Section2 Module7 Direct Integration

8/11/2019 Section2 Module7 Direct Integration

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Static Analysis:Direct Integration


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Objectives

This module will present the equations and numerical methods used tosolve the equations of motion directly. Although more

computationally intensive, this method can be used to solve problems

that are not characterized by constant mode shapes.

In Module 6, the Modal Superposition method of solving the equations of

motion was presented. This method required the determination of the mode

shapes and natural frequencies of the system and then used them to

transform the coupled equations into uncoupled modal equations of motion.

Problems having gaps, surface contact, and non-linearities can be solvedusing the method presented in this module.

Section IIStatic Analysis

Module 7Direct Integration

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Governing Equations

The governing equations developed for static problems in Module 4are

Inertial forces and viscous damping forces can be introduced as

external force terms, resulting in

Note that the displacement increment {Du} in going from time, t, to

time, t+Dt, and the acceleration and velocity at t+Dt are unknowns.

unbextT FRFuK D int

tttttttT RuCuMFuK D DDD



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Equations of Motion

The previous equation can be rewritten as

One of the most commonly used numerical methods for solving thisset of equations is the Newmark-b method.

The Newmark-bmethod assumes a linear variation of acceleration

during the time interval, Dt, and uses two interpolation parametersto select the acceleration used in the solution.

tttTtttt RFuKuCuM D DDD



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First Acceleration Approximation

The acceleration during the time interval, t+Dt, can be estimatedusing the equation

The parameter, g, is used to select the acceleration used in the

numerical integration procedure.

The selected value of the parameter, g, affects the accuracy andstability of the resulting numerical integration scheme.

The Newmark-bmethod is stable, provided .

tttttt uu

t

uuu D

D D

ggg 1

2

1g



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Graphical Illustration

ttu D

tu t

gu

tu

ttu D

g

0g

1g

If gis equal to zero, then

the acceleration at time, t,

is used.

If gis equal to one, then

the acceleration at time,

t+Dt, is used.

If gis equal to , then the

acceleration at the middleof the time interval is used.

tttttt uutuu

u DD

D

ggg 1



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Kinematic Relationships

The kinematic equations for acceleration are

If ais a constant, this equation can be integrated to yield2

2

1attuuu oo

2

2

dt

uda

where and are initial conditions.o

uo

u



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Second Acceleration Approximation

Newmark based the second acceleration approximation on thiskinematic relationship, via the following equation:

22

1tutuuu tttt DDD b

where

ttt uuu D bbb 221

and

2

10 b

bis an interpolation parameter that,

like g,is used to select the

acceleration used in the numerical

integration procedure.

The Newmark- bmethod uses two

parameters for accelerations used in

the procedure



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Governing Approximation Equations

The Newmark-bmethod is based on the two equations

The second of these equations can be rearranged to yield

tutuuu tttttt DD DD gg1

and

22212

1tututuuu ttttttt DDD DD bb

tttt uut

ut

u b

b

bb 2

21112

DD

DD



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Governing Approximation Equations

Substituting the last equation on the previous slide into the topequation on the previous slide yields

These last two equations provide equations for and in

terms of the displacement increment and the velocity and

accelerations at the beginning of the time interval.

The velocity and acceleration at the beginning of the time intervalare known.

The only unknown is the displacement increment, .

tttt utuu

tu D

D

DD

b

g

b

g

b

g

211

ttu D ttu D

uD



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Combination of Equations

The three equations used to determine the displacement incrementusing the Newmark-bmethod are:

Equations of Motion

Acceleration at the end of the time step

Velocity at end of the time step

tttTtttt RFuKuCuM D DDD

tttt uut

ut

u b

b

bb 2

21112

DD

DD

tttt utuut

u D

D

DD

b

g

b

g

b

g

211



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Combined Equations

These three equations can be combined to yield the followingequation

The right hand side of the equation yields an effective load vector

based on quantities at time, t, that are known.

The left hand side of the equation is an effective tangent stiffness

matrix that includes mass and viscous damping terms.

tttt

tttex tT

uCtuMuCuMt

RFuKCt

Mt

D

D

D

D

D D

b

bg

b

b

b

bg

b

b

g

b

2

2

2

211

1int2



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Equivalent Static Problem

The equation on the previous slide can be written as

These show that finding the displacement increment in a dynamic

analysis is equivalent to solving a static problem using an effective

tangent stiffness matrix and internal restoring force vector.

D

D TeffT KC

tM

tK

b

g

b 21

teffttexteffT

RFuK D D

where

tttttteff uCtuMuCuMtRR D

D

b

bg

b

b

b

bg

b 2

2

2

211int



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Stability and Accuracy

The Newmark-bmethod is unconditionally stable for linear problemswhen gand bsatisfy the equations

Values of g=1/2 and b=1/4 are frequently used.

The method is generally stable for nonlinear problems if these same

criteria for gand bare used and equilibrium iterations are used to

improve accuracy.

2

1g and .

2

1

4

1 2

gb



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Time Step Size

A sufficiently small time step must be used to ensure solutionaccuracy.

A Dt of around one-tenth of the period of the highest natural

frequency of interest is commonly used.

The time step does not have to be constant for all time steps and it is

common for variable time step methods to be used.

Autodesk Simulation 2012 uses a variable time step in the

Mechanical Event Simulation module.



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Rayleigh Damping

Rayleigh damping is a mathematically convenient way of describingviscous damping.

Rayleigh damping is defined by the equation

The constants aand bmust be determined from experimental data.

This is a convenient form because the damping matrix can beuncoupled along with the mass and stiffness matrices using the

mode shapes.

.KMC ba



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Rayleigh Damping

The transformation of the Rayleigh damping equation to the modeshape domain takes the form

The i th equation can be written as

where ziis the critical damping ratio for thei

th mode.

2baba IKMC TTT



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iiiic zba 22


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Finding and b

aand bcan be found from thisequation if zis known for two

modes.

A least squares approximation toaand bcan be found if zis

known for more than two modes.

22

11

2

2

21

2

2

1

1

b

a

CBB T

b

a

2

2

2

2

1

1

1

1

n

B

nn

C

2

2

2

22

11



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Example Problem

5 lb. force distributed

over the 17 nodes onthe upper edge of the

free end

Fixed

End

1 inch wide x 12 inch long x 1/8 inch thick.

Material - 6061-T6 aluminum.

Brick elements with mid-side

nodes are used to improve the

bending accuracy through the

thin section. 0.0625 inchelement size.

Simulation is used to compute the step response of

the cantilevered beam shown in the figure. This is the

same beam used in Module 6: Modal Superposition.



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ExampleAnalysis Parameters

Same as in

Module 6



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Values for the

Rayleigh damping

factors are

presented on a

following slide.

Forces can be

applied here or

through the FE

Editor. The FE

Editor was used in

this example.


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ExampleLoad Curve Factor



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The load curve is

zero until 0.05seconds. At that

time, it goes to one

in 0.0001 seconds

to simulate a step

input.


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ExampleForce Magnitude

Nodes selected

along upper edge

5 lb./17

nodes acting

in negative y-

direction

Load Curve 1 is defined inAnalysis Parameters



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ExampleLoad Summary

1

Time0.05 seconds

Load Curve Factor

F(t) = Load Curve Factor * Magnitude

-0.294 lb.

Time

F(t)

0.05 seconds



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ExampleRayleigh Damping Factors

Damping for each mode is estimated to be 0.5 percent of critical.

Modes associated with bending about the weak axis will be used to

determine aand b.

The first three weak axis bending modes were computed in Module 5.They are:

Mode 1 28 Hz = 176 rad/sec,

Mode 2175 Hz = 1100 rad/sec,

Mode 4492 Hz = 3091 rad/sec.

303,556,91

000,210,11

976,301

B

91.30

00.11

76.1

C

0

7676.41CBBB

TT

b

a



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ExampleMATLAB Program

This MATLAB program finds the Rayleigh damping coefficients forthis example. The critical damping ratio for each mode is 0.005 or

0.5%.



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S i II S i A l i


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ExampleClamped End Stress

zz is plotted

Note that this

curve is the

same as thatcomputed

using modal

superposition

in Module 6.



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S ti II St ti A l i


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ExampleFree End Tip Displacement

This curve is much

smoother than the

stress curve. Thestress curve is

based on strains

that are computed

from the

derivatives of the

displacements.



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Section II Static Analysis


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Module Summary

This module has presented the equations used to perform a directintegration of the equations of motion used for a linear or non-linear

dynamic analysis.

It was shown that the Newmark-bmethod for integrating the

equations of motion reduces the dynamic problem to a sequence of

static analyses that uses an effective tangent stiffness matrix and

internal restoring force vector.

The Newmark-bmethod is unconditionally stable for linear problems

and generally stable for non-linear problems that use equilibrium

iterations. A sufficiently small time step must be used to ensure accurate results.

Results from an example were the same as those obtained using the

modal superposition method in Module 6.



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Section2 Module7 Direct Integration

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