computational dynamics edited
TRANSCRIPT
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COMPUTATIONAL DYNAMICS
Jesan MoralesME 195
Supervised by Dr. GoyalUniversity of California Merced
Dec 22 2013
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Pendulum problem• Forward Euler Method • Simulink• Linear Statespace • Backward Euler • Newton methodParticle problem• Euler methods• Newton method• Non-linear Statespace• Generalized Alpha methodStatic Rod Model
Overview
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Pendulum problem
�̈�=−𝑔𝐿 sin (𝜃)
𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒖𝒄𝒕𝒊𝒐𝒏Ѳ
Figure 1. Pendulum.
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Forward Euler Method
𝑦 𝑖+1=𝑦 𝑖+ �̇� 𝑖h
�̇� 𝑖+1= �̇� 𝑖+ �̈� 𝑖h
Graph 1…
=.2
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Step size
• The step size h was increased to h=0.002 Smoother and no speed loss
Graph 2…
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Simulink
Figure 2. Simulink model.
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Simulink Statespace
[ �̇�1�̇�2]=[ 0 1−𝑔𝑙 0 ] [𝑥1𝑥2]+[00][𝑢1𝑢2]
y+0
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Comparing error with different methodshold on• plot(time,theta,'r'); >>>>>>>>>>>>>>>>>>>>>> euler• plot(timesimulink,pendulumsimulink,'g');>>>> Simulink• plot(time,real,'b');>>>>>>>>>>>>>>>>>>>>>>>> by hand• plot(timesimulink,Statespace,‘dot'); >>>>>>> state space
Graph 4. Method Comparison Graph 5. Method comparison (close-up)
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Particle problem• A particle is traveling with an acceleration described with this
non-linear second order differential equation = The initial conditions of (0) = 0 and y(0)=0.2 are given
• Find the position of the particle at any given time t
Figure 3. www.wpclipart.com
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Damping
Graph 6.Damping. www.splung.com
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Damping
• Critical damping (ζ = 1)
• Over-damping (ζ > 1)
• Under-damping (0 ≤ ζ < 1)
==.034021
• Under-damped}
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Under-Damped
Graph 7. Underdamped Oscillations. http://commons.wikimedia.org
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Forward Euler Method
Image 4. Forward Euler Method
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Forward Euler MethodResults : h=2
Graph 8. Step 2
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Forward Euler Methodh=1
Graph 9. Step 1
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h=.02
Results (cont.) h=0.2
Forward Euler Method
Graph 10.Step 0.2
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Results (Cont.) h= 0.02
Forward Euler Method
Graph 11. Step 0.02
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Results (cont.) h= 0 .002
Forward Euler Method
Graph 12. Step 0.002
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Forward Euler MethodResults (cont.) h= 0 .002
Graph 13. Step 0.002 Zoomed-out
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Backwards Euler method
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Backwards Euler Method (cont.)
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Newton Method f(x) = f’(x)=
• Guess a value of Iterate with a tolerance of
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Symbolic vs. Discretize • Symbolic functions
• Takes about 5 minutes
• Anonymous functions• About 20 seconds
• Discretized • Takes a few seconds
Graph 12.
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Non-Linear Statespace
y’’ =( -y’/3 - 8sin(y) +.2)/3
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Euler Statespace
• g(x)==
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Euler Statespace• g’(x)=• g’(x)=
•
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General Alpha Method
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Euler Statespace
Image 5. Euler Satespace h=.002
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General Alpha Method (cont.)
• =
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Non-Linear Statespace
y’’ =( -y’/3 - 8sin(y) +.2)/3
f(x)=
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General Alpha Method (Cont.)g(x)=
=
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General Alpha Method (Cont.)
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Newton with General Alpha Method
• h=.001
Image 6.Newton with General Alpha Method
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Different
(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
Graph 14. Different .
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General Alpha Method
Graph 15. Generalized Alpha Method
(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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Origin error
Graph 17. Origin Error
(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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Step h= 0.001
(
Magenta = (0.5, 0.5, 0.5)
Red = (0.3, 0.1, 0.3)
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General Alpha methodThe second-order accuracy for the generalized-α method requires
• Unconditionally stable
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General Alpha method
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Forward Euler and Generalized Alpha Method
• If 0
• and
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Forward Euler and Generalized Alpha Method
• If and • Then • Therefore it is not second order accurate
• Since
• Is not true then it is not unconditionally stable
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Backward Euler and Generalized Alpha Method
• If
• and if
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Backward Euler and Generalized Alpha Method
• If and • Then • Therefore it is not second order accurate• If • Then is satisfied and
• Backward Euler is unconditionally stable
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Static Rod Model
• The following equation describe the rod model
• Non-linear differential equations govern the formation of the beam and lead to loop deformation
Image 6.
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• These equation represent the following system• Where s is along the rod• Unshearable and inextensible
Image 7.
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Vectors
• Internal force along the cross section fixed reference
• Moment vector applied to the cross section
• Curvature third component is twist
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Constitutive Relationship
• These equations show the relationship between the moment and the curvature which will be helpful in solving for the linear and non-linear equations:
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Pure Torsion
Image 8. Pure Torsion
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Pure Moment
Image 9. Pure Moment
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Pure Shear Force
Image 10. Pure Shear Force
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All Applied Equally
Image 11. All applied equally
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Static Rod Model• Here are the step taken to derive the equations.
• Linearized equations about :
X= >>>>>>> =X= >>>>>>> =
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Linear Rod Model=
=
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Static Rod Model
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Static Rod Model• ,
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Static Rod Model
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Linear Rod Model
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Results•
•
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Thank you for your timeAny Questions?