Rotating Beam - Natural Frequencies

Please excuse my probable over simplification here.

This sounds more like a basic rotational moment of inertia problem. Here is also a nice link, too. And to complete the set a Physics source. Without adding a second energy storage component (flexing of the stiff beam), I see no natural resonant frequency expected. Don't forget that the Coriolis force is technically a fictitious force.

As you know, this type of a mechanics problem requires a good understanding of the three dimensional movement of all components. I enjoy trying to tackle this kind of a problem but I do not grasp from your description what is happening. So with the exception of a few reference links, sorry but I can't help you.

I am sorry to contradict you. The Coriolis acceleration (which appears when a vector is changed as direction) and correlated force is REAL.

You can see it in the way rivers errodate more one shore depending on flow direction and latitude. It is used in the non rotating gyroscope (mostly mems to day) where a bending moment generated by Coriolis forces is measured.

I presume that RHABE thinks of the gyroscopic effects of a rotating inertia in case of transversal vibrations. This is a "dy/dt" proportional resistance if "y" is the radial shaft deformation.

Hi, some addendum:

non-flexible beam - nonrotating - will be described by the equation:

Jq*Φ_x^..+k_Φ=T_x(t) Jq*Φ_y^..+k_Φ=T_y(t)

Jq= moment of inertia vertical to length axis.

Φ = angular movement

k_Φ = angular stiffness

T(t) = time variable Torque

Angular movement in x- and y-direction. These are the angular movements, oscillations.

In reality there is also the possibility of a linear movement, oscillation (also) perpendicular to the beam. This will introduce 2 more (mass-spring) equations.

m*x^.. + k * x = F_x(t) m* y^.. + k * y = F_y(t)

As the center of mass usually is not in the center of elasticity there is a cross-coupling between radial and angular (tilting) movements. (To be introduced into above equations, but negelected here for clarity.)

Rotating beam, including groscopic torques:

Jq*Φ_x^..+ H*Φ_y^. +k_Φ=T_x(t) Jq*Φ_y^..- H*Φ_x^. +k_Φ=T_y(t)

H is angular momentum H = J_z * ω_z

_{z-direction is along beam-axis.}

So now the behaviour of the rotating beam is described by 4 differential equations, giving 4 natural frequencies. (No stiffness -k-non-symmetry assumed.)

2 are tilting motions, 2 are radial motions (natural frequencies).

At standstill there is only one tilting and one radial. These are split into two by the gyroscopic (Coriolis) torques if there is rotation. This will be detectable only if rotation speed is fast.

Next problem: bending.

If the beam is flexible (L/D>?) 2 new equations have to be introduced: one in x one in y.

For this situation I am searching for a closed (maybe series) solution.

This shall be introduced into a program: first calculate the bearings (air- or fluid- bearings) or introduce the known stiffnesses of ball-bearings. Then add the mass distribution of the rotor, then calculate the first 6 natural freqiencies to be sure not to get big problems there.

Gas- and fluid bearings will be destructively unstable at rotation speed above twice the first natural frequency (resonance).

10,000 rev/s would be very good, 5,000 achieved today.

RHABE

I may be wrong but I have the feeling that the equations do not show the coupling of inertias by the shaft stiffness according to the way the equations for the 2 inertia have to be written it should exist a term with the angle difference which is not present in your system. Further more those torsional vibrations can occur even if the shaft turns.

Torsional vibrations:

If 2 inertia are on same shaft there are 2 equations one for each inertia as follows:

J₁*(d²θ₁/dt²)=M₁(t) – η₁* (dθ₁/dt) – k(θ₁-θ₂) (1)

J ₂*(d²θ₂/dt²)=- M₂(t) – η₂* (dθ₂/dt) + k(θ₁-θ₂) (2)

Where

J_1,2 = Inertia of the rotating masses with respect to rotation axis

Θ_1,2 = Rotation angles of the masses

η_1,2 = Damping coefficients of the masses

k = Stiffness of the shaft connecting the 2 masses.

Assuming a linear behaviour we obtain after a Laplace transform a linear equations sytem as follows:

Lθ₁*(J₁*s² + η₁*s + k) – Lθ₂*k = LM₁

-Lθ₁*k + Lθ₂*(J₂*s² + η₂*s + k) = -LM₂

Where

L = is the symbol for the Laplace transform of the function

S = Laplace operator

The solutions in the Laplace form can be deducted with the Cramer rule the system being linear and considering Lθ₁ and Lθ₂ as the unknown the 2 moments are the perturbation functions.

Lθ₁ = [LM₁* P₂(s²) + k*LM₂] / [P₁(s²) * P₂(s²) –k²]

Lθ₂= [LM₂* P₁(s²) + k*LM₁] / [P₁(s²) * P₂(s²) –k²]

With

P₁(s²) = J₁*s² + η₁*s + k and P₂(s²) = J₂*s² + η₂*s + k.

The critical frequencies are the roots of the denominator D = [P₁(s²) * P₂(s²) –k²] = 0.

By replacing "s" with "j*ω" we obtain an equation of the 4^th degree in "ω" whose results are the critical pulsations of the system. In above j= (-1)^0.5. For further analysis ω=2*π*f and allows a direct computation of the frequencies.

The reason M₂ has sign "-" is that M₁ was considered as driving moment and M₂ as resistant.

Please comment since it can be also a notation problem I did not understand.

That's my fault,

I did not state that I am only thinking about radial movements/oscillations of the shaft and attached masses.

Your equations consider torsion - ok. To be treated separately. Same with axial movements.

Radial movements:

if at both bearings in phase: radial movement/resonance

if at the bearings 180° phase difference: tilting movement/resonance

So 2 resonances (at standstill), movements in x or y direction (if rotation axis is z).

(Rotational symmetry assumed!)

If gyroscopic torques included, both resonances split into 2, so 4 resonances. May be not distinguishable as splitting may be too small and any two have a tendency to synchronise -depending on coupling.

Above problem to solve for a rigid shaft is not a big problem.

My problem: the bending of the shaft with 2 end-masses and its related natural frequency (first one only), -also split into 2- may have a resonance within the above mentioned resonances.

Or: how thick the shaft has to be selected? There is a need for a thin shaft, as in airbearings the viscosity of the air will cause severe drag if high-speed (3rd power of diameter if constant length.)

Either: all these resonances to be designed to have the same value, or only the radial and tilting resonance to have nearly the same value and the bending resonance to be placed well above. (2.2 times the lowest)

Any damping can be neglected.

Thank you for your interest.

RHABE

As usual I shall start with the sentence "I may be wrong" since I consider, as engineer, that nobody detains the whole truth.

A first question concerns the arrangement of the "discs" and "bearings" since the shaft/beam deformation and resonances depend on it. It will not be the same if masses are between bearings or outside of bearings.

A second question concerns the bearing design. Depending on it the bearing could or could not manifest a "tilting stiffness". It will have a radial stiffness anyway depending on several factors but if the design does not generate it a tilting stiffness which is according to my opinion determinant for the results will not be present.

The 3^rd aspect is that I think the gyroscopic moments to be due to a disc rotation speed around X or Y if Z is the rotation one. This means the effect will oppose an angular movement but proportional to the angular velocity of the movement (bending or tilting whichever) so that it will be equivalent to a damping and will stabilize increasing the logarithmic decrement.

Please give the information so that I can as promised have a correct look at your problem and correct me if I am wrong.

Hi,

"discs" are at the extreme left and right, outside the bearings.

"bearings" may have a tilting stiffness, depending on construction.

If ball-bearings and X-configuration then no tilting stiffness.

If ball-bearings and O-configuration, then high tilting stiffness.

If air-bearings or fluid-bearings, then maybe high tilting stiffness depending on construction.

So tilting stiffness has to be included.

"The 3^rd aspect is that I think the gyroscopic moments to be due to a disc rotation speed around X or Y if Z is the rotation one."

Yes, exactly.

"This means the effect will oppose an angular movement but proportional to the angular velocity of the movement" Yes, like a stiffness, proportional to ω_z.

"so that it will be equivalent to a damping and will stabilize increasing the logarithmic decrement" No, this is not a damping as total energy remains constant. This is a cross-coupling: rotation-speed ω_x (around x-axis) results in torques T_y around y-axis and vice-versa.

These torques may come from inertias and/or stiffnesses (or damping, here to be neglected).

Any more questions: no problem.

RHABE

Here is a first attempt to answer to your question.

Assumption: bearings have only a radial stiffness no tilting one. In case they have also a tilting one the frequencies are even higher, what you are interested first is the lowest frequency.

The model :

For this analysis the shaft is considered as stiff. I am aware of the influence of its own stiffness but I think that the approach has to be done step by step.

The perturbing force is either at M1 or at M2 and has a pulsation "ω".

Under the forces the shaft will have due to the bearings compliance a plane movement which can be considered as the sum of a plane translation and a plane rotation.

The translation will have the critical frequency as : ft=(1/2π)*[M1*(C1+C2)/(1+µ)]^0.5

µ=M2/M1

The rotation behaviour will be dependent on two parameters:

-stiffness with respect to an angular movement Cr= C1*(a+m)^2+C2*(b+l-m)^2

-inertia with respect to rotation centre Jr= M1*(a+m)^2+M2*(b+l-m)^2

We are interested in the minimal value which is obtained for a value of "m" satisfying the equation:

E= d( ωr^2)/dm=0 where

ωr^2= Cr/Jr

The 2^nd order equation to determine the "m" value is:

E= m² *{(C1+C2)*[ a-µ*(b+l)]+C2*l*(1+µ)} + m* {(C1+C2)*[a^2-µ*(b+l)^2]-C2*l^2*(1+µ)}

-C2*l*[ a^2-µ*(b+l)^2] – C2*l*[ a-µ*(b+l)] = 0

The equation can be verified by making the assumption of full symmetry:

a=b; C1=C2;M1=M2 (µ=1)

In this case m = l/2. You may check it, I did, but if I copied with errors it can only this way be discovered.

The frequency of this second oscillation will be : fr= (1/2*π)*(Cr/Jr)^0.5

both values- Cr and Jr- computed with the "m" value obtained with the above equation "E".

This approach delivers already two critical frequencies from the whole spectrum of possibilities.

There are some remarks to be made:

- If the bearings have a high "tilting stiffness" the frequencies will be higher

- The gyroscopic moment is proportional to 2 rotational speeds, first the ωz if we assume as "Z" the rotation axis and second the oscillation speed of shaft with respect to the rotational centre which has the same frequency as the ωz. It is thus proportional to ωz^2 if and only if perturbations are originated only by disbalances. With respect to the shaft oscillation it will be proportional to the first derivative of angular displacement thus it will act as damping since it is opposed to the rotation of the "discs" (opposed to the change of rotation plane of discs), at least it is the way I see the problem. But as usual I can be wrong at least 5% (=100-95%).

I will look at the other influences but I still think that above frequencies are the lowest.

I would appreciate comments.

Hope it will help,

Nick Name

Hi, yes this helped!

"bearings have only a radial stiffness no tilting one"

ok for the moment later tilting stiffness to be added.

The model is ok, except I am searching for the solution with flexible shaft.

This will be necessary in order to have a thin shaft (but still with enough stiffness) in order to limit bearing friction.

"This approach delivers already two critical frequencies"

One is the radial- the other one the tilting-frequency.

The gyroscopic moment is proportional to 2 rotational speeds, first the ωz if we assume as "Z" the rotation axis (yes) and second the oscillation speed of shaft with respect to the rotational centre which has the same frequency as the ωz (no, at any frequency). It is thus proportional to ωz^2 if and only if perturbations are originated only by disbalances (any frequency from grinding or milling operation). With respect to the shaft oscillation it will be proportional to the first derivative of angular displacement (yes) thus it will act as damping (no) since it is opposed to the rotation of the "discs" (opposed to the change of rotation plane of discs), at least it is the way I see the problem. But as usual I can be wrong at least 5% (=100-95%).

This is not a damping as the gyroscopic effect draws energy from the x-tilting and transfers this to the y-tilting. No energy is dissipated (as required for damping)!

I will post a hand-drawn sketch with my model and some basics.

RHABE

I'm not certain how this applies to the present situation, but coupling the modes often does increase the effective damping. This depends on the k-Q product. In fact for a given loss in each mode there is a specific coupling factor at which the effective damping reaches a peak. I would have imagined the combination of the inter-mode coupling being minimised and (possibly) a peak in parametric amplification would be the reason the problem is exhibited at the speeds you see.

A silly question, perhaps: Is there any possibility to place bearings outside the (presumed) turbines? Then you could have a fat rod and slim bearings.

BTW, I assume that as the resonance frequency is the critical parameter you are already using a low-density alloy - or is there an insuperable corrosion issue?

Hi,

damping may be inherently high but limited by a compression effect - if gas-bearings are used. High damping is achieved if the gap can be design to be an efficient squeeze-film damper.

RHABE

Agree. The bending is present in the second attempt to answer your question and it considers the presence of the 2 masses as you see.

I analysed the 2 effects uncoupled one from the other but they can be coupled if needed.

Now you have both aspects.

Are the two parts to be considered as corresponding to your expectations ?

Even if the gyroscopic aspect is discarded?

I made the assumption that the perturbing frequency can be or cannot be equal to the rotation since I did not know the application. The presence of the grinding wheel justifies your concern due to very high generated frequencies.

Coming back to the gyroscopic effect, if it transfers energy from the "X" to "Y" axis this means that the energy on the "X" axis will be less. Is this not equivalent to a "damping" for the "X" axis? There is the dynamic damping which uses a secondary resonator to reduce the main amplitude. Could it not be considered as similar?

"Are the two parts to be considered as corresponding to your expectations ?"

Yes, very helpful, as tackling my problem from a different view than I tried. This helps in understanding.

"Coming back to the gyroscopic effect, if it transfers energy from the "X" to "Y" axis this means that the energy on the "X" axis will be less. Is this not equivalent to a "damping" for the "X" axis?"

This transfer from x into y is perpetuated as the same is true for y back to x. So this is a circular motion that may be without any damping. Only if bearing- or air-damping is significant, then the energy is dissipated. Maybe some in bending of the shaft too.

If the rotor is not constrained in bearings (bearings with stiffness = 0) then this circular motion is the nutation, occurring at ω_N=ω_z*Jz/Jq. z = rotation axis, J = moment of inertia, around axis z or q= x or y.

If the rotor is constrained then the gyroscopic stiffness adds and subtracts from the other stiffnesses (I am not sure if this model is really valid, but it helps). If at very high speed the gyroscopic stiffness is much bigger than the bearing stiffness then there is the asymptotic behaviour of natural frequency towards nutation (and precession-) frequency.

"There is the dynamic damping which uses a secondary resonator to reduce the main amplitude. Could it not be considered as similar?"

This is working by high amplitudes and sufficient damping in the secondary resonator. But very difficult to tune and not working if any changes in natural frequency of primary or secondary oscillator not matching each other.

RHABE

I come again to the second model

Assumption: bearings have only a tilting stiffness.

The model is same as the one from precedent comment.

All other conditions same as in precedent model.

I made, when looking at the gyroscopic influences, a limited sketch which was the reason for the wrong statement. Sorry.

I checked once more the system and the results and came to a modified equation due to an algebraic error made in previous computing:

Notations:

a= a/l β=b /l γ_1,2=CT_1,2/CL CT_1,2 = bearing tilting stiffness µ=M₂/M₁

CL = shaft bending stiffness = E*J/l for the model above.

γ_Σ = 1+γ₁ + γ₂

Dimensionless parameters are introduced to allow a better optimization analysis.

The equation is "s" is:

E= s⁴ * Ω^-4 *µ* [α³*(1+3/( α *γ_Σ))* β³*(1+3/( β *γ_Σ) -9* α² β²]

+s²* Ω^-2*{[α³*(1+3/( α *γ_Σ))]+µ*[β³*(1+3/( β *γ_Σ)) ]} + 1 = 0

M₁*[l³/(3*E*J)] = Ω^-2 is a frequency characteristic for the basic system dynamics.

I would like to make a comment with respect to bearings design:

Air bearings (aerostatic)- have a very low friction but also a low specific carrying capacity. I do not know how big are the maximal radial forces but the smaller the bearing the lower its stiffness. A double bearing is compulsory in order to generate a tilting stiffness. It could be the right choice from the point of view of power losses but could demand a higher diameter and length for the stiffness. The damping is, I believe, very low.

Hydraulic bearings (hydrostatic): Higher friction but also a lot higher radial stiffness and a more important damping effect. The problem could be the heat evacuation. Double bearings are required. The small flow can be a problem for the upstream restrictor.

Hydrodynamic bearings: A double bearing could be a solution since the oil wedge has an own stiffness but I do not know how important could be the damping.

In all 3, above concepts a bidirectional axial bearing is necessary to take care of the possible axial components either from the grinding tool or from the driving motor field.

Ball bearings with angular contact: In the "O" assembly offer an excellent stiffness as well in radial as in tilting directions, can accept as well radial as axial forces, accept rpm even in excess of 10.000 ( checked up to 26.000!) but have no damping and will require a very well designed lubrication system (starving but not dry!). It could be possible that due to the high stiffnesses (especially with an optimized preload) the critical frequencies are so high that the ratio ω/ω _critic will be small enough to avoid a need of damping (working in under critical domain very far from resonance).

If one bearing has to be able to maintain an axial position the other should allow an axial sliding free to compensate for thermal expansion. Needle bearings are available for similar rpm. They present as well a radial (important) as tilting (less important) stiffness. The inconvenient aspect of the ball and needle bearings is the variable stiffness versus rotation angle. It can be source of vibrations.

Electrodynamic bearings: Several advantages as controlled damping, as angular stiffness controlled independently of radial stiffness, very low friction but a high complexity and I do not know if it is justified although in modern milling (high speed) and grinding heads those bearings are more and more integrated. If what you design is an only prototype it is not justified.

Hi,

I succeeded in calculating the split of the tilting frequency by the gyroscopic/Coriolis torques. Plot is below.

The limited performance or my limited knowledge of Mathcad does not allow me this to be extended to radial and bending natural frequencies. I will try some more approach.

"Air bearings (aerostatic)- have a very low friction but also a low specific carrying capacity. I do not know how big are the maximal radial forces but the smaller the bearing the lower its stiffness."

I am just in half-way to make a program that can calculate any needed data of outlet-restricted airbearings (grooved types).

These rotating frequency calculation shall be part of this.

Load capacity of air-bearings: first estimate is 10% of projected bearing area times pressure. Stiffness is load capacity divided by bearing gap. So limited by technology and viscous drag.

"The damping is, I believe, very low"

Damping may be very high but not at zero frequency nor at high frequency. This is a squeeze-film-damper action.

Hydraulic: next to calculate, not a big problem, inlet-restrictors are all problematic in air- and hydro-bearings.

Angular contact ball bearings: more sensitive to geometric tolerances than anybody will believe.

Professional Instruments is offering now a tapered roller bearing with only 75nm runout. (see www.airbearings.com)

I was in a project of a 100mm rotor in 22mm OD bearings with 60,000rpm and minimum expected lifetime of 4 years continuous running. That was back in 1986.

Ball-bearing grinding spindles now routinely operate at 2000 rev/sec. (www.grinders.com)

I have developed and built three different types of starved lubrication systems for use in high precision ball-bearings. These are driven by capillary forces, and transport oil at a rate of (adjustable) 1 to 1000 µg per day.

And a measurement system that can measure (contactless) the resulting lubricant film thickness.

An additional positive feature is the ability to wash down any wear particles and maybe dirt to a filter, so operation is much cleaner and predictable than without oil recirculation.

No needle bearings at intended rotation speed of 5000 to 10,000 rps ! possible.

Electrodynamic bearing: not possible with very small diameter shaft, as magnetic force is limited at maximum 1 Tesla equivalent to 4 bar, very likely only 0.5 Tesla available, so 1 bar. Not enough to compete with airbearings. And cost is typically a factor of 5 higher compared to ball-baring or airbearing spindles - so no way.

RHABE

This is interesting, and as expected the problems seem to occur when the difference between the red and blue curves is an even multiple of the resonance of interest - which represents the condition where the planar vibrational modes stop coupling to each other. But I'm not clear what this tells you that you didn't already know?

Hi,

"as expected the problems seem to occur when the difference between the red and blue curves is an even multiple of the resonance of interest"

Why do you think so? The red and the blue curves are two independent resonances (as function of speed).

If both are excited by broadband excitation (from grinding noise for example) then there will be some difference frequency generation by the nonlinearity of the bearings. But the other way?

"planar vibrational modes"

This too I do not understand. In any of the resonances at rotational speed there is conical motion. So what are you thinking is a planar vibrational mode?

"But I'm not clear what this tells you that you didn't already know?"

You are right, nothing new. But a first step towards the final solution. Next will be incorporating the radial motions and the coupling of radial to tilting.

Then (hopefully) the bending, and its coupling to radial and tilting. Maybe by a brute force approach of two isolated flexibilities that couple three pieces of the shaft?

RHABE

RHABE

We are clearly looking from different perspectives. I'm sure they end up being equivalent; the sole "benefit" of mine is that it may be possible to get some insight based on the behaviour of coupled resonators (on the other hand, it could be pure red herring ).

Before starting on mine, a comment on your notes:
A rotational asymmetry that still retains planar symmetry will change the cones of the two modes from being circular to elliptical. I believe this effect will be small enough to be unimportant, however.
What will be important is if the centre of mass is off-axis**, as this will force a vibration (coherent with the environment) at the rotation rate. That is never the same as either frequency, but it is the same as the frequency difference. We clearly need an element of nonlinear coupling between the modes (easily generated by the rotation) for this to be effective. So the issue is why the situation peaks near twice the resonance. I believe it has to be the phase of the back-coupled signal between the two resonant frequencies - and this is easier to access if we can convert the situation into the form of coupled resonators rather than as a pair of uncoupled rotations.

It's probably easier to explain my viewpoint starting at zero rotation. It you have perfect rotational symmetry, you can consider the fundamental bending mode frequency either as being modes a pair of modes in vibrating a plane or a pair of modes rotating around a cone - it makes no difference. You may therefore say "why bother even mentioning this"?

So my next step is to look at low rates of rotation. Here, we can either consider the modes to be a pair of uncoupled conical modes with opposite rotations and at different frequencies, or a pair of planar modes that couple to each other, with a coupling coefficient K that depends on the rotational frequency. Again it makes no difference to the results - except that maybe it simplifies consideration of a forcing vibration that is stationary with respect to the environment*. For clarity (although it makes no difference in the end) I will consider only a forcing that is in the plane of vibration. Now we see forced power going into one of the planar modes, and then being coupled into the other ("secondary") mode. Now, for coupled modes it turns out that build-up of power in the secondary mode and subsequent feed-back to the forced mode can considerably reduce the effective resonant Q of the primary mode, even when both modes are subject to the same damping (the "stealth" guys use an equivalent effect quite a lot to kill radar reflections, but that's quite another story).

If now we look at a pair of coupled modes under these conditions, we find that the apparent Q of each mode falls as the coupling increases and rises again. So the "new" questions might be:
1) Where does the effect of non-linear coupling into the modes reach its peak. Well, with ordinary lowish-loss coupled resonators, we generally get constructive interference in the second-coupled signal at about the point where the split is approximately equal to twice the uncoupled frequency (Sorry, no, I couldn't either find a reference - it's not even a general result, as the optimum depends on losses - plus I'm not certain that the effective centre frequency remains constant in your case)
2) Where is the loss of a mode approaching the stationary case - again, it generally reaches a peak somewhere below unity split (depending on the actual Q's) and decays thereafter.

This isn't of course answering your question any more than what you have done already - but maybe it points at useful places to look - off-axis mass loading, and nonlinear inter-modal coupling via the rotationally invariant line of the bearings.

*It's the same if we consider the rotational modes, of course; the only difference is that it is easier to visualise a planar mode excitation being stationary w.r.t. the environment than a rotating one - and the coupling viewpoint which allows stealing ideas from other fields isn't so obvious.
**I imagine this is so "motherhood" that you didn't bother to highlight it - so apologies for the egg-sucking bit of the discussion.

Hi, some nuts to crack, thank you for giving me a start of new thinking!

"A rotational asymmetry that still retains planar symmetry will change the cones of the two modes from being circular to elliptical."

Pure bending, radial or tilting motion (without rotation around z-axis) may have x- and/or y-components that add up to linear motion if in phase and circular motion if 90 degrees phase difference (circular left or right depending on + or - 90 °) or elliptical if not having the same magnitude. Any of these are good examples of low damping nearly linear oscillators.

Pure gyroscopic motion (nutation or precession are pure conical movements). In solving the determinant of the two equations of motion precession comes up with positive sign and nutation with negative sign. This is a linear oscillator that can oscillate only by transferring energy between x- and y-axis tilt.

What I never heard is how a rotating shaft is moving in - one or more of - bending-, radial or tilting movement by influence of the gyroscopic torques.

I understand your comment that there is a coupling but this is weak so the oscillation may stay in x-axis for some periods then y-axis movement comes sup then x-axis again. As in coupled oscillators. I am not sure but very likely this will happen. To be sure the equations of motion to be solved with an impulse (or step) input.

"centre of mass is off-axis"

Certainly this is existing in any rotating shaft. Balancing is never ideal.

This will excite either the upper or the lower of the 2 (per principle movement), so 6 in total resonances if the speed is identical to one of these resonances.

Any other mechanism may also excite one of these resonances if the frequencies match.

For clarity I added the line Omega = rotation-speed into the diagrams.

At the crossovers of this line with the lines of natural frequencies there will be critical speeds: excitation of resonance by unbalance.

"but it is the same as the frequency difference"

This I still do not understand. Please explain if possible.

"So the issue is why the situation peaks near twice the resonance."

This is quite another issue as at half-frequency whirl the tangential load-capacity of hydro- or aero-dynamic bearings vanishes. And at rotation rate of twice the first natural frequency also the radial load-capacity vanishes - with the result of nearly immediate failure.

This is not a problem to explain in pure dynamic bearings. But I have measured the same effect in inlet-restricted aero-static bearings and cannot explain why. Measurement of response with vibrational excitation at base of spindle shows 730Hz first resonance with very high damping (D>20). And at 1450 rev/s rotation speed the 730 is highly excited growing from 10nm amplitude at 1445 Hz to 10 µm at 1450Hz and abrupt destruction at 15 µm amplitude between 1450 and 1451 rev/s.

As the load and stiffness generation in aero-static bearings is totally different from aerodynamic bearings I do not know what happened. One speculation is that we had a high percentage of aerodynamic action in this spindle.

I have no measurements in the newer one - no possibility to measure the movement of the shaft at whatever rotation rate. So I do not know if the upper speed limit is also this and only this effect - but I suppose this as no other effect is known.

"Now, for coupled modes it turns out that build-up of power in the secondary mode and subsequent feed-back to the forced mode can considerably reduce the effective resonant Q of the primary mode"

Would be nice to use here too. Looks promising, but needs high damping and this is existing either in both modes or in none.

"Well, with ordinary lowish-loss coupled resonators, we generally get constructive interference in the second-coupled signal at about the point where the split is approximately equal to twice the uncoupled frequency"

I have to think about. I estimate that we will not reach this region as the half frequency whirl will limit earlier the rotation speed.

Thank you fro commenting

RHABE

Pure bending, radial or tilting motion (without rotation around z-axis) may have x- and/or y-components that add up to linear motion if in phase and circular motion if 90 degrees phase difference (circular left or right depending on + or - 90 °) or elliptical if not having the same magnitude. Any of these are good examples of low damping nearly linear oscillators
Perhaps I should have mentioned that these two elliptical modes have different frequencies?

I understand your comment that there is a coupling but this is weak so the oscillation may stay in x-axis for some periods then y-axis movement comes sup then x-axis again. As in coupled oscillators. I am not sure but very likely this will happen. To be sure the equations of motion to be solved with an impulse (or step) input.
Coupling between planar modes is small when the rotational rate is zero, and rises rapidly with increasing rotation rate. At the same time, the rotation suppresses direct coupling between counter-rotating conical modes. But coupling via the stationary environment is different again, and depends on asymmetries and on nonlinearities. It can be significant when non-rotating, fall, and rise again. The slow/non-rotating situation is not an issue, because there is minimal forcing at the resonance.

"but it is the same as the frequency difference"
This I still do not understand. Please explain if possible.
Assuming that I am interpreting the graph correctly (I can't be certain because I can't read any of the labels - on the other hand it correlates with expectations), the red and the blue curves represent the rotational resonant frequencies in the two directions, and the horizontal axis is the rate of rotation. In this case (unless I have something wrong) sum of the signed rotational frequencies and the difference between the unsigned frequencies (of rotation) are always equal to the rotation rate. I wrote difference frequency on the basis of the usual coupled resonator descriptions (though not the mathematical representations) usually ignore sign - but for what it is worth there could be a physical justification as well:
A major relevant interaction with the stationary part of the structure will obviously be rotationally asymmetric - and this is insensitive to rotational direction (and therefore sign); so for this interaction it we are looking at effective differences.

at half-frequency whirl the tangential load-capacity of hydro- or aero-dynamic bearings vanishes
I have absolutely zero knowledge here, and I'm not sure I'm interpreting this correctly. Tangential to what? If the rotation then it's not obvious to this naive observer that it is an issue? If (for whatever reason) it is an issue, I think you are saying that the relevant stiffness at half the rotational frequency becomes zero. If this is right, based on your graph the system should fail when the rotation rate is about 1.2x the static resonant frequency, which is where the lower of the two whirl modes is half the rate of rotation.

Regards

Fyz

Thank you Nick Name,

"You have a better approach using the Castigliano theorem"

I never heard of Castiglianos method used in dynamics. Do you know an introductory book or can give a very first introduction (two sentences may suffice).

I prefer Castigliano on elasticity problems - nearly ideal.

"Limitations by: E*J*d²y/dx²=-M"

This is not a problem as length to diameter ratio will be near 10.

You are absolutely right that I need to solve the problem myself - that is why I asked and I greatly appreciate your and others help.

RHABE

I'm not sure what you intend when you say that the standard (static) from of Castigliano only applies to thin beams. SFIK the standard forms apply to all elastic structures operating in the regions where the elasticity (not the form of the structure) is linear.

Maybe you are referring to an approximation similar to Euler's? If so, that is fine provided that both the KE of the supporting beam and the elastic energy stored in the end masses are negligible. (I.e. the structure approximates a rigid mass on the end of a light spring).

But whether or not this is the end point, I think that the simplified dynamic problem (using massless spring and rigid mass + disturbing forces and interactions) is more than complex enough to start with.

The reason I consider it as an approximation is the way how the angle is accepted to be.

In fact there is an energy balance on one side the energy delivered by the vector "V" and on the other side the deformation energy given by the integral over the beam length. The first is E= V*f/2 where V is the vector amplitude and f the corresponding travel of same vector. The second is E= ∫(M*θ/2)*ds where M is the bending moment and θ the rotation angle of M. If it is accepted that θ = d²y/ds²=M/EJ then we obtain the form E=∫M²*ds/EJ. The derivation of both sides with respect to V leads to the form I mentioned above. Here is the approximation valid only under certain circumstances among other the relative length of the beam. The shaft subject of all discussions is NOT slim so that applying this method to compute the critical frequency is NOT any more than a GOOD approximation. With respect to the shaft own mass it is possible to take it into account depending on the beam deformation as a reduced mass at same place as the carried mass. I shall give an example with a spring.

Spring mass Msp is uniform over the length msp=Msp/H. The stroke being "f" at a distance "z" from the spring end the displacement is f(z)= f*z/H. The equivalent mass placed at spring end will be Mspeq=f^-2*∫[(f/H)*z]*msp*dz (the integral from 0 to H). The result is Mspeq=Msp/3. So that if the frequency has to be more accurate or the dynamic simulation nearer to reality it is better to add at the Mass put on the spring Msp/3. In most cases this term is totally neglectable. same approach can be used for other forms of springs. And to came back to the shaft since as RHABE wrote the tool can be very low inertia it could be perhaps good to consider the shaft mass in the computations.

What you need is an equation putting together inertia, stiffness, and loads. The use of the mentioned theorem is a way to obtain this equation.

For a beam the displacement due to a vector "V" is f(V)= 1/EJ*∫M*∂M/∂s*ds applied on the whole beam length "L". M-being the bending moment generated by all vectors applied to the beam. You have 4 unknowns: -displacement (radial) of mass 1, same of mass 2 and the 2 angular movement in the bearings with tilting stiffness. Depending how you introduce the vectors you can get the connecting system of above mentioned parameters and solve for the critical pulsation.

I made a computation for a single mass since I think that for what you intend the working tool end is a lot less inertial than the motor rotor (and I was too lazy to do for the 2 masses) and it works pretty well. It is possible to bring the result in a dimensionless form with real advantages for optimization and extrapolation.

Guest's comment is OK.

Hi Nick Mame,

I have done this today for radial-, tilting- and gyroscopic motions - so until now not including any flexibility of the shaft. (4x4 matrix)

I think that I will have more unknowns if including flexibility of the shaft: radial and angular displacements of the 2 masses in x and y, then ex, ey and fix, fiy = radial and angular displacements at the location of the bearings. So in total 16 unknowns. May be reduced to 8th order by symmetry as nominally identical in x and y.

This will take some time.

At the moment I will try to get to work the 4x4 solution with Mathcad, seems to be a problem, the first three attempts did not work. Mathcads "polyroot" function will not find any roots.

Thank you for your help.

RHABE

No shaft flexibility? All in the bearings then? But I don't think it matters so long as you can treat the shaft as massless, as all the flexibility in each plane can calculated separately and lumped into a single item.

BTW, I hate, loathe and detest mathcad with all my heart (something I would never say about animates)

Here is a second attempt to answer to your question.

Assumption: bearings have only a torsional stiffness or tilting one.

The model :

For this analysis the shaft is not any more considered as stiff.

The perturbing force is either at M1 or at M2 and has a pulsation "ω".

Under the forces the shaft will have due to the bearings zero radial compliance and finite angular compliance a deformation which can be considered as the sum of a all influences. If we consider for instance the active force on mass M1 the shaft movement will move also mass M2 and induce an inertial resistance at its level which will modify the beam/shaft deformation at the mass M1 position.

Taking all those effects into consideration the equation for the critical pulsations is:

-ω^4*M1*M2*[α1*(β1+β2)+α2*β1] + ω^2*M1*[α1+α2-µ*(β1+β2)] -1 =0

Where:

α1- a^3/(3*E*Jx) α2= a^2/CΣ β1=b^3/(3*E*Jx) β2=b^2/ CΣ

CΣ = CT1+CT2+E*Jx/l

E= Young modulus

Jx= inertia moment in bending (thus as reference the axis "X" perpendicular to the axis "Z").

As a control of the equation if CΣ is very big (only due to the CT1&CT2 values) α2 and β2 go to zero and the equation becomes:

-ω^4*M1*M2*[α1*β1] + ω^2*M1*[α1-µ*β1] -1 =0 being thus dependent only on the cantilever compliances (was expected).

The gyroscopic moments are:

MG1= -K*J1*ω*(dθ1/dt) MG2=-K*J2*ω*(dθ2/dt)

Where θ1 & θ2 are the rotation (tilting) angles around the "X" axis. "ω" is the shaft rotation speed.

Since θ and y (linear displacement) are in phase those moments act as dampers.

Or at least it seems to me that they act as dampers.

For instance the gyroscopic moment of mass M1 which has the inertia moment "J1" will be:

MG1= k*J1*ω²*{[-F*cosωt-M1*ω*(dy1/dt)]*[a²/(2*E*Jx)+a/CΣ]+M2*ω*(dy2/dt)*b/ CΣ}

One sees that:

- a 1^st term opposes the movement and the generating force

- a 2^nd term compensates partly this damping, since y2<y1 the damping effect will be preponderant.

Of course those equations have to be adapted to the real expected conditions but I think they are a good starting point.

Comments will be welcome.

I will look at the other influences but I still think that the frequencies computed in previous comment are the lowest and thus have to be avoided.

Hope it will help,

Nick Name

I don't pretend to be confident about what your physical system is, but if its a rotating shaft with bearings at the ends and some point masses at the end/s near the bearings, then you are mostly talking about whirling of shafts - as distinct from torsional vibration. The stiffness of the bearings will have an effect and the masses probably very little.

If you are talking about whirling of shafts, then a Google should turn up something useful. I've long forgotten the maths, but basically the critical whirling speeds or orders 1,2,3...etc will correspond exactly (I think) with the vibrations frequencies that would otherwise occur if the shaft were forced/stroked laterally (like a violin/guitar string) when there was no rotation.

Dear Trevor M.,

this is a rotating shaft, but not at usual (60Hz or near) speed.

Whirling of shafts is ok, but gyroscopic torques to be included as speed will be very high, near the instability limit of the bearings. So any frequency is split into two nearly identical frequencies at low speed (so not to be detected because of synchronisation), but the two frequencies lying widely apart if speed is really high.

So the guitar-string solution is no longer valid.

I try to make a revival and extension of my mechanics calculation experiences - never did this as I found everything necessary in the textbooks.

RHABE

Given that you cite Timishenko, I assume you have a short, fat beam. I hate to say this, but if the masses at the ends are significant they are probably flexible as well. In addition, the beam is rotating at an angular frequency comparable with the resonance frequency. I don't believe that even Timishenko covers this in his publications.
On this last point, in my ignorance I would have thought that the critical resonance frequency in the direction of spin would be the actual resonance frequency, and nonlinear coupling to other directions would become significant when the rotational frequency was half the lowest resonance frequency - can you provide a reference so I can educate myself?

Returning to topic, you have unevenly distributed mass on a short beam. Analytic solutions exist for some modes with uniform continuations at a different impedance, but not SFIK for the case you describe. I think you know my penchant for analytic solutions - nevertheless, Iwould (regrettably) be inclined to develop design curves using FEA.

If you are desperate for an analytic method, the best lead I know of would be Dimitri Vassiliev (now at UCL), who used to work in related areas, and would probably know exactly where to look or whom to ask; but you would have to define the structure and rotation a bit more tightly so that even I could visualise it.

Fyz

Hi,

I will try Timoshenko next week.

As I have to go up with speed as high as possible I have to include gyroscopic torques. The equations for a fast gyroscope in elastic constraints I took from Savet: Gyroscopes, Theory and Design, (1965 ?).

The theory for a rigid non-rotating beam including coupling of radial to tilting motions I took from a book by Magnus (in German).

I do not include here any nonlinearity nor axial or torsional resonance. As these are decoupled (not really) I try first this way.

I do not include damping as damping in bearings is a really difficult topic.

In ball-bearings with starved lubrication there is no significant damping. Any more lubrication will either overload the motor or overheat the oil - by too much power.

In air-bearings there is (maybe) very high damping but this is dependent on frequency and I can make only an educated guess about the magnitude of damping. (See PHD work of Roblee at LLNL 1987 (?))

In one experiment with 12mm shaft and 100mm length we reached 1400 rev/s at onset of half-frequency whip. Dimensionless damping below instability was near 20.

Half frequency whip is one of the speed limits in an airbearing: if the rotor is near twice the lowest natural frequency it will start whirling circular motions - either radial or tilting or bending. This is a nonlinear action but I have no clear description which mechanism is transferring the energy of rotation into the whirling (forward) motion.

In hydro- or aero-dynamic bearings this is more simple as the radial load capacity vanishes at a speed of twice the first resonance. But the same mechanism is working in hydro- or aero-static bearings too.

So to design an optimum rotor I have to calculate and control these three resonance modes.

I do not want to include FEM, this is giving only pointwise solutions, and I have to rely on outside resources to do the calculations.

RHABE

Φ=12, L=100 is a relatively thin shaft, so the lowest resonance will be a beam mode, and (from the aspect of flexural restorative forces) the end loads (turbos?) probably can be treated as points. However, their moment of inertia could be important. The non-rotating solution for this is a bit messy - but I think straightforward enough?

Now to some (possibly incorrect) arm-waving:

From here on I'm assuming that the intended axis of rotation is the axis of the shaft (I hadn't originally appreciated this - my own stupidity, as you gave enough clues).
On the other hand, the net centripetal acceleration rises as the square of the displacement, so if you can avoid a vibration building in the first instance I believe that you can ignore centripetal effects. So it seems that the issues are just basic strain and the Coriolis effect.

I'm afraid that I've only ever considered the Coriolis effect on a vibrating beam at low rotational frequencies (for use as gyros), where variational analysis is good enough to do the job. I imagine from what you have already written that the reason for the peak when the rotation frequency is twice the resonance frequency is that the precession equates to half a vibrational cycle per rotational cycle. Intuition (often wrong) suggests that the resonant frequency under these conditions should be the same as for the non-rotating shaft. If correct, analysis of a single case would provide this information and allow subsequent calculations to be made for the simpler non-rotating case.

Please let me/us know how wrong I am whenever you get some results.

Regards (and thanks)

Fyz

Hi, this is my knowledge at the moment, analytical formulations to be added,

will give 4 equations/natural frequencies if rotational symmetry is assumed and rotor is rigid.

What I need is an extension (analytically) to include bending of the shaft, so then 6 equations/natural frequencies.

If no solution including gyroscopic torques is suitable then a solution at standstill including bending of the shaft is useful too.

RHABE

I'm probably misreading as I think you imply that only the modes that are asymmetric horizontally will precess. You will get precession of symmetrical bending modes as well. But at the critical rotational frequency (Frotate=2xFres) I believe that this will amount to 180-degrees per cycle of Frotate, so effectively zero (this applies to all non-torsional/non-extensional modes.).

BTW, in case you haven't already: I would avoid placing the bearings near the nodes of the first bending mode.

Hi, both modes can be excited indepently if no rotation.

If rotation then x and y are coupled by the gyroscopic torques.

See post 18.

"I would avoid placing the bearings near the nodes of the first bending mode."

I do not have any freedom to change this situation and I do today not know where the bearings will be.

As the structure is open to be discussed today: single or double ball-bearings at both sides? Or single or double air-bearings at bot sides? Or may be hydrodynamic-bearings.

Choice will depend on the results of this calculation among other arguments

Why to avoid placing the bearings near the nodes?

RHABE

"Why to avoid placing the bearings near the bending nodes?"

If the stiffness of the bearings can interact with the vibration the resonance frequency will be increased. If the loss of the bearing can interact with the vibration it will reduce the Q. Both desirable.

If the bearings are at the bending nodes, then at small excursions the only movement within the bearing will be angular. So the coupling between the vibration and the bearing will be relatively small.

Thank you,

"If the stiffness of the bearings can interact with the vibration"

this argument helps in designing the bearings to have tilting stiffness.

"If the loss of the bearing can interact with the vibration"

Unfortunately there are no loss-data for ball-bearings nor a calculation procedure for hydro- or air-bearings known to me (at tilting).

"The grinding wheel is so large/heavy that first-order you can almost ignore the inertia of the shaft."

This is only my exaggerated sketch, in reality the grinding wheel may have - depending on application - a diameter that is smaller than the shaft-diameter. I have to go through a variation of possible diameter and mass examples.

Until now I did not calculate the allowable-diameter/speed relation that is imposed by the centrifugal limit, density and safety-factor.

RHABE

This is getting painfully multivariate. I now see why you want a near-analytic solution.

Fortunately, my comment on the applicability of Euler's stiffness formula (for the shaft) this application still holds good.

PS/NB I see that the situation is rather different from what I'd previously imagined. The grinding wheel is so large/heavy that first-order you can almost ignore the inertia of the shaft. A classical gyroscopic approach should be adequate - and treat the stiffness of the (thin) shaft exactly according to the Euler formula.