See attachment
Question:
Consider the equation for free mechanical vibration, my'' + by' + ky=0, and assume the motion is overdamped. Suppose y(0) > 0 and y'(0) > 0. Prove that the mass will never mass throught its equilibrium at any positive time.
Really stuck on this question, all i know is for over damped:
b^2 > 4mk
General solution: y(t)=C1e^(r1t) + C2e^(r2t) --> y(0) = C1 + C2 > 0
y'(t)=C1r1e^(r1t) + C2r2e^(r2t) --> y'(0) = C1r1 + C2r2 > 0
Any help with this question would be really appreciated thanks.
A + B >0 comes from y(0) > 0
A - B >0 comes from y ' (0) > 0
So A and B can't both be negative further A > B so A > 0
A > -B
1> -B/A
I then set y equal to 0
e^(-bt/2m) > 0
so if y = 0 then Ae^(rt) +Be^(-rt) = 0
multiply by e^rt
A e^(2rt) + B = 0
e^(2rt) = -B/A
take logs
2rt = ln(-B/A) if B s also postive no solutions.
if B < 0 -B/A < 1 from above and ln(x) < 0 if x < 1
hope this clears things up