1. Response of linear SDOF systems to general
loading – use of superposition.
Mass-Spring Damper System
y is the displacement, in the downward direction, as
a result of the force p(t) acting on m
p(t)
k c
m
y

2. An approach to constructing the
general solution for arbitrary forcing

3. Linear systems satisfy the principle of superposition
(in the Time and Frequency Domains)
?

4. Two special free-decay response functions
The Unit Amplitude Free-Decay Function

5. So what does the UAFD function look like?

6. Two special free-decay response functions The
Unit Velocity Free-Decay Function

7. So what does the UVFD function look like?

8. The free response to any set of initial
conditions (from superposition) is:
( ) ( ) ( )
y y y t y y t
cf UAFD UVFD
=  + 
0 0
 ( )

9. The Impulse Response function h(t)

10. Impulse response
Mass-Spring Damper System
Consider a unit impulse h(t) applied to a mass-
spring-damper system, with light (subcritical)
damping parameter ξ (i.e. ξ << 1) and natural
frequency ωn . How would it respond?
h(t)
2ξωn
y

11. The Impulse Response function (irf) h(t)
(identical to the UVFD function)

12. To recap
Two functions have been defined: the Unit
Amplitude Free Decay Function:
𝑦(𝑡) = 𝑦UAFD(𝑡)
and the Impulse Response Function h(t) (which
is identical to the Unit Velocity Free Decay
Function):
𝑦 𝑡 = h t = 𝑦UVFD 𝑡
We will see in the next lecture that only these two
functions are needed to generate the general solution
to the forced response of a SDOF system.