Example Analysis of MDOF Forced Damped Systems Analysis of MDOF Forced Damped Systems ASEN 3112 Lecture 22 – Slide 1 ASEN 3112 - Structures Objective This Lecture introduces damping

ASEN 3112 - Structures

22Example Analysisof MDOF ForcedDamped Systems

ASEN 3112 Lecture 22 – Slide 1


Objective

This Lecture introduces damping within the context of modal analysis.To keep the exposition focused we will primarily restrict the kind ofdamping 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.



Good and Bad News

Accounting for damping effects brings good and bad news. All realdynamical syste, experience damping because energy dissipation is likedeath and taxes: inevitable. Hence inclusion makes the dynamic modelmore physically realistic.

The bad news is that it can seriously complicate the analysis process.Here the assumption of light viscous damping helps: it allows thereuse of major parts of the modal analysis techniques coveredin the previous three Lectures.



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 motiongenerates 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 offriction. In structural systems, damping is more complex, appearing inseveral forms. These may be broadly categorized into

internal versus external distributed versus localized



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 representsenergy dissipated per unit volume of material and per stress cycle. Closely linked to cyclic motions.



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 mediasuch as soil. This form is often modeled as Coulomb damping, whichdescribes 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 asviscosity, convection or turbulence. A well known instance isa vehicle shock absorber: a fluid (liquid or air) is forcedthrough a small opening by a piston.



Distributed versus Localized Damping

All damping ultimately comes from frictional effects, which may takeplace at several scales. If the effects are distributed over volumes orsurfaces at macro scales, we speak of distributed damping.

But occasionally the engineer uses damping devices intended to producebeneficial 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 aspoint forces or moments, and said to produce localized damping. The distinction beteen distributed and localized appears at the modelinglevel, since all motion-damper devices ultimately work as a result ofsome kind of internal energy dissipation at material micro scales.



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, whichare beyond the scope of the course.



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 moststructures is light. In addition the presence of damping isusually beneficial to safety in the sense that resonance effectsare 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



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, whichusually will be assumed to be light.


Matrix EOM of Two-DOF ExampleASEN 3112 - Structures

m1 00 m2

u1

u2+ c1 + c2 −c2

−c2 c2

u1

u2+ k1 + k2 −k2

−k2 k2

u1

u2= p1

p2

Consider again the two-DOF mass-spring-dashpot example systemof Lecture 19. This is reproduced on the right for convenience.

The physical-coordinate EOM derived in that Lecture are,in detailed matrix notation:

��

c

k

p (t) 1

2

k1

u = u (t)22

u = u (t)11

1

p (t) 2

c2

1Mass m

2Mass m

Static equilibriumposition

Static equilibriumposition



Matrix EOM of Two-DOF Example (cont')

Passing to compact matrix notation,

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).

. ..

...M u + C u + K u = p



EOM Using Undamped ModesThis 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 U = ω M U . Normalize the vibration mode shapes U into φ so that they are orthonormal wrt M:

in which δ denotes the Kronecker delta. Let Φ be the modal matrixconstructed with the orthonormalized mode shapes as columns, and denoteby 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

φ Ti

i i

i i i

M φj = δ ij

ij

u = Φη

2

..



Matrix EOM of Damped System (cont')

Mg = ΦT M Φ Cg = ΦT C Φ K

g

g

g = ΦT K Φ f = ΦT p

Mg = Iγ Kg = diag[ω2i ]γ

Following the same scheme as in previous Lectures, the transformedEOM in modal coordinates are:

Define the generalized mass, damping, stiffness and forces as

Of these the generalized mass matrix M and the generalized stiffness matrix K were introduced in Lecture 20. If Φ contains mode shapes orthonormalized wrt M, it was shown there that

are diagonal matrices. The generalized forces f were introducedin the previous Lecture.

ΦT M Φ u + ΦT C Φ u + ΦT K Φ u = ΦT p(t)



Matrix EOM of Damped System (cont')

The new term in the modal EOM is the generalized damping matrix C ,also called the modal damping in the literature. Substituting the definitions we arrive at the modal EOM for the damped system:

Here we run into a major difficuty: C 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.

η(t) (t) ([ t)] (t)+ Cg

g

g

η + diag ω2i η = f



Three Ways Out

There are three ways out of the dead end:

• Diagonalization. Stay with the modal EOM, but make C 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.)

g



Diagonalization

Advantages

Diagonalization allow straightforward reuse of undamped frequenciesand mode shapes, which are fairly easy to obtain with standard eigensolver software. The uncoupled modeal equations often havestraightforward physical interpretation, allowing comparison with experiments. Only real arithmetic is necessary.

Disadvantages

We don't solve the original EOM, so some form of approximation isgenerally 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.



When Diagonalization Fails

There are problems, however, in which diagonalization cannot adequatelyrepresent damping effects within engineering accuracy. Three suchscenarios:

(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.



Complex Eigensystem

The complex eigensystem approach is mathematically irreproachable andcan solve the original EOM in physical coordinates without additionalapproximations. 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.



Direct Time Integration (DTI)

DTI has the advantages of being completely general. Numerical timeintegration can in fact handle not only the linear EOM, but nonlinearsystems, which occur for other types of damping (e.g. Coulombfriction, turbulent fluid drag). No transformation to mode coordinatesis 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) lieoutside the scope of an introductory course (they are usually taught at the graduate level) our choice is easy: diagonalization it is.



Example System With Specific Data& 3 Free Parameters

[m1 00 m2

] [u1

u2

]+

[c1 + c2 −c2

−c2 c2

] [u1

u2

]+

[k1 + k2 −k2

−k2 k2

] [u1

u2

]=

[p1

p2

]

m1 = 2, m2 = 1, k1 = 6, k2 = 3, c2 = c1 = c, p1 = 0, p2 = F2 cos �t

[2 00 1

] [u1

u2

]+

[2c −c−c c

] [u1

u2

]+

[9 −3

−3 3

] [u1

u2

]=

[0

F2 cos �t

]

M =[

2 00 1

], C =

[2c −c−c c

], K =

[9 −3

−3 3

], p =

[0

F2 cos �t

]



Generalized Mass, Damping, Stiffness and ForcesIn Terms of The Undamped Modal Matrix Φ

ω21 = 3

2, ω2

2 = 6, Φ = [ φ1 φ2 ] = 1√

61√3

2√6

− 1√3

=

[0.4082 0.5773

0.8165 −0.5773

]

Mg = ΦT M Φ =[

1 00 1

]= I, Cg = ΦT C Φ =

c3 − c

3√

2

− c3√

25c2

Kg = ΦT K Φ =[

3/2 00 6

]= diag[3/2, 6], f = ΦT p = F2 cos �t

[ 2√6

1√3

](t) (t)



Modal EOMs Are Coupled Through C

η + Cg η + diag[3/2, 6] η = f

Cg = ΦT C Φ =

c3 − c

3√

2

− c3√

25c2

g

(t) (t) (t) (t)



Diagonalization Device: Rayleigh Damping

g g gT

Often used in Civil Engineering structures. Assume that the dampingmatrix is a linear combination of the mass and stiffness matrices:

in which a and a 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):

which is a diagonal matrix. If the modes in Φ are orthonormal with respect to the mass matrix, the diagonal entries of C are

C = a M + a K0

0

0

0

1

1

12

1

C = Φ C Φ = a M + a K

C = a + a ω gii i

assume



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

where ξ is the dimensionless damping ratio and ω the natural frequency.By analogy the i diagonal entry of the Rayleigh-damping diagonalized C can be taken to be 2 ξ ω . Equating that to C = a + a ω (from previous slide) shows that the i damping ratio is

The assignment of values to a and a is often done by matching thedamping ratios of two modes. For example, matching ξ for mode 1and ξ for mode 2 gives two linear equations

from which a and a can be determined.

1

1

1

1

1 12

2

2

211

1

1

i ii

ii i2

0

0

0

0

0

0

12

giig

2 ξ ω

ξ = + a ω aω

( )

12

ξ = + a ω aω

( )12

ξ = + a ω aω

( )

th

th



A Second Diagonalization Device Also Bears The Name Rayleigh

(but for a different purpose)

The Rayleigh Quotient (RQ) was introduced by Lord Rayleighas a device to approximate the fundamental frequency of alinear 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 cyclewhen the damping matrix is diagonalized



"Rayleigh Quotient" (abbrev. RQ) Diagonalization Of Damped Matrix C

CRQg = diag[C RQ

1 , C RQQ

Q

2 ] =[

C R1 00 C R

2

]

C RQ1 = φT

1 Cφ1

φT1 φ1

= 2c

5, C RQ

2 = φT2 Cφ2

φT2 φ2

= 5c

2

ξRQ1 = 2c

5(2ω1)= 0.1633 c ξ

RQ2 = 5c

2(2ω2)= 0.5103 c

The effective modal damping factors are



RQ-Damped Modal EOM Now Decouple

But an approximation has been introduced(Is it serious?)

η1 + 2c

5η1 + 3

2η1 = 2√

6F2 cos �t

η2 + 5c

2η2 + 6η2 = − 1√

3F2 cos �t

(t) (t)

(t)(t)

(t)

(t)



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

��

c

k

p (t) 1

2

k1

u (t)2

u (t)1

1

p (t) 2

c2

1Mass m

2Mass m

m1 = 2, m2 = 1, k1 = 6, k2 = 3, c2 = c1 = c,

p1 = 0, p2= F2 cos �t


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Example Analysis of MDOF Forced Damped Systems Analysis of MDOF Forced Damped Systems ASEN 3112 Lecture 22 – Slide 1 ASEN 3112 - Structures Objective This Lecture introduces damping