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5/26/2011
1
CAESAR II: Calculating Modes of Vibration A Quick Overview26 May 2011Presented by David Diehl
Quick Agenda
Modal Extraction, a brief introductionDynamic Input Review Results ReviewModel AdjustmentsModel AdjustmentsUse as Acceptance Criteria Close
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INTRODUCTIONModal Extraction / Eigen Solution
Modal Extraction / Eigen Solution – the “Start of It All”
ω is the angular frequency (radians/second) of this free oscillation
tFC
tFKxxCxM
=
=++
)(0
)(&&&
let
be harmonicThere is a matching “shape” to this oscillation
There is no magnitude to this shape
This is important:Think of a mode of vibration (the ω & mode shape pair) as a single degree of freedom system xMK
tF
tFKxMxxtAx
tAx
tF
=−
=
=+−
−=−=
=
ω
ω
ωωω
ω
0)(0)(
)(sin
sin
)(
2
2
22&&
let
be harmonicso
MK
MK
x
=
=−
=
ω
ω 0
0
2
so
or
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Examples of Modes of Vibration
Two examples of a One Degree of Freedom
(DOF) System A two DOF System An n DOF System
Mode 1 Mode 2 Mode 1
Mode 2
Mode 3Mode 3
Mode n
Mode 4…
These are NOT circumferential modes
We are following nodal displacement – distortion of the pipe centerline
The pipe also has modes of vibration associated with shell distortion:
:From Piping Vibration Analysisby J.C. Wachel, Scott J. Morton and Kenneth E. Atkins of Engineering Dynamics, IncorporatedSan Antonio, TX
A Tutorial from theProceedings of 19th Turbomachinery SymposiumCopyright 1990
CAESAR II does NOT calculate these circumferential or axial modes
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DYNAMIC INPUT REVIEWControlling the Analysis
Starting the Dynamic Input Processor
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Starting the Dynamic Input Processor
Starting the Dynamic Input Processor
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General Comments on Data Entry
Delete selected
Add a new line below
current selected line(s)
current
Save, Error Check
Comment (do not process)
Error Check, Run
Modifying Mass
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Modifying Mass
The signed magnitude is
A zero li i t th
X, Y, Z or ALLor
The affected Node number
Or a range of Nodesmagnitude is
summed with the calculated mass.
Node Node Node
eliminates the mass.
RX, RY, RZ or RALL
Calculated Mass:
Adding Snubbers
Remember, damping was eliminated from the equation of motion (C=0). Point damping is simulated with a stiff spring.
Mechanical Hydraulic
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Control Parameters
Def=Default;this is a button
Entry cell(use F1 for help)
Nonlinear Considerations
Our equation of motion insists on a linear system – that is, the stiffness, K, is constant.
But our static model allows nonlinear conditions.
0)( 2 =− xMK ω
The dynamic model must “linearize” those nonlinear conditions.
In many cases, the operating state of nonlinear boundary conditions can serve as the linear state for the dynamic evaluation.
An example will help…
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Nonlinear Considerations (Liftoff)
: Cold Position
A +Y (resting) restraint
Nonlinear Considerations (Liftoff)
: (Static) Operating Position 1: (Static) Operating Position 1
Liftoff
Dynamic Model(no restraint)
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Nonlinear Considerations (Liftoff)
: (Static) Operating Position 2
No liftoff
Dynamic Model (double-acting Y)
Nonlinear Considerations (Friction)
Y
: (Static) Operating Position
Friction defined; Normal Load = N
Dynamic Model
X
Z
X
K=Stiffness Factor for Friction*μ*N
K
K
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Stiffness Factor for Friction
This “Stiffness Factor for Friction” is not a physical parameter; it is a modeling tool.
Larger normal loads (N) will produce greater restraintg ( ) p g
This is NOT a 0 or 1! I use 1000 but values as low as 200 produce similar results for the models I run.
This value will knock out frequencies associated with frictionless surfaces.
ASCE 7-10 para. 15.5.2.1: "Friction resulting from gravity loads shall not be considered to provide resistance to seismic forces“ (But we’re not running a seismic analysis here )(But we re not running a seismic analysis here.)
Use it as a tuning parameter in forensic engineering.
How right is it?
Control Parameters (nonlinear issues)
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Stopping the eigensolver
A system with n degrees of freedom will have n modes of vibration.
Are all mode important?
N t f– No, not for our purposes.
– The lower (frequency) modes contribute the greatest structural response of the system.
CAESAR II extracts modes starting with the lowest mode (lowest frequency).
Piping modes of vibration above 33 Hertz do not show resonant response to seismic motion. This is the default CAESAR II cutoff frequency.
Piping modes of higher frequency (100+ Hz) may play a role in fast-acting events such as fluid hammer.
Piping modes at lower frequencies respond to many “environmental” harmonic loads (equipment vibration, acoustic vibration & pulsation).
Stopping the eigensolver
Two parameters are checked to stop the eigensolution:
– A maximum frequency.
– The total count of calculated modes (count = 0 ignores this check)
First limit reached stops the solution.
Frequency cutoff is typically used alone.
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Control Parameters (to stop the eigensolution)
Lumped Mass versus Consistent Mass
For many years CAESAR II (like most analysis tools) ignored rotational inertia and off-diagonal mass terms.
This is what we call “lumped mass”.p
Today’s bigger and faster PCs can handle the fully-developed, complete mass matrix.
This is the “consistent” mass approach.
Consistent mass will more accurately determine the frequencies of natural vibration without adding more nodes (mass points) to the static model.
BUT… more mass points may still be required to establish a proper mode shape in the frequency/mode shape pair.
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Lumped Mass versus Consistent Mass
Lumped mass matrix Consistent mass matrix
Control Parameters (mass model)
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Confirming the calculation
The Sturm sequence check is a back check on the calculated frequencies
View the eigensolver as a search routine that finds system natural frequencies from lowest to highest.g
At times these frequencies may be “discovered” out of sequence.
The Sturm sequence check as a separate calculation of the total number of modes below the last frequency produced. If this count doesn’t match the eigensolver total, the program will state that the check has failed.
A cheap (time-wise) insurance that no mode is missing.
Not so much a problem with today’s PCs
Control Parameters (confirming the modal solution)
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RESULTS REVIEWWhat Does It All Mean?
The Output Menu
No Load
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Results – Frequency Report
f ω t
cycles per secondradians per secondseconds per cycle
Mode shapes (mass & unity normalized)
– Modes Mass Normalized – the tendency of that mode’s contribution to the overall response to a quickly-applied load, all other things being equal (i.e. DLF and point of load application)
Results – Mode Shapes
of load application).
– Model Unity Normalized – the typical mode shape. This is the same shape but normalized to one.
Same shape;different magnitude
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Results – Mass Model
: Lumped Mass
Consistent Mass :
Results – Active Boundary Conditions
InputOperating Position (Liftoff 30, Resting 40)
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Results – Animation
MODEL ADJUSTMENTSIs the Static Model Sufficient?
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Is the static model adequate?
More mass points may be required to approximate the continuous mass beam
Reality:
CAESAR II:
Adding more nodes improves the calculation
continuous mass throughout
half of total mass at end
2010
g p
Is the static model adequate?
Mode2 node lumped
2 node consistent
10 node lumped
10 node consistent
100 node lumped
hand calculation (continuous)
OD=4.5 int=0.237 inlength=50 ft
1 0.328 0.473 0.469 0.479 0.471 0.4712 1.51 2.902 2.971 2.948 2.953 4.658 8.039 8.235 8.248 8.264 57.339 15.572 16.005 16.1435 25.415 26.377 26.646
Consistent mass will develop better frequencies***BUT***
density=0.283 lb/cu.inE=29.5E6 psi
More mass points may be needed to develop the mode shapes
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Suggested mass spacing
Some simple suggestions:
– Add nodes (break pipe) so that the maximum node spacing is no more than one foot (300mm) per nominal inch of pipe( ) p p p
– Use half this spacing into anchors
– Have a node between restraints
– Have a node between bends
– from the paper “On Mass-Lumping Technique for Seismic Analysis of Piping” - John K Lin & Adolph T Molin of United Engineers &Piping - John K. Lin & Adolph T. Molin of United Engineers & Constructors and Eric N. Liao of Stone & Webster
4 3 )(2.9 WtDL =
USE AS ACCEPTANCE CRITERIA An End in Itself
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Use as an acceptance criteria
The lowest natural frequency can be used to assess the risk of failure associated with dynamic response
DNV-RP-D101 recommends the first mode of vibration be no less than 4-5 Hz
You typically increase frequency by adding stiffness
Adding stiffness will increase cost
Adding stiffness may impact thermal flexibility
CLOSE
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Closing Points
Many systems are built for static loads (deadweight and thermal strain) by providing Y supports alone, leaving great flexibility in the horizontal plane –modal analysis will uncover such oversights.
Modal evaluation is a quick and easy tool to learn more about your piping system response.
The topic for June’s webinar is not established.
Next dynamic session – response to harmonic loads.
PDH Certificate
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Intergraph @ Hexagon 2011
www.hexagonconference.com/ppm
Join us for Intergraph @ Hexagon 2011Intergraph’s International Users’ ConferenceOrlando, FL, USA | June 6-9, 2011
CADWorx & Analysis University
www.cau2011.com