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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
An approach to constructing the
general solution for arbitrary forcing
Linear systems satisfy the principle of superposition
(in the Time and Frequency Domains)
?
Two special free-decay response functions
The Unit Amplitude Free-Decay Function
So what does the UAFD function look like?
Two special free-decay response functions The
Unit Velocity Free-Decay Function
So what does the UVFD function look like?
The free response to any set of initial
conditions (from superposition) is:
( ) ( ) ( )
y y y t y y t
cf UAFD UVFD
=  + 
0 0
 ( )
The Impulse Response function h(t)
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
The Impulse Response function (irf) h(t)
(identical to the UVFD function)
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.

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DOMV No 2 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (2).pdf

  • 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.