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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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.
Section II – Static Analysis
Module 7 – Direct Integration
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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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Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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.
Section II – Static Analysis
Module 7 – Direct Integration
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Example – Analysis Parameters
Same as in Module 6
Section II – Static Analysis
Module 7 – Direct Integration
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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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Example – Load Curve FactorSection II – Static Analysis
Module 7 – Direct Integration
Page 21
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
Section II – Static Analysis
Module 7 – Direct Integration
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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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Module 7 – Direct Integration
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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
Section II – Static Analysis
Module 7 – Direct Integration
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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%.
Section II – Static Analysis
Module 7 – Direct Integration
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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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Module 7 – Direct Integration
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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.
Section II – Static Analysis
Module 7 – Direct Integration
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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.
Section II – Static Analysis
Module 7 – Direct Integration
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