Harmonic Analysis - CAESAR II - Help

CAESAR II Users Guide

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CAESAR II
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CAESAR II Version
12

The response of a system to a dynamically applied load is generally expressed through the dynamic equation of motion:

Where:

M = system mass matrix

= acceleration vector, as a function of time

C = system damping matrix

= velocity vector, as a function of time

K = system stiffness matrix

x(t) = displacement vector, as a function of time

F(t) = applied load vector, as a function of time

The harmonic solver is most commonly used to analyze low frequency field vibrations due to fluid pulsation or out-of-round rotating equipment displacements. This differential equation cannot be solved explicitly, except in a few specific cases. Harmonic analysis looks at one of these cases—the set of dynamic problems where the forces or displacements (such as pulsation or vibration) acting on the piping system take sinusoidal forms. When damping is zero under harmonic loading, the dynamic equation of the system can be reduced to

M (t) + K x(t) = F0 cos (w t + Q)

Where:

F0 = harmonic load vector

w = angular forcing frequency of harmonic load (radian/sec)

t = time

Q = phase angle (radians)

This differential equation is solved directly for the nodal displacements at any time. From there the system reactions, forces and moments, and stresses are calculated.

The equation has a solution of the form

x (t) = A cos (w t + Q)

Where:

A = vector of maximum harmonic displacements of system

Because acceleration is the second derivative of displacement with respect to time,

(t) = -A w2 cos w t

Inserting these equations for displacement and acceleration back into the basic harmonic equation of motion yields,

-M A w2 cos (w t + Q) + K A cos (w t + Q) = Fo cos (w t + Q)

Dividing both sides of this equation by cos (w t + Q),

-M A w2 + K A = Fo

Reordering this equation,

(K - M w2) A = Fo

This is the same form of the equation as is solved for all linear (static) piping problems. The solution time for each excitation frequency takes only as long as a single static solution, and, when there is no phase relationship to the loading, the results directly give the maximum dynamic responses. Due to the speed of the analysis, and because the solutions are so directly applicable, you should make as much use of this capability as possible. Keep two considerations in mind:

  • When damping is not zero, the harmonic equation can only be solved if the damping matrix is defined as the sum of multiples of the mass and stiffness matrix (Rayleigh damping), that is

    [C] = a [M] + b [K]

    On a modal basis, the relationship between the ratio of critical damping Cc and the constants a and b is

    Where:

    w = Undamped natural frequency of mode (rad/sec)

    For practical problems, a is extremely small, and can be ignored. The definition of b reduces to

    b = 2 Cc/w

    CAESAR II uses this implementation of damping for its harmonic analysis, but two problems exist. First, for multi-degree-of-freedom systems, there is not really a single b, but there must be only a single b in order to get a solution of the harmonic equation. The second problem is that the modal frequencies are not known prior to generation of the damping matrix. Therefore, the w used in the calculation of b is the forcing frequency of the load, instead of the natural frequency of a mode. When the forcing frequency of the load is in the vicinity of a modal frequency, this gives a good estimation of the true damping.

  • If multiple harmonic loads occur simultaneously and are not in phase, system response is the sum of the responses due to the individual loads

    x(t) = S Ai cos (w t + Qi)

    Where:

    Ai = displacement vector of system under load i

    Qi = phase angle of load i

    In this case, an absolute maximum solution cannot be found. Solutions for each load, and the sum of these, must be found at various times in the load cycle. These combinations are then reviewed in order to determine which one causes the worst load case. Alternatively, CAESAR II can select the frequency/phase pairs which maximize the system displacement.

Damped harmonics always cause a phased response.

The biggest use by far of the harmonic solver is in analyzing low frequency field vibrations resulting from either fluid pulsation or out-of-round rotating equipment displacements. The approach typically used is described briefly below:

  1. A potential dynamic problem is first identified in the field. Large cyclic vibrations or high stresses (fatigue failure) are present in an existing piping system, raising questions of whether this represents a dangerous situation. As many symptoms of the problem (such as quantifiable displacements or overstress points) are identified as possible for future use in refining the dynamic model.

  2. A model of the piping system is built using CAESAR II. This should be done as accurately as possible, because system and load characteristics affect the magnitude of the developed response. In the area where the vibration occurs, you should accurately represent valve operators, flange pairs, orifice plates, and other in-line equipment. You may also want to add additional nodes in the area of the vibration.

  3. Assume the cause of the load, and estimate the frequency, magnitude, point, and direction of the load. This is difficult because dynamic loads can come from many sources. Dynamic loads may be due to factors such as internal pressure pulses, external vibration, flow shedding at intersections, and two-phase flow. In almost all cases, there is some frequency content of the excitation that corresponds to (and therefore excites) a system mechanical natural frequency. If the load is caused by equipment, then the forcing frequency is probably some multiple of the operating frequency. If the load is due to acoustic flow problems, then the forcing frequency can be estimated using of Strouhal’s equations (from fluid dynamics). Use the best assumptions available to estimate the magnitudes and points of application of the dynamic load.

  4. Model the loading using harmonic forces or displacements, normally depending upon whether the cause is assumed to be pulsation or vibration. Perform several harmonic analyses, sweeping the frequencies through a range centered about the target frequency to account for uncertainty. Examine the results of each of the analyses for signs of large displacements, indicating harmonic resonance. If the resonance is present, compare the results of the analysis to the known symptoms from the field. If they are not similar, or if there is no resonance, this indicates that the dynamic model is not a good one. It must then be improved, either in terms of a more accurate system (static) model, a better estimate of the load, or a finer sweep through the frequency range. After the model has been refined, repeat this step until the mathematical model behaves just like the actual piping system in the field.

  5. At this point, the model is a good representation of the piping system, the loads and the relationship of the load characteristics to the system characteristics.

  6. Evaluate the results of this run, in order to determine whether they indicate a problem. Because harmonic stresses are cyclic, they should be evaluated against the endurance limit of the piping material. Displacements should be reviewed against interference limits or esthetic guidelines.

  7. If the situation is deemed to be a problem, its cause must be identified. The cause is normally the excitation of a single mode of vibration. For example, the Dynamic Load Factor for a single damped mode of vibration, with a harmonic load applied is

    Where:

    DLF = dynamic loading factor

    Cc = ratio of system damping to "critical damping,"

    where "critical damping" =

    wf = forcing frequency of applied harmonic load

    wn = natural frequency of mode of vibration

    A modal extraction of the system is done; one or more of these modes should have a natural frequency close to the forcing frequency of the applied load. The problem mode can be further identified as having a shape very similar to the shape of the total system vibration. This mode shape has been dynamically magnified far beyond the other modes and predominates in the final vibrated shape.

  8. The problem mode must be eliminated. You typically want to add a restraint at a high point and in the direction of the mode shape. If this cannot be done, the mode may also be altered by changing the mass distribution of the system. If no modification of the system is possible, it may be possible to alter the forcing frequency of the load. If the dynamic load was assumed to be due to internal acoustics, you should reroute the pipe to change the internal flow conditions. This may resolve or amplify the problem, but in either case avoids CAESAR II’s "good model" of the system. After modifying the system, the harmonic problem is re-run using the single forcing frequency determined as a "good model." The stresses and displacements are then re-evaluated.

  9. If the dynamic problem has been adequately solved, the system is now re-analyzed statically to determine the effects of any modifications on the static loading cases.

    Adding restraint normally increases expansion stresses, while adding mass increases sustained stresses.

Process output from a harmonic analysis in two ways:

  • Use the output processor to review displacement, restraint, force, or stress data either graphically or in report form.

  • Animate the displacement pattern for each of the frequency load cases.

The results of harmonic dynamic loads cannot be combined using the Static/Dynamic Combination option.