Unit-IV; Professional Sales Representative (PSR).pptx
SDEE: Lectures 1 and 2
1. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Structural Dynamics
& Earthquake Engineering
Lecture #1: Introduction
+ Equations of Motion for SDoF Oscillators
Dr Alessandro Palmeri
Architecture, Building & Civil Engineering @ Loughborough University
Tuesday, 3rd October 2017
2. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Why bother about Structural Dynamics?
http://www.youtube.com/watch?v=
XggxeuFDaDU
http://en.wikipedia.org/wiki/
Tacoma_Narrows_Bridge_(1940)
The 1940 Tacoma
Narrows Bridge
It was a steel suspension bridge
in the US state of Washington.
Construction began in 1938, with
the opening on 1st July 1940
From the time the deck was built,
it began to move vertically in
windy conditions (construction
workers nicknamed the bridge
Galloping Gertie).
The motion was observed even
when the bridge opened to the
public.
Several measures to stop the
motion were ineffective, and the
bridge’s main span finally
collapsed under 64 km/h wind
conditions the morning of 7th
3. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Why bother about Structural Dynamics?
http://www.youtube.com/watch?v=
eAXVa__XWZ8
http://en.wikipedia.org/wiki/
Millennium_Bridge,_London
The Millennium Bridge
It is an iconic steel suspension
bridge for pedestrians crossing
the River Thames in London.
Construction began in 1998, with
the opening on 10th June 2000.
Londoners nicknamed the bridge
the Wobbly Bridge after
participants in a charity walk to
open the bridge felt an
unexpected and uncomfortable
swaying motion.
The bridge was then closed for
almost two years while
modifications were made to
eliminate the wobble entirely.
It was reopened in 2002.
4. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Why bother about Structural Dynamics?
http:
//dx.doi.org/10.1063/1.2963929
http://www.enea.it/en/en/
research-development/
new-technologies/
materials-technologies
Shaking Table Tests
The video shows the last,
destructive shaking table test
conducted in 2007 on a 1:2
scale model of a masonry
building resembling a special
type (the so-called Tipo Misto
Messinese), which is proper to
the reconstruction of the city of
Messina after the 1783 Calabria
earthquake.
The model, incorporating a novel
timber-concrete composite slab,
has been tested on the main
shaking table available at the
ENEA Research Centre
Casaccia in Rome, both before
and after the
reinforcement with FRP materials.
6. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Aim of the Module
This module is aimed at:
developing knowledge and understanding of
vibrational problems in structural engineering
and to provide the basic analytical and numerical tools
to assess the dynamic response of structures
with special emphasis on the
seismic analysis and design to Eurocode 8
7. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Module Description: Parts 1 and 2
Dynamics of single-degree-of-freedom (SDoF) oscillatiors
Equations of motion for SDoF oscillators
Modelling and evaluation of damping in SDoF oscillators
Dynamic response of SDoF oscillators in both time and
frequency domains
Dynamics of multi-degree-of-freedom (MDoF) structures
Equations of motion for MDoF structures
Modal analysis and mode superposition method
Classically and non-classically damped MDoF structures
Dynamic response of MDoF structures in both time and
frequency domains
8. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Module Description: Part 3
Earthquake engineering
Engineering seismology (basics) and seismic action on
structures
Elastic response spectrum and ductility-dependent
design spectra
Lateral force method
Response spectrum method
Design options to improve the seismic performance of
existing and new buildings
A look into the (near) future: performance-based
earthquake engineering (PBEE)
9. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Assessment of Module
2-hour examination (65% of the module)
Section A (50% of the exam marks) on Parts 1 and 2
Solving 2 out of 3 questions (each one weighting 25%)
Section B (50% of the exam marks) on Part 3
Solving 1 out of 2 questions (including some
discursive reasonings
Coursework comprising a group written assignment
based on Part 1 (35% of the module)
Issued in Week 3 (Tuesday 17th
October 2017)
Handed in Week 9 (Monday 27th
November 2016)
Returned in Week 12 (Tuesday 9th
January 2018)
10. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Schedule of the Module
Lecture #1 Lecture #2 Tutorial
Week 1 Introduction Equation of motion for SDoF
oscillators
Part 1 (Tue)
Fourier analysis for structural
dynamics applications
Frequency response function
for SDoF oscillators
—
Week 2 State-space formulation for
SDoF oscillators
Energy dissipation devices for
vibration control
Part 1 (Tue)
Equations of motion for
undamped MDoF structures
Modal analysis —
Week 3?
Classically and non-classically
damped MDoF structures
Newmark- method Part 2 (Tue)
State-space formulation for
MDoF structures
Coursework brief Part 2 (Fri)
4th
-order Runge-Kutta method — —
Week 7 Basics of engineering
seismology
Strong motion characteristics Part 2
Week 8 Elastic response spectrum Ductility µ, behaviour factor q
and design spectra
Part 3
Week 9†
Lateral force method Part 3
Week 10 Response spectrum method Part 3
Week 11 Performance-based earthquake engineering Part 3
Christmas break
Week 12 ‡
Revision —
? Coursework brief is issued in Week 3
† Group report is handed in Week 9 – On Monday 27th
November 2017
‡ Feedback is given in Week 12 (before the Exams)
11. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Timetable of the module
Extra sessions in weeks 1 to 3, to support the development of the
coursework
No lectures and tutorials in weeks 4 to 6
Weekly bookable one-hour surgery sessions throughout the term
12. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Changes introduced this year
1 Slightly simplified the coursework, without changing the
eight (still 35%)
2 REVIEW Lecture Capture
3 One-hour surgery session every week
Starting from Week 2
Typically, Wednesday 11:30am to 12:30pm, in my office
(RT.0.15)
Typically, 10 minutes per student/group (requires
pre-booking)
Can be rearranged in some weeks, e.g. Friday 3:00 to
4:00pm in Week 4
14. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Newton’s Second Law of Motion
Sir Isaac Newton
(25 Dec 1642 – 20 Mar 1726)
The acceleration
!
a of a body
is parallel and directly
proportional to the net force
!
F acting on the body, is in the
direction of the net force, and
is inversely proportional to the
mass m of the body, i.e.
!
F = m
!
a (1)
15. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Newton’s Second Law of Motionr
What are the implications of Newton’s Second Law of
Motion from the point of view of a structural engineer?
Structures are not rigid, therefore they always
experience deformations when subjected to loads
The acceleration a is the rate of variation of the velocity
v, which in turn is the rate of variation of the
displacement u, that is:
a(t) =
d
dt
v(t) =
d2
dt2
u(t) (2)
It follows that, if the displacements vary rapidely, the
inertial forces (proportional to the masses) have to be
included in the dynamic equilibrium of the structure
16. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
u(t)
k
m
The figure on the left-hand
side shows a typical graphical
representation of an unforced
and undamped Single-
Degree-of-Freedom (SDoF)
oscillator, which consists of:
A rigid mass, ideally moving
on frictionless wheels,
whose value m is usually
expressed in kg (or Mg=
103
kg= 1 ton)
An elastic spring, whose
stiffness k is usually
expressed in N/m (or kN/m)
17. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
When the oscillator is displaced (u(t) is positive if directed to the
right), then an elastic force fk(t) = k u(t) arises in the spring at the
generic time t, and is applied to the left on the mass
According to Newton’s Second Law of Motion, the acceleration a(t)
is given by the net force acting on the mass m at time t, divided by
the mass itself, i.e. a(t) = fk(t)/m, and is directed to the left as well
The acceleration a(t) is therefore negative, and the inertial force
fm(t) = m a(t) felt by the oscillator is directed to the left in order to
reconcile the sign conventions
18. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
The equation of motion for this unforced and undamped SDoF oscillator
reads:
fm(t) + fk(t) = 0
) m a(t) + k u(t) = 0
) m ¨u(t) + k u(t) = 0
where the overdot means derivation with respect to time t:
v(t) =
d
dt
u(t) = ˙u(t); a(t) =
d
dt
v(t) = ¨u(t)
19. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
The free (i.e. unforced) vibration of an undamped SDoF
oscillator is ruled by a 2nd-order homogeneous linear
ordinary differential equation (ODE) with
constant coefficients:
m ¨u(t) + k u(t) = 0 (3)
This equation can be rewritten as:
¨u(t) + !2
0 u(t) = 0 (4)
where !0 is the so-called undamped natural circular
frequency of vibration for the SDoF oscillator. This new
quantity is expressed in rad/s and is given by:
!0 =
r
k
m
(5)
20. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
Similarly, the undamped natural period of vibration
(measured in s) is:
T0 =
2 ⇡
!0
(6)
and the undamped natural frequency of vibration (measured
in Hz= 1/s):
⌫0 =
1
T0
=
!0
2 ⇡
(7)
The adjective natural is used because such quantities only
depend on the natural mechanical properties of the
oscillator, i.e. the mass m and the stiffness k
21. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
Golden Gate Bridge
San Francisco, California
The fundamental natural
periods of vibration of this
iconic suspension bridge,
having the main span of
1,300 m, are:
18.2 s for the lateral
movement
10.9 s for the vertical
movement
4.43 s for the
torsional movement
22. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
We have then to solve this homogenous differential
equation:
¨u(t) + !2
0 u(t) = 0 (4)
and the solution can be expressed as:
u(t) = C1 cos(!0 t) + C2 sin(!0 t) (8)
where C1 and C2 are two arbitrary integration constants,
whose values are usually determined by assigning the initial
condition in terms of displacement u(0) = u0 and velocity
v(0) = ˙u(0) = v0 at the initial time instant t = 0
23. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
The first initial condition leads to:
u(0) = C1 cos(!0 ⇥ 0) + C2 sin(!0 ⇥ 0) = u0
) C1 ⇥ 1 + C2 ⇥ 0 = u0 ) C1 = u0
(9)
By differentiating the displacement function u(t), given by Eq. (8),
and taking into account the identity above, one gets:
˙u(t) =
d
dt
u(t) = u0 !0 sin(!0 t) + C2 !0 cos(!0 t)
We can then apply the second initial condition:
˙u(0) = u0 !0 sin(!0 ⇥ 0) + C2 !0 cos(!0 ⇥ 0) = v0
) u0 ⇥ 0 + C2 !0 ⇥ 1 = v0 ) C2 =
v0
!0
(10)
24. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
Substituting the values of constants C1 and C2, given by
Eqs. (9) and (10), the mathematical expressions of
displacement and velocity of a SDoF experiencing free
vibration can be obtained:
u(t) = u0 cos(!0 t) +
v0
!0
sin(!0 t) (11)
˙u(t) = v0 cos(!0 t) u0 !0 sin(!0 t) (12)
25. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
By using the trigonometric identities, one can prove that:
u(t) = ⇢0 cos(!0 t + '0) (13)
where ⇢0 is the amplitude of the motion and '0 is the phase
at the initial time instant t = 0:
⇢0 =
q
C2
1 + C2
2 =
s
u2
0 +
✓
v0
!0
◆2
(14)
tan('0) =
C2
C1
=
v0/!0
u0
=
v0
!0 u0
(15)
26. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Undamped SDoF Oscillator
u(t)
t
m
a)
u0 ρ0
T0=2π/ω0
t
b)
u(t)
T0=2π/ω0
u0
ρ0
u0>0
c)
u(t)
T0+ϕ0/ω0
u0
t
ϕ0/ω0
u0<0
t
d)
ϕ0/ω0
u0<0
u(t)
u0>0
ϕ0/ω0
T0 /2+ϕ0/ω0
Free undamped vibration of a SDoF oscillator for given initial conditions: a) u0 > 0
and ˙u0 = 0; b) u0 > 0 and ˙u0 > 0; c) u0 > 0 and ˙u0 < 0; d) u0 < 0 and ˙u0 > 0
27. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
The unrealistic feature of the solution presented in the
previous slides is that the motion of the SDoF oscillator
appears to continue indefinitely
The reason is that the effects of energy dissipation
phenomena are not considered
We must include a damping force in the equation of motion,
which acts in parallel with the elastic force fk (t) = k u(t),
proportional to the displacement, and the inertial force
fm(t) = m ¨u(t), proportional to the acceleration
Damping mechanics are very difficult to model, and the
simplest approach is to assume a viscous type of energy
dissipation, so that the damping force is proportional to the
velocity:
fc(t) = c ˙u(t) (16)
where c is the so-called viscous damping coefficient, usually
expressed in N⇥s/m (or kN⇥s/m)
28. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
The equation of motion (with viscous damping) becomes:
m ¨u(t) + c ˙u(t) + k u(t) = 0 (17)
Normalisng all terms with respect to the mass m, one
obtains:
¨u(t) + 2 ⇣0 !0 ˙u(t) + !2
0 u(t) = 0 (18)
where ⇣0 is the so-called equivalent viscous damping ratio
for the SDoF oscillator, given by:
⇣0 =
c
2 m !0
=
c
ccr
(19)
in which ccr is the so-called critical damping coefficient:
ccr = 2 m !0 = 2
p
k m =
2 k
!0
(20)
29. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
One can mathematically prove that, assuming initial
conditions u(t) = u0 > 0 and v0 = 0:
If c = ccr (or ⇣0 = 1), then the system is said to be
critically damped, and the mass returns to the
equilibrium configuration (u = 0) without oscillating
If c > ccr (or ⇣0 > 1), then the system is said to be
overdamped, and the mass returns to the equilibrium
position at a slower rate, again without oscillating
If c < ccr (or ⇣0 < 1), then the system is said to be
underdamped, and the mass oscillates about the
equilibrium position with a progressively decreasing
amplitude
31. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
The vast majority of civil engineering structures are
underdamped, and the range of values for the viscous
damping ratio ⇣0 is between 0.01 and 0.07, ⇣0 = 0.05 being
for instance the assumed value for reinforced concrete
frames and steel frames with bolted joints
The solution of Eq. (18) can be posed in the form:
u(t) = e ⇣0 !0 t
h
C1 cos(!0 t) + C2 sin(!0 t)
i
(21)
In this case, the displacement function u(t) is a harmonic
wave of circular frequency !0 and exponentially decaying
amplitude. Again, the integration constants C1 and C2 have
to be determined by imposing the initial conditions
32. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
For the free vibration of an underdamped SDoF system, the
time history of the deflection takes the expression:
u(t) = e ⇣0 !0 t
u0 cos(!0 t) +
v0 + ⇣0 !0 u0
!0
sin(!0 t)
(22)
or, equivalently:
u(t) = ⇢0 e ⇣0 !0 t
cos(!0 t + '0) (23)
in which:
⇢0 =
q
C
2
1 + C
2
2 =
s
u2
0 +
✓
v0 + ⇣0 !0 u0
!0
◆2
(24)
tan('0) =
C2
C1
=
v0 + ⇣0 !0 u0
!0 u0
(25)
33. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
u0
t
u(t) u0 > 0
ρ0e-ζ0ω0
t
−ρ0e-ζ0ω0
t
T0=2π/ω0
ϕ0 /ω0
Free vibration of an underdamped SDoF oscillator with initial
conditions u0 > 0 and v0 = ˙u0 > 0
34. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion:
Unforced Damped SDoF Oscillator
t
u(t)
0 1 2 3
-50
-25
0
25
50
ζ0=0.05
ζ0=0.1
ζ0=0.2
ζ0=0.4
t
u(t)
0 1 2 3
-50
-25
0
25
50
ζ0=0.05
t
u(t)
0 1 2 3
-50
-25
0
25
50 ζ0=0.1
t
u(t)
0 1 2 3
-50
-25
0
25
50
ζ0=0.2
t
u(t)
0 1 2 3
-50
-25
0
25
50
ζ0=0.4
35. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
u(t)
k
c
m
F(t)=F0 sin(ωf t)
a)
u(t)
ku(t)
cu(t)
.
m
b)
mu(t)
..
F(t)=F0 sin(ωf t)
Let us now consider the forced vibration of a SDoF oscillator
subjected to an harmonic input F(t) = F0 sin(!f t), where F0
and !f are amplitude and circular frequency of the force
The figure above shows the sketch of the oscillator (a) and
the forces acting on the mass m at a generic time instant t
(b)
36. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
u(t)
k
c
m
F(t)=F0 sin(ωf t)
a)
u(t)
ku(t)
cu(t)
.
m
b)
mu(t)
..
F(t)=F0 sin(ωf t)
The equations of motion in terms of forces is:
m ¨u(t) + c ˙u(t) + k u(t) = F0 sin(!f t) (26)
and in terms of accelerations is:
¨u(t) + 2 ⇣0 !0 ˙u(t) + !2
0 u(t) =
F0
m
sin(!f t) (27)
37. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
The general solution to this inhomogeneous 2nd-order ODE
is given by:
u(t) = uh(t) + up(t) (28)
where:
up(t) is the particular integral, i.e. any function
satisfying Eq. (26) (or equivalently Eq. (27)), which in
turn depends on the particular forcing function F(t)
uh(t) is the general solution of the related
homogeneous ODE (Eqs. (17) and (18)), which is given
by the time history of free vibration of the SDoF
oscillator with generic integration constants C1 and C2
(see Eq. (21))
uh(t) = e ⇣0 !0 t
h
C1 cos(!0 t) + C2 sin(!0 t)
i
(29)
38. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
For a sinusoidal load, F(t) = F0 sin(!f t), the particular
integral takes the expression:
up(t) = ⇢p sin(!f + 'p) (30)
That is, up(t) is again a sinusoidal function with the same
circular frequency !f as the input, while amplitude ⇢p and
phase angle 'p are given by:
⇢p =
F0
k
1
q
(1 2)
2
+ (2 ⇣0 )2
(31)
tan('p) =
2 ⇣0
1 2
(32)
in which = !f/!0
39. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
ζ
0
=0
0.0 2.5 5.0 7.5 10.0
-0.50
-0.25
0.00
0.25
0.50
a)
t 0.0 2.5 5.0 7.5 10.0
-0.50
-0.25
0.00
0.25
0.50
b)
t
ζ
0
=0.05
Dynamic response u(t) of a SDoF oscillator (solid line) compared to the
particular integral up(t) (dashed line) for a sinusoidal excitation
( = !f/!0 = 0.2; !0 = 7 rad/s; F0/m = 10 m/s2
; u0 = 0; v0 = 1.5 m/s)
and two different values of the viscous damping ratio: a) ⇣0 = 0; b)
⇣0 = 0.05
40. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
Alternatively, the particular integral for a sinusoidal
excitation can be expressed as:
up(t) = ust D( ) sin(!f + 'p) (33)
where:
ust = F0
k is the static displacement due to the force F0
D( ) is the so-called dynamic amplification factor,
which depends on the frequency ratio = !f/!0:
D( ) =
⇢p
ust
=
1
q
(1 2)
2
+ (2 ⇣0 )2
(34)
41. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
β
D
0 1 2 3
0
1
2
3
4
5
6
7
8
a)
ζ0=0.0
ζ0=0.1
ζ0=0.2
ζ0=0.3
ζ0=0.5
ζ0=0.7
ζ0=1.0
0 1 2 3
0
ϕp
b)
π/4
π/2
3/4 π
π
β
ζ0=0.0
ζ0=0.1
ζ0=0.2
ζ0=0.3
ζ0=0.5
ζ0=0.7
ζ0=1.0
ζ0=0.0
Dynamic amplification factor D (a) and phase angle 'p (b) as functions of
the frequency ratio for different values of the viscous damping ratio ⇣0
42. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
For ! 0, the static case is approached (i.e. D ! 1), as the force
F(t) varies very slowly and therefore the mass m can follows it
For ! 1 and ⇣0 = 0 (undamped case), the system experience an
unbounded resonance, meaning that D ! +1 and (theoretically)
the amplitude of the dynamic response keeps increasing with time
If = 1 and 0 ⇣0 < 0.5, then D > 1, i.e. the amplitude of the
steady-state dynamic response ⇢p is larger than the static
displacement ust; moreover, the less ⇣0, the larger is D
If > 1.41, then D < 1, i.e. (independently of ⇣0) the amplitude of
the steady-state dynamic response ⇢p is less than the static
displacement ust
For ! +1, the dynamic amplification factor goes to zero (i.e.
D ! 0), as the force F(t) varies so rapidely that the mass m cannot
follows it
If ⇣0 0.707, then D < 1, meaning that (independently of the
frequency ratio ) the amplitude of the steady-state dynamic
response is less than the static displacement
43. Structural
Dynamics
& Earthquake
Engineering
Dr Alessandro
Palmeri
Motivations for
this module
Introduction to
the module
Equations of
motion for
SDoF
oscillators
Equation of Motion: Forced SDoF Oscillator
0 1 2 3
-3
-2
-1
0
1
2
3
β=0.1
t
a)
β=0.5
t0 1 2 3
-3
-2
-1
0
1
2
3
b)
β=1
t0 1 2 3
-3
-2
-1
0
1
2
3
c)
β=2
t0 1 2 3
-3
-2
-1
0
1
2
3
d)
Normalised particular integral up(t)/ust (solid line) compared to the normalised
pseudo-static response F(t)/(k ust) (dashed line) for a damped (⇣0 = 0.20) SDoF
oscillator subjected to a sinusoidal force F(t) = F0 sin(!0 t), with !f = 2 ⇡, and
for four different values of the frequency ratio = !f/!0
