Answer:
a) The system is under-damped
b) The steady state response, Â [tex]x_{p} = \frac{8 sin(5t)}{15}[/tex]
c) The transient response, [tex]x_{h} = \frac{ie^{(-3-4i)t}(-1+e^{8it} }{3}[/tex]
d) The total response, [tex]x(t) = \frac{8 sin(5t)}{15} +\frac{ie^{(-3-4i)t}(-1+e^{8it} ) }{3}[/tex]
x(t) was plotted on matlab
Explanation:
a) The equation is : [tex]4\frac{d^{2}x }{dt} + 24\frac{dx}{dt} + 100x = 16 cos 5t[/tex]............(1)
The general equation of a spring-mass-damper system is given by:
[tex]\frac{d^{2} x}{dt^{2} } + 2 \zeta w_{n} \frac{dx}{dt} + w_{n} ^{2} x = 0[/tex]....................(2)
Comparing equation (1) with equation (2)
[tex]w_{n} ^{2} = 100/4\\w_{n} = 52 \zeta w_{n} = 6\\2 * 5 * \zeta = 6\\\zeta = 0.6[/tex]
Since [tex]\zeta < 1[/tex], the system is under-damped
b) Take the inverse laplace transform of equation (1)
[tex](4s^{2} + 24s + 100) X(s) = \frac{16s}{s^{2} + 25 }[/tex]
[tex](4s^{2} + 24s + 100)(s^{2} + 25) X(s) = 16s\\X(s) = \frac{16s}{(4s^{2} + 24s + 100)(s^{2} + 25)}[/tex]
Taking the Inverse laplace transform of X(s)
[tex]x(t) = \frac{16 sin(5t)}{30} + 16 \frac{ie^{(-3-4i)t}(-1+e^{8it} }{48}[/tex]
The steady state response is:
[tex]x_{p} = \frac{16 sin(5t)}{30}[/tex]
c) The transient response is:
[tex]x_{h} = \frac{16ie^{(-3-4i)t}(-1+e^{8it} ) }{48}\\x_{h} = \frac{ie^{(-3-4i)t}(-1+e^{8it} }{3}[/tex]
d) The total response:
[tex]x(t) = x_{h} + x_{p} \\[/tex]
[tex]x(t) = \frac{8 sin(5t)}{15} +\frac{ie^{(-3-4i)t}(-1+e^{8it} ) }{3}[/tex]
