may 2011 webinar

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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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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


: (Static) Operating Position 1: (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

: (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

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Join us for Intergraph @ Hexagon 2011Intergraph’s International Users’ ConferenceOrlando, FL, USA | June 6-9, 2011

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may 2011 webinar

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