Can anybody help?
A system is modelled by Tx' + x = Ky
given the following results deduce the parameters T and K
. When the input y(t) of 5 units was applied the output x(t) ultimately settled with a value of 20 units
. when a sinusoidal input was applied at an input frequency of 10 rad/s the output lagged behind the input by exactly -45 degrees.
ii) if y(t) was a step change drive, at what time would x(t) reach half of its final steady state value ( T & K deduced previously)
You substitute the solutions given into the DE's and solve for what you don't know:
Sub x = 20 into the first:
20 = 5K => K = 4.
Sub K = 4 into the second:
Now sub (expand using compound angle formula) into the DE. Equate coefficients of sin and cos to get two simultaneous equations in A and T. Solve those equations to get the value of A and T.
My knowledge of the compound angle formula is a bit rusty. So i've cheated and used my calculator. This has returned a very long expansion.
A(sqrt2/2 sin(t) - sqrt2/2 Cos(t)) x {512(sqrt2/2 cos(t) + sqrt2/2 sin(t))^9 - 1024(sqrt2/2 Cos(t) + sqrt2/2 sin(t))^7 + 672(sqrt2/2 Cos(t) + sqrt2/2 sin(t))^5 - 160(sqrt2/2 Cos(t) + sqrt2/2 sin(t))^3 + 10(sqrt2/2 Cos(t) + sqrt2/2 sin(t))}
this is not the smallest expresion to put into a DE. is my calculator right? i suspect it is! as i've tested it out on some smaller equations, or is there an easier method of solving the original problem.
Actually I was anticipating you'd do something like the following:
.
You could also use symmetry and the complementary angle formula to get the last line.
Substituting a solution where the argument of the trig function is 10t is pretty essential since the term on the right hand side is a a trig with an argument of 10t. Hard to justify equating coefficients of sin and cos on each side if they are sining and cosing different arguments ......