PROBLEM SHEET 2
1.
Experiments on an oven show that when full power is switched on the temperature rises
according to a first order differential equation with a time constant of 2 minutes to 100°C
above ambient. What will be the temperature of the oven above ambient after it has been
switched on for 1, 2, 4, 8 and 20 minutes?
2. A second order system has damping ratios ζ of
(a) 0.5
(b) 0.25.
What will be the % overshoot to a step input in both cases?
3.
The first overshoot for a unit step into a second order system has been shown, using the
standard notation, to be exp(-ζπ/(1-ζ2)1/2). Derive the relationship given in the first
laboratory sheet that
1/ 2
 2   %OS   2 
 π +  ln
  


100

  


where % OS is the percentage overshoot.
 %OS 
ζ = − ln

 100 
4. Find the Laplace transforms of the following functions:(a) δ(t-1)
(b) u(t-2)
(c ) e −5t
(d) sin2t
(e) cos(t-π/2)
(f) t u(t)
(g) t k u (t )
(h ) sint sin2t
(i) e-t sin2t
(j) t2u(t-a)
5. Find the inverse Laplace transforms of
(a) 2
(b)
3
s
(c)
4
2s + 7
(d) 3e −2 s
(e)
s + 13
s +s−6
(f)
s−3
s + 2s + 5
(g)
1
s ( s + 1) 2
(h)
1 − e−s
( s + 2) 2
2
2
1
(i)
2s + 1
s+2
(j)
5s + 13
s s + 4 s + 13
(
2
)
6. Find lim f(t) for F(s) equal to
t→0
(a) (s + 2)/s2
(b) (s + 1)(s + 2)/s(s + 3)
7. Find lim f(t) for F(s) equal to
t→∞
(a) 10(s + 2)/s2 + 25s + 4
(b) 10(s + 2)/s(s2 – 25s + 4)
8. Show from the defining integral of the Laplace transform that (e-atsin ω t ) is
ω /[(s+a)2 + ω2]. Confirm this result by using the fact that (sin ω t ) = ω/(s2+ω2) and the
complex shifting theorem.
1
s + 4s + 3
(Note: find the roots of s2 + 4s + 3 and then put F(s) into partial fractions.)
9. Use the universe Laplace transform to find f(t) if F ( s ) =
10. Find f(t) if F ( s ) =
2
s+2
by putting F(s) into partial fractions as in problem 9.
s + 4s + 3
2
F(s) may also be written as
s
2
+ 2
s + 4s + 3 s + 4s + 3
2
and s may be interpreted as differentiation. Use these facts and the result of problem 9 to
check your solution for f(t).
11. What will be the response of a system with transfer function 1/(s2+s+1) to a unit impulse
input?
12. Check the answer you have obtained in 11. by using the solution derived in class for the unit
step response of a second order system by differentiating the result. Note:- since the
derivative of a unit step is a unit impulse the derivative of the response of a system to a unit
step is its response to a unit impulse.
13. A system has a transfer function of 5/(s+1)(s+2). What will be its output for a unit step
input? Check the initial and final values of your solution using the initial and final value
theorems.
2
Answers:
5. (e) x(t ) = 2δ (t ) − 3e −2t u (t )
1
1
(f) x(t ) = u (t ) − (1 + j )e ( −2+ j 3) t u (t ) − (1 − j )e ( −2− j 3) t u (t )
2
2
3