Momento de impulsión:
Articulaciones robóticas y de vehículo.
Ley:
Sea m·d_{t}[x]·d = L(t) ==>
Si d[L(t)] = Mv·d[x] ==>
x(t) = re^{(M/m)·(v/d)·t}
Deducción:
L(t) = int[ d[L(t)] ] = int[ Mv·d[x] ] = Mv·int[ d[x] ] = Mvx
m·d_{t}[x]·d = m·d_{t}[ re^{(M/m)·(v/d)·t} ]·d = mr·d_{t}[ e^{(M/m)·(v/d)·t} ]·d = ...
... mre^{(M/m)·(v/d)·t}·(M/m)·(v/d)·d = Mvre^{(M/m)·(v/d)·t} = Mvx
Ley:
Sea m·d_{t}[x]·d = L(t) ==>
Si d[L(t)] = Mgt·d[x] ==>
x(t) = re^{(M/m)·(g/d)·(1/2)·t^{2}}
Deducción:
L(t) = int[ d[L(t)] ] = int[ Mgt·d[x] ] = Mgt·int[ d[x] ] = Mgtx
m·d_{t}[x]·d = m·d_{t}[ re^{(M/m)·(g/d)·(1/2)·t^{2}} ]·d = ...
... mr·d_{t}[ e^{(M/m)·(g/d)·(1/2)·t^{2}} ]·d = ...
... mre^{(M/m)·(g/d)·(1/2)·t^{2}}·(M/m)·(1/d)·gt·d = Mgt·re^{(M/m)·(g/d)·(1/2)·t^{2}} = Mgtx
Ley:
Sea m·d_{t}[x]·d = L(t) ==>
Si d[L(t)] = Mav·2x·d[x] ==>
x(t) = ( (-1)·(M/m)·(v/d)·at )^{(-1)}
Ley:
Sea m·d_{t}[x]·d = L(t) ==>
Si d[L(t)] = Magt·2x·d[x] ==>
x(t) = ( (-1)·(M/m)·(g/d)·a·(1/2)·t^{2} )^{(-1)}
Momento de inercia:
Motores de rotación.
Ley:
Sea I_{c}·(1/2)·d_{t}[w]^{2} = E ==>
Si d[I_{c}] = Mrv·d[t] ==>
w(t) = ( (8/M)·(1/(rv))·E )^{(1/2)}·t^{(1/2)}
Deducción:
I_{c} = int[ d[I_{c}] ] = int[ Mrv·d[t] ] = Mrv·int[ d[t] ] = Mrvt
I_{c}·(1/2)·d_{t}[w]^{2} = Mrvt·(1/2)·d_{t}[ ( (8/M)·(1/(rv))·E )^{(1/2)}·t^{(1/2)} ]^{2} = ...
... Mrvt·(1/2)·( ( (8/M)·(1/(rv))·E )^{(1/2)}·d_{t}[ t^{(1/2)} ] )^{2} = ...
... Mrvt·(1/2)·( (8/M)·(1/(rv))·E )·d_{t}[ t^{(1/2)} ]^{2} = 4Et·d_{t}[ t^{(1/2)} ]^{2} = ...
... 4Et·( (1/2)·(1/t)^{(1/2)} )^{2} = E
Ley:
Sea I_{c}·(1/2)·d_{t}[w]^{2} = E ==>
Si d[I_{c}] = Mrgt·d[t] ==>
w(t) = ( (4/M)·(1/(rg))·E )^{(1/2)}·ln(ut)
Deducción:
I_{c} = int[ d[I_{c}] ] = int[ Mrgt·d[t] ] = Mrg·int[ t·d[t] ] = Mrg·(1/2)·t^{2}
I_{c}·(1/2)·d_{t}[w]^{2} = Mrgt^{2}·(1/4)·d_{t}[ ( (4/M)·(1/(rg))·E )^{(1/2)}·ln(ut) ]^{2} = ...
... Mrgt^{2}·(1/4)·( ( (4/M)·(1/(rg))·E )^{(1/2)}·d_{t}[ ln(ut) ] )^{2} = ...
... Mrgt^{2}·(1/4)·( (4/M)·(1/(rg))·E )·d_{t}[ ln(ut) ]^{2} = Et^{2}·d_{t}[ ln(ut) ]^{2} = ...
... Et^{2}·(1/t)^{2} = E
Fuerzas de amortiguación y de pistones de fluido:
Horizontal:
Ley:
m·d_{tt}^{2}[x] = (-k)·x
x(t) = re^{(k/m)^{(1/2)}·it}
Deducción:
m·d_{tt}^{2}[x] = m·d_{t}[ d_{t}[x] ] = m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ] ] = ...
... m·d_{t}[ r·d_{t}[ e^{(k/m)^{(1/2)}·it} ] ] = m·d_{t}[ re^{(k/m)^{(1/2)}·it}·(k/m)^{(1/2)}·i ] = ...
... mr·(k/m)^{(1/2)}·i·d_{t}[ e^{(k/m)^{(1/2)}·it} ] = ...
... mr·( (k/m)^{(1/2)}·i )^{2} e^{(k/m)^{(1/2)}·it} = ...
... (-k)·re^{(k/m)^{(1/2)}·it} = (-k)·x
Ley
m·d_{tt}^{2}[x] = (-b)·d_{t}[x]
d_{t}[x] = ve^{(-1)·(b/m)·t}
Deducción:
m·d_{tt}^{2}[x] = m·d_{t}[ d_{t}[x] ] = m·d_{t}[ ve^{(-1)·(b/m)·t} ] = ...
... mv·d_{t}[ e^{(-1)·(b/m)·t} ] = mv·e^{(-1)·(b/m)·t}·(-1)·(b/m) = ...
... (-b)·ve^{(-1)·(b/m)·t} = (-b)·d_{t}[x]
Vertical:
Ley:
m·d_{tt}^{2}[y] = (-k)·y+qg
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·qg
Deducción:
m·d_{tt}^{2}[y] = m·d_{t}[ d_{t}[y] ] = m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it}+(1/k)·qg ] ] = ...
... m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ]+d_{t}[ (1/k)·qg ] ] = ...
... m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ]+0 ] = m·d_{t}[ d_{t}[ re^{(k/m)^{(1/2)}·it} ] = ...
... m·d_{tt}^{2}[ re^{(k/m)^{(1/2)}·it} ] = (-k)·re^{(k/m)^{(1/2)}·it} = ...
... (-k)·re^{(k/m)^{(1/2)}·it}+(-1)·qg+qg = (-k)·( re^{(k/m)^{(1/2)}·it}+(1/k)·qg )+qg = (-k)·y+qg
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+qg
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·qg
Deducción:
m·d_{tt}^{2}[y] = m·d_{t}[ d_{t}[y] ] = m·d_{t}[ ve^{(-1)·(b/m)·t}+(1/b)·qg ] = ...
... m·( d_{t}[ ve^{(-1)·(b/m)·t} ]+d_{t}[ (1/b)·qg ] ) = ...
... m·( d_{t}[ ve^{(-1)·(b/m)·t} ]+0 ) = m·d_{t}[ ve^{(-1)·(b/m)·t} ] = (-b)·ve^{(-1)·(b/m)·t} = ...
... (-b)·ve^{(-1)·(b/m)·t}+(-1)·qg+qg = (-b)·( ve^{(-1)·(b/m)·t}+(1/b)·qg )+qg = (-b)·d_{t}[y]+qg
Ley:
Sea I_{c}·(1/2)·d_{t}[w]^{2} = E ==>
Si d[I_{c}] = Mr^{2}·i·(k/m)^{(1/2)}·e^{(k/m)^{(1/2)}·it}·d[t] ==>
w(t) = ( (8/M)·(1/r)^{2}·E )^{(1/2)}·(m/k)^{(1/2)}·e^{(-1)·(1/2)·(k/m)^{(1/2)}·it}
Ley:
Sea I_{c}·(1/2)·d_{t}[w]^{2} = E ==>
Si d[I_{c}] = Mrve^{(-1)·(b/m)·t}·d[t] ==>
w(t) = ( (8/M)·(1/(rv))·E )^{(1/2)}·i·(m/b)^{(1/2)}·e^{(1/2)·(b/m)·t}
Obertura de hombros y caderas robótica:
Por amortiguador de retorno con empuje de obertura de fluido.
Ley:
m·d_{tt}^{2}[y] = (-k)·y+( sin(w)·qg+sin(s)·pg )
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·( sin(w)·qg+sin(s)·pg )
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+( sin(w)·qg+sin(s)·pg )
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·( sin(w)·qg+sin(s)·pg )
Estiramiento de rodillas y codos robótica:
Por amortiguador de retorno con empuje de obertura de fluido.
Ley:
m·d_{tt}^{2}[y] = (-k)·y+( F+(-1)·qg )
y(t) = re^{(k/m)^{(1/2)}·it}+(1/k)·( F+(-1)·qg )
Ley:
m·d_{tt}^{2}[y] = (-b)·d_{t}[y]+( F+(-1)·qg )
d_{t}[y] = ve^{(-1)·(b/m)·t}+(1/b)·( F+(-1)·qg )
Ley: [ de rezo al Mal ]
Los hombres no están atacando,
a los xtraterrestres.
Los xtraterrestres no están atacando,
a los hombres.
Ley:
Se matan entre ellos en su mundo.
Cometen adulterio entre ellos en su mundo.
Principio:
U(x,y,z) = Potencial[ Q(x,y,z) ]
U(yz,zx,xy) = Anti-Potencial[ Q(yz,zx,xy) ]
Principio:
div-exp[ U(x,y,z) ] = sum[k = 1]-[3][ d_{xyz}^{3}[ e^{U_{k}(x,y,z)} ]
Si div-exp[ U(x,y,z) ] = 0 ==>
div-exp[ U(x,y,z) ] = d_{xyz}^{3}[ e^{sum[k = 1]-[3][ U_{k}(x,y,z) ]} ]
Principio:
Anti-div-exp[ U(yz,zx,xy) ] = sum[k = 1]-[3][ d_{kij}^{2}[ e^{U_{k}(yz,zx,xy)} ] ]
Si Anti-div-exp[ U(yz,zx,xy) ] = 0 ==>
Anti-div-exp[ U(yz,zx,xy) ] = d_{kij}^{2}[ e^{sum[k = 1]-[3][ U_{k}(yz,zx,xy) ]} ]
Ley:
Si Q(x,y,z) = U·< (1/x),(1/y),(1/z) > ==>
U(x,y,z) = U·( ln(ax)+ln(ay)+ln(az) )
div-exp[ U(x,y,z) ] = Ua^{3}
F(z) = int-int[ div-exp[ U(x,y,z) ] ]d[x]d[x]+int-int[ div-exp[ U(x,y,z) ] ]d[y]d[y] = ...
... Ua^{3}·(1/2)·( x^{2}+y^{2} )
Ley:
Si Q(yz,zx,xy) = U·< (1/(yz)),(1/(zx)),(1/(xy)) > ==>
U(yz,zx,xy) = U·( ln(byz)+ln(bzx)+ln(bxy) )
Anti-div-exp[ U(yz,zx,xy) ] = Ub^{3}·4xyz
F(z) = int-int[ Anti-div-exp[ U(yz,zx,xy) ] ]d[x]d[y]+int-int[ Anti-div-exp[ U(yz,zx,xy) ] ]d[y]d[x] = ...
... Ub^{3}·2z·(xy)^{2}
Ley:
Si Q(x,y,z) = aU·< ((y+z)/x),((z+x)/y),((x+y)/z) > ==>
U(x,y,z) = U·( (ay+az)·ln(ax)+(az+ax)·ln(ay)+(ax+ay)·ln(az) )
div-exp[ U(x,y,z) ] = Ua^{3}·( ...
... (ax)^{ay+az+(-1)}·ln(ax)·( 2+(ay+az)·ln(ax) )+...
... (ay)^{az+ax+(-1)}·ln(ay)·( 2+(az+ax)·ln(ay) )+...
... (az)^{ax+ay+(-1)}·ln(az)·( 2+(ax+ay)·ln(az) ) )
Ley:
E(x_{k}) = int-int[ div-exp[ U_{k}(x,y,z) ] ]d[(1/a)^{2}·(i+j)] = ...
... U·(ak)^{ai+aj+(-1)}·[o(ai+aj)o] ( 2+(1/2)·(ai+aj)·ln(ak) )·(ai+aj)
x_{k}(t) = ...
... (1/a)·Anti-[ ( s /o(s)o/ ...
... int[ (as)^{ai+aj+(-1)}·[o(ai+aj)o] ( 2+(1/2)·(ai+aj)·ln(as) )·(ai+aj) ]d[s] )^{[o(s)o] (1/2)} ]-( ...
... ( (2/m)·U )^{(1/2)}·at )
Deducción:
d_{ax}[ (ax)^{ay+az} ] = (ay+az)·(ax)^{ay+az+(-1)}
d_{ay}[ (ay+az)·(ax)^{ay+az+(-1)} ] = (ax)^{ay+az+(-1)}·( 1+ay·ln(ax)+az·ln(ax) )
d_{az}[ (ax)^{ay+az+(-1)}·( 1+(ay+az)·ln(ax) ) ] = (ax)^{ay+az+(-1)}·ln(ax)·( 2+(ay+az)·ln(ax) )
Ley:
Si Q(yz,zx,xy) = bU·< ((zx+xy)/(yz)),((xy+yz)/(zx)),((yz+zx)/(xy)) > ==>
U(yz,zx,xy) = U·( (bzx+bxy)·ln(byz)+(bxy+byz)·ln(bzx)+(byz+bzx)·ln(bxy) )
Anti-div-exp[ U(yz,zx,xy) ] = Ub·( ...
... (byz)^{bzx+bxy+(-1)}·(bz+by)·( 1+ln(byz) )+...
... (bzx)^{bxy+byz+(-1)}·(bx+bz)·( 1+ln(bzx) )+...
... (bxy)^{byz+bzx+(-1)}·(by+bx)·( 1+ln(bxy) ) )
Ley:
E(x_{k}) = int[ Anti-div-exp[ U_{k}(yz,zx,xy) ] ]d[(1/b)·k] = U·(bij)^{bik+bkj+(-1)}·( ( 1/ln(bij) )+1 )
x_{k}(t) = (a/b)·Anti-[ ( s /o(s)o/ int[ (bij)^{ais+asj+(-1)}·( ( 1/ln(bij) )+1 ) ]d[s] )^{[o(s)o] (1/2)} ]-( ...
... ( (2/m)·U )^{(1/2)}·(b/a)·t )
Deducción:
d_{byz}[ (byz)^{bzx+bxy} ] = (bzx+bxy)·(byz)^{bzx+bxy+(-1)}
d_{x}[ (bzx+bxy)·(byz)^{bzx+bxy+(-1)} ] = (byz)^{bzx+bxy+(-1)}·(bz+by)·( 1+ln(byz) )
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