CAESAR II: Calculating Modes of Vibration

Transcription

5/26/2011
CAESAR II: Calculating Modes of Vibration
A Quick Overview
26 May 2011
Presented by David Diehl
Quick Agenda

Modal Extraction, a brief introduction
Dynamic Input Review
Results Review
Model Adjustments
Use as Acceptance Criteria
Close
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INTRODUCTION
Modal Extraction / Eigen Solution
Modal Extraction / Eigen Solution – the “Start of It All”
M &x& + C x& + Kx = F ( t )

ω is the angular frequency
(radians/second) of this free
oscillation

There 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
let
C =0
F (t )
be harmonic
so
x = A sin ω t
&x& = − ω 2 A sin ω t = − ω 2 x
− ω 2 Mx + Kx = F ( t )
let
F (t ) = 0
(K − Mω 2 )x = 0
so
x =0
or
K − Mω 2 = 0
ω =
K M
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Examples of Modes of Vibration
Two examples of a One
Degree of Freedom
(DOF) System
A two DOF System
Mode 1
Mode 2
An n DOF System
Mode 1
Mode 2
Mode 3
Mode 4
…
Mode n
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 Analysis
by J.C. Wachel,
Scott J. Morton and
Kenneth E. Atkins of
Engineering Dynamics, Incorporated
San Antonio, TX
A Tutorial from the
Proceedings of 19th Turbomachinery Symposium
Copyright 1990

CAESAR II does NOT calculate these circumferential or axial modes
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DYNAMIC INPUT REVIEW
Controlling the Analysis
Starting the Dynamic Input Processor
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General Comments on Data Entry
Add a new
line below
current
Delete
selected
line(s)
Save,
Error Check
Check,
Run
Comment
(do not process)
Modifying Mass
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Modifying Mass
X, Y, Z or ALL
The affected
Or a range
or
Node number
of Nodes
RX, RY, RZ or RALL
The
A signed
zero
magnitude
is
eliminates
li i t the
th
summed
with
mass. the
calculated mass.
Calculated Mass:
Node
Node
Node
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. ( K − M ω 2 ) x = 0

But our static model allows nonlinear conditions.

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
: (Static) Operating Position 1
Liftoff
Dynamic Model
(no restraint)
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: (Static) Operating Position 2
No liftoff
Dynamic Model
(double-acting Y)
Nonlinear Considerations (Friction)
Y
X
: (Static) Operating Position
Friction defined;
Normal Load = N
Dynamic Model
K
X
Z
K
K=Stiffness Factor for Friction*μ*N
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Stiffness Factor for Friction

This “Stiffness Factor for Friction” is not a physical parameter; it is a modeling
tool.

Larger
g normal loads ((N)) will p
produce g
greater restraint

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
we re not running a seismic analysis here
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?
– No,
N nott for
f 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
p mass”.

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 REVIEW
What Does It All Mean?
The Output Menu
No Load
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Results – Frequency Report
f
ω
t
cycles perradians
secondper second
seconds per cycle
Results – Mode Shapes

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)
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
Input
Operating Position (Liftoff 30, Resting 40)
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Results – Animation
MODEL ADJUSTMENTS
Is 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:
continuous mass throughout

CAESAR II:
half of total mass at end
10

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Adding
g more nodes improves
p
the calculation
Is the static model adequate?
2 node lumped
Mode
1
2
3
4
5

0.328
2 node consistent
0.473
1.51
4.658
57.339
OD=4.5 in
t=0.237 in
length=50 ft
density=0.283 lb/cu.in
E=29.5E6 psi
10 node 10 node lumped consistent
0.469
2.902
8.039
15.572
25.415
0.479
2.971
8.235
16.005
26.377

100 node lumped
0.471
2.948
8.248
16.143
26.646
hand calculation (continuous)
0.471
2.95
8.26
Consistent mass will develop
better frequencies
***BUT***
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)) p
per nominal inch of p
pipe
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
Piping
K. Lin & Adolph T
T. Molin of United Engineers &
Constructors and Eric N. Liao of Stone & Webster
L = 4 9.2( D 3 t W )
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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