Hi All,

I have got some quries regarding the Dynamic Analysis

1. DLF SPECTRUM GENERATOR
For carrying out the spectrum analysis, we give the time Vs. Force data & the DLF generator generates freq Vs Load factor table.

What is the formula used to calculate the freq. & Load factor.
What is the effect of the input “ ENTER THE MAXIMUM TABLE FREQUENCY(Hz)” on the analysis.

What is the effect of the input “ ENTER THE DESIRED NUMBER OF POINTS IN THE TABLE” on the analysis.

2. What is the equation of motion for Spectrum analysis (Seismic/Water Hammer/Safety Valve Relief).
Like for Harmonic analysis the equation of motion is

M*d2x/dt2 + C* dx/dt + K*x = F Sin (wt+Q)

Similarly, What is the equation of motion for spectrum analysis.

The dynamic input to do the spectrum analysis, we give the time Vs. Force data & the DLF generator generates freq Vs Load factor table.

How these generated freq. And Load factors are applied on the system( pl. explain by means of the equation of motion, similar to the one shown for the harmonic analysis).

3. What is Mass Participation Factor. What is its significance with respect to the analysis.

4. Can you pl. Suggest some good reference book regarding dynamic analysis of piping system.

Regards,

1a) There is no formula to convert from a time pulse to a DLF. An integration technique is used. This technique is explained in a newsletter article. If you don't have a hard copy of the November 1994 issue, you can find it in the Newsletters section of this web site. Here is the direct link to that issue. The article begins on page 9.

1b) The maximum table frequency is the frequency where the DLF curve (you are about to generate) stops. This should coincide with your analysis cut-off frequency.

1c) The DLF curve is a generated set of points. This setting determines how many points make up the curve. You want enough so that the interpolation (between points) is accurate. You don't have to go crazy here, 25 to 40 points is usually enough.

2a) MU + KU = -Mu_g from Gupta eq 1.13.

2b) For a seismic analysis, the loading on the system is taken from "u_g", at each natural frequency of the system. For all other spectrum types, the DLF value at the natural frequency is used instead, because there is no Mu_g.

3) The mass participation factor simply gives an indication of the importance of the modes (natural frequencies) relative to one another, in the overall system response.

4) Check the reference section of this website. There are a number of texts listed in the "General Engineering Reference" section.

Anu,

You can read an article on Dynamic Analysis of Pipe Stress published in Feb 2003 issue of Hydrocarbon Processing.I was one of the authors of that paper.


The fundamental concept used in Dynamic Analysis is the concept of degree of freedom which for dynamic analysis is defined as " no. of independent coordinates to describe the displaced position of the masses w.r.t their original position".By "independent" we mean that they are "mathematically independent" i.e. one cannot be expressed in terms of the other one as fixed ratio.

For dynamic analysis typically we ignore the rotational degrees of freedom .One the degrees of freedom are determined we write down the differential equations of motion for the system. This will result in set of linear differential equations for linear elastic systems.

Using the concept of lumped mass i.e. mass concentrated at degree of freedoms, we have no mass couplings in these equations but there will be stiffness and damping couplings.

Usng energy methods ( Betti's reciprocity theorem) we can arrive at what is known as orthogonality of modes which physically mean" the work done by nth mode inertia force in going through rth mode displacements is zero". This is a very important concept which can be used to "uncouple" the set of differential equations i.e. they can be solved independently of each other.

For linear systems the principle of superposition is valid. Hence the solution of equations ( now uncoupled) for different degrees of freedom which will give you the displacements can be added ( the solutions of the uncoupled equations will give you modal coordinates which are to be multiplied by the non dimensional eigenvectors ) to give the total diaplacement.Once the displacements are computed , stresses , forces . moments etc. can be computed. This in essence is the dynamic analysis. The main difference w.r.t static analysis is that the displacements are time dependent.

For Harmonic analysis the exciting force is a sine/cosine function and the equations can have closed form solution.For Spectrum analysis where the loading is arbitary we use the concept of response spectrum where the peak values ( displacements., accelerations etc.)are computed for each natural frequency and a plot is made for the maximas for all such natural frequencies which is the response spectrum.

Remember that the accelaration term in inertia force is absolute accelaration.

The force vs time input you give in Caesar II for the DLF generator uses the "Duhamel's integral" to compute the DLF ( dynamic displanemnt w.r.t static displacement ) . In spectrum method the output is not time dependent but an addition of the modal results based on statistical methods like ABS, SRSS etc. CQC( complete quadratic combination ) is currently not available in Caesar II.

The participation factor is the extent to which the mode is excited during loading and is a measure of the modes contribution in the final result.Mass and mode particitation have the same meaning for seismic excitation.

In time History- the equations ( after uncouling) are solved by numerical method like Wilson theta method.The output is time dependent.

You can read the following books on Structural Dynamics which will give you a broad overview on this subject:

1) Structural Dynamics-Theory and Computation by Mario paz

2) Dynamics of Structures- By Anil Chopra

3) 3 Dimensional Static and Dynamic analysis of Structures by E.L.Wilson.This book has an excellent treatise on the concept of mass participation.


Anindya Bhattacharya