European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards and Catastrophic Events
520121-1-2011-1-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
10/03/2014
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 mu Df 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
tIf mu
0tmu cu ku
( ) ( ) ( )tgu 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
gmu gu
( ) ( )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( )ppu tk
( ) cos sinc n nu t A t B t
n k m
0( ) cos sinn npu t A t B tk
(0) 0u (0) 0u
0 0pA Bk
0( ) (1 cos )npu t tk
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
nn
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 )npu t tk
0 (1 cos )np tk
00
0 0
cos ( )1( ) sin[ ( )]tt
nn
n n n
p tu t p t dm 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 nn
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
2n
nf
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
1 22
stn n n
st st
g gf Tg
st mg k
22
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 cm c
22 2cr nn
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 nD 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
22 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 t Tu t T
2
nn
T
21n
DTT
2
( ) 2exp( ) 1D
u tu t T
21
2exp1
i
i
uu
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
2exp1
i
i
uu
21
2ln1
i
i
uu
21 1 2
31 1 2
1 2 3 4 1
j j
j j
uuu u u eu 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 2i i
i j i j
u usauj 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. şi Penzien, J. (2003). "Dynammics of structures", Third edition, Computers & Structures, Inc., Berkeley, USA
http://steel.fsv.cvut.cz/suscos