# Describe the force of disconector, dumper, (C-element), c - dissipative property and this formula describes velocity with respect to c, dissipative(drag, ressistant) force - # force which acts in opposite direction with respect to the main force.# If the velocity is higher the higher drag(dumper) force# spring tries to keep system equivalent,# dumper just transfers the energyEq(F_c,c*x(t).diff(t))
# As x might represent the distance, to find the speed we need just to find derivative, Formula for kinetic energy, laborT=S.One/2*m*x(t).diff(t)**2V=Integral(k*x(t),x(t))T
# https://en.wikipedia.org/wiki/Work_(physics)#formula for potential energyV.doit()
# wrong approach - just for explanationW=F_k*x(t)#application of Hooke's lawW.subs(eq_hook.lhs,eq_hook.rhs)
# amount of motion is kinetic energy, capacity of energy is the potential energy.
v=x(t).diff(t)V=S.One/2*k*x(t)**2+S.One/2*k*(-1*sin(Omg*t)+x(t))**2#Solution using the lagrangian mechanics application, difference between kinetic energy and potential energy solution for the out with the first picture(out 8), picture with single degreeL=(T-V).doit()#dissipative potentialD=S.One/2*c*v**2Eq(L.diff(v).diff(t)-L.diff(x(t))+D.diff(v),0).expand().doit()
# in this case u is the displacement, in general it is given by the data, and aslo it can be understood from the case for the lettereom_u0=Eq(m*x(t).diff(t,2),-x(t)*k-c*x(t).diff(t)+k*(1*sin(Omg*t)-x(t)))eom_u0.expand()