dampers )

Dear Sir,

Thanks for the response.

Can you point a picture of the blade vibrating in an edge-wise mode?

Also, is the term nodal line same as nodal diameter? Is there any difference?

Vishal

AlephZero
Homework Helper
Can you point a picture of the blade vibrating in an edge-wise mode?
http://saam.mech.upatras.gr/saam/activities/specialtyfedamp.html [Broken]

Also, is the term nodal line same as nodal diameter?
They are two different ideas.

A nodal line is where there is no movement of the vibratiing structure. For a circular plate the nodal lines are either radial lines or circles. For a general cyclic symmetric structure they can be more general shapes.

The number of nodal diameters is the number of "repeats" of a cyclic vibration pattern around the wheel.

Last edited by a moderator:
THis is one way to do it, either as a transient analyisis or to generate a linearised damper model to include in a steady state dynamic analysis. http://web.itu.edu.tr/sanliturk/Publications/GTP00919.pdf (You may remember that reference from another thread about underplatform dampers )
Sir, regarding the above;

I have read in a reference (copying and pasting below);

a stiff constraining spring (1E7) is attached to the mid point of the right hand side of the platform.A spring constraint is used in preference to a friction damping element, as the response remains linear, and it is therefore possible to observe the effects of the constraint completely from a single response condition. The constraining spring will have similar behaviour to a damper in the ‘stuck condition’, where the influence on the response modeshape will be maximised. The localised constraint provided by the stiff spring changes the first flexural modeshape such that bending is concentrated in the aerofoil section above the platform, and also gives rise to local deformation and stress concentration in the platform

IS the spring attached here, simulating a damper simply because it bending gets concentrated in the aerofoil section above the platform thus giving rise to ,local distortion and stress concentration in the platform- which is similar to damper behaviour? Any thoughts Sir?