02 Section1 Fundamentals 012904

7/28/2019 02 Section1 Fundamentals 012904

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NAS122, Section 1, January 2004

Copyright 2004 MSC.Software Corporation

SECTION 1

REVIEW OF FUNDAMENTALS


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TABLE OF CONTENTS

Page

SINGLE DOF SYSTEM 1-3

UNDAMPED FREE VIBRATIONS SDOF SYSTEM 1-6

SINGLE DOF SYSTEM - UNDAMPED FREE VIBRATIONS 1-8

DAMPED FREE VIBRATION SDOF 1-9

DAMPING WITH FORCED VIBRATION 1-13

MSC.NASTRAN DOCUMENTATION 1-23

TEXT REFERENCES ON DYNAMIC ANALYSIS 1-25


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This section will introduce the basics of DynamicAnalysis by considering a Single Degree of Freedom(SDOF) problem

Initially a free vibration model is used to describe the

natural frequency Damping is then introduced and the concept of

critical damping and the undamped solution is shown

Finally a Forcing function is applied and the response

of the SDOF is explored in terms of time dependencyand frequency dependency and compared to theterms found in the equations of motion

SINGLE DOF SYSTEM


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SINGLE DOF SYSTEM (Cont.)

Consider the System Shown

m = mass (inertia)

b = damping (energy dissipation)

k = stiffness (restoring force)

p = applied force

u = displacement of mass

= velocity of mass

= acceleration of mass

u, , and p are time varying in general.

m, b, and k are constants.

m

k b

p(t)

u(t)

u

u

u

u


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Some Theory: The equation of motion is:

In undamped,free vibration analysis, the SDOF equation ofmotion reduces to:

Has a solution of the form:

This form defines the response as being HARMONIC, with a resonant

frequency of:

)()()()( tptkutubtum

0)()( tkutum

tBtAtunn

cossin)(

n

SINGLE DOF SYSTEM (Cont.)


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UNDAMPED FREE VIBRATION SDOFSYSTEM

For an SDOF system the resonant, or natural frequency, isgiven by:

Solve for the constants:

m

kn

n

nn

nnnn

n

tuA

tBt

tBtAtu

u(tBtt

)0(

thus0)sin(,0When

sincos)(

:solutionatingDifferenti

)0thus0)sin(,0When

tutu

tunn

n

cos)0(sin)0(

)(


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UNDAMPED FREE VIBRATION SDOFSYSTEM (Cont.)

The response of the Spring will be harmonic, but the actual form of theresponse through time will be affected by the initial conditions:

If there is no response

If response is a sine function magnitude

If response is a cosine function (180

phase change), magnitude If response is phase and magnitude

dependent on the initial values

0)0(and0)0( uu

0)0(and0)0( uu n

u

0

0)0(and0)0( uu 0u

0)0(and0)0( uu


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SINGLE DOF SYSTEM UNDAMPEDFREE VIBRATIONS

The graph is from a transient analysis of a spring mass system with Initial

velocity conditions only

Time

Disp.

k= 100

m = 1

T = 1/f = 0.63 secs

Hz59.12/f

rad/s10

n

nm

k

10 u

T

Amp

1.0/Amp 0 nu


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DAMPED FREE VIBRATION SDOF

If viscous damping is assumed, the equation of motionbecomes:

There are 3 types of solution to this, defined as: Critically Damped Overdamped

Underdamped

A swing door with a dashpot closing mechanism is a

good analogy If the door oscillates through the closed position it is underdamped

If it creeps slowly to the closed position it is overdamped.

If it closes in the minimum possible time, with no overswing, it iscritically damped.

0)()()( tkutubtum


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DAMPED FREE VIBRATION SDOF (Cont.)

For the critically damped case, there is no oscillation, just adecay from the initial conditions:

The damping in this case is defined as:

A system is overdamped when b > bcr

Generally only the final case is of interest - underdamped

ncr mkmbb 22

mbteBtAtu

2/)()(


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For the underdamped case b < bcr and the solution is the form:

represents the Damped natural frequency of the system

is called the Critical damping ratio and is defined by:

In most analyses is less than .1 (10%) so

)cossin()( 2/ tBtAetudd

mbt

d

21 nd

crb

b

nd


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The graph is from a transient analysis of the previous spring mass system with

damping applied

Frequency and

period as before

Amplitude is a

function of damping

2% Damping

5% Damping


Time

Disp.


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DAMPING WITH FORCED VIBRATION

Apply a harmonic forcing function: note that is the DRIVING or INPUT frequency

The equation of motion becomes

The solution consists of two terms:

The initial response, due to initial conditions which decays rapidly in the presence of

damping

The steady-state response as shown:

This equation is described on the next page

tp sin

tptkutubtum sin)()()(

22

2

2

)/2()1(

)sin(/)(

n

n

tkptu


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DAMPING WITH FORCED VIBRATION(Cont.)

This equation deserves inspection as it shows several importantdynamic characteristics:

At = n this term = (2 )^2 and controlsthe scaling of the response

From this is derived the Dynamic

Magnification Factor 1/2

22

2

2 )/2()1(

)sin(/)(

n

n

tkptu

This is the static loading

and dominates as tendsto 0.0

At = n this term = 0.0With no damping present this

results in an infinite response

Phase lead of the response relative to the input

(see next page)

At >> n both terms drive theresponse to 0.0


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is defined as a phase lead in Nastran :

2

21

1/2tan

n

n



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Summary:

For

Magnification factor 1 (static solution)

Phase angle 360 (response is in phase with the force)

For

Magnification factor 0 (no response)

Phase angle 180 (response has opposite sign of force)

For

Magnification factor 1/2

Phase angle 270

1

n

1

n

1

n



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A Frequency Response Analysis can be used to explore the

response of our spring mass system to the forcing function.

This method allows us to compare the response of the

spring with the input force applied to the spring over a wide

range of input frequencies

It is more convenient in this case than running multiple

Transient Analyses, each with different input frequencies

Apply the input load as 1 unit of force over a frequency

range from .1 Hz to 5 Hz

Damping is 1% of Critical



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Magnification Factor = 1/2 = 1/G = 50Static Response = p/k = .01

Peak Response = .5 at 1.59 Hz

Note:

Use of a Log scale helps identify loworder response

Displacement

Frequency (Hz)



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There are many important factors in setting up a Frequency Response Analysis

that will be covered in a later section

For now, note the response is as predicted by the equation of motion

At 0 Hz result is p/k

At 1.59 Hz result is p/k factored by Dynamic Magnification

At 5 Hz result is low and becoming insignificant

The Phase change is shown here:

In phase up to 1.59 Hz

Out of phase180Degrees after 1 .59 Hz



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Try a Transient analysis with a unit force applied to the spring at 1.59 Hz Again damping of 1% Critical is applied

The result is shown on the next page:

The response takes around 32 seconds to reach a steady-state solution

After this time the displacement response magnitude stays constant at .45

units The theoretical value of .5 is not reached due to numerical inaccuracy (see

later) and the difficulty of hitting the sharp peak



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Transient analysis with a unit force applied to the spring at 1.59 Hz

Displacement

Time



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MSC.NASTRAN DOCUMENTATION

Manuals MSC.NASTRAN Quick Reference Guide

MSC.NASTRAN Reference Manuals

Users Guides

Getting Started with MSC.NASTRAN MSC.NASTRAN Linear Static Analysis

MSC.NASTRAN Basic Dynamic Analysis

MSC.NASTRAN Advanced Dynamic Analysis

MSC.NASTRAN Design Sensitivity and Optimization

MSC.NASTRAN DMAP Module Dictionary MSC.NASTRAN Numerical Methods

MSC.NASTRAN Aeroelastic Analysis

MSC.NASTRAN Thermal Analysis


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MSC.NASTRAN DOCUMENTATION (Cont.)

Other Documentation MSC.NASTRAN Common Questions and Answers

MSC.NASTRAN Bibliography

Documentation available in online form for

workstations and PCs


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TEXT REFERENCES ON DYNAMICANALYSIS

1. W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, Prentice-Hall, 1964.

2. R. W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, 1975.

3. S. Timoshenko, D. H. Young, and W. Weaver, Jr., Vibration Problems in Engineering,4th Ed., John Wiley & Sons, 1974.

4. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, 1976.

5. J. S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hill, 1968.

6. C. M. Harris and C. E. Crede, Shock and Vibration Handbook, 2nd Ed., McGraw-Hill,1976.

7. L. Meirovitch,Analytical Methods in Vibrations, MacMillan, 1967.

8. L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill, 1975.

9. M. Paz, Structural Dynamics Theory and Computation, Prentice-Hall, 1981.

10. W. T. Thomson, Theory of Vibrations with Applications, Prentice-Hall, 1981.

11. R. R. Craig, Structural Dynamics: An Introduction to Computer Methods, John Wiley& Sons, 1981.

12. S. H. Crandall and W. D. Mark, Random Vibration in Mechanical Systems, AcademicPress, 1963.

13. J. S. Bendat and A. G. Piersel, Random Data Analysis and Measurement Techniques,2nd Ed., John Wiley & Sons, 1986.


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