2c09 design for seismic and climate changes - upt · 2014-10-29 · l3 – dynamic response of...
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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 03: Dynamic response of single-degree-of-freedom systems II
Daniel Grecea, Politehnica University of Timisoara
11/03/2014
L3 – Dynamic response of single-degree-of-freedom systems II
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards and Catastrophic Events
L3.1 – Response of SDOF systems to step, ramp and harmonic forces.
2C09-L3 – Dynamic response of single-degree-of-freedom systems II
Response to step force Step force:
Duhamel integral
0 0p t p t
00
2( ) 1 cos 1 cosst nn
p tu t u tk T
Response to step force Maximum displacement (undamped system): The system vibrates with a period Tn about the static
position Effect of damping:
– a smaller overshoot over the static response – a more rapid decay of motion
0 02 stu u
Response to ramp force Ramp force Response of an undamped system:
The system vibrates with a period Tn about the static position
0 0r
tp t p tt
0 0
sin sin 2( )2
n n nst st
r n r n r r n
t T t Tt tu t u ut t T t t T
Response to step force with finite rise time Force (ramp phase and constant phase):
Response of an undamped system:
– ramp phase
– constant phase
0
0
0r r
r
p t t t tp t
p t t
0sin( ) n
st rr n r
ttu t u t tt t
0
1( ) 1 sin sinst n n r rn r
u t u t t t t tt
Response to step force with finite rise time Ramp phase: system
vibrates with a period Tn about the static position
Constant phase: idem the
system does not vibrate for t>tr
Small tr/Tn response similar to the one under a step force
Large tr/Tn response similar to the static one
0ru t
Harmonic vibrations of undamped systems Harmonic force: or
– amplitude p0 – circular frequency
0( ) sinp t p t 0( ) cosp t p t
Harmonic vibrations of undamped systems Equation of motion: Initial conditions
Particular solution
Complementary solution
Complete solutions
Final solution
0 sinmu ku p t (0) (0)u u u u
0
2
1( ) sin1
p nn
pu t tk
( ) cos sinc n nu t A t B t
0
2
1( ) cos sin sin1
n nn
pu t A t B t tk
0 02 2
0 / 1( ) 0 cos sin sin1 1
nn n
n n n
u p pu t u t t tk k
transient response steady-state response
Harmonic vibrations of undamped systems
0.2n
(0) 0u (0) /n ou p k
Harmonic vibrations of undamped systems Steady-state response: due to applied force; is not
influenced by the initial conditions Transient response: depends on initial displacement and
velocity, as well as properties of SDOF and exciting force
Neglecting dynamic response static response
Steady-state response:
0 02 2
0 / 1( ) 0 cos sin sin1 1
nn n
n n n
u p pu t u t t tk k
0 sinstpu t tk
00st
puk
20
1( ) sin1
stn
u t u t
transient response steady-state response
Harmonic vibrations of undamped systems <n displacement
u(t) and exciting force p(t) have the same algebraic sign. Displacement is in phase with the applied force.
>n displacement u(t) and exciting force p(t) have different algebraic signs. Displacement is out of phase with the applied force.
20
1( ) sin1
stn
u t u t
Harmonic vibrations of undamped systems Steady-state response:
Alternative representation of steady-state response:
0 0( ) sin sinst du t u t u R t
20
1( ) sin1
stn
u t u t
0
20
011
nd
nst n
uR andu
Displacement response factors Displacement response
factor – small <n: amplitude of
dynamic response close to the static deformation
– /n>2: amplitude of dynamic response smaller then the static deformation
– /n 1: amplitude of dynamic response much larger than static deformation
Resonant frequency - frequency for which the response factor Rd is maximum (=n)
0
0d
st
uRu
Resonance Solution for the equation of motion when =n:
– particular solution
– total solution
0 cos2p n n npu t t tk
01( ) cos sin2 n n npu t t t tk
(0) (0) 0u u
Harmonic vibrations of damped systems Equation of motion
Initial conditions
Particular solution
Complementary solution
Complete solution
0 sinmu cu ku p t
(0) (0)u u u u
( ) sin cospu t C t D t
20 0
2 22 22 2
1 2
1 2 1 2
n n
n n n n
p pC Dk k
( ) cos sinntc D Du t e A t B t
( ) cos sin sin cosntD Du t e A t B t C t D t
21D n
transient response steady-state response
Harmonic vibrations of damped systems
transient response steady-state response
0.2n 0.05
(0) 0u (0) /n ou p k
( ) cos sin sin cosntD Du t e A t B t C t D t
Harmonic vibrations of damped systems: =n For =n response of a damped SDOF system is:
0 2
1( ) cos sin cos2 1
ntst D D nu t u e t t t
0
1( ) 1 cos2
ntst nu t u e t
0.05
(0) (0) 0n
u u
Harmonic vibrations of damped systems: =n
Small damping: – Larger amplitude – More cycles to attainment of a certain ratio of the steady-state
response
(0) (0) 0n
u u
Harmonic vibrations of damped systems: Rd and Steady-state response can be written as:
Displacement response factor Rd
0 0sin sinst du t u t u R t
0
2 220
1
1 / 2 /d
stn n
uRu
12
2 /tan
1 /n
n
Harmonic vibrations of damped systems: Rd and
0.2
Harmonic vibrations of damped systems: Rd and : amplitude of dynamic
response close to the static deformation (Rd 1) and almost independent of damping. Response controlled by stiffness of the system.
: amplitude of dynamic response approaches 0 (Rd 0) and almost independent of damping. Response controlled by mass of the system.
: amplitude of dynamic response larger than the static deformation (Rd max) and sensible to damping. Response controlled by damping of the system.
1n 0
0 0stpu uk
1n
2
00 2 20
nst
pu um
1n 0 00 2
st
n
u puc
Harmonic vibrations of damped systems: Rd and : phase angle close to 0,
displacement in phase with the applied force.
: phase angle close to , displacement out of phase with the applied force.
: phase angle equal to /2 for any value of , displacement maximum when force equals 0.
1n
1n
1n
Resonance Resonant frequency:
frequency for which the maximum response in terms of displacement (or velocity or acceleration) is obtained
Displacement resonant frequency:
Maximum response:
21 2n
21 2 1dR
Half-power bandwidth Difference
between circular frequencies for which the displacement response factor is times smaller than the resonant response
1 2
2b a
n
Damping for engineering structures stress level structural type (%)
stress level below 0.5 times the yield strength
welded steel structures, prestressed concrete structures, strongly reinforced concrete structures (limited cracks) 2-3
reinforced concrete structures with significant cracking 3-5
steel structures with bolted or riveted connections, wood structures connected with screws or nails 5-7
stresses close to the yield strength
welded steel structures, prestressed concrete structures (without total loss of prestress) 5-7
prestressed concrete structures with total loss of prestress 7-10
reinforced concrete structures 7-10
steel structures with bolted or riveted connections, wood structures connected with screws 10-15
wood structures connected with nails 15-20
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
