1. In considering the general response y(t) of a SDOF system to
arbitrary forcing, we have previously defined three special functions:
The Unit Amplitude Free Decay Function 𝑦UAFD 𝑡
The Unit Velocity Free Decay Function 𝑦UVFD 𝑡 , and
The Impulse Response Function h(t)
But h(t) is identical to yUVFD t so really we only have two unique functions.
We will see in this lecture that only these functions are needed to
generate the general response y(t) of a SDOF system excited by
arbitrary forcing from any set of initial conditions.
The general solution of a SDOF
system to arbitrary forcing

2. Reminder of what a unit impulse is

3. A reminder of what the Impulse Response
function h(t) typically looks like

4. Four Scenarios involving the
Impulse response function h(t)
To build-up the solution to general forcing, I am going
to consider four scenarios i) – iv) involving the
Impulse response function.
Note: Time-delayed ‘firing’ of the impulse function
When h(t) is an impulse response function, and we
consider h(T-t), then the ‘spike’ of the impulse occurs
when the argument of h(t) is zero, i.e. when T-t=0 or
t=T. This is effectively a time-delay of magnitude T.
h(T-t) is therefore an impulse that is delayed until t=T
rather than occurring at t=0.

5. Scenario i) involving the Impulse
response function h(t)

6. Scenario ii) involving the Impulse
response function h(t)

7. Four Scenarios involving the Impulse
respone function - Scenario iii):
Scenario iii) involving the Impulse
response function h(t)

8. Scenario iv) involving the Impulse
response function h(t)

9. The General time domain solution for
arbitrary excitation

10. Frequency Domain Solution
Method

11. Relationship between the Impulse Response
Function h(t) and the Frequency Response
Function H(jω)
Note: for nonzero initial conditions we have to add the complementary solution

12. Some general comments about
Time and Frequency Domain
analysis of Linear Systems
We can generate the response of a SDOF system to
any input function.
Inverse Fourier Transform is usually difficult analytically
since requires contour integration in the complex plane.
Practical approach involves use of numerical discrete
Fourier Transform algorithms i.e. the FFT (Fast Fourier
Transform) to convert input to frequency domain, and
the use of the IFFT (Fast Inverse Fourier Transform) to
obtain the response Y(t).
Time and frequency domain approaches can be
extended to MDOF systems, in fact, for general types of
damping, frequency domain methods are usually used.