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2C09 Design for seismic and climate changes · 2014-10-29 · European Erasmus Mundus Master Course...
Transcript of 2C09 Design for seismic and climate changes · 2014-10-29 · European Erasmus Mundus Master Course...
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 05: Dynamic analysis of multi-degree-of-freedom systems II
Daniel Grecea, Politehnica University of Timisoara
11/03/2014
L6 – Dynamic analysis of multi-degree-of-freedom systems II
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards and Catastrophic Events
L5.1 – Free vibration response. L5.2 – Damping matrix. L5.3 – Modal analysis.
2C09-L5 – Dynamic analysis of multi-degree-of-freedom systems II
Free vibrations of MDOF systems with damping Natural modes of the damped system identical to those of
the undamped system - {}n
Displacements similar to those of the undamped system, but amplitudes decrease with time
Response of each mass is harmonic, similarly to that of a SDOF system
Free vibrations of MDOF systems with damping For each natural mode n, equation of motion in modal
coordinates is
Dividing by Mn one gets: where
The same form as the equation of motion in the case of damped free vibrations of SDOF systems
Combining modal contributions:
0n n n n nM q C q Kq
22 0n n n n n nq q q
2n
nn n
CM
(0) (0)( ) (0)cos sinn nt n n nn n nD nD
nD
q qq t e q t t
21nD n n
1
(0) (0)(0)cos sinn n
Nt n n n
n nD nDnn nD
q qu t e q t t
Modal analysis Equation of motions of a MDOF system with damping
excited by dynamic forces:
Displacements {u} can be expanded as:
Replacing {u} in 4.82:
Multiplying 4.84 to the left by we obtain:
Which, considering orthogonality of natural modes becomes:
m u c u k u p t
1
N
rrr
u q q
4.82
1 1 1
N N N
r r rr r rr r r
m q t c q t k q t p t
4.84
Tn
1 1 1
N N NT T T T
r r rn r n r n r nr r r
m q t c q t k q t p t
n n n n n n nM q t C q t K q t P t 4.86
Modal analysis Dynamic response analysis of damped MDOF system
Equation of motions of a MDOF system with damping excited by dynamic forces:
Displacements {u} can be expanded as:
Replacing {u} in 4.82:
Multiplying 4.84 to the left by we obtain:
Which, considering orthogonality of natural modes becomes:
m u c u k u p t
1
N
rrr
u q q
4.82
1 1 1
N N N
r r rr r rr r r
m q t c q t k q t p t
4.84
Tn
1 1 1
N N NT T T T
r r rn r n r n r nr r r
m q t c q t k q t p t
n n n n n n nM q t C q t K q t P t 4.86
Modal analysis Dividing by Mn one gets:
Solving a system of N differential equations was reduced
to solution of N independent equations Direct estimation of the damping matrix [c] not necessary The same form with the equation of motions of a SDOF
system same solution methods Solution: modal coordinate qn(t) for mode n Contribution of mode n to total displacement {u(t)}:
Total displacements (combination of the contribution of
all modes):
22 nn n n n n n
n
P tq q q
M
nnnu t q t
1 1
N N
nnnn n
u t u t q t
Modal analysis Analysis procedure is called modal analysis and is
applicable only to linear systems with classical damping Element forces can be obtained using 2 methods: 1. Contributions rn(t) in n-th mode are obtained from
imposing displacements {u(t)}n Total forces are obtained by superposition of modal contributions
2. Equivalent static forces from the n-th mode are determined: Static analysis modal contributions rn(t) from the n-th mode
1
N
nn
r t r t
2 2n n nnn n n
f t k u t m u t m q t
1
N
nn
r t r t
Modal analysis: summary Define the structural properties
- mass [m] and stiffness [k] matrices - critical damping ratio n
Determine natural circular frequencies n and natural modes of vibrations {}n
Compute response in each mode following the sequence: - set up equation of motion - compute modal displacements {u(t)}n - compute element forces rn(t) from the n-th mode
Combine modal contributions to obtain the total response
Note: generally it is NOT necessary to consider ALL modes of vibration
1 1
N N
nnnn n
u t u t q t
22 nn n n n n n
n
P tq q q
M
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