Point Mechanics in rigid body rotation. Vectors relationship.

[czarodziej_snow]

Hi

Sorry for my language but I dont speak english very well and probably I have trouble understanding comments.

Over a year ago Professor Jadczyk was interested me Dzanibekov's effect.

https://www.youtube.com/watch?v=BGRWg4aV2mw

 

Currently I can calculate and simulate a lot. I will use the equations because they are a universal language and they are understandable to all science on over the world.

 

Vector product.

a x b = c

 

Perpendicular axis

i ┴ j ┴ k

Proportions of vectors
c_k=a_i*b_j
a_i=c_k/b_j
b_j=c_k/a_i

 

The inverse of the vector.

a(a_x,a_y,a_z)=√(a_x²+a_y²+a_z²)
1/a=a/a²=(a_x/a², a_y/a², a_z/a²)

 

Example

d (1,1,1)    1/d (1/3, 1/3, 1/3)
e (1,1,0)    1/e (1/2, 1/2, 0)
f (1,0,0)    1/f (1,0,0)

 

Easy vector product equations.

c = a x b
a = 1/b x c
b = c x 1/a

 

Vector product equations for velocity, angular velocity and position. These equations are correct only for a free point.

v = ω x r = 1/s * m
ω = 1/r x v = 1/m * m/s
r = v x 1/ω = m/s * s

 

In rigid body rotation usually Angular velocity is not stable.

https://youtu.be/CAVXGDMbquk

 

How to calculate temporary angular velocity for rigid body rotation?

Rigid body elements

m1x`(√2, √2, 0);       v1(0,0,-2)
m2x`(-√2, -√2, 0);     v2(0,0,2)
m3y`(-√2, √2, 0);      v3(0,0,-2)
m4y`(√2, -√2, 0);      v4(0,0,2)

m1,m2 is x` main axis. m3,m4 is y` main axis. The center of mas is the center of the coordinate system.

We calculate angular velocity for main axis

ωx`=(-√2/2, √2/2, 0)
ωy`=(-√2/2, -√2/2, 0)

 

These are components temporary angular velocity for rigid body rotation

Ω = ωx` + ωy` =(√2, 0, 0)
 

https://m.salon24.pl/051c2bf7373b20e081ac0f25ababa170,750,0,0,0.jpg

 

How to calculate velocity?

Property vector product:

a x b = c ---> c=absinα

 

If

Ω (x,0,0) and r (x,y,0)

 

this give

ry┴Ω   i    rx║Ω
v = Ω x ry

 

Angular momentum for the free point

L = r x p

 

Angular momentum for rigid body show that equations using tensor moment of inertia.

L_x= ω_xΣm_n(r_n²-x_n²) + ω_yΣm_nx_ny_n + ω_zΣm_nx_nz_n
L_y= ω_xΣm_ny_nx_n + ω_yΣm_n(r_n²-y_n²) + ω_zΣm_ny_nz_n
L_z= ω_xΣm_nz_nx_n + ω_yΣm_nz_ny_n + ω_zΣm_n(r_n²-z_n²)

 

r is the position point to the center of mass.

 

Easier equations using moment of inertia for main axis

I_x=m_xr² + m_-xr²;     I_y=m_yr²=m_-yr² 

 

L_x = (r₁ x p₁)_x + (r₂ x p₂)_-x
L_y = (r₃ x p₃)_y + (r₄ x p₄)_-y
L_z = (r₅ x p₅)_z + (r₆ x p₆)_-z
L = L_x + L_y + L_z

 

Centripetal acceleration a_d for point.

a_d=rω²  --> rω=v
a_d=v²/r  --> v/r=ω
a_d=ωv

 

a_d║-r

 

Vector product equations for centripetal acceleration

a_d = ω x v
ω = 1/v x a_d
v =a_d x 1/ω

 

Equations temporary angular velocity for rigid body rotation

Ω=ωx`+ωy`+ωz`=(1/v x ad)x` + (1/v x ad)y` + (1/v x ad)z`

 

Centripetal forces  for point in rigid body rotation, three possibilities.

First version

F=am
ω = 1/v x a_d
ω = 1/mv x ma_d
ω = 1/p x F_d
F_d = ω x p
p = F_d x 1/ω

 

Second version

Ω = (x, 0, 0)
r = r_x +r_y = r┴ + r║
F_d=mω² r_y

 

Third version

Ω = (x, 0, 0)
r = rx +ry = r┴ + r║
Fd=mω2 r
Fd=m(1/ry x v)2 r

 

Everything shows my animation

https://youtu.be/oz1uw9x13kA

 

Another animation shows the angular acceleration vector for the effect

https://youtu.be/exwM5bTuO6Q

https://youtu.be/v2kwwzLA3aM

 

Another animation shows the vectors in no inertial frame

https://youtu.be/LZ9YwG9cVBE

 

code for the first simulation Vpython.

 

from visual import *

 

 

mx=0.5 #masy x`,y`

my=1.

 

x1=1 #pozycja m1,m2 na osi x`

y1=0

z1=0

 

x2=0 #pozycja m3,m4 na osi y`

y2=1

z2=0

 

r=vector(x1,y1,z1) #promien

R=mag(r)

#print R

 

W=vector(0.9,0,0) #omega

 

 

v1=W.x*y1 #predkosci

v2=W.x*-y1

v3=W.x*y2

v4=W.x*-y2

 

p1=mx*v1 #ped

p2=mx*v2

p3=my*v3

p4=my*v4

 

ax=(v1*v1) #przyspieszenia a=(v^2)/r; r=1

ay=(v3*v3)

 

#print "v",v1,v2,v3,v4

 

 

TIxx=(2*mx*((y1*y1)+(z1*z1)))+(2*my*((y2*y2)+(z2*z2))) #mx1 i mx2 --> 2*mx

TIyy=(2*mx*((x1*x1)+(z1*z1)))+(2*my*((x2*x2)+(z2*z2))) #elementy tensora

TIzz=(2*mx*((x1*x1)+(y1*y1)))+(2*my*((x2*x2)+(y2*y2)))

TIxy=(2*mx*x1*y1)+(2*my*x2*y2)

TIxz=(2*mx*x1*z1)+(2*my*x2*z2)

TIyx=(2*mx*y1*x1)+(2*my*y2*x2)

TIyz=(2*mx*y1*z1)+(2*my*y2*z2)

TIzx=(2*mx*z1*x1)+(2*my*z2*x2)

TIzy=(2*mx*z1*y1)+(2*my*z2*y2)

 

Fdx=mx*W.x*W.x*y1 #sily dosrodkowe na osiach glownych z Fd=mW^2ry

Fdy=my*W.x*W.x*y2

#print "F", Fd1,Fd2

 

Fdax=mx*ax #sily dosrodkowe na osiach glownych z F=ma

Fday=my*ay

 

#L=vector((W.x*TIxx-W.y*TIxy-W.z*TIxz),(-W.x*TIyx+W.y*TIyy-W.z*TIyz),(-W.x*TIzx-W.y*TIzy+W.z*TIzz))

#print "W",W

 

omega=arrow(axis=vector(W.x,W.y,0), color= color.blue, shaftwidth=0.05) #omega startowa

#kret=arrow(axis=vector(0,0,0), color= color.red, shaftwidth=0.04)

kret2=arrow(axis=vector(0,0,0),color=vector(1,0.4,0.4), shaftwidth=0.04)

#kret2x=arrow(axis=vector(0,0,0),color=vector(1,1,0.3), shaftwidth=0.04)

#kret2y=arrow(axis=vector(0,0,0),color=vector(1,1,0.3), shaftwidth=0.04)

dv1=arrow(axis=vector(0,0,0), color=vector(0.3,0.6,0), shaftwidth=0.05) #przyspieszenie punktow

dv2=arrow(axis=vector(0,0,0), color=vector(0.3,0.6,0), shaftwidth=0.05)

dv3=arrow(axis=vector(0,0,0), color=vector(0.3,0.6,0), shaftwidth=0.05)

dv4=arrow(axis=vector(0,0,0), color=vector(0.3,0.6,0), shaftwidth=0.05)

dvg=arrow(pos=vector(0,1,0),axis=vector(0,0,0), color=vector(0.6,0.6,0), shaftwidth=0.05)

dvd=arrow(pos=vector(0,-1,0),axis=vector(0,0,0), color=vector(0.6,0.6,0), shaftwidth=0.05)

 

#omegax=arrow(axis=vector(0,0,0), color= vector(0,0,0.01), shaftwidth=0.02)

#omegay=arrow(axis=vector(1,0,0), color= vector(0,0,0.01), shaftwidth=0.02)

 

masa1x=sphere(pos=vector(x1,y1,0),radius=0.05) #bryla sztywna

masa2x=sphere(pos=vector(-x1,-y1,0),radius=0.05)

masa1y=sphere(pos=vector(x2,y2,0),radius=0.05)

masa2y=sphere(pos=vector(-x2,-y2,0),radius=0.05)

promien1=arrow(pos=masa2x.pos, axis=masa1x.pos-masa2x.pos, color= color.yellow, shaftwidth=0.005)

promien2=arrow(pos=masa2y.pos, axis=masa1y.pos-masa2y.pos, color= color.yellow, shaftwidth=0.005)

 

vm1x=arrow(pos=masa1x.pos, axis=vector(0,0,v1), shaftwidth=0.01) #wektory predkosci punktow

vm2x=arrow(pos=masa2x.pos, axis=vector(0,0,v2), color=color.green, shaftwidth=0.01)

vm1y=arrow(pos=masa1y.pos, axis=vector(0,0,v3), color=color.green, shaftwidth=0.01)

vm2y=arrow(pos=masa2y.pos, axis=vector(0,0,v4), color=color.green, shaftwidth=0.01)

#orbita1=ring(pos=vector(x1,0,0), axis=vector(1,0,0), radius=y1, thickness=0.01) #orbita

os=arrow(pos=vector(-2,0,0), axis=vector(4,0,0),color=vector(0.3,0.3,0.3),shaftwidth=0.005) #os obrotu

 

 

 

sila1=arrow(pos=masa1x.pos,axis=-vector(x1,y1,0)*Fdx, color= vector(1,1,0), shaftwidth=0.05) #wektory sil dosrodkowych punktow

sila2=arrow(pos=masa1x.pos,axis=-vector(-x1,-y1,0)*Fdx, color= vector(1,1,0), shaftwidth=0.05)

sila3=arrow(pos=masa1y.pos,axis=-vector(x2,y2,0)*Fdy, color= vector(1,1,0), shaftwidth=0.05)

sila4=arrow(pos=masa1y.pos,axis=-vector(-x2,-y2,0)*Fdy, color= vector(1,1,0), shaftwidth=0.05)

sumasilag=arrow(color= vector(0.8,0.5,0), shaftwidth=0.05) #suma sil dosrodkowych gora z Fd=mW^2ry

sumasilad=arrow(color= vector(0.8,0.5,0), shaftwidth=0.05) #suma sil dosrodkowych dol z Fd=mW^2ry

#sila1a=arrow(pos=masa1x.pos,axis=-vector(x1,y1,0)*Fdax, color= vector(1,1,0.5), shaftwidth=0.05) #wektory sil dosrodkowych punktow

#sila2a=arrow(pos=masa1x.pos,axis=-vector(x1,y1,0)*Fdax, color= vector(1,1,0.5), shaftwidth=0.05) #wektory sil dosrodkowych punktow

#sila3a=arrow(pos=masa1x.pos,axis=-vector(x1,y1,0)*Fday, color= vector(1,1,0.5), shaftwidth=0.05) #wektory sil dosrodkowych punktow

#sila4a=arrow(pos=masa1x.pos,axis=-vector(x1,y1,0)*Fday, color= vector(1,1,0.5), shaftwidth=0.05) #wektory sil dosrodkowych punktow

#sumasilga=arrow(color= vector(0.8,0.5,0), shaftwidth=0.05) #suma sil dosrodkowych dol F=am

#sumasilda=arrow(color= vector(0.8,0.5,0), shaftwidth=0.05) #suma sil dosrodkowych dol F=am

 

 

 

#sumaF=arrow(axis=-vector(0,Fd1+Fd2,0), color= vector(1,0.5,0), shaftwidth=0.03)

 

 

 

t=0

 

 

while t<20:

rate(3)

 

 

# print t,L

# print TIxx-TIxy-TIxz,-TIyx+TIyy-TIyz,-TIzx-TIzy+TIzz

# print TIxx,TIxy,TIxz," I ",TIyx,TIyy,TIyz," I ",TIzx,TIzy,TIzz

 

x1=x1-0.1 #nowe pozycje punktow

y1=sqrt(1-(x1*x1))

y2=y2-0.1

x2=-sqrt(1-(y2*y2))

 

v1=W.x*y1 # v = W x ry

v2=W.x*-y1

v3=W.x*y2

v4=W.x*-y2

 

ov1=1/v1 #1/v

ov2=1/v2

ov3=1/v3

ov4=1/v4

# print t,v1*ov1

 

r1=vector(x1,y1,0) #promienie

r2=vector(-x1,-y1,0)

r3=vector(x2,y2,0)

r4=vector(-x1,-y1,0)

# print mag(r1),mag(r2),mag(r3),mag(r4)

 

p1=mx*v1 #ped

p2=mx*v2

p3=my*v3

p4=my*v4

 

op1=1/p1 #1/p

op2=1/p2

op3=1/p3

op4=1/p4

# print "v",v1,v2,v3,v4, "p",p1,p2,p3,p4

 

wx=vector(y1*v1,-(x1*v1),0) # wx` = r x v12; r=1

wy=vector(y2*v3,-(x2*v3),0) # wy` = r x v34; r=1

Wk=wx+wy # omega koncowa Wk=wx`+wy`

# print t, "Ws=", W, "Wk=", Wk

 

a1=(v1*v1)/mag(r1) #przyspieszenia a=(v^2)/r; r=1

a2=(v2*v2)/mag(r2)

a3=(v3*v3)/mag(r3)

a4=(v4*v4)/mag(r4)

a1v=vector(x1,y1,0)*-a1 #wektory przyspieszen

a2v=vector(-x1,-y1,0)*-a2

a3v=vector(x2,y2,0)*-a3

a4v=vector(-x2,-y2,0)*-a4

# a1r=-r1*(v1*v1)

# f1am=a1*mx

# print t, f1am

# print t,a1,a2

 

if t<10: #suma par przyspieszen dosrodkowych gora, dol z Fd=mW^2ry

adg=a1v+a3v

add=a2v+a4v

else:

adg=a1v+a4v

add=a2v+a3v

 

Wax=vector(-ov1*a1v.y,ov1*a1v.x,0) # w = 1/v x a

Way=vector(-ov3*a3v.y,ov3*a3v.x,0)

Wa=Wax+Way #W=wx`+wy`

# print t, Wa

 

 

# r1=sqrt((x1*x1)+(y1*y1))

 

TIxx=(2*mx*((y1*y1)+(z1*z1)))+(2*my*((y2*y2)+(z2*z2))) #mx1 i mx2 --> 2*mx

TIyy=(2*mx*((x1*x1)+(z1*z1)))+(2*my*((x2*x2)+(z2*z2)))

TIzz=(2*mx*((x1*x1)+(y1*y1)))+(2*my*((x2*x2)+(y2*y2)))

TIxy=(2*mx*x1*y1)+(2*my*x2*y2)

TIxz=(2*mx*x1*z1)+(2*my*x2*z2)

TIyx=(2*mx*y1*x1)+(2*my*y2*x2)

TIyz=(2*mx*y1*z1)+(2*my*y2*z2)

TIzx=(2*mx*z1*x1)+(2*my*z2*x2)

TIzy=(2*mx*z1*y1)+(2*my*z2*y2)

 

L=vector((W.x*TIxx-W.y*TIxy-W.z*TIxz),(-W.x*TIyx+W.y*TIyy-W.z*TIyz),(-W.x*TIzx-W.y*TIzy+W.z*TIzz))

 

if y1<0: #wartosc sily dosrodkowe Fd=mW^2ry

Fdx=mx*W.x*W.x*y1

else:

Fdx=-mx*W.x*W.x*y1

 

if y2<0:

Fdy=my*W.x*W.x*y2

else:

Fdy=-my*W.x*W.x*y2

# print t, Fdy, x2,y2

 

Lpr1=vector(y1*p1,-x1*p1,0) # L = r x p

Lpr2=vector(-y1*p2,x1*p2,0)

Lpr3=vector(y2*p3,-x2*p3,0)

Lpr4=vector(-y2*p4,x2*p4,0)

Lprx=Lpr1+Lpr2

Lpry=Lpr3+Lpr4

Lpr=Lprx+Lpry #suma kretow

# print t, L-Lpr

 

Fd1=vector(x1,y1,0)*Fdx #wektory sil dosrodkowch z Fd=mW^2ry

Fd2=vector(-x1,-y1,0)*Fdx

Fd3=vector(x2,y2,0)*Fdy

Fd4=vector(-x2,-y2,0)*Fdy

 

if t<10: #suma par sil dosrodkowych gora, dol z Fd=mW^2ry

Fdg=Fd1+Fd3

Fdd=Fd2+Fd4

else:

Fdg=Fd1+Fd4

Fdd=Fd2+Fd3

 

Fd1a=a1v*mx #F=am

Fd2a=a2v*mx

Fd3a=a3v*my

Fd4a=a4v*my

 

WFpx=vector(-op1*Fd1a.y,op1*Fd1a.x,0) #w = (1/p) x F

WFpy=vector(-op3*Fd3a.y,op3*Fd3a.x,0)

WFp=WFpx+WFpy

# print t, WFp, W

 

if t<10: #suma par sil dosrodkowych gora, dol z F=ma

Fdag=Fd1a+Fd3a

Fdad=Fd2a+Fd4a

else:

Fdag=Fd1a+Fd4a

Fdad=Fd2a+Fd3a

print t,Fdag, Fdad

 

 

# kret.axis=vector(L.x,L.y,L.z+0.01)

kret2.axis=vector(Lpr.x,Lpr.y,Lpr.z+0.01)

# kret2x.axis=vector(Lprx.x,Lprx.y,Lprx.z+0.01)

# kret2y.axis=vector(Lpry.x,Lpry.y,Lpry.z+0.01)

masa1x.pos=vector(x1,y1,0)

masa2x.pos=vector(-x1,-y1,0)

masa1y.pos=vector(x2,y2,0)

masa2y.pos=vector(-x2,-y2,0)

promien1.pos=masa2x.pos

promien1.axis=masa1x.pos-masa2x.pos

promien2.pos=masa2y.pos

promien2.axis=masa1y.pos-masa2y.pos

vm1x.pos=masa1x.pos

vm1x.axis=vector(0,0,v1)

vm2x.pos=masa2x.pos

vm2x.axis=vector(0,0,v2)

vm1y.pos=masa1y.pos

vm1y.axis=vector(0,0,v3)

vm2y.pos=masa2y.pos

vm2y.axis=vector(0,0,v4)

# orbita1.pos=vector(x1,0,0)

# orbita1.radius=y1

# omegax.axis=wx

# omegay.axis=wy

 

# dv1.pos=vector(x1,y1,0)

# dv1.axis=vector(a1v.x,a1v.y,a1v.z)

# dv2.pos=vector(-x1,-y1,0)

# dv2.axis=vector(a2v.x,a2v.y,a2v.z)

# dv3.pos=vector(x2,y2,0)

# dv3.axis=vector(a3v.x,a3v.y,a3v.z)

# dv4.pos=vector(-x2,-y2,0)

# dv4.axis=vector(a4v.x,a4v.y,a4v.z)

# dvg.axis=vector(adg.x,adg.y,adg.z)

# dvd.axis=vector(add.x,add.y,add.z)

 

sila1.pos=vector(x1,y1,0)

# sila1.axis=vector(0,-mag(Fd1),0)

sila1.axis=vector(Fd1.x,Fd1.y,Fd1.z)

sila2.pos=vector(-x1,-y1,0)

# sila2.axis=vector(0,mag(Fd2),0)

sila2.axis=vector(Fd2.x,Fd2.y,Fd2.z)

sila3.pos=vector(x2,y2,0)

sila3.axis=vector(Fd3.x,Fd3.y,Fd3.z)

sila4.pos=vector(-x2,-y2,0)

sila4.axis=vector(Fd4.x,Fd4.y,Fd4.z)

 

# if t<10:

# sila3.axis=vector(0,-mag(Fd3),0)

# sila4.axis=vector(0,mag(Fd4),0)

# else:

# sila3.axis=vector(0,mag(Fd3),0)

# sila4.axis=vector(0,-mag(Fd4),0)

 

sumasilag.axis=vector(Fdg.x,Fdg.y,Fdg.z)

sumasilad.axis=vector(Fdd.x,Fdd.y,Fdd.z)

 

# sila1a.pos=vector(x1,y1,0)

# sila1a.axis=vector(Fd1a.x,Fd1a.y,Fd1a.z)

# sila2a.pos=vector(-x1,-y1,0)

# sila2a.axis=vector(Fd2a.x,Fd2a.y,Fd2a.z)

# sila3a.pos=vector(x2,y2,0)

# sila3a.axis=vector(Fd3a.x,Fd3a.y,Fd3a.z)

# sila4a.pos=vector(-x2,-y2,0)

# sila4a.axis=vector(Fd4a.x,Fd4a.y,Fd4a.z)

# sumasilga.axis=vector(Fdag.x,Fdag.y,Fdag.z)

# sumasilda.axis=vector(Fdad.x,Fdad.y,Fdad.z)

# sumaF.axis=sila1.axis+sila2.axis

 

# LxF=sumaF.axis.x*L.x+sumaF.axis.y*L.y+sumaF.axis.z*L.z

# print t,"L x F",LxF

 

t=t+1

[swansont]

Is there a question here?

 

A basic explanation is found here:
http://www.sciencealert.com/watch-wtf-is-going-on-with-this-object-spinning-in-zero-gravity

and it's discussed in the embedded video starting at ~4:20

[czarodziej_snow]

@swansont

Thanks for the link, I'm trying to understand this deeper

 

" Is there a question here?"

There are many answers in this note but there are still many questions. The main question,  how do the centripetal forces work in this effect?

[J.C.MacSwell]

On 9/24/2017 at 10:11 AM, czarodziej_snow said:

@swansont

Thanks for the link, I'm trying to understand this deeper

 

" Is there a question here?"

There are many answers in this note but there are still many questions. The main question,  how do the centripetal forces work in this effect?

They vary in direction with respect to the centre of mass, with the net forces small enough and close enough toward it to allow it to spin seemingly freely at times, and at other times applying enough torque to shift it to the next spinning mode. I hope that makes some sense.

[swansont]

On 9/24/2017 at 9:11 AM, czarodziej_snow said:

@swansont

Thanks for the link, I'm trying to understand this deeper

 

" Is there a question here?"

There are many answers in this note but there are still many questions. The main question,  how do the centripetal forces work in this effect?

I don't think centripetal forces are the primary issue here. The rotation about the axis is unstable, so that any small amount of perturbation will cause the change in orientation. Almost every instance of rotation there will be some rotation about another axis, but if the primary rotation is stable this isn't an issue. 

https://en.wikipedia.org/wiki/Tennis_racket_theorem

[czarodziej_snow]

The mechanics of rigid bodies describe the Euler equations.

I_x(dω_x /dt) + (I_z - I_y)ω_zω_y   = 0

I_y(dω_y /dt)+ (I_x - I_z)ω_zω_x = 0

I_z(dω_z /dt)+ (I_y - I_x)ω_xω_y   = 0

Important is the difference in moments of inertia on the main axes and angular velocity vector components .

After easy transformations we have

I_xɛ_x = (I_y - I_z)ω_zω_y   

I_yɛ_y = (I_z - I_x)ω_zω_x

I_zɛ_z = (I_x - I_y)ω_xω_y    

Iωω = Iɛ = M - momentum forces

Its easily visualize these vectors

 

 

The problem is that Euler's equations work only in non-inertial systems.