1. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Structural Dynamics
& Earthquake Engineering
Lectures #6 and 7: State-space equation of motion
and Transition matrix for SDoF oscillators
Dr Alessandro Palmeri
Civil and Building Engineering @ Loughborough University
Tuesday, 25th February 2014

2. State-Space Formulation
Structural
Dynamics
& Earthquake
Engineering
The equation of motion for a SDoF oscillator reads:
2
¨
˙
u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) =
Dr Alessandro
Palmeri
1
f (t)
m
(1)
and can be posed in the alternative matrix form:
˙
y(t) = A · y(t) + b f (t)
(2)
where y(t) is the array of the state variables (displacement
and velocity) for the oscillator:
y(t) =
u(t)
˙
u(t)
(3)
while:
A=
0
1
2 −2 ζ ω
−ω0
0 0
, b=
0
1/m
(4)

3. Duhamel’s Solution
Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Let us consider a scalar first-order inhomogeneous ODE:
˙
y (t) = A y(t) + b f (t)
(5)
The integral solution of Eq. (5) can be formally written as:
t
Θ(t − τ ) b f (τ ) dτ
y(t) = Θ(t) y(0) +
(6)
0
where y(0) is the initial condition at time t = 0, while the
transition function Θ(t) is so defined:
Θ(t) = eA t
(7)

4. Duhamel’s Solution
Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
This integral solution can be extended to systems with many
state variables as:
t
y(t) = Θ(t) · y(0) +
Θ(t − τ ) · b f (τ ) dτ
(8)
0
where the array y0 = y(0) collects the initial conditions at
time t = 0, and the transition matrix Θ(t) is evaluated as the
exponential matrix of [A t]:
Θ(t) = eA t
(9)

5. Step-by-Step Numerical Solution
Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
For t = ∆t, and assuming a linear variation of the forcing
term f (t) in the time interval [0, ∆t]:
f (t) = f0 +
f1 − f0
t
∆t
(10)
one can mathematically prove that the Duhamel’s integral
gives:
y1 = Θ(∆t) · y0 + Γ0 (∆t) · {b f0 } + Γ1 (∆t) · {b f1 }
(11)
where y1 = y(∆t) collects the state variables at t = ∆t,
while the integration matrices Γ0 (∆t) and Γ1 (∆t) can be
computed from the transition matrix Θ(∆t) and the matrix of
coefficients A.

6. Step-by-Step Numerical Solution
Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
That is:
Γ0 (∆t) = [Θ(∆t) − L(∆t)] · A−1
(12)
Γ1 (∆t) = [L(∆t) − I2 ] · A−1
(13)
in which I2 is the 2-dimensional identity matrix, while the
loading matrix L(∆t) is given by:
L(∆t) =
1
[Θ(∆t) − I2 ] · A−1
∆t
(14)

7. Step-by-Step Numerical Solution
Structural
Dynamics
& Earthquake
Engineering
Moreover:
Dr Alessandro
Palmeri

Θ(∆t) = e−ζ0 ω0 ∆t 
ζ0 ω0
ω0
2
ω0
− ω0 S
C+
S
C
1
ω0 S
0
− ζωω0
0

S
 (15)
in which C = cos(ω 0 ∆t), S = sin(ω 0 ∆t) and
ω0 =
2
1 − ζ0 ω0 , while:
A−1 =
− 2 ζ00
ω
1
− ω2
1
0
0
(16)

8. Step-by-Step Numerical Solution
Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
The incremental solution offered by Eq. (11) for the time
interval [0, ∆t] can be extended to a generic time instant
tn = n ∆t as:
yn+1 = y(tn+1 ) =Θ(∆t) · yn
+ Γ0 (∆t) · {b f (tn )}
+ Γ1 (∆t) · {b f (tn+1) }
for n = 1, 2, 3, · · ·
(17)