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 )
