1) The document discusses single degree of freedom (SDOF) systems undergoing forced, damped vibrations, with a focus on resonance and dynamic amplification.
2) It presents equations to describe the total, transient, and steady state responses of damped SDOF systems subjected to harmonic forcing inputs.
3) At resonance, when the forcing frequency matches the natural frequency of the system, the response is amplified. This amplification is quantified by the resonance response factor.
2. 2
SDOF: Forced, Undamped & Damped
( ) ( )
t
Sin
p
t
p F
0
= frequency
forcing
F =
⚫ Damped
Response
⚫ Transient Response = difference between
Total Response and
Steady State Response
⚫ Undamped Response
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
+
−
−
+
+
−
−
+
+
+
= −
t
Cos
r
r
r
t
Sin
r
r
r
u
t
Sin
u
u
t
Cos
u
e
t
u
is
solution
TOTAL
The
F
F
Static
D
D
N
D
t
N
2
2
2
2
2
2
2
0
0
0
2
1
2
2
1
1
:
⚫ Transient Response decays
3. 3
SDOF: Forced, damped, Resonance
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) ( )
( ) ( )
t
Cos
u
t
u
t
t
Cos
u
t
u
Since
t
Cos
u
e
t
u
t
Cos
t
Sin
u
t
Sin
e
t
u
u
u
conditions
rest
at
initially
For
t
Cos
t
Sin
u
t
Sin
u
u
t
Cos
u
e
t
u
r
thus
sonance
At
t
Cos
r
r
r
t
Sin
r
r
r
u
t
Sin
u
u
t
Cos
u
e
t
u
is
solution
TOTAL
The
N
Static
F
N
Static
N
F
F
Static
t
F
F
Static
D
D
t
F
F
Static
D
D
N
D
t
N
F
N
F
F
F
Static
D
D
N
D
t
N
N
N
N
2
1
2
:
2
0
2
0
1
1
2
0
0
0
0
0
0
:
2
0
1
1
2
2
0
1
1
0
1
1
1
:
Re
2
1
2
2
1
1
:
2
2
0
0
2
2
2
2
0
0
0
2
2
2
2
2
2
2
0
0
0
−
=
=
−
=
=
−
+
=
+
=
−
−
+
+
+
+
=
=
=
+
=
−
−
+
+
=
−
=
−
+
+
+
=
=
=
=
+
−
−
+
+
−
−
+
+
+
=
−
−
−
−
Resonance time
history
N
F
=
Resonance
4. 4
SDOF: Forced, damped, Dynamic Amplification
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
−
=
−
+
=
+
−
=
+
−
−
=
−
=
+
−
=
+
−
−
+
+
−
−
+
−
=
−
+
−
+
−
=
+
−
−
+
+
−
−
+
=
+
−
−
+
+
−
−
+
+
+
=
−
−
−
t
Sin
R
u
t
u
t
Cos
Sin
t
Sin
Cos
R
u
t
u
r
r
r
Sin
r
r
r
Cos
r
r
Tan
Angle
Lag
Phase
Define
r
r
R
Factor
ion
Amplificat
sponse
nt
Displaceme
Dynamic
Define
t
Cos
r
r
r
t
Sin
r
r
r
r
r
u
t
u
t
Cos
r
t
Sin
r
r
r
u
t
u
t
Cos
r
r
r
t
Sin
r
r
r
u
small
very
t
u
small
becomes
e
time
with
decays
solution
Transient
t
Cos
r
r
r
t
Sin
r
r
r
u
t
Sin
u
u
t
Cos
u
e
t
u
for
solution
d
generalize
is
solution
TOTAL
The
F
d
Static
F
F
d
Static
d
F
F
Static
F
F
Static
F
F
Static
t
F
F
Static
D
D
N
D
t
N
F
N
N
2
2
2
2
2
2
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
2
1
2
2
1
1
1
2
:
:
2
1
1
:
Re
:
2
1
2
2
1
1
2
1
2
1
2
1
2
1
2
2
1
1
:
2
1
2
2
1
1
:
( )
r
2
( )
2
1 r
−
( ) ( )2
2
2
2
1 r
r
+
−
5. 5
SDOF: Forced, damped, Dynamic Amplification
( )
( ) ( )
( ) ( )
sponse
nt
Displaceme
Static
Max
sponse
nt
Displaceme
Dynamic
State
Steady
R
Factor
ion
Amplificat
sponse
nt
Displaceme
Dynamic
r
r
R
Angle
Lag
Phase
r
r
Tan
t
Sin
R
u
t
u
for
is
solution
State
Steady
Final
The
d
d
F
d
Static
N
F
Re
Re
Re
2
1
1
:
1
2
:
2
2
2
2
1
=
=
+
−
=
=
−
=
−
=
−
⚫ Rd depends on:
⚫ Forcing Function:
⚫ System natural characteristics:
( )
r
2
( )
2
1 r
−
( ) ( )2
2
2
2
1 r
r
+
−
Damping
Frequency
Natural
m
k
N
=
=
=
Frequency
Forcing
F =
6. 6
SDOF: Forced, damped, Dynamic Amplification
( ) ( )
( ) ( )
sponse
nt
Displaceme
Static
Max
sponse
nt
Displaceme
Dynamic
R
Factor
ion
Amplificat
sponse
Dynamic
r
r
R
Angle
Lag
Phase
r
r
Tan
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
d
d
F
d
Static
Re
Re
Re
2
1
1
:
1
2
:
2
2
2
2
1
=
=
+
−
=
=
−
=
−
=
−
⚫ Max Static response
occurs a same time as
max forcing function
⚫ Dynamic response lags
by
2
7. 7
SDOF: Forced, damped, Dynamic Amplification, r<<1
( ) ( )
( ) ( )
( ) ( )
k
p
k
p
u
u
ku
p
ku
F
R
r
For
r
r
R
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
Static
Static
d
d
F
d
Static
0
0
max
0
2
2
2
1
1
1
1
1
2
1
1
:
=
=
=
=
=
+
−
=
−
=
⚫ At r << 1
⚫ Forcing Frequency << Natural Frequency
⚫ Rd ≤1
⚫ Little or no amplification of
response
⚫ Response controlled by
stiffness.
Ratio
Frequency
r
N
F
=
=
Rd
=
Dynamic
Displacement
Response
Amplification
Factor
N
F
Ratio
Frequency
r F
=
=
8. 8
SDOF: Forced, damped, Dynamic Amplification, r>>1
( ) ( )
( ) ( )
( )
2
0
2
0
2
max
0
2
2
2
2
2
2
2
2
1
1
1
2
1
1
:
F
F
F
Static
Static
F
F
N
N
F
d
d
F
d
Static
m
p
m
k
k
p
m
k
u
u
ku
p
ku
F
m
k
r
R
r
For
r
r
R
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
=
=
=
=
=
=
=
=
+
−
=
−
=
⚫ At r >> 1
⚫ Forcing Frequency >> Natural Frequency
⚫ Rd <<1
⚫ Response de-amplification.
⚫ Response approaches zero.
⚫ Response controlled by mass.
Ratio
Frequency
r
N
F
=
=
Rd
=
Dynamic
Displacement
Response
Amplification
Factor
N
F
Ratio
Frequency
r F
=
=
9. 9
SDOF: Forced, damped, Dynamic Amplification, r ≈ 1
( ) ( )
( ) ( )
( ) ( )
( )
N
N
N
Static
Static
d
d
F
d
Static
c
p
u
m
c
m
p
r
k
p
r
u
u
ku
p
ku
F
r
r
R
r
For
r
r
R
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
0
max
2
0
0
max
0
2
2
2
2
2
2
2
1
1
2
1
2
1
2
1
2
0
1
1
2
1
1
:
=
=
=
=
=
=
=
=
=
+
+
−
=
−
=
⚫ At r ≈ 1, Resonance
⚫ Forcing Frequency ≈ Natural Frequency
⚫ Rd = Very Large
⚫ Large Response amplification.
⚫ Response controlled by damping.
Ratio
Frequency
r
N
F
=
=
Rd
=
Dynamic
Displacement
Response
Amplification
Factor
Ratio
Frequency
r F
=
=
N
F
10. 10
SDOF: Forced, damped, Dynamic Amplification, phase angle
⚫ Phase Angle
⚫ r<<1, ≈ 0, response in
phase with forcing function.
⚫ Response peaks at same time as forcing function
⚫ r>>1, ≈ 180˚, response out of
phase with forcing function.
⚫ Response has negative peak at the time when
forcing function has positive peak.
⚫ r ≈ 1, ≈ 90˚, response peaks
when forcing function at zero.
Ratio
Frequency
r
N
F
=
=
( ) ( )
:
1
2
:
2
1
Angle
Phase
r
r
Tan
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
F
d
Static
=
−
=
−
=
−
Rd
=
Dynamic
Displacement
Response
Amplification
Factor
Ratio
Frequency
r F
=
=
11. 11
SDOF: Dynamic Response Amplification: Velocity
⚫ Dynamic Response Amplification for
Velocity:
⚫ Rv = Very Large at Resonance, r=1
⚫ Rv = 0 at r ≈ 0 (Rd ≈ 1)
⚫ Rv approaches zero at r = large
(Rd ≈ 1/r2) Ratio
Frequency
r
N
F
=
=
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
−
=
−
=
−
=
−
=
−
=
=
=
−
=
=
+
−
=
−
=
−
=
t
Cos
R
km
p
t
u
t
Cos
R
km
p
t
Cos
R
m
k
p
t
u
t
Cos
R
m
k
k
p
t
Cos
R
k
p
t
u
rR
R
R
R
Define
t
Cos
R
k
p
t
u
sponse
Static
Max
sponse
nt
Displaceme
Dynamic
State
Steady
r
r
R
t
Sin
R
k
p
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
F
v
F
v
F
v
F
v
F
v
N
d
v
d
F
v
N
F
d
F
d
F
d
F
d
Static
0
0
2
1
2
1
0
2
1
2
1
0
0
0
2
2
2
0
:
Re
Re
2
1
1
:
d
v rR
R =
12. 12
SDOF: Dynamic Response Amplification: Acceleration
⚫ Dynamic Response Amplification for
Acceleration:
⚫ Ra = Very Large at Resonance, r = 1
⚫ Ra = 0 at r ≈ 0 (Rd ≈ 1)
⚫ Ra approaches 1 at r = large
(Rd ≈ 1/r2)
Ratio
Frequency
r
N
F
=
=
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
−
−
=
−
−
=
−
=
−
−
=
=
=
−
−
=
=
+
−
=
−
=
−
=
t
Sin
R
m
p
t
u
t
Sin
R
m
p
t
u
t
Sin
R
m
p
t
Sin
R
k
p
t
u
R
r
R
R
R
Define
t
Sin
R
k
p
t
u
sponse
Static
Max
sponse
nt
Displaceme
Dynamic
State
Steady
r
r
R
t
Sin
R
k
p
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
F
a
F
a
F
a
N
N
F
a
N
d
a
d
F
a
N
F
d
F
d
F
d
F
d
Static
0
0
2
2
0
2
0
2
2
2
2
0
2
2
2
0
:
Re
Re
2
1
1
:
d
a R
r
R 2
=
13. 13
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
⚫ Will use this
relationship later,
in development of
design code
Response Spectra
( ) ( )
( ) ( )
( ) ( )
sponse
nt
Displaceme
Static
Max
sponse
nt
Displaceme
Dynamic
R
r
r
R
t
Sin
R
k
p
t
u
t
Sin
R
u
t
u
is
solution
State
Steady
Final
The
d
d
F
d
F
d
Static
Re
Re
2
1
1
:
2
2
2
0
=
+
−
=
−
=
−
=
d
v
a
rR
R
r
R
=
=
Response Spectra for ASCE 7-05
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Period, T (sec)
Sa
(g)
TL=8, Lancaster, CA
Ratio
Frequency
r
N
F
=
=
14. 14
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
Find the frequency at which, Rd ,
Response Displacement is max
Take the derivative of Rd with
respect to r; set derivative = 0.
Ratio
Frequency
r
N
F
=
=
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( ) max
Re
2
1
2
1
2
1
4
4
2
1
4
1
2
4
0
4
1
2
4
1
2
2
1
2
1
0
1
2
2
1
2
1
1
2
2
2
2
3
2
3
2
3
2
2
3
4
2
2
2
2
1
4
2
2
2
2
2
=
=
−
=
=
−
=
−
=
−
+
−
=
+
−
+
−
+
−
=
=
=
=
+
−
+
=
+
−
=
−
−
d
F
N
N
F
d
d
a
d
v
d
R
for
Frequency
sonant
r
r
r
r
r
r
brackets
two
first
the
in
terms
the
out
Factor
r
r
r
r
dr
dR
R
r
R
rR
R
r
r
r
r
R
15. 15
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
Find the frequency at which, Rv ,
Response Velocity is max
Take the derivative of Rv with
respect to r; set derivative = 0.
Ratio
Frequency
r
N
F
=
=
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
max
Re
1
1
:
2
1
2
2
1
2
2
1
1
2
2
1
2
2
1
1
2
2
4
4
1
2
4
2
1
2
2
1
)
1
(
1
2
2
1
4
1
2
4
1
2
2
1
2
1
2
2
1
)
1
(
4
1
2
4
1
2
2
1
2
1
2
2
1
0
1
2
2
1
1
2
2
1
2
1
1
4
4
2
2
4
2
2
2
2
2
4
2
2
2
2
2
3
2
4
2
2
2
3
4
2
2
3
2
2
3
4
2
2
2
1
4
2
2
3
2
2
3
4
2
2
2
1
4
2
2
2
1
4
2
2
2
1
4
2
2
2
2
2
=
=
=
=
=
+
−
=
+
−
+
+
−
=
+
−
+
+
−
−
=
+
−
−
=
+
−
+
−
+
−
+
+
−
+
−
+
−
=
+
−
+
−
+
−
+
−
+
−
+
+
−
+
=
=
+
−
+
=
=
+
−
+
=
+
−
=
−
−
−
−
−
−
N
F
v
v
d
v
d
R
for
Frequency
sonant
r
r
is
solution
the
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
by
sides
both
Multiply
r
r
r
r
r
r
r
r
r
r
r
r
r
r
dr
dR
r
r
r
R
rR
R
r
r
r
r
R
16. 16
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
Find the frequency at which, Ra ,
Response Acceleration is max
Take the derivative of Ra with
respect to r; set derivative = 0.
Ratio
Frequency
r
N
F
=
=
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
max
Re
2
1
2
1
1
1
2
1
1
2
1
1
2
1
2
1
2
2
1
1
2
2
4
4
1
2
4
2
1
2
2
1
)
2
(
1
2
2
1
4
1
2
4
1
2
2
1
2
1
2
2
1
)
2
(
4
1
2
4
1
2
2
1
2
1
2
2
1
2
0
1
2
2
1
2
2
2
2
2
2
2
2
4
2
2
2
2
2
4
2
2
2
2
3
3
2
2
4
2
2
2
3
4
2
2
3
2
2
3
4
2
2
2
2
1
4
2
2
3
2
2
3
4
2
2
2
2
1
4
2
2
2
1
4
2
2
2
2
=
=
−
=
=
−
=
−
−
−
−
=
+
−
=
+
−
=
+
−
+
+
−
−
=
+
−
−
=
+
−
+
−
+
−
+
+
−
+
−
+
−
=
+
−
+
−
+
−
+
−
+
−
+
+
−
+
=
=
+
−
+
=
=
−
−
−
−
−
a
F
N
N
F
a
a
d
a
R
for
Frequency
sonant
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
by
sides
both
Multiply
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
dr
dR
r
r
r
R
R
r
R
17. 17
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
Find Max Dynamic Response
Displacement.
It will occur at the
resonant frequency for Rd
Ratio
Frequency
r
N
F
=
=
( )
( )
( )
( )( ) ( )
( ) ( )
( )( )
( )
2
2
1
2
2
2
1
4
2
2
1
4
2
4
2
2
1
4
2
2
4
2
2
1
2
2
2
2
2
2
1
4
2
2
2
Re
2
1
2
1
1
4
4
4
4
4
1
2
8
8
1
4
4
1
2
1
4
2
2
1
2
1
2
1
1
2
2
1
1
2
2
1
2
1
max
Re
2
1
−
=
−
=
−
=
+
−
+
−
−
+
=
+
−
+
+
−
−
+
=
−
+
−
−
+
=
=
=
+
−
+
=
−
=
=
−
=
−
−
−
−
−
−
MAX
MAX
MAX
MAX
MAX
MAX
d
d
d
d
d
d
d
a
d
v
d
sonant
d
N
F
R
R
R
R
R
R
R
r
R
rR
R
r
r
R
r
R
for
Frequency
sonant
18. 18
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
Find Max Dynamic Response Velocity.
It will occur at the resonant frequency
for Rv
Ratio
Frequency
r
N
F
=
=
( )
( )( ) ( )
( )
( )
2
1
2
4
1
2
4
1
1
1
2
2
1
1
1
1
2
2
1
1
1
2
2
1
1
max
Re
1
2
1
2
2
1
2
2
1
2
2
1
4
2
2
2
2
1
4
2
2
Re
=
=
=
+
−
+
=
+
−
+
=
=
+
=
−
+
=
=
=
=
+
−
+
=
=
=
=
−
−
−
−
−
−
MAX
MAX
MAX
MAX
MAX
MAX
v
v
v
v
v
v
d
a
d
v
d
sonant
v
N
F
R
R
R
R
R
r
r
r
R
R
r
R
rR
R
r
r
R
r
R
for
Frequency
sonant
19. 19
SDOF: Dynamic Response Amplification: Displacement, Velocity,
Acceleration
Find Max Dynamic Response
Acceleration.
It will occur at the
resonant frequency for Ra
Ratio
Frequency
r
N
F
=
=
( )
( ) ( )
( )
( )
( )( ) ( )
( ) ( )( ) ( )
( ) ( )( )
( )
( )( )
( )
2
2
1
2
2
2
1
4
2
2
1
4
2
4
2
2
1
2
4
2
4
2
2
1
1
2
2
2
2
2
1
2
2
1
2
2
2
2
2
1
2
2
1
2
2
1
2
2
2
2
1
4
2
2
2
1
2
2
Re
2
1
2
1
1
4
4
4
2
8
8
4
4
2
1
2
1
4
2
2
4
4
1
1
2
1
1
2
2
2
1
2
1
2
1
1
2
2
1
2
1
2
1
2
1
1
2
2
1
2
1
1
2
2
1
2
1
2
1
1
max
Re
2
1
−
=
−
=
−
+
=
−
−
+
+
−
=
+
+
−
−
+
+
−
=
+
−
−
+
−
=
−
+
−
−
+
−
=
−
+
−
−
+
−
=
=
=
=
+
−
+
=
−
=
−
=
=
=
−
=
−
−
−
−
−
+
+
−
−
−
+
−
−
−
−
−
−
MAX
MAX
MAX
MAX
MAX
MAX
a
a
a
a
a
a
d
a
d
v
d
N
F
sonant
a
N
F
R
R
R
R
R
r
R
R
r
R
rR
R
r
r
R
r
R
for
Frequency
sonant
