Forced and Unforced systems Flashcards
Unforced systems ODE
linear homogeneous ODEs with constant coefficients that can be solved using the exponential trial solution method: y= Ce^(lamba*t)
Repeated root in general solution
need to multiply one part of solution by t due to repeated real roots
e.g y= C1e^3t + C2e^3t
y’ =3C1e^3t +C2e^3t + 3C2e^3t
Order
highest power or derivative in equation
Linearity
whether dependent variable has unusual non linear functions, such as y^2, or siny or 1/y
Homogeneity
Take all of the terms in the equation containing dependent variable to LHS, if RHS = 0 , then equation is homogeneous, If there are terms involving t (or constants) remaining on the RHS the equation is nonhomogeneous
Car suspension system
Overdamping: After being released from the initial disturbance, the suspension system is slow to return to the undisturbed state. No oscillation.
Critical damping: After being released from the initial disturbance, has large displacement initially, but isquick to return to the undisturbed state. No oscillation.
Underdamping: Oscillates around an equilibrium point slowly returning to the undisturbed state.
No Damping: The car will keep oscillating around the equilibrium point, at the same magnitude forever.
Car suspension system table
Overdamping c>4mk two real roots
Critical damping c= 4mk two equal roots
Underdamping c<4mk two complex roots
No damping c = 0 two purely imaginary roots