Question

/////// Givens: • g = 9.81 m/s Initial Conditons: • $\theta_1$ = 1/3$\pi$ rad • $\dot{\theta}_1$ = 0 rad/s Initial Conditons: • Time step interval: [0.1, 0.01] seconds • Time span: 5 seconds L1 • $m_1$ = 1 kg $m_2$ = 1 kg • $\theta_2$ = 2/3$\pi$ rad m $\theta_2$ L2 m • $L_1$ = 1 m • $\dot{\theta}_2$ = 0 rad/s • $L_2$ = [0.25, 0.5] m Figure 1: A Double Pendulum Dynamic System Equations of Motion: $\ddot{\theta}_1$ = \frac{-g(2m_1 + m_2)sin\theta_1 - m_2gsin(\theta_1 - 2\theta_2) - 2sin(\theta_1 - \theta_2)m_2(\dot{\theta}_2^2L_2 + \dot{\theta}_1^2L_1cos(\theta_1 - \theta_2))}{L_1(2m_1 + m_2 - m_2cos(2\theta_1 - 2\theta_2))} $\ddot{\theta}_2$ = \frac{2sin(\theta_1 - \theta_2)(\dot{\theta}_1^2L_1(m_1 + m_2) + g(m_1 + m_2)cos\theta_1 + \dot{\theta}_2^2L_2m_2cos(\theta_1 - \theta_2))}{L_2(2m_1 + m_2 - m_2cos(2\theta_1 - 2\theta_2))}

          ///////
Givens:
• g = 9.81 m/s
Initial Conditons:
• $\theta_1$ = 1/3$\pi$ rad
• $\dot{\theta}_1$ = 0 rad/s
Initial Conditons:
• Time step interval: [0.1, 0.01] seconds
• Time span: 5 seconds
L1
• $m_1$ = 1 kg
$m_2$ = 1 kg
• $\theta_2$ = 2/3$\pi$ rad
m
$\theta_2$
L2
m
• $L_1$ = 1 m
• $\dot{\theta}_2$ = 0 rad/s
• $L_2$ = [0.25, 0.5] m
Figure 1: A Double Pendulum Dynamic System
Equations of Motion:
$\ddot{\theta}_1$ = \frac{-g(2m_1 + m_2)sin\theta_1 - m_2gsin(\theta_1 - 2\theta_2) - 2sin(\theta_1 - \theta_2)m_2(\dot{\theta}_2^2L_2 + \dot{\theta}_1^2L_1cos(\theta_1 - \theta_2))}{L_1(2m_1 + m_2 - m_2cos(2\theta_1 - 2\theta_2))}
$\ddot{\theta}_2$ = \frac{2sin(\theta_1 - \theta_2)(\dot{\theta}_1^2L_1(m_1 + m_2) + g(m_1 + m_2)cos\theta_1 + \dot{\theta}_2^2L_2m_2cos(\theta_1 - \theta_2))}{L_2(2m_1 + m_2 - m_2cos(2\theta_1 - 2\theta_2))}

Show more…

///////
Givens:
• g = 9.81 m/s
Initial Conditons:
• θ1 = 1/3π rad
• θ̇1 = 0 rad/s
Initial Conditons:
• Time step interval: [0.1, 0.01] seconds
• Time span: 5 seconds
L1
• m1 = 1 kg
m2 = 1 kg
• θ2 = 2/3π rad
m
θ2
L2
m
• L1 = 1 m
• θ̇2 = 0 rad/s
• L2 = [0.25, 0.5] m
Figure 1: A Double Pendulum Dynamic System
Equations of Motion:
θ̈1 = (-g(2m1 + m2)sinθ1 - m2gsin(θ1 - 2θ2) - 2sin(θ1 - θ2)m2(θ̇2^2L2 + θ̇1^2L1cos(θ1 - θ2)))/(L1(2m1 + m2 - m2cos(2θ1 - 2θ2)))
θ̈2 = (2sin(θ1 - θ2)(θ̇1^2L1(m1 + m2) + g(m1 + m2)cosθ1 + θ̇2^2L2m2cos(θ1 - θ2)))/(L2(2m1 + m2 - m2cos(2θ1 - 2θ2)))

Added by Francisco Jose A.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Adi S

07/18/2023

Step 1

81 m/s^2 - m1 = 1 kg - m2 = 1 kg - L1 = 1 m - L2 = [0.25, 0.5] m - Initial conditions: - θ1 = 1/3 π rad - θ1' = 0 rad/s - θ2 = 2/3 π rad - θ2' = 0 rad/s - Time step interval: [0.1, 0.01] seconds - Time span: 5 seconds Show more…

Show all steps

lock

Thanks for your feedback!

Simulate a Doublr Pendulum Dynamical System using three integration methods: Euler Method, Trapezoid Method, & Runge-Kutta Method Givens: Initial Conditons: Initial Conditons: g=9.81(m)/(s) heta _(1)=(1)/(3)pi rad Time step interval: 0.1,0.01 seconds m_(1)=1kg heta _(1)^(˙)=0ra(d)/(s) Time span: 5 seconds m_(2)=1kg heta _(2)=(2)/(3)pi rad L_(1)=1m heta _(2)^(˙)=0ra(d)/(s) L_(2)=[0.25,0.5]m Figure 1: A Double Pendulum Dynamic System Equations of Motion: heta _(1)^('')=(-g(2m_(1)+m_(2))sin heta _(1)-m_(2)gsin( heta _(1)-2 heta _(2))-2sin( heta _(1)- heta _(2))m_(2)( heta _(2)^('2)L_(2)+ heta _(1)^('2)L_(1)cos( heta _(1)- heta _(2))))/(L_(1)(2m_(1)+m_(2)-m_(2)cos(2 heta _(1)-2 heta _(2)))) heta _(2)^('')=(2sin( heta _(1)- heta _(2))( heta _(1)^('2)L_(1)(m_(1)+m_(2))+g(m_(1)+m_(2))cos heta _(1)+ heta _(2)^('2)L_(2)m_(2)cos( heta _(1)- heta _(2))))/(L_(2)(2m_(1)+m_(2)-m_(2)cos(2 heta _(1)-2 heta _(2))))Givens: g=9.81(m)/(s) m_(1)=1kg m_(2)=1kg L_(1)=1m L_(2)=[0.25,0.5]m Initial Conditons: Initial Conditons: heta _(1)=(1)/(3)pi rad Time step interval: 0.1,0.01 seconds heta _(1)^(˙)=0ra(d)/(s) Time span: 5 seconds heta _(2)^(˙)=0ra(d)/(s) Givens: Initial Conditons: Initial Conditons: g=9.81m/s =1/3rad =0rad/s Time step interval:[0.1,0.01]seconds m=1kg Time span5 seconds m2=1kg =2/3T rad L=1m =0rad/s L=[0.25,0.5]m m -2 Figure 1:A Double Pendulum Dynamic System Equations of Motion: L12m1+m2-mcos(20-20 L(2 m1+m2-mcos(201-20

Play audio

Feedback

Powered by NumerAI

Adi S and 92 other subject Physics 103 educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Recommended Videos

Recommended Textbooks

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later