COMPUTATIONAL DYNAMICS

Jesan MoralesME 195

Supervised by Dr. GoyalUniversity of California Merced

Dec 22 2013

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

Pendulum problem

�̈�=−𝑔𝐿 sin (𝜃)

𝑭𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒖𝒄𝒕𝒊𝒐𝒏Ѳ

Figure 1. Pendulum.

Forward Euler Method

𝑦 𝑖+1=𝑦 𝑖+ �̇� 𝑖h

�̇� 𝑖+1= �̇� 𝑖+ �̈� 𝑖h

Graph 1…

=.2

Step size

• The step size h was increased to h=0.002 Smoother and no speed loss

Graph 2…

Simulink

Figure 2. Simulink model.

Simulink Statespace

[ �̇�1�̇�2]=[ 0 1−𝑔𝑙 0 ] [𝑥1𝑥2]+[00][𝑢1𝑢2]

y+0

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)

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

Damping

Graph 6.Damping. www.splung.com

Damping

• Critical damping (ζ = 1)

• Over-damping (ζ > 1)

• Under-damping (0 ≤ ζ < 1)

==.034021

• Under-damped}

Under-Damped

Graph 7. Underdamped Oscillations. http://commons.wikimedia.org

Forward Euler Method

Image 4. Forward Euler Method

Forward Euler MethodResults : h=2

Graph 8. Step 2

Forward Euler Methodh=1

Graph 9. Step 1

h=.02

Results (cont.) h=0.2

Forward Euler Method

Graph 10.Step 0.2

Results (Cont.) h= 0.02

Forward Euler Method

Graph 11. Step 0.02

Results (cont.) h= 0 .002

Forward Euler Method

Graph 12. Step 0.002

Forward Euler MethodResults (cont.) h= 0 .002

Graph 13. Step 0.002 Zoomed-out

Backwards Euler method

Backwards Euler Method (cont.)

Newton Method f(x) = f’(x)=

• Guess a value of Iterate with a tolerance of

Symbolic vs. Discretize • Symbolic functions

• Takes about 5 minutes

• Anonymous functions• About 20 seconds

• Discretized • Takes a few seconds

Graph 12.

Non-Linear Statespace

y’’ =( -y’/3 - 8sin(y) +.2)/3

Euler Statespace

• g(x)==

Euler Statespace• g’(x)=• g’(x)=

•

General Alpha Method

Euler Statespace

Image 5. Euler Satespace h=.002

General Alpha Method (cont.)

• =

Non-Linear Statespace

y’’ =( -y’/3 - 8sin(y) +.2)/3

f(x)=

General Alpha Method (Cont.)g(x)=

=

General Alpha Method (Cont.)

Newton with General Alpha Method

• h=.001

Image 6.Newton with General Alpha Method

Different

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

Graph 14. Different .

General Alpha Method

Graph 15. Generalized Alpha Method

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

Origin error

Graph 17. Origin Error

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

Step h= 0.001

(

Magenta = (0.5, 0.5, 0.5)

Red = (0.3, 0.1, 0.3)

General Alpha methodThe second-order accuracy for the generalized-α method requires

• Unconditionally stable

General Alpha method

Forward Euler and Generalized Alpha Method

• If 0

• and

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

Backward Euler and Generalized Alpha Method

• If

• and if

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

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.

• These equation represent the following system• Where s is along the rod• Unshearable and inextensible

Image 7.

Vectors

• Internal force along the cross section fixed reference

• Moment vector applied to the cross section

• Curvature third component is twist

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:

Pure Torsion

Image 8. Pure Torsion

Pure Moment

Image 9. Pure Moment

Pure Shear Force

Image 10. Pure Shear Force

All Applied Equally

Image 11. All applied equally

Static Rod Model• Here are the step taken to derive the equations.

• Linearized equations about :

X= >>>>>>> =X= >>>>>>> =

Linear Rod Model=

=

Static Rod Model

Static Rod Model• ,

Static Rod Model

Linear Rod Model

Results•

•

Thank you for your timeAny Questions?