1) The document discusses the state-space formulation and numerical solution of the equation of motion for a single-degree-of-freedom oscillator subjected to earthquake ground motion. 2) It presents the state-space representation using state variables of displacement and velocity, and defines the transition matrix used to solve the equation of motion incrementally over time. 3) The solution is obtained by dividing the time interval into steps of size Δt, and computing the state variables at each step using the transition matrix and integration matrices dependent on the transition matrix.

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)