Developing the Integration sequence to the Duhamel Integral.

1. What is Duhamel’s Integral?
(whimsical)

2. • Considering the interstitial binding of
earthquake regions:
• If there is a simultaneous existence of earthquake
activity, it becomes apparent that these are related in a
greater realm as to velocity spectrum as velocity is the
achievement of a uniform displacement matter over a
time threshold.
• The velocity spectrum will be conceived if within the
sequence of earthquake events there is a similitude in
damage potential/aaclimated.
• If the damage potential/acclimation is differentiable,
the determination of acceleration related or
displacement spectra could be addressed.
ˉ 2016 Seismicity 8 – 9th Month

3. M@
--->
---------
| |
| |
---------
| |
| |
---------
| |
| |
_______ +-------+ _______
}}}} / {{{{
/
love @@@@@@@@@@@@@@@@@@@@@@@@@@@
p ->> ->> ->> ->> ->> ->> ->> ->>
s %> %> %> %> %> %> %> %> %>
Example is a frame
with inclined
foundation piling at
ring beam. 2D.

4. single degree of
freedom.
harmonic
motion. free
vibration.
(equation of
motion)
(displace
ment)
u = Asin wt +
Bcos wt + C
-mũ = ku + củ -
> mũ + củ + ku
= 0
(velocity)
ủ = -Aw cos wt
+ Bw sin wt
Forced
Vibration
mũ + củ + ku =
pú(t)
(acceler
ation)
ũ = -Aw² sin wt -
Bw² cos wt
Linear Elastic
Systems
forms of
Forced
Vibration
pú(t) = ?
singularity type (also
short duration
uniform)
k = 24 EIc /h³
elastic torque or
torsional moment
short duration (linear)
graphed form (ie. seismic wave)
harmonic

5. infinitesimally short
impulses.
u(t) = 1/(m wn)§ p(tau) sin [ wn * (t - tau) ] d|tau|
wn = ’/´k/m ?
undamped
SDF System.
using Duhamel's integral, determine SDF system initially at rest.
pú(t) = for t >= 0 , pồ
convolution
integral : u(t) = pồ / (m wn) * §(0..t)
Frequency
Domain :
P(w) = FF [p(t)] = §(*-8..*8)[ pú(t) * e ^(-iwt)
]dt
complex frequency - response function. H(w)
describes response of system to harmonic
excitation. U(w) = H(w)*P(w)
Hú(w) motion to be generally varied with time
may include harmonic components for wide range of frequencies.
Measurement
of
Acceleration
u(t) = -(1/wn²)Rd ũg( t - phi/w )
Recorded u(t) is base acceleration modified according to {-Rd/ wn²}
at time lag: {phi/w}.
make Rd and phi/w independent of frequency of excitation. then ea. {h} component will
record with modifying factor and time lag.
if the motion to be recorded consists of h components :
(h = harmonic ¿large?)
Stage the
records of time.

6. Measurement
of
Acceleration
u(t) = -(1/wn²)Rd ũg( t - phi/w )
Recorded u(t) is base acceleration modified according to {-Rd/ wn²}
at time lag: {phi/w}.
make Rd and phi/w independent of frequency of excitation. then ea. {h} component will
record with modifying factor and time lag.
if the motion to be recorded consists of h components :
(h = harmonic ¿large?)
+____________+______________+___________+_______
_______+__+
closed-form results can be obtained
only if U(w) = H(w)*P(w) ||
p(t) is a simple function. app of F[] ie. §(*-8..*8)[ pú(t) * e ^(-iwt)
]dt
contour
integration in
complex plane
u(t) = 1/(2pi)§(*-
8..*8)H(w)*P(w)*e^(iwt)
dw
e^(îwt)
= cos(wt) + i*sin(wt)
Fourier transform is
feasible for dynamic
analysis.
Fourier transform is feasible for dynamic analysis of linear
systems to complicated excitations p(t) or ũg(t)
integralsevaluated via fFT
displacement u(t) due to external force p(t) =
pồ
Stage the
records of time..
Initiating the fFT..

7. motion to be generally varied with time, may include harmonic components for wide range of frequencies.
Measurement of Acceleration
u(t) = -(1/wn²)Rd ũg( t - phi/w )
recorded u(t) is base acceleration modified according to {-Rd/ wn²}
at time lag:
{phi/w}.
make Rd and phi/w independent of frequency of excitation.
then ea. {h} component will record with modifying factor and time lag.
if the motion to be recorded consists of h components
(h = harmonic ¿large?)
+____________+______________+___________+______________+__+
U(w) = H(w)*P(w) || closed-form results can be obtained only if
p(t) is a simple function. app of F[]
ie §(*-8..*8)[ pú(t) * e ^(-iwt) ]dt
contour integration in complex plane
u(t) = 1/(2pi)§(*-8..*8)H(w)*P(w)*e^(iwt)dw
e^(îwt) = cos(wt) + isin(wt)
Fourier transform is feasible for dynamic analysis.
Fourier transform is feasible for dynamic analysis of linear systems to
complicated excitations p(t) or ũg(t)
integrals evaluated via fFT
displacement u(t) due to external force p(t) = pồ
H(w) = e^(îwt)
sin x = 1/2i * {eix - e(-ix)}
cos x = 1/2 * {eix + e(-ix)} , x = j*wồ*t
sum of two terms expressed as:
Pj*e^î(j*wồ*t) + P-j*e^-î(j*wồ*t)
u(t) = €(*-8..*8) H(j*wồ*t) Pj e^î(j*wồ*t)
where Fourier coefficients Pj are defined
Pj = 1/Tồ §(0..Tồ)pú(t)
complex frequency-response function
describes SS response to force, harmonic force of unit amplitude.
H
|H(w)|/(Ust)ồ = inv[
(Ust)ồ is the static initial
displacement
Real and Imaginary Parts of Hú(w)
-Im[Hú(w)]/Re[Hú(w)] = 2£(w/wn)/(1-(w/wn)²)
u(t) = u¿(t) + iu¡(t) ¿=real part solution
¡=imaginary part solution
SDF System with Rate-Independent Damping
mü + nu*k/w * ú + ku = p(t)
mü + nu*k/w * ú + ku = p(t)

8. motion to be generally varied with time, may include harmonic components for wide
range of frequencies.
Measurement of Acceleration
u(t) = -(1/wn²)Rd ũg( t - phi/w )
recorded u(t) is base acceleration modified according to {-Rd/ wn²}
at time lag:
{phi/w}.
make Rd and phi/w independent of frequency of excitation.
then ea. {h} component will record with modifying factor and time lag.
if the motion to be recorded consists of h components
(h = harmonic ¿large?)
+____________+______________+___________+______________+__+
U(w) = H(w)*P(w) || closed-form results can be obtained only if
p(t) is a simple function. app of F[]
ie §(*-8..*8)[ pú(t) * e ^(-iwt) ]dt
Repetition along
Fourier Construct
Constraints to
p(t).

9. contour integration in complex plane
u(t) = 1/(2pi)§(*-8..*8)H(w)*P(w)*e^(iwt)dw
e^(îwt) = cos(wt) + isin(wt)
Fourier transform is feasible for dynamic analysis.
Fourier transform is feasible for dynamic analysis of linear systems to
complicated excitations p(t) or ũg(t)
integrals evaluated via fFT
displacement u(t) due to external force p(t) = pồ
H(w) = e^(îwt)
sin x = 1/2i * {eix - e(-ix)}
cos x = 1/2 * {eix + e(-ix)} , x = j*wồ*t
sum of two terms expressed as:
Pj*e^î(j*wồ*t) + P-j*e^-î(j*wồ*t)
u(t) = €(*-8..*8) H(j*wồ*t) Pj e^î(j*wồ*t)
where Fourier coefficients Pj are defined
Pj = 1/Tồ §(0..Tồ)pú(t)
Expression of
Duhamel

10. Pj = 1/Tồ §(0..Tồ)pú(t)
complex frequency-response function
describes SS response to force, harmonic force of unit amplitude.
H
|H(w)|/(Ust)ồ = inv[
(Ust)ồ is the static initial displacement
Real and Imaginary Parts of Hú(w)
-Im[Hú(w)]/Re[Hú(w)] = 2£(w/wn)/(1-(w/wn)²)
u(t) = u¿(t) + iu¡(t) ¿=real part solution
¡=imaginary part solution
SDF System with Rate-Independent Damping
mü + nu*k/w * ú + ku = p(t)
mü + nu*k/w * ú + ku = p(t)
The Real and
Imaginary
Solution
Rate-
Independent
Damping

11. Response of linear system,
Steady-State Response
the response of a linear system to the periodic force may be
determined with combining responses to determine individual
responses.
S(*-8..*8)
pj(t) = Pj * ei(jwồt)
w
jth term in Fourier
series
replacing w by jwo
u(t)
= Uj
H(w)=inv { 1-(w/wn)^2) + 2£(w/wn) }
substitute w = jwo
H(jwo
)
inv {k*[ 1-(j*wo/wn)^2
+ î2£(j*wo/wn) ]}
§3.1
3
Response to Periodic
Force :: we are interested in finding the SS response.
Periodic Excitation
uc
(t) + îus
(t)
A Step Force uc(t) =
( (1-w/wn)^2
*cos(w*t) + (2*£*(w/wn))*sin(w*t)
)*H(w)
§4.3
u(t) = (ust)o*(1-cos(wnt)) (ust)o*(1-cos(2pt/Tn))
us(t)
=
( (1-w/wn)^2
*sin(w*t) - (2*£*(w/wn))*cos(w*t)
)*H(w)
(ust)o= po/k
, static deformation due to force
po
|Hu(w)| / (Ust)o
complex frequency-response
function
wnto
= j * PI() com (( (1-w/wn)^2
)^2
- (2*£*(w/wn))^2
) )^0.5
(Ust)o = po / k
u(t)
=
p(t) varying arbitrarily with time, shall be represented as a sequence of infinitesimally short
impulses.
the response of a linear dynamic system to one of these impulses.
du(t) = [p(t)*dt]*h(t-t) t > t
p(t)
is a simple
function. due to superposition:
response at time t §[0..t] [p(t)*h(t-t)*dt] unit impulse response function Duhamel's integration method may not apply to
plastic zone deformation.(1/mw ) §[0..t] p(t) e-£wn(t-t)sin[w (t-t)]dt
Response
Superimposition
Steady State
Response
formulation