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European Erasmus Mundus Master Course
Sustainable Constructions
under Natural Hazards and Catastrophic Events520121-1-2011-1-CZ-ERA MUNDUS-EMMC
2C09
Design for seismic and climate changes
Lecture 02: Dynamic response of single-degree-of-freedom systems I
Daniel Grecea, Politehnica University of Timisoara
03/04/2014
L2 – Dynamic response of single-degree-of-freedom systems I
European Erasmus Mundus Master Course
Sustainable Constructions under Natural
Hazards and Catastrophic Events
L2.1 – Introduction to dynamics of structures.
L2.2 – Solutions methods for the equation of motion.
L2.3 – Free vibration analysis of SDOF systems.
2C09-L2 – Dynamic response of single-degree-of-
freedom systems I
Dynamics of structures
Dynamics of
structures
determination of
response of
structures under
the effect of
dynamic loading
Dynamic load is
one whose
magnitude,
direction, sense
and point of
application changes
in time
t
p(t)
p(t)
p(t)
t
t
t
üg(t)
Single degree of freedom systems
Simple structures:
– mass m
– stiffness k
Objective: find out response of SDOF system under the
effect of:
– a dynamic load acting on the mass
– a seismic motion of the base of the structure
The number of degree of
freedom (DOF) necessary
for dynamic analysis of a
structure is the number of
independent displacements
necessary to define the
displaced position of
masses with respect to
their initial position
k
m
Single degree of freedom
systems (SDOF)
Single degree of freedom systems
One-storey frame =
– mass component
– stiffness component
– damping component
Number of dynamic degrees of freedom = 1
Number of static degrees of freedom = ?
Force-displacement relationship
Force-displacement relationship
Linear elastic system:
– elastic material
– first order analysis
Inelastic system:
– plastic material
– First-order or second-order analysis
Sf k u
,S Sf f u u
Damping force
Damping: decreasing with time of amplitude of vibrations
of a system let to oscillate freely
Cause: thermal effect of elastic cyclic deformations of the
material and internal friction
Damping
Damping in real structures:
– friction in steel connections
– opening and closing of microcracks in r.c. elements
– friction between structural and non-structural elements
Mathematical description of these components
impossible
Modelling of damping in real structures
equivalent viscous damping
Damping
Relationship between damping force
and velocity:
c - viscous damping coefficient
units: (Force x Time / Length)
Determination of viscous damping:
– free vibration tests
– forced vibration tests
Equivalent viscous damping modelling of the energy
dissipated by the structure in the elastic range
Df c u
Equation of motion for an external force
Newton’s second law of motion
D'Alambert principle
Stiffness, damping and mass components
Equation of motion: Newton’s 2nd law of motion
Forces acting on mass m:
– external force p(t)
– elastic (or inelastic) resisting force fS
– damping force fD
External force p(t), displacement u(t), velocity and
acceleration are positive in the positive direction of
the x axis
Newton’s second law of motion:
( )u t( )u t
S Dp f f mu
S Dmu f f p
( )mu cu ku p t
Equation of motion: D'Alambert principle
Inertial force
– equal to the product between force and acceleration
– acts in a direction opposite to acceleration
D'Alambert principle: a system is in equilibrium at each
time instant if al forces acting on it (including the inertia
force) are in equilibrium
I S Df f f p
If mu
S Dmu f f p
( )mu cu ku p t
Equation of motion:
stiffness, damping and mass components Under the external force p(t), the system state is
described by
– displacement u(t)
– velocity
– acceleration
System = combination of three pure components:
– stiffness component
– damping component
– mass component
External force p(t) distributed to the three components
( )u t
( )u t
If muDf c u
Sf k u
I S Df f f p
SDOF systems: classical representation
Equation of motion: seismic excitation
Dynamics of structures in the case of seismic motion
determination of structural response under the effect of
seismic motion applied at the base of the structure
Ground displacement ug
Total (or absolute) displacement of the mass ut
Relative displacement between mass and ground u
( ) ( ) ( )t
gu t u t u t
Equation of motion: seismic excitation
D'Alambert principle of dynamic equilibrium
Elastic forces relative displacement u
Damping forces relative displacement u
Inertia force total displacement ut
0I S Df f f
Df c u
Sf k u
t
If mu
0tmu cu ku
( ) ( ) ( )t
gu t u t u t
gmu cu ku mu
Equation of motion: seismic excitation
Equation of motion in the case of an external force
Equation of motion in the case of seismic excitation
Equation of motion for a system subjected to seismic
motion described by ground acceleration is identical
to that of a system subjected to an external force
Effective seismic force
gmu cu ku mu
( )mu cu ku p t
gmugu
( ) ( )eff gp t mu t
Problem formulation
Fundamental problem in dynamics of structures:
determination of the response of a (SDOF) system under
a dynamic excitation
– a external force
– ground acceleration applied to the base of the structure
"Response" any quantity that characterizes behaviour
of the structure
– displacement
– velocity
– mass acceleration
– forces and stresses in structural members
Determination of element forces
Solution of the equation of motion of the SDOF system
displacement time history
Displacements forces in structural elements
– Imposed displacements forces in structural elements
– Equivalent static force: an external static force fS that produces
displacements u determined from dynamic analysis
Forces in structural elements by static analysis of the structure
subjected to equivalent seismic forces fS
( )u t
( ) ( )sf t ku t
Combination of static and dynamic response
Linear elastic systems:
superposition of effects possible
total response can be determined
through the superposition of the
results obtained from:
– static analysis of the structure under
permanent and live loads, temperature
effects, etc.
– dynamic response of the structure
Inelastic systems: superposition of
effects NOT possible dynamic
response must take account of
deformations and forces existing in
the structure before application of
dynamic excitation
Solution of the equation of motion
Equation of motion of a SDOF system
differential linear non-homogeneous equation of second
order
In order to completely define the problem:
– initial displacement
– initial velocity
Solution methods:
– Classical solution
– Duhamel integral
– Numerical techniques
( ) ( ) ( ) ( )mu t cu t ku t p t
(0)u
(0)u
Classical solution
Complete solution u(t) of a linear non-homogeneous
differential equation of second order is composed of
– complementary solution uc(t) and
– particular solution up(t)
u(t) = uc(t) +up(t)
Second order equation 2 integration constants initial
conditions
Classical solution useful in the case of
– free vibrations
– forces vibrations, when dynamic excitation is defined analytically
Classical solution: example
Equation of motion of an undamped (c=0) SDOF system
excited by a step force p(t)=p0, t≥0:
Particular solution:
Complementary solution:
where A and B are integration constants and
The complete solution
Initial conditions: for t=0 we have and
the eq. of motion
0mu ku p 0( )p
pu t
k
( ) cos sinc n nu t A t B t
n k m
0( ) cos sinn n
pu t A t B t
k
(0) 0u (0) 0u
0 0p
A Bk
0( ) (1 cos )n
pu t t
k
Duhamel integral
Basis: representation of the dynamic excitation as a
sequence of infinitesimal impulses
Response of a system excited by the force p(t) at time t
sum of response of all impulses up to that time
Applicable only to "at rest" initial conditions
Useful when the force p(t)
– is defined analytically
– is simple enough to evaluate analytically the integral
0
1( ) ( )sin[ ( )]
t
n
n
u t p t dm
(0) 0u (0) 0u
Duhamel integral: example
Equation of motion of an undamped (c=0) SDOF system,
excited by a ramp force p(t)=p0, t≥0:
Equation of motion
0mu ku p
0( ) (1 cos )n
pu t t
k
0 (1 cos )n
pt
k
00
0 0
cos ( )1( ) sin[ ( )]
tt
nn
n n n
p tu t p t d
m m
Undamped free vibrations
General form of the equation of motion:
Equation of motion in the case of undamped free
vibrations:
Vibrations the system disturbed from the static
equilibrium position by
– initial displacement
– initial velocity
Classical solution
where
( )mu cu ku p t
0mu ku
(0)u
(0)u
(0)( ) (0)cos sinn n
n
uu t u t t
n k m
Undamped free vibrations
Simple harmonic motion(0)
( ) (0)cos sinn n
n
uu t u t t
Undamped free vibrations
Natural period of vibration - Tn - time needed for
an undamped SDOF system to perform a
complete cycle of free vibrations
Natural circular frequency
Natural frequency of vibration fn represents the
number of complete cycles performed by the
system in one second
mass stiffness
"Natural" - depends only on the properties of the
SDOF system
2n
n
T
1n
n
fT
2
nnf
n k m
n k m
Undamped free vibrations
Alternative expressions for n, fn, Tn:
elastic deformation of a SDOF system under
a static force equal to mg
Amplitude: magnitude of oscillations
12
2
stn n n
st st
g gf T
g
st mg k
2
2
0
00
n
uu u
Damped free vibrations
General form of the eq. of motion:
Equation of motion in the case of damped free vibrations:
Dividing eq. by m we obtain
with the notations:
Critical damping coefficient
Damping coefficient c - a measure of the energy
dissipated in a complete cycle
- critical damping ratio: a non-dimensional measure of
damping, which depends on the stiffness and mass as
well
( )mu cu ku p t
0mu cu ku 22 0n nu u u
n k m 2 n cr
c c
m c
22 2cr n
n
kc m km
Types of motion
c=ccr or = 1 the system returns to the position of
equilibrium without oscillation
c>ccr or > 1 the system returns to the position of
equilibrium without oscillation, but slower
c<ccr or < 1 the system oscillates with respect to the
equilibrium position with progressively decreasing
amplitudes
Types of motion
ccr - the smallest value of the damping coefficient that
completely prevents oscillations
Most engineering structures - underdamped (c<ccr)
Few reasons to study:
– critically damped systems (c=ccr)
– overdamped systems (c>ccr)
Underdamped systems
Solution of the eq. for systems with
c<ccr or < 1:
with the notation:
(0) (0)( ) (0)cos sinnt n
D D
D
u uu t e u t t
0mu cu ku
21D n
The effect of damping in underdamped systems
Envelope of damped vibrations
Lowering of the circular frequency from n to D
Lengthening of he period of vibration form Tn to TD
nte
2
2 0 (0)0
n
D
u uu
Attenuation of motion
Ratio between displacement at time t and the one after a
period TD is independent of t:
Using and
( )
exp( )
n D
D
u tT
u t T
2
n
n
T
21
nD
TT
2
( ) 2exp
( ) 1D
u t
u t T
21
2exp
1
i
i
u
u
Attenuation of motion
Natural logarithm of the ratio
is called logarithmic decrement
and is denoted with :
For small values of the damping
Determination of logarithmic
decrement based on peaks several
cycles apart
21
2exp
1
i
i
u
u
21
2ln
1
i
i
u
u
21 1 2
31 1 2
1 2 3 4 1
j j
j j
uuu u ue
u u u u u
1 11 ln 2jj u u
Free vibration tests
Determination of damping in structures: free vibration
tests
1 1ln ln
2 2
i i
i j i j
u usau
j u j u
References / additional reading
Anil Chopra, "Dynamics of Structures: Theory and
Applications to Earthquake Engineering", Prentice-Hall,
Upper Saddle River, New Jersey, 2001.
Clough, R.W. and Penzien, J. (2003). "Dynamics of
structures", Third edition, Computers & Structures, Inc.,
Berkeley, USA
