static analysis: 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 to solve 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 solved using the method presented in this module.

Section II – Static Analysis

Module 7 – Direct Integration


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Governing Equations

The governing equations developed for static problems in Module 4 are

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 this set of equations is the Newmark-b method.

The Newmark-b method assumes a linear variation of acceleration during the time interval, Dt, and uses two interpolation parameters to 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 estimated using 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 and stability of the resulting numerical integration scheme.

The Newmark-b method is stable, provided .

tttttt uu

tuuu D

D D

ggg 1

21

g




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Graphical Illustration

ttu Dtu t

gutu

ttu Dg0g

1g

If g is equal to zero, then the acceleration at time, t, is used.

If g is equal to one, then the acceleration at time, t+Dt, is used.

If g is equal to ½, then the acceleration at the middle of the time interval is used.

tttttt uu

tuuu D

D D

ggg 1




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Kinematic Relationships

The kinematic equations for acceleration are

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

21 attuuu oo

2

2

dtuda

where and are initial conditions.ou ou




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Second Acceleration Approximation

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

2

21 tutuuu tttt DDD b

where

ttt uuu D bbb 221

and

210 b

b is an interpolation parameter that, like g, is used to select the acceleration used in the numerical integration procedure.

The Newmark- b method uses two parameters for accelerations used in the procedure




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Governing Approximation Equations

The Newmark-b method is based on the two equations

The second of these equations can be rearranged to yield

tutuuu tttttt DD DD gg1

and

222121 tututuuu ttttttt DDD DD bb

tttt uut

ut

u bb

bb 22111

2

DD

DD




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Governing Approximation Equations

Substituting the last equation on the previous slide into the top equation 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 interval are known.

The only unknown is the displacement increment, .

tttt utuut

u D

D

DD b

gbg

bg

211

ttu D ttu D

uD




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Combination of Equations

The three equations used to determine the displacement increment using the Newmark-b method 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 bb

bb 22111

2

DD

DD

tttt utuut

u D

D

DD b

gbg

bg

211




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Combined Equations

These three equations can be combined to yield the following equation

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

tttextT

uCtuMuCuMt

RFuKCt

Mt

D

D

D

D

D D

bbg

bb

bbg

b

bg

b

22

2211

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

bg

b 2

1

teffttexteffT RFuK D D

where

tttttteff uCtuMuCuMt

RR D

D

bbg

bb

bbg

b 22

2211

int




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Stability and Accuracy

The Newmark-b method is unconditionally stable for linear problems when g and b satisfy 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 g and b are used and equilibrium iterations are used to improve accuracy.

21

g and .21

41 2

gb




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Time Step Size

A sufficiently small time step must be used to ensure solution accuracy.

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 describing viscous damping.

Rayleigh damping is defined by the equation

The constants a and b must be determined from experimental data.

This is a convenient form because the damping matrix can be uncoupled 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 mode shape domain takes the form

The i th equation can be written as

where zi is the critical damping ratio for the i th mode.

2baba IKMC TTT



iiiic zba 22


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Finding a and b

a and b can be found from this equation if z is known for two modes.

A least squares approximation to a and b can be found if z is known for more than two modes.

22

1122

21

22

11

ba

CBB T

ba

2

22

21

1

11

n

B

nn

C

2

22

22

11




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Example Problem

5 lb. force distributed over the 17 nodes on the 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 inch element 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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Example – Analysis Parameters

Same as in Module 6



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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Example – Load Curve FactorSection II – Static Analysis


The load curve is zero until 0.05 seconds. At that time, it goes to one in 0.0001 seconds to simulate a step input.


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Example – Force Magnitude

Nodes selected along upper edge

5 lb./17 nodes acting in negative y-direction

Load Curve 1 is defined in Analysis Parameters




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Example – Load 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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Example – Rayleigh 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 a and b.

The first three weak axis bending modes were computed in Module 5. They are: Mode 1 28 Hz = 176 rad/sec, Mode 2 175 Hz = 1100 rad/sec, Mode 4 492 Hz = 3091 rad/sec.

303,556,91000,210,11976,301

B

91.3000.1176.1

C

07676.41

CBBB TT

ba




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Example – MATLAB Program

This MATLAB program finds the Rayleigh damping coefficients for this example. The critical damping ratio for each mode is 0.005 or 0.5%.




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Example – Clamped End Stress

zz is plotted

Note that this curve is the same as that computed using modal superposition in Module 6.




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Example – Free End Tip Displacement

This curve is much smoother than the stress curve. The stress curve is based on strains that are computed from the derivatives of the displacements.




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Module Summary

This module has presented the equations used to perform a direct integration of the equations of motion used for a linear or non-linear dynamic analysis.

It was shown that the Newmark-b method 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-b method 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.



static analysis: direct integration

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