The output of the "Modal Analysis" load case in CAEPIPE has been enhanced. In addition to the modal frequencies and mode shapes, it now includes two new outputs called "dynamic stresses" and "dynamic susceptibility". The dynamic stresses are the dynamic bending stresses associated with vibration in a natural mode. That is to say, the modal analysis result has been generalized to include the alternating bending stresses associated with the vibration in a natural mode. The dynamic susceptibility for any mode is the ratio of the maximum alternating bending stress to the maximum vibration velocity. This "susceptibility ratio" provides an indicator of the susceptibility of the system to large dynamic stresses. Also, the associated animated mode shapes include color-spot-markers identifying the respective locations of maximum vibration and maximum dynamic bending stress. The susceptibility ratio and the graphics feature provides incisive insights into the reasons for high susceptibility and how to make improvements. This new feature is illustrated by application to the CAEPIPE "sample problem" system.
When addressing vibration issues, the piping designer does not have the specific requirements, nor the analytical tools and technical references typically available for other plant equipment such as rotating machinery. Typically, piping vibration problems only become apparent at the time of commissioning and early operation, after a fatigue failure or degradation of pipe supports. Discovery of a problem is then followed by an ad hoc effort to assess, diagnose and correct as required. The "Dynamic Susceptibility" analysis, now included in CAEPIPE, provides a new analytical tool to assist the piping designer at any stage, from preliminary layout to resolution of field problems.
CAEPIPE's Dynamic Susceptibility feature utilizes the "Stress per Velocity" method, an incisive analytical tool for "screening" the vibration modes of a system. It readily identifies which modes, if excited, could potentially cause large dynamic stresses. Furthermore, it reveals which features of the system layout and support are responsible for the susceptibility to large dynamic stresses. At the design stage, the method allows the designer to quickly identify and correct features that could lead to large dynamic stresses at frequencies likely to be excited. Where problems are encountered in the field, the method provides quick and incisive support to efforts of observation, measurement, assessment, diagnosis and correction.
The technical foundation of this method lies in an underlying fundamental relationship between the kinetic energy of vibratory motion, and the corresponding potential energy stored in elastic stresses. That is to say, the kinetic energy at zero displacement and maximum system velocity must equal the stored elastic energy at zero velocity and maximum displacement. This implies a fundamental relationship between vibration velocity and dynamic bending stresses, which is the foundation of the stress per velocity approach for 'susceptibility screening' of vibration modes.
The key analytical step is to determine, mode by mode, the ratio of maximum dynamic stress to maximum vibration velocity. This ratio will lie in a lower "baseline range" for uncomplicated systems such as classical uniform-beam configurations. For more complex systems, the stress / velocity ratio will increase due to typical complications such as three-dimensional layout, discrete heavy masses, changes of cross-section and susceptible branch connections. System modes with large stress-velocity ratios are the potentially susceptible modes. The Stress / Velocity method, implemented in CAEPIPE as the Dynamic Susceptibility feature, automatically and quickly finds these modes and quantifies the susceptibility. Evaluation of the results, including special-purpose color animation, helps to identify which details of layout and support are responsible for the large stresses.
This technical note is to present and explain the "dynamic susceptibility" outputs now included in the modal analysis load case, and to illustrate by application to the standard CAEPIPE "sample problem" system.
2.1 Kinetic Energy and Potential Energy; Vibration Velocity and Dynamic Stresses
The underlying theoretical basis for the Stress / Velocity method is a deceptively straightforward but universally-applicable relationship between kinetic energy and potential (elastic) energy for vibrating systems. Stated simply, for vibration at a system natural frequency, the kinetic energy at maximum velocity and zero displacement must then be stored as elastic (strain) energy at maximum displacement and zero velocity. Since the strain energy and kinetic energy are respectively proportional to the squares of stress and velocity, it follows that dynamic stress, s , will be proportional to vibration velocity, v. For idealized straight-beam systems, consisting of thin-walled pipe and with no contents, insulation or concentrated mass, the ratio s / v is dependent primarily upon material properties, (density r and modulus E) ,and is remarkably independent of system-specific dimensions, natural-mode number and vibration frequency.
For real continuous systems of course, the kinetic and potential energies are distributed over the structure in accordance with the respective modes shapes. However, integrated over the structure, the underlying energy- equality holds true. Provided the spatial distributions are sufficiently similar, ie harmonic functions, the rms or maximum stress will still be directly related to the rms or maximum vibration velocity.
2.2 The "Screening" Approach
As stated above, for idealized pure beam systems the stress-velocity ratio will depend primarily upon material properties.
For real systems, the spatial patterns of the mode shapes will depart from the idealized harmonic functions, and the stress-velocity ratios accordingly increase above the theoretical minimum or baseline value. System details causing the ratios to increase would include the three-dimensional layout, large unsupported masses, high-density contents in thin-walled pipe, susceptible branch connections, changes of cross section etc. The more "unfavorable" the system layout and details, the larger the s / v ratios for some modes.
Thus, the general susceptibility of a system to large dynamic stresses can be assessed by determining the extent to which the s / v ratios for any mode exceed the baseline range. Furthermore, by determining which particular modes have the high ratios, and whether these modes are known or likely to be excited, the at-risk vibration frequencies and mode shapes are identified for further assessment and attention. This is the basis of the Stress / Velocity method of analysis and it's implementation as the "dynamic susceptibility" feature in CAEPIPE.
2.3 Relation to Velocity-based Vibration Acceptance Criteria
There are various general and application-specific acceptance criteria based upon vibration velocity as the quantity of record. Some, in order to cover the worst case scenarios, are overly conservative for many systems. Others are presented as being applicable only to the first mode of simple beams, leading to the misconception that the stress / velocity relationship does not apply at all to higher modes. In any case, there are real and perceived limitations on the use of screening acceptance criteria based upon a single value of vibration velocity.
The dynamic susceptibility method turns this apparent limitation into a useful analytical tool! Specifically, large stress / velocity ratios, well above the baseline values, are recognized as a 'warning flag'. Large values indicate that some feature(s) of the system make it particularly susceptible to large dynamic stresses in specific modes.
3.1 General Approach
The Dynamic Susceptibility method is essentially a post processor to fully exploit the modal analysis results of the system. Mode shape tables of dynamic bending stress and vibration velocity are searched for their respective maxima. Dividing the maximum stress by the maximum velocity yield the "s/v ratio" for each mode. That ratio is the basis for assessing the susceptibility to large dynamic stresses. Larger values indicate higher susceptibility associated with specific details of the system.
3.2 Specific Implementation in CAEPIPE
The Stress / Velocity method has been implemented as additional analysis and output of the CAEPIPE modal analysis. The modal analysis load case now includes additional outputs and features as follows:
Dynamic Stresses This output provides the 'mode shapes' of dynamic bending stresses, tabulated along with the conventional mode shape of vibration magnitude.
Dynamic Susceptibility This output is a table of s/v ratios, in psi / ips, mode by mode, in rank order of decreasing magnitude. In addition to modal frequencies and s/v ratios, the table also includes the node locations of the maxima of vibration amplitude and bending stresses.
With the dynamic susceptibility output selected, the animated graphic display of the vibration mode shape includes the added feature of color spot markers showing the locations of maximum vibration and maximum dynamic bending stress.
These outputs will assist the designer through a more-complete understanding of the system's dynamic characteristics. They provide incisive quantified insights into how specific details of components, layout and support could contribute to large dynamic stresses, and into how to make improvements.
The Stress / Velocity method of assessment, and it's implementation in CAEPIPE as dynamic susceptibility, is based entirely upon the system's dynamic characteristics per se. Thus the vibration velocities and dynamic stresses employed in the analysis, although directly related to each other, are of arbitrary magnitude. There is no computation of the response to a prescribed forcing function, and no attempt to calculate actual dynamic stresses. Thus the dynamic susceptibility results do not factor directly into a pass-fail code compliance consideration. Rather, they assist the designer to assess and reduce susceptibility to large dynamic stresses if necessary, in order to meet whatever requirements have been specified.
The "dynamic susceptibility" feature of CAEPIPE will be illustrated here by application to the standard CAEPIPE Example system. The modal analysis was performed for frequencies up to 200 Hz, resulting in a reporting-out for 12 modes. The frequencies range from mode 1 at 14.5 Hz to mode 12 at 192 Hz. In two instances, very similar horizontal and vertical modes appear in pairs, ie modes 3&4 and 7&8.
The relevant features of this system can be readily identified and understood, by reference to the dynamic-susceptibility table and the animated graphic display of mode shape. Results will be considered here in order of decreasing susceptibility.
5.1 Axial movement of long pipe run (large added mass in motion)
From the dynamic susceptibility table the top of the list is mode 2 at 20.8 Hz, having a dynamic susceptibility of 649 psi / ips. From the animated graphic display, note that the maximum dynamic bending stresses are at the anchored point, node 50. Note also that the dominant motion is a "Z" motion of the straight run between nodes 20 and 40 (ie in effect an axial motion of that run as a rigid body). The designer's interpretation here is that the vertical rise from node 50 to node 40 is effectively a cantilevered beam with an effective large added mass at the tip; that feature of layout accounts for the high susceptibility.
5.2 Effects associated with the valve (local rigidity to bending, and added mass)
The next-highest values of susceptibility are for the two pairs of modes, modes 7&8 at 129 and 133 Hz, and modes 3&4 at 27.8 and 31.2 Hz. As will be shown here, these are associated with effects of the valve,
The susceptibility for modes 7&8, respectively 594 and 589 psi / ips, is attributable to the rigidity of the valve element within an otherwise flexible pipe run. This can be seen from a close look at the animated graphic. Notice that these relatively high frequency modes feature a reversal of bending curvature along the run between nodes 30 and 80. Notice also that there is a stronger localized curvature on approach to the valve body. The designer's interpretation here is that, since there cannot be any curvature of the rigid valve itself, there must be a more concentrated curvature of the adjacent pipe.
The dynamic susceptibility of modes 3&4, respectively 521 and 526 psi / ips, is associated with the more straightforward 'concentrated mass' effect of the valve. From the animated graphic, these modes feature a large amplitude vibration at the valve. The kinetic energy of this added mass must be stored as strain energy in the flexing (ie spring) element, resulting in elevated dynamic stresses.
5.3 Beam modes with 'moderate' added mass effects of adjacent spans)
Modes 1,5 and 6, with frequencies of 14.5, 47.4 and 52.4 Hz, show progressively decreasing 'intermediate to low' values of susceptibility at respectively 456, 383 and 339 psi / ips. Reference to the animated graphics shows that these modes involve predominantly transverse vibration (as contrasted with the prominent axial movement of mode 2) and involve little participation at the valve (which accounted for the elevated susceptibility of modes 7&8 and 3&4). Notice that these modes, 1, 5, and 6, involve varying degrees of the influence of effective added mass of adjacent spans, and of length of the cantilevered span contributing most to stiffness.
Mode 2 animation (note color spot markers [for max vibration and max. dynamic stress] )
Mode 2 animation
Dynamic Susceptibility table from CAEPIPE output
Dynamic Susceptibility table
5.4 Modes approaching the 'simple-beam baseline' behavior
Modes 10, 11 and 12 have significantly higher frequencies, 164 to 192 Hz, and correspondingly short wavelengths. Consequently, the vibration pattern tends to be transverse beam vibration 'within the span', with little or no effect from connected spans or the valve. For these modes, the susceptibility ratios range from 256 to 272 psi / ips. These values are approaching the baseline values for uncomplicated mode shapes of the pipe section and pipe contents of this system.
NOTE: Mode 9, at 138 Hz, is clearly an exception, with a susceptibility of only 104 psi / ips, well below the baseline level. From the animated display, it can be seen that this is not really a 'bending' mode; rather, the spring effect for this mode is an axial stretching of the run between nodes 80 and 30. Consequently, the bending stresses are low, as reflected in the abnormal susceptibility ratio. In effect, this mode lies outside the intended application of the dynamic susceptibility approach. Notice however, that the low susceptibility ratio has in effect 'flagged' this mode as 'not a bending mode'; that in itself provides the designer additional insight into system characteristics and behavior.
5.5 Summary Comment
As per paragraphs 5.1 to 5.4, the dynamic susceptibility method has incisively identified the key features of the 'sample' system, with respect to potentially large dynamic stresses. This of course is a relatively simple system. An experienced designer, with some appreciation of dynamics, might view the results as obvious. However, the method will do the same job, automatically and directly, on any larger or more complex system for which nothing is obvious!
The stress / velocity method, implemented in CAEPIPE as the "Dynamic Susceptibility" feature, provides quantified insights into the stress versus vibration characteristics of the system layout per se.
In particular, the dynamic susceptibility table identifies specific modes that are susceptible to large dynamic stresses for a given level of vibration. The larger the stress / velocity ratio, the stronger the indication that some particular feature of layout, mass distribution, supports, stress raisers etc is causing susceptibility to large dynamic stresses.
The animated mode-shape display identifies, by the color-spot-markers, the locations of the respective maxima in dynamic stress and vibration velocity. Review of these animated plots will reveal the offending pattern of motion, and provide immediate insight into what features of the system are responsible for the large dynamic stresses.
Finally, the "dynamic stresses" table provides the distribution of dynamic stresses around the system, ie in effect, the mode shape of dynamic stresses to go along with the conventional mode shape of vibration. This information allows identification of other parts of the system, if any, with dynamic stresses comparable to the identified maximum.
7.1 At the Design Stage
At the design stage, the dynamic susceptibility feature allows the designer to quickly determine whether the system may be susceptible to very large dynamic stresses. This could be a broad look at all frequencies, or could be focused on particular frequencies where excitation is likely to occur. On identifying high susceptibility, the designer can then make changes to improve the design. It is important to note that this method is based upon the dynamic-stress versus vibration-velocity characteristics of the system per se. There is no need to specify a forcing function and perform a response calculation and stress / fatigue analysis. However, where such analysis is a requirement, the dynamic susceptibility module can assist the designer to achieve a system layout that will meet the requirements and criteria.
7.2 Commissioning, Acceptance Testing
The dynamic susceptibility feature can also contribute to planning acceptance testing and associated measurements where these are undertaken whether by formal requirement or by choice. Locations for measurement of vibration or dynamic strain can be selected based upon knowing the locations of the maxima and the distribution of vibration and dynamic stress. Reference to the dynamic susceptibility results can help assure that the modes of most potential concern are well covered by the minimum set of practically-achievable measurements. Furthermore, mode-specific acceptance criteria can be readily established to avoid the restrictions of generally over-conservative guideline type criteria, while providing assurance that any highly-susceptible situations are identified and addressed.
7.3 Troubleshooting and Correction
As mentioned earlier, when vibration and/or fatigue problems are recognized at start up or early operation, there is typically an ad hoc program of observation, measurement, assessment, diagnosis and correction. It is not uncommon for there to be some uncertainty about what to measure and what is acceptable. The dynamic susceptibility module can contribute very effectively in these situations.
Normally, the overall symptoms, approximate frequency and pattern of vibration are known to some extent from observation and/or a few measurements. After modelling the system, and obtaining the dynamic susceptibility results, the subsequent steps can be highly focused on specific frequencies and locations, the optimum measurements, and system-specific acceptance criteria.
Equally or more importantly, the proposed solution options can be modeled and evaluated to make sure they will achieve the required improvement.
7.4 General
The dynamic susceptibility module does not apply directly to meeting code or other formal stress analysis requirements. However, it is an incisive analytical tool to help the designer understand the stress / vibration relationship, assess the situation and to decide how to modify the design if necessary. It can be used for design, planning acceptance tests, and troubleshooting and correction.
The Stress /Velocity method for screening piping system modes was developed and brought to the attention of SST Systems by Dr. R.T Hartlen of Plant Equipment Dynamics Inc.
The background material provided here is intended to provide only a concise summary of the underlying fundamentals, the universality for idealized systems, and the expected detail-dependent variations for real systems. The stress / velocity method, although not yet widely known and applied, is fundamentally theoretically sound. However, complete theoretical rigor is beyond the scope of this note
For users who may wish to independently examine and validate the underlying theoretical fundamentals, a few key references are provided. References 1, 2 and 3 deal with fundamentals. References 4 and 5 deal with application to piping. The CEA research projects reported in References 3 and 4 were initiated and guided by Dr. Hartlen.
1 F.V Hunt, Stress and Strain Limits on the Attainable Velocity in Mechanical Systems, JASA, 32(9) 1123-1128, 1960
2 E.E Ungar, Maximum Stresses in Beams and Plates Vibrating at Resonance, ASME Journal of Engineering for Industry, v84, n1, pp149-155, 1962
3 R Elmaraghy et al, Correlation of Vibratory Stress, Velocity and Sound Canadian Electrical Association Project, G197, Feb 1982
4 J.D. Tulk, Correlation Between Dynamic Stress and Vibration Velocity in Complex Piping Systems, Canadian Electrical Association Project G521, March 1988
5 Michael P. Norton, Acoustically Induced Structural Vibration and Fatigue - A Review
Third International Congress on Air-and Structure-borne Sound and Vibration, June 1994, Montreal, Canada.
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