ASEN 3112 - Structures
22
Example Analysis
of MDOF Forced
Damped Systems
ASEN 3112 Lecture 22 – Slide 1
ASEN 3112 - Structures
Objective
This Lecture introduces damping within the context of modal analysis.
To keep the exposition focused we will primarily restrict the kind of
damping considered to be linearly viscous, and light.
Linearly viscous damping is proportional to the velocity. Light
damping means a damping factor that is small compared to unity.
In the terminology of Lecture 17, lightly damped mechanical systems
are said to be underdamped.
ASEN 3112 Lecture 22 – Slide 2
ASEN 3112 - Structures
Good and Bad News
Accounting for damping effects brings good and bad news. All real
dynamical syste, experience damping because energy dissipation is like
death and taxes: inevitable. Hence inclusion makes the dynamic model
more physically realistic.
The bad news is that it can seriously complicate the analysis process.
Here the assumption of light viscous damping helps: it allows the
reuse of major parts of the modal analysis techniques covered
in the previous three Lectures.
ASEN 3112 Lecture 22 – Slide 3
ASEN 3112 - Structures
What is Mechanical Damping?
Damping is the (generally irreversible) conversion of mechanical
energy into heat as a result of motion.
For example, as we scratch a match against a rough surface, its motion
generates heat and ignites the sulphur content. When shivering under cold,
we rub palms against each other to warm up.
Those are two classical examples of the thermodynamic effect of
friction. In structural systems, damping is more complex, appearing in
several forms. These may be broadly categorized into
internal versus external
distributed versus localized
ASEN 3112 Lecture 22 – Slide 4
ASEN 3112 - Structures
Internal versus External Damping
Internal damping is due to the structural material itself.
Various sources: microstructural defects, crystal grain slip,
eddy currents (in ferromagnetic materials), dislocations in metals,
chain movements in polymers.
Key macroscopic effect: a hysteresis loop. Loop area represents
energy dissipated per unit volume of material and per stress cycle.
Closely linked to cyclic motions.
ASEN 3112 Lecture 22 – Slide 5
ASEN 3112 - Structures
Internal versus External Damping (cont'd)
External damping comes from boundary effects. An important form
is structural damping, produced by rubbing friction, stick and
slip contact or impact. May take place beween structural components
such as joints, or between a structural surface and non-structural media
such as soil. This form is often modeled as Coulomb damping, which
describes the energy dissipation of rubbing dry friction.
Another form of external damping is fluid damping. When a material
is immersed in a fluid such as air or water and there is relative motion
between the structure and the fluid, a drag force appears. This force
causes energy dissipation through internal fluid mechanisms such as
viscosity, convection or turbulence. A well known instance is
a vehicle shock absorber: a fluid (liquid or air) is forced
through a small opening by a piston.
ASEN 3112 Lecture 22 – Slide 6
ASEN 3112 - Structures
Distributed versus Localized Damping
All damping ultimately comes from frictional effects, which may take
place at several scales. If the effects are distributed over volumes or
surfaces at macro scales, we speak of distributed damping.
But occasionally the engineer uses damping devices intended to produce
beneficial effects. For example:
shock absorbers, airbags, parachutes
motion mitigators for structures in seismic or hurricane zones
active piezoelectric dampers for space structures
These devices can be sometimes idealized as lumped objects, modeled as
point forces or moments, and said to produce localized damping.
The distinction beteen distributed and localized appears at the modeling
level, since all motion-damper devices ultimately work as a result of
some kind of internal energy dissipation at material micro scales.
ASEN 3112 Lecture 22 – Slide 7
ASEN 3112 - Structures
Distributed versus Localized Damping (cont')
Localized damping devices may in turn be classified into
passive: no feedback
active: responding to motion feedback
But this would take us too far into control systems, which
are beyond the scope of the course.
ASEN 3112 Lecture 22 – Slide 8
ASEN 3112 - Structures
Modeling Damping in Structures
In summary: damping is complicated business. It often is
nonlinear, and level may depend on fabrication or construction
details that are not easy to predict.
Balancing those complications is the fact that damping in most
structures is light. In addition the presence of damping is
usually beneficial to safety in the sense that resonance effects
are mitigated, This gives the structural engineer some leeway:
o A simple model, such as linear viscous damping, can be assumed
o Mode superposition is applicable because the EOM is linear.
Frequencies and mode shapes for the undamped system can
be reused if additional assumptions, such as Rayleigh damping
or modal damping, are made
ASEN 3112 Lecture 22 – Slide 9
ASEN 3112 - Structures
Modeling Damping in Structures (cont')
It should be stressed that the foregoing simplifications are not
recommended if precise modeling of damping effects is important
to safety and performance. This occurs in the following scenarios:
o
Damping is crucial to function or operation. Think, for
instance, of a shock absorber, airbag, or parachute.
o
Damping may destabilize the system by feeding energy instead
of removing it. This can happen in active control systems
and aeroelasticity.
The last two scenarios are beyond the scope of this course. In this
Lecture we focus attention on linear viscous damping, which
usually will be assumed to be light.
ASEN 3112 Lecture 22 – Slide 10
;;
;;
ASEN 3112 - Structures
Matrix EOM of Two-DOF Example
Consider again the two-DOF
mass-spring-dashpot example system
of Lecture 19. This is reproduced on
the right for convenience.
k1
c1
Static equilibrium
position
u1 = u1(t)
Mass m1
p1(t)
The physical-coordinate EOM
derived in that Lecture are,
in detailed matrix notation:
Static equilibrium k2
position
c2
u 2 = u2(t)
Mass m 2
p2 (t)
m1
0
0
m2
ü 1
c + c2 −c2
+ 1
ü 2
−c2
c2
u̇ 1
k + k 2 −k 2
+ 1
u̇ 2
−k 2
k2
ASEN 3112 Lecture 22 – Slide 11
u1
u2
=
p1
p2
ASEN 3112 - Structures
Matrix EOM of Two-DOF Example (cont')
Passing to compact matrix notation,
.
..
Mu+Cu +Ku=p
Here M, C and K denote the mass, damping and stiffness matrices,
..
.
respectively, p, u, u and u are the force, displacement, velocity and
acceleration vectors, respectively. The latter four are functions of time:
u = u(t), etc., but the time argument will be often omitted for brevity.
As previously noted, matrices M, C and K are symmetric, whereas M
is diagonal. In addition we will assume that M is positive definite
(PD) whereas K is nonnegative definite (NND).
ASEN 3112 Lecture 22 – Slide 12
ASEN 3112 - Structures
EOM Using Undamped Modes
This technique attempts to reuse modal analysis methods covered in
Lecture 19-21. Suppose that damping is removed so that C = 0.
Get the natural frequencies and mode shapes of the undamped and
..
unforced system governed by M u + K u = 0, by solving the
eigenproblem K Ui = ω2i M U i. Normalize the vibration mode
shapes U i into φ i so that they are orthonormal wrt M:
φiT M φj = δ ij
in which δ ij denotes the Kronecker delta. Let Φ be the modal matrix
constructed with the orthonormalized mode shapes as columns, and denote
by η the array of modal amplitudes, also called generalied coordinates.
As before, assume modal superposition is valid, so that physical DOF
are linked to mode amplitudes via
u = Φη
ASEN 3112 Lecture 22 – Slide 13
ASEN 3112 - Structures
Matrix EOM of Damped System (cont')
Following the same scheme as in previous Lectures, the transformed
EOM in modal coordinates are:
ΦT M Φ ü + ΦT C Φ u̇ + ΦT K Φ u = ΦT p(t)
Define the generalized mass, damping, stiffness and forces as
Mg = ΦT M Φ
Cg = ΦT C Φ
Kg = ΦT K Φ
f = ΦT p
Of these the generalized mass matrix Mg and the generalized stiffness
matrix Kg were introduced in Lecture 20. If Φ contains mode shapes
orthonormalized wrt M, it was shown there that
Mg = Iγ
Kg = diag[ωi2 ]γ
are diagonal matrices. The generalized forces f were introduced
in the previous Lecture.
ASEN 3112 Lecture 22 – Slide 14
ASEN 3112 - Structures
Matrix EOM of Damped System (cont')
The new term in the modal EOM is the generalized damping matrix Cg ,
also called the modal damping in the literature. Substituting the
definitions we arrive at the modal EOM for the damped system:
η̈(t) + Cg η̇ (t) + diag [ωi2] η ( t) = f (t)
Here we run into a major difficuty: Cg generally will not be diagonal.
If that happens, the above modal EOM will not decouple. We seem
to have taken a promising path, but hit a dead end.
ASEN 3112 Lecture 22 – Slide 15
ASEN 3112 - Structures
Three Ways Out
There are three ways out of the dead end:
• Diagonalization. Stay with the modal EOM, but make Cg
diagonal through some artifice
• Complex Eigensystem. Set up and solve a different eigenproblem
that diagonalizes two matrices that comprise M, C and K as
submtarices. [The name comes from the fact that it generally leads
to frequencies and mode shapes that are complex numbers.]
• Direct Time Integration, or DTI. Integrate numerically the
EOM in physical coordinates.
Each approach has strengths and weaknesses. (Obviously,
else we would mention only one.)
ASEN 3112 Lecture 22 – Slide 16
ASEN 3112 - Structures
Diagonalization
Advantages
Diagonalization allow straightforward reuse of undamped frequencies
and mode shapes, which are fairly easy to obtain with standard
eigensolver software. The uncoupled modeal equations often have
straightforward physical interpretation, allowing comparison with
experiments. Only real arithmetic is necessary.
Disadvantages
We don't solve the original EOM, so some form of approximation is
generally inevitable. This is counteracted by the fact that structural
damping is often difficult to quantify since it can come from many sources.
Thus the approximation in solving the EOM may be tolerable in view of
modeling uncertainties. This is particularly true if damping is light.
ASEN 3112 Lecture 22 – Slide 17
ASEN 3112 - Structures
When Diagonalization Fails
There are problems, however, in which diagonalization cannot adequately
represent damping effects within engineering accuracy. Three such
scenarios:
(1) Structures with localized damper devices: shock absorbers,
piezoelectric dampers, ...
(2) Structure-media interaction: building foundations, tunnels,
aeroelasticity, parachutes, marine structures, surface ships, ...
(3) Active control systems
In those situations one of the two remaining approaches: complex
arithmetic or direct time integration (DTI) must be taken.
ASEN 3112 Lecture 22 – Slide 18
ASEN 3112 - Structures
Complex Eigensystem
The complex eigensystem approach is mathematically irreproachable and
can solve the original EOM in physical coordinates without additional
approximations. No assumptions as to light versus heavy damping
are needed.
However, it involves a substantial amount of preparatory work because
the EOM must be transformed to the so-called state-space form. For a
large number of DOF, solving complex eigensystems is unwieldy.
Physical interpretation of complex frequencies and modes is
less immediate and may require substantial expertise in math as well as
engineering experience. Finally, it is restricted in scope to linear dynamic
systems unless some convenient form of linearization is available.
ASEN 3112 Lecture 22 – Slide 19
ASEN 3112 - Structures
Direct Time Integration (DTI)
DTI has the advantages of being completely general. Numerical time
integration can in fact handle not only the linear EOM, but nonlinear
systems, which occur for other types of damping (e.g. Coulomb
friction, turbulent fluid drag). No transformation to mode coordinates
is necessary and no complex arithmetic emerges.
The main disadvantage is that requires substantial expertise
in computational handling of ODE, which is a hairy topic onto itself.
Since DTI can only handle numerically specified models, the approach
is not particularly useful during preliminary design stages,
when many design parameters float around.
Beacuse the last two approcahes (complex arithmetic and DTI) lie
outside the scope of an introductory course (they are usually taught
at the graduate level) our choice is easy: diagonalization it is.
ASEN 3112 Lecture 22 – Slide 20
ASEN 3112 - Structures
Example System With Specific Data
& 3 Free Parameters
m1
0
0
m2
ü 1
c + c2
+ 1
ü 2
−c2
−c2
c2
u̇ 1
k + k2
+ 1
u̇ 2
−k2
−k2
k2
2
0
2
M=
0
0
1
2c
ü 1
+
ü 2
−c
0
,
1
2c
C=
−c
−c
c
u̇ 1
9
+
u̇ 2
−3
−c
,
c
9
K=
−3
−3
3
u1
u2
−3
,
3
ASEN 3112 Lecture 22 – Slide 21
u1
u2
=
p1
p2
p2 = F2 cos t
m 1 = 2, m 2 = 1, k1 = 6, k2 = 3, c2 = c1 = c, p1 = 0,
=
0
F2 cos t
p=
0
F2 cos t
ASEN 3112 - Structures
Generalized Mass, Damping, Stiffness and Forces
In Terms of The Undamped Modal Matrix Φ
ω12
3
= ,
2
ω22
= 6,
Φ = [ φ1
Mg = ΦT M Φ =
1
0
 1
√
φ2 ] =  6
√2
6
0
= I,
1

√1
3  = 0.4082
0.8165
− √1
3

Cg = ΦT C Φ = 
3/2
0
T
Kg = Φ K Φ =
= diag[3/2, 6],
0 6
c
3
c
− √
3 2
0.5773
−0.5773
c 
− √
3 2
5c
2
2 √
6
f (t) = ΦT p (t) = F2 cos t
1
√
3
ASEN 3112 Lecture 22 – Slide 22
ASEN 3112 - Structures
Modal EOMs Are Coupled Through Cg
η̈ (t) + Cg η̇ (t) + diag[3/2, 6] η (t) = f (t)

Cg = ΦT C Φ = 
c
3
c
− √
3 2
c 
− √
3 2
5c
2
ASEN 3112 Lecture 22 – Slide 23
ASEN 3112 - Structures
Diagonalization Device: Rayleigh Damping
Often used in Civil Engineering structures. Assume that the damping
matrix is a linear combination of the mass and stiffness matrices:
assume
C = a0 M + a1 K
in which a 0 and a 1 are numerical coefficients with physical dimensions
of (1/T) and T, respectively, T being a time unit. Transforming to modal
coordinates gives the generalized damping matrix (a.k.a. modal matrix):
T
C g = Φ C Φ = a 0 Mg + a1 K g
which is a diagonal matrix. If the modes in Φ are orthonormal with
respect to the mass matrix, the diagonal entries of C are
Cgii = a 0 + a1 ωi2
ASEN 3112 Lecture 22 – Slide 24
ASEN 3112 - Structures
Choosing the Rayleigh Damping Coefficients
Recall that for the single-DOF oscillator with viscous damping, the
coefficient of the velocity term in the canonical form is
2ξω
where ξ is the dimensionless damping ratio and ω the natural frequency.
By analogy the i th diagonal entry of the Rayleigh-damping diagonalized
Cg can be taken to be 2 ξ i ω i . Equating that to C gii = a 0 + a 1 ω i2
(from previous slide) shows that the i th damping ratio is
1 a0
ξi =
+ a1 ωi
2 ωi
The assignment of values to a 0 and a1 is often done by matching the
damping ratios of two modes. For example, matching ξ1 for mode 1
and ξ 2 for mode 2 gives two linear equations
1
ξ1 =
2
a0
ω1 + a1 ω1
ξ2 = 1
2
a0
ω 2 + a 1 ω2
from which a 0 and a 1 can be determined.
ASEN 3112 Lecture 22 – Slide 25
ASEN 3112 - Structures
A Second Diagonalization Device
Also Bears The Name Rayleigh
(but for a different purpose)
The Rayleigh Quotient (RQ) was introduced by Lord Rayleigh
as a device to approximate the fundamental frequency of a
linear acoustic system if an approximate mode shape is known.
The RQ is frequently used in linear algebra just for
eigenvalue calculations. Here we will use it in another
context: conservation of dissipation energy over a cycle
when the damping matrix is diagonalized
ASEN 3112 Lecture 22 – Slide 26
ASEN 3112 - Structures
"Rayleigh Quotient" (abbrev. RQ)
Diagonalization Of Damped Matrix C
CgR Q
C1R Q
=
=
diag[C1R Q , C2R Q ]
φ1T Cφ1
φ1T φ1
2c
= ,
5
C1R Q
=
0
C2R Q
=
0
C2R Q
φ2T Cφ2
φ2T φ2
=
5c
2
The effective modal damping factors are
ξ1R Q =
2c
= 0.1633 c
5(2ω1 )
ξ2R Q =
5c
= 0.5103 c
2(2ω2 )
ASEN 3112 Lecture 22 – Slide 27
ASEN 3112 - Structures
RQ-Damped Modal EOM Now Decouple
η̈1(t) +
2c
3
2
η̇1 (t) + η1(t) = √ F2 cos t
5
2
6
η̈2 (t) +
5c
1
η̇2 (t) + 6η2 (t) = − √ F2 cos t
2
3
But an approximation has been introduced
(Is it serious?)
ASEN 3112 Lecture 22 – Slide 28
ASEN 3112 - Structures
;;
We Will Investigate This Issue
By Comparing Two Solutions
Exact responses of original EOM of example
system in physical coordinates (obtained by
Direct Time Integration, aka DTI)
Approximate responses obtained by solving
the RQ modal EOM and transforming to
physical coordinates by the undamped
modal matrix Φ
k1
u1(t)
c1
Mass m1
p1(t)
k2
c2
u2(t)
Mass m2
p2 (t)
m 1 = 2, m 2 = 1,
k1 = 6, k2 = 3,
p1 = 0,
ASEN 3112 Lecture 22 – Slide 29
c2 = c1 = c,
p2 = F2 cos t