00:01
So here we have to solve the differential equation of the motion of the damped harmonic oscillator which is given as f external of t that is equals to f naught multiplied by the e raised to the power minus alpha t cos of omega t.
00:15
So this equation is given.
00:16
So here we have to express the motion for the damped harmonic equation that is equals to mx double naught plus c of x double naught plus k of x that is equals to f naught e raised to the power minus of alpha t cos of omega t.
00:30
Here we are having f naught e raised to the power minus alpha t cos of omega t which is a external damped harmonic force.
00:40
So this is a external damped harmonic force where mass is mc is the damping coefficient k is the stiffness of the system.
00:51
So f external of t become equals to f naught e raised to the power minus of alpha t multiplied by the cos of omega t.
00:59
So this one here is equals to f naught multiplied by the e raised to the power minus alpha t r of e multiplied e raised to the power iota of omega t.
01:08
So this from here equals to f naught r e multiplied by the e raised to the power alpha t plus iota of omega t.
01:15
So here we are assuming that is minus alpha plus iota omega t is equals to beta.
01:21
So f of external multiplied by the t become equals to f naught r e e raised to the power beta t.
01:29
So this is the value from here.
01:31
Now we are considering about the equation of motion for the damped harmonic here.
01:36
So x mx double dash plus c of x double dash plus kx that is equals to f naught r e raised to the power e raised to the power beta t.
01:45
So assuming the here.
01:47
So x of t from here is equals to a raised to the power beta of t minus iota of phi.
01:53
Phi is the phase angle here so m multiplied by the d square a raised to the power of beta of t minus iota of phi which is divided by the dt plus cd r raised to the power beta t minus iota of phi which is divided by the dt.
02:08
Plus k multiplied by the a e raised to the power beta of t minus iota phi that is equals to f0 re e raised to the beta of t...