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Rayleigh frequency

 
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sukanta.adhikari
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PostPosted: Mon May 30, 2011 6:21 am    Post subject: Rayleigh frequency Reply with quote

Dear Sefians,

I am trying to understand Rayleigh frequency in detail.Could not find good literature on the same.

On going through NZS 1170 I could find the formula for calculating time period by Ralyleigh method.

The equation for calculating Rayleigh frequency is dependent on horizontal displacement,the magnitude of horizontal force,weight of the structure and the number of storey.

My question is natural frequency of a structure is independent of the horizontal force...it is the function of mass and stiffness of structure...but the frequency calculation from Rayleigh method is dependent on the magnitude of horizontal force and the horizontal displacement of structure.

Can someone explain this.

Regards,
S.Adhikari
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suresh_sharma
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PostPosted: Mon May 30, 2011 8:15 am    Post subject: Reply with quote

Dear Mr. Adhikari,
Like you, I am also trying to understand the Rayleigh frequency. It appears to me that you you are very close to the correct understanding. The horizontal force that you have spoken to in your posting, according to me, is the weight of the sturcture applied in horizntal direction and the horizontal deflection of the joint resulting from the above horizontal load is to be accounted for.
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sukanta.adhikari
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PostPosted: Mon May 30, 2011 8:50 am    Post subject: Reply with quote

Dear Suresh Sir,

Might be you are correct because when we calculate Rayleigh frequency in STAAD we only provide the mass of the structure.

Let us get one more conformation from expert's.

I have seen your profile.You have 35 years of experience in this field.Nice to see ur active participation in the forum.

Regards,
S.Adhikari



suresh_sharma wrote:
Dear Mr. Adhikari,

Like you, I am also trying to understand the Rayleigh frequency. It appears to me that you you are very close to the correct understanding. The horizontal force that you have spoken to in your posting, according to me, is the weight of the sturcture applied in horizntal direction and the horizontal deflection of the joint resulting from the above horizontal load is to be accounted for.
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Radha krishna
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PostPosted: Mon May 30, 2011 12:13 pm    Post subject: Reply with quote

Dear Adhikari,

Rayliegh frequency is the function of mass and stiffness (stiffeness in the direction in which we are calculating frequancy)

T= 2 pi() sqrt(m/k) or (delta/g)
what staad or other sowftware does is applying the load in that particular direction and  ( say for example  self weight in X ) take the maximum deflection of the structure ( say for example max X )
and calculates the rayleight time period by using T= 2 PI() Sqrt(delta/g)
that is the reason why we get higher time period in structures where we have flexible parts in structure like cantilever with less lateral stiffness, Even though whole structure is stiff

You can observe what staad help says.
From Staad: ( for the command to calculate rayliegh frequancy)
This command is based on the Rayleigh method of iteration using 1 iteration. The frequency calculated estimates the frequency as if the structure were constrained to vibrate in the static deflected shape generated by the loads in the load case.

In many instances, the forces should be in one global direction to get the mode and frequency associated with that direction.

Radha Krishna Gavarasana.


Last edited by Radha krishna on Mon May 30, 2011 12:33 pm; edited 1 time in total
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mrbabu77
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PostPosted: Mon May 30, 2011 12:33 pm    Post subject: Rayleigh’s method for calculating natural frequency Reply with quote

Let me summarise my understanding on this.<xml><o></o>
Rayleigh’s method derives natural frequency of a structure by equating potential energy and kinetic energy. Usually, potential energy and kinetic energy calculations are done by applying a force/s on the structure at discrete points. For a simple cantilever beam, the point could be at the tip of the cantilever and for a multi-story building; points could be at floor levels. For a distributed mass system, the concept can be applied by assuming discrete points on the structure.<o></o>

Now, we look at the question of horizontal force and displacement- For any structural dynamic problem, the starting point is assuming a deflected curve ( shape function) of the structure, and calculating velocity, acceleration and from acceleration the resulting force by applying Newton’s second law of motion-F = ma. We are doing the same procedure in Rayleigh’s method by equating potential energy and kinetic energy. However, the final answer of natural frequency is independent of horizontal force and displacement. It is just the formula that creates confusion.<o></o>

Let me explain this further. Most codes provides formula of the type<o></o>

Period T = 2*p*SQRT((w1*(y1)2+w2*(y2)2+….)/g*(f1*y1+f2*y2+…))<o></o>

Where w1,w2 etc are effective seismic weights at discrete points (floor levels), f1,f2 etc are horizontal forces at discrete points and y1,y2 etc are corresponding displacement due to f1,f2 etc. and g is the acceleration due to gravity.The horizontal forces are derived from distributing base shear vertically.<o></o>

Now, look at the basic equation for period T = 2p*SQRT(m/k), where m is the effective mass and k is the effective stiffness of the structure. Note that, I have used the word effective because not all masses are participating in particular vibration modes. By comparison, we can see that, this is the same equation of time period by codes based on Rayleigh’s method. Only difference is that w=m/g is used in the code equations. Also remember that, how to calculate stiffness k ?-this done by applying forces at specified points-the forces could be any arbitrary value. From the basics we know that k=P/y and above procedure is nothing more than applying fundamentals to solve substantially large systems with reasonable accuracy.<o></o>

I hope, this clarifies the question.<o></o>

Regards,<o></o>

Babu M. Raghavan<o></o>
Chartered Structural Engineer<o></o>
Director<o></o>
ENGCEPTS Engineering and Consultancy Pvt Ltd<o></o>
Thrissur, Kerala<o></o>
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manohar
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PostPosted: Mon May 30, 2011 2:51 pm    Post subject: Rayleigh frequency Reply with quote

Dear Adhikari

Here is my bit of explanation.

The Rayleigh’s method provides a quick way to estimate the fundamental
natural frequency starting with a crude approximation to the mode shape.
For example, if an estimate of the fundamental natural frequency of a
cantilever beam is required, one can use the deflected profile of the beam
under self-weight as an approximation to the mode shape and arrive at an
approximation to the first natural frequency using the expression
w=sqrt(u*Ku/u*Mu) where u is the assumed mode shape, u* its transpose, and
K and M are the structure mass and stiffness matrices respectively.  The
assumed mode shape here satisfies the boundary conditions at the fixed end
and at the free end and it is not a bad approximation to the first mode
shape. I am not aware of the details of the Staad programme. The initial
calculation is possibly being made to arrive at a guess on the mode shape.

The interesting thing about this method is that the true natural frequency
would be always less than ratio sqrt(u*Ku/u*Mu ) for alternative choices
of u. This would mean that between two alternative choices for u leading
to two alternative values of w, the one which that leads to lesser w is to
be accepted.

An intuitive explanation for this could be as follows: Imagine, two
postures of a soldier: one standing-at-ease and the other
standing-in-attention. Let the stand-at-ease posture be the true normal
mode and stand-in-attention be an approximation. To stand in attention the
soldier needs to stiffen his body which naturally leads to enhanced
natural frequency. Similarly, any approximation to the mode shape that is
different from the true mode shape, additional forces need to be applied
to maintain the assumed shape. This would increase the potential energy of
the system and hence increase the estimate of the natural frequency.

I hope this helps in your admirable desire to understand things.

Best regards
Manohar


[quote]Dear Adhikari,

Rayliegh frequency is the function of mass and stiffness (stiffeness in
the direction in which we are calculating frequancy)

T= 2 pi() sqrt(m/k) or (delta/g)
what staad or other sowftware does is applying the load in that particular
direction and  ( say for example  self weight in X ) take the maximum
deflection of the structure ( say for example max X )
and calculates the rayleight time period by using T= 2 PI() Sqrt(delta/g)
that is the reason why we get higher time period in structures where we
have flexible parts in structure like cantilever with less lateral
stiffness, Even though whole structure is stiff

You can observe what staad help says.
Quote:
From Staad: ( for the command to calculate rayliegh frequancy)
This command is based on the
Rayleigh[/color:1f15ffdb9a]
method of iteration using 1 iteration. The frequency calculated estimates
the frequency as if the structure were constrained to vibrate in the
static deflected shape generated by the loads in the load case.


In many instances, the forces should be in one
global direction to get the mode and frequency associated with that
direction.


<!--kadov_tag{{<implicit_p>}}--><Metadata>

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sukanta.adhikari
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PostPosted: Mon May 30, 2011 4:18 pm    Post subject: Re: Rayleigh’s method for calculating natural frequency Reply with quote

Dear Raghavan sir,

Thanks you very much for providing this crystal clear explanation on Rayleigh frequency.

Regards,
S.Adhikari

mrbabu77 wrote:
Let me summarise my understanding on this.<xml><o></o>

Rayleigh’s method derives natural frequency of a structure by equating potential energy and kinetic energy. Usually, potential energy and kinetic energy calculations are done by applying a force/s on the structure at discrete points. For a simple cantilever beam, the point could be at the tip of the cantilever and for a multi-story building; points could be at floor levels. For a distributed mass system, the concept can be applied by assuming discrete points on the structure.<o></o>

Now, we look at the question of horizontal force and displacement- For any structural dynamic problem, the starting point is assuming a deflected curve ( shape function) of the structure, and calculating velocity, acceleration and from acceleration the resulting force by applying Newton’s second law of motion-F = ma. We are doing the same procedure in Rayleigh’s method by equating potential energy and kinetic energy. However, the final answer of natural frequency is independent of horizontal force and displacement. It is just the formula that creates confusion.<o></o>

Let me explain this further. Most codes provides formula of the type<o></o>

Period T = 2*p*SQRT((w1*(y1)2+w2*(y2)2+….)/g*(f1*y1+f2*y2+…))<o></o>

Where w1,w2 etc are effective seismic weights at discrete points (floor levels), f1,f2 etc are horizontal forces at discrete points and y1,y2 etc are corresponding displacement due to f1,f2 etc. and g is the acceleration due to gravity.The horizontal forces are derived from distributing base shear vertically.<o></o>

Now, look at the basic equation for period T = 2p*SQRT(m/k), where m is the effective mass and k is the effective stiffness of the structure. Note that, I have used the word effective because not all masses are participating in particular vibration modes. By comparison, we can see that, this is the same equation of time period by codes based on Rayleigh’s method. Only difference is that w=m/g is used in the code equations. Also remember that, how to calculate stiffness k ?-this done by applying forces at specified points-the forces could be any arbitrary value. From the basics we know that k=P/y and above procedure is nothing more than applying fundamentals to solve substantially large systems with reasonable accuracy.<o></o>

I hope, this clarifies the question.<o></o>

Regards,<o></o>

Babu M. Raghavan<o></o>
Chartered Structural Engineer<o></o>
Director<o></o>
ENGCEPTS Engineering and Consultancy Pvt Ltd<o></o>
Thrissur, Kerala<o></o>
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sukanta.adhikari
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PostPosted: Tue May 31, 2011 4:59 am    Post subject: Rayleigh frequency Reply with quote

Respected Manohar Sir,

Thank you very much for the explanation on Rayleigh frequency.A year before also you have helped me in understanding mode shapes.

Regards,
S.Adhikari
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aditya
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PostPosted: Sun Jul 31, 2022 11:47 am    Post subject: Reply with quote

Dear Sirs,
I have confusion regarding the meaning of the term "lateral deflection" at each story level while using the formula for Rayleigh frequency. Is the lateral deflection for using the formula to be calculated for:
1) "lateral load only" i.e. lateral deflection due to "lateral load alone acting on the structure"
or
2) lateral load+ gravity load both acting in the structure, i.e lateral deflection due to "lateral load+gravity load combined"?. Please kindly clarify on which is the correct approach.
with regards,
Aditya
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