Quiz 6

10/10

Should know: Euler & RK2

quiz:

Review

How to simulate

$\dot v=-\frac{c}{m}v$

using RK2

Euler Integration

$v(t_{k+1})=v(t_k)+\dot v(t_k)\Delta t=v(t_k)-\frac{c}{m}v(t_k)\Delta t$

$x(t_{k+1})=x(t_k)+v(t_k)\Delta t$

RK2: $v(t_{k+1})=v(t_k)+\frac{\dot v(t_k)+\dot v_p(t_{k+1})}{2}\Delta t$

What is $\dot v_p(t_{k+1})$?

$v_p(t_{k+1})=v(t_k)+\dot v(t_k)\Delta t$

$\dot v_p(t_{k+1})=-\frac{c}{m}v_p{t_{k+1}}$

Mass-Spring-damper (vertical case)

Free Body Diagram:

• Gravity $mg$
• Spring force $-ky$
• Damper force $-k\dot y$

Equations of motion

$$m\ddot x=0\\ m\ddot y=f_{total}=mg-ky-c\dot y\\ \ddot y+ky+c\dot y=g$$

System dynamics: $\ddot y=g-ky-c\dot y$

Highest derivatives

$\frac{force}{torqus}$->System Dynamics->(Highest derivatives)-> Integration->(position, velocity)->system Dynamics

Steady state: when derivatives are equal to zero

$$\ddot y=0,\dot y=0\\ y_{ss}=\frac{mg}{k}$$

It end up to be steady state

Add $f_{ext}$ and make anywhere we want

Assume I want $y_{ss}=y_d$=>desired value of Y

$\ddot y+\frac{c}{m}+\frac{k}{m}y=g+\frac{1}{m}f_{ext}$

$f_{ext}=ky_d-mg$

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Under damped: <8

Over damped: >8

$\zeta=$damping ratio

$0<\zeta<1\Rightarrow$under damped

$\zeta=1\Rightarrow$critical damped

$\zeta>1\Rightarrow$over damped:

$\omega_n\Rightarrow$ natural frequency

$\ddot y$

$e^{-\zeta\omega_nt}$

Analytical solution

$y(t)=[1-e^{-\zeta\omega_nt}(\cos\omega_nt+\frac{\zeta}{1-\zeta^2}\sin\omega_nt)]y_d$

$\omega_d=w_n\sqrt{1-\zeta^2}$ damped natural frequency

transient response

determine by $\zeta$ and $\omega_n$ parameters

$e^{-\zeta\omega_nt}\Rightarrow$

※Time constant=$t_c=\frac{1}{\zeta\omega_m}$

※Settling time $T_{settle}=4T_c$

if T settle =1, then $T_c=0.25$

Don't want any oscillation $\zeta=1, \omega_n=4$

$\alpha_1=2\zeta\omega_n=8$

$\alpha_0=\omega^2=16$

key things: exponetial parameters

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Have a equation

$\ddot y+\frac{c}{m}\dot y+\frac{k}{m}y=\frac{k}{m}y_d$

We want to control

$\ddot y+\frac{c}{m}\dot y+\frac{k}{m}y=g+\frac{1}{m}f_{ext}$

want desired dynamics

$\ddot y+20\dot y+100y=100y_d$

feedback control

PD Controller

f, m

$m\ddot x=f$ want

$\ddot x=\frac{1}{m}f$

$\ddot x+\frac{c}{m}\dot x+\frac{k}{m}x=\frac{k}{m}x_d$

choose $f=-kx-c\dot x+kx_d$ PD controller

Euler Method