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
European Erasmus Mundus Master Course
Sustainable Constructions
under Natural Hazards and Catastrophic Events
520121-1-2011-1-CZ-ERA MUNDUS-EMMC
European Erasmus Mundus Master Course
Sustainable Constructions under Natural
Hazards and Catastrophic Events
2C09-L3 – Dynamic response of single-degree-offreedom systems II
L3.1 – Response of SDOF systems to step, ramp and
harmonic forces.
L3 – Dynamic response of single-degree-of-freedom systems II
Response to step force
 Step force:
 Duhamel integral 
p  t   p0
t0
p0 
2 t 
u (t )   ust 0 1  cos n t   1  cos

k 
Tn 
Response to step force
 Maximum displacement (undamped system): u0  2  u st 0
 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
Response to ramp force
t
p  t   p0
t0
 Ramp force
tr
 Response of an undamped system:
 t sin nt 
 t Tn sin 2 t Tn 
u (t )   ust 0  

   ust 0 

n tr 
2 tr Tn 
 tr
 Tn tr
 The system vibrates with a period Tn about the static
position
Response to step force with finite rise time
 Force (ramp phase and constant phase):
 p0  t tr 
0  t  tr
p t   
t  tr
 p0
 Response of an undamped system:
– ramp phase
 t sin nt 
u (t )   ust 0  

t

t
n r 
 r
t  tr
– constant phase


1
u (t )   ust 0 1 
sin nt  sin n  t  tr   
 n t r

t  tr
Response to step force with finite rise time
 Ramp phase: system
vibrates with a period
Tn about the static
position
 Constant phase: idem
 u  tr   0  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
Harmonic vibrations of undamped systems
 Harmonic force: p (t )  p0 sin  t or
– amplitude p0
– circular frequency 
p (t )  p0 cos  t
Harmonic vibrations of undamped systems
  ku  p0 sin  t
 Equation of motion: mu
 Initial conditions u  u (0) u  u (0)
p0
1
sin t   n
 Particular solution u p (t ) 
2
k 1   n 
 Complementary solution uc (t )  A cos n t  B sin n t
p0
1
sin  t
 Complete solutions u (t )  A cos n t  B sin n t 
2
k 1    n 
 Final solution
 u  0  p
 / n 
p0
1
0
u (t )  u  0  cos n t  

sin t
 sin n t 
2
2
k 1   n  
k 1   n 
 n
transient response
steady-state response
Harmonic vibrations of undamped systems
 n  0.2
u (0)  0
u (0)  n po / 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
 u  0  p
 / n 
p0
1
0
u (t )  u  0  cos n t  

sin t
 sin n t 
2
2
k 1   n  
k 1   n 
 n
steady-state response
transient response
 Neglecting dynamic response  static response
p0
ust  t   sin t
k
 ust 0
p0

k


1
 sin t
 Steady-state response: u (t )   ust 0 
2
1   n  
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.


1
u (t )   ust 0 
 sin t
2
1   n  
Harmonic vibrations of undamped systems
 Steady-state response:


1
u (t )   ust 0 
 sin t
2
1   n  
 Alternative representation of steady-state response:
u (t )  u0 sin  t      u st 0 Rd sin  t   
u0
1
Rd 

 ust 0 1   n 2
and
0   n

   n
Displacement response factors
 Displacement response
u0
factor
Rd 
 ust 0
– 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)
Resonance
 Solution for the equation of motion when =n:
p0
– particular solution u p  t   
n t cos n t n  
2k
1 p0
– total solution
u (t )  
n t cos nt  sin nt 
2 k
u (0)  u (0)  0
Harmonic vibrations of damped systems
 Equation of motion
mu  cu  ku  p0 sin  t
 Initial conditions u  u (0)
u  u (0)
 Particular solution u p (t )  C sin t  D cos t
2
1    n 
p
C 0
k 1     2  2   2     2
n
n 

 
D
2  n 
p0
k 1    2  2   2     2
n
n 

 
 Complementary solution uc (t )  e
n t
 A cos D t  B sin D t 
 D  n 1   2
 Complete solution
u (t )  e nt  A cos D t  B sin D t   C sin t  D cos t
transient response
steady-state response
Harmonic vibrations of damped systems
u (t )  ent  A cos D t  B sin D t   C sin t  D cos t
transient response
steady-state response
 n  0.2   0.05
u (0)  0 u (0)  n po / k
Harmonic vibrations of damped systems: =n
 For =n response of a damped SDOF system is:
u (t )   ust 0
u (t )
 ust 0
1
2





 n t
e
 cos D t 
sin D t   cos n t 
2




1




1 n t
e
 1 cos n t
2


  0.05   n
u (0)  u (0)  0
Harmonic vibrations of damped systems: =n
 Small damping:
– Larger amplitude
– More cycles to attainment of a certain ratio of the steady-state
response
  n
u (0)  u (0)  0
Harmonic vibrations of damped systems: Rd and
 Steady-state response can be written as:
u  t   u0 sin t      ust 0 Rd sin t   
 Displacement response factor Rd
u0
1
Rd 

 ust 0 1   /  2  2   2  /    2
n
n 

 
  tan
1
2  / n 
1    / n 
2
Harmonic vibrations of damped systems: Rd and
  0.2
Harmonic vibrations of damped systems: Rd and
  n  1 : amplitude of dynamic
p0
u0   ust 0 
response close to the static
k
deformation (Rd 1) and almost
independent of damping. Response
controlled by stiffness of the system.
   n  1 : amplitude of dynamic
response approaches 0 (Rd 0) and
n2
p0
u0   ust 0 2 
almost independent of damping.
2

m

Response controlled by mass of the
system.
   n  1 : amplitude of dynamic
 ust 0 p0
response larger than the static
u0 

deformation (Rd  max) and sensible to
2
cn
damping. Response controlled by
damping of the system.
Harmonic vibrations of damped systems: Rd and
  n  1 : phase angle  close to 0,
displacement in phase with the applied
force.
   n  1 : phase angle  close to ,
displacement out of phase with the
applied force.
   n  1 : phase angle  equal to /2 for
any value of , displacement maximum
when force equals 0.
Resonance
 Resonant frequency:
frequency for which the
maximum response in
terms of displacement
(or velocity or
acceleration) is obtained
 Displacement resonant
frequency:
  n 1  2 2
 Maximum response:

Rd  1 2 1   2

Half-power bandwidth
 Difference
between circular
frequencies for
which the
displacement
response factor is
1 2 times
smaller than the
resonant
response
b  a

2n
Damping for engineering structures
stress level
structural type
welded steel structures, prestressed concrete structures,
strongly reinforced concrete structures (limited cracks)
stress level
below 0.5 times reinforced concrete structures with significant cracking
the yield strength
steel structures with bolted or riveted connections, wood
structures connected with screws or nails
 (%)
2-3
3-5
5-7
welded steel structures, prestressed concrete structures
(without total loss of prestress)
5-7
prestressed concrete structures with total loss of
prestress
7-10
stresses close to reinforced concrete structures
the yield strength
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
[email protected]
http://steel.fsv.cvut.cz/suscos