.

25Modal Analysis

of MDOF ForcedDamped Systems

25–1

Lecture 25: MODAL ANALYSIS OF MDOF FORCED DAMPED SYSTEMS 25–2

TABLE OF CONTENTS

Page§25.1. What is Mechanical Damping? 25–3

§25.1.1. Internal vs. External Damping . . . . . . . . . . . . . 25–3§25.1.2. Distributed vs. Localized Damping . . . . . . . . . . . 25–3

§25.2. Modeling Structural Damping 25–4

§25.3. Matrix Equations of Motion 25–4§25.3.1. Equations of Motion Using Undamped Modes . . . . . . . 25–4§25.3.2. Three Ways Out . . . . . . . . . . . . . . . . . 25–6

§25.4. Diagonalization by Modal Damping 25–6§25.4.1. Guessing Damping Factors . . . . . . . . . . . . . . 25–7§25.4.2. Energy Equivalent Damping Factor . . . . . . . . . . 25–7

§25.5. Diagonalization by Rayleigh Damping 25–7

25–2

25–3 §25.1 WHAT IS MECHANICAL DAMPING?

The present lecture introduces damping within the context oif modal analysis. To keep the expositionfocused 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 ratio that issmall compared to unity.

Accounting for damping effects brings good and bad news. All real dynamical systems experiencedamping because energy dissipation mechanisms are like death and taxes:* inevitable. Hence inclusionmakes the dynamical model more physically realistic. The bad news is that it can seriously complicatethe analysis process. Here is where the assumption of light viscous damping helps: it allows the reuseof key parts of the modal analysis techniques introduced in the previous 3 lectures.

§25.1. 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 thesulphur content. When shivering under cold, we rub palms against each other to warm up. Those aretwo classical examples of the thermodynamical effect of friction. In structural systems, damping is morecomplex, appearing in several forms. These may be broadly categorized into internal versus external,and distributed versus localized.

§25.1.1. Internal vs. External Damping

Internal damping is due to the structural material itself. Sources are varied: microstructural defects,crystal grain boundaries, eddy currents in ferromagnetic media, dislocations in metals, and chain motionin polymers. The key macroscopic effect is the production of a hysteresis loop in stress-strain plots. Thehysteresis loop area represents the energy dissipated per unit volume of material and per stress cycle.This kind of damping is intimately related to cyclic motions such as vibration.

External damping come from boundary effects. An important form is structural damping, which isproduced by rubbing friction: stick-and-slip contact or impact. That may happen between structuralcomponents such as joints, or between a structural surface and non-structural solid media such as soil.This form is often modeled by Coulomb damping, which describes the energy dissipation of rubbingdry-friction.

Another form of external damping is fluid damping. When a material is immersed in a fluid, such asair or water, and there is relative motion between the structure and the fluid, a drag force appears. Thisforce causes an energy dissipation thorugh internal fluid mechanisms such as viscosity, convection orturbulence. This is collectively known as fluid damping. One well known example is a vehicle shockabsorber: a fluid (liquid or air) is forced through a small opening by a piston.

§25.1.2. Distributed vs. Localized Damping

All damping ultimately comes from frictional effects, which may however take place at different scales.If the effects are distrbuted over volumes or surfaces at macro scales, we speak of distributed damping.But occasionally the engineer uses damping devices intended to produce benefical damping effects.For example: shock absorbers or airbags in cars, motion mitigators for buildings or bridges in seismiczones, or piezoelectric dampers for space structures. Those devices can be often treated in lumped form,modeled as point forces or moments, and said to produce localized damping. The distinction appearsprimarily at the modeling level, since all motion-damper devices ultimately work as a result of some kind

* One exception: tax-exempt vampires.

25–3

Lecture 25: MODAL ANALYSIS OF MDOF FORCED DAMPED SYSTEMS 25–4

of internal energy conversion. Localized damping devices are in turn classified into passive or active,but that would take us into control theory.

§25.2. Modeling Structural Damping

The foregoing summary should make clear that damping is a ubiquitous but complicated business. Instructures containing joints, for example, Coulomb (dry friction) damping often dominates; this modelis partly nonlinear because the damping force depends on the sign of the velocity. Fluid damping tendsto be highly nonlinear if the interacting flow is turbulent, since if so the drag is nonlinear in the relativevelocity. Another modeling complication is that friction may depend on fabrication or constructiondetails that are not easy to predict.

Balancing those complications is the fact that damping in most structures, especially metallic ones, islight in the sense that the damping factor introduced in Lecture 21 is much smaller than one. In addition,the presence of damping is usually beneficial to safety in the sense that resonance effects are mitigated.This gives structural engineers leeway to simplify the dynamic analysis as follows.

• A simple damping model, such as linear viscous damping, can be assumed without much concern.

• Mode superposition is applicable because the EOM is linear. Moreover the frequencies and modeshapes for the undamped system can be reused if additional assumptions, such as Rayleigh damping,or modal damping, are made.

It should be stressed that such simplifications are not recommended if precise modeling of dampingeffects is important to safety and performance. That occurs in the following scenarios:

• Damping is crucial to function or operation. Think, for instance, of a shock absorber. Fortunatelydamper devices can be modeled more accurately than, say, dry friction.

• Damping may destabilize the system by feeding energy instead of removing it. This is true in activecontrol systems and aeroelasticity.

The last two scenarios are beyond the scope of this course. In this and the next lecture we shall focus onlinear viscous damping, which will be usually assumed to be light.

§25.3. Matrix Equations of Motion

Consider again the two-DOF mass-spring-dashpot example system of Lecture 22. This is reproduced inFigure 25.1 for convenience. The physica-coordinate EOM in detailed matrix notation are

[m1 00 m2

] [u1

u2

]+

[c1 + c2 −c2

−c2 c2

] [u1

u2

]+

[k1 + k2 −k2

−k2 k2

] [u1

u2

]=

[p1

p2

]. (25.1)

In compact notation,

M u + C u + K u = p. (25.2)

where M, C and K are the mass, damping and stiffness matrix, respectively, p, u, u and u are theforce, displacement, velocity and acceleration vectors, respectively. The latter four are function of time:u = u(t), etc, but the time argument will be often omitted for brevity.

In the sequel it will be assumed that M, C and K are symmetric. Furthermore M is positive definite (PD)while K is nonnegative definite (NND).

25–4

25–5 §25.3 MATRIX EQUATIONS OF MOTION

��

F = m u..

F = k us1

s2

s2

.F = c ud1

c

k

(a) (b)

p (t) 1p (t) 1

1 1

2

k1

u = u (t)22

u = u (t)11

1

1 11

I1 1 1

F = m u..

Ip (t) 2p (t) 2

2 2 2

1

F = k (u −u ) . .F = c (u −u )d2 222

Fd2

2

F c2

1Mass m

2Mass m

Static equilibriumposition

Static equilibriumposition

x

Figure 25.1. Two-DOF, forced, damped spring-mass example system: (a) configuration, (b) DFBD.

§25.3.1. Equations of Motion Using Undamped Modes

This technique attempts to reuse the modal analysis methods introduced in Lectures 22-24. Supposethat damping is negligible whence C = 0. Get the natural frequencies and mode shapes of the unforced,undamped system governed by M u + K u = 0, by solving the eigenproblem K U = ω2 M U. Orthonor-malize the vibration mode shapes Ui ⇒ φi so that they are orthonormal with respect to the mass matrix:φT

i Mφi = δi j , where δi j denotes the Kronecker delta: δi j = 1 if i = j , else zero (the Kronecker delta).Let Φ be the modal matrix constructed with the orthonormalized mode shapes φi , and denote by η thearray of modal amplitudes ηi , also called generalized coordinates.

As before, assume mode superposition: u = Φη. Following the same scheme as in the previous twoLectures, the transformed EOM in generalized coordinates are

ΦT M Φ + ΦT C Φ + ΦT KΦ = ΦT p(t). (25.3)

Define the generalized versions of mass, damping, stiffness and forces as

Mg = ΦT M Φ, Cg = ΦT C Φ, Kg = ΦT K Φ, f = ΦT p(t). (25.4)

Of these, the generalized mass matrix Mg and the generalized stiffness matrix Kg were introduced inLecture 23. If Φ is built by stacking mass-orthonormalized vibration modes as columns, it was shownthere that

Mg = I, Kg = diag[ω2i ]. (25.5)

That is, Mg reduces to the identity matrix while Kg becomes a diagonal matrix with squared frequenciesstacked along its diagonal. The modal forces f(t), also called generalized forces was introduced in

25–5

Lecture 25: MODAL ANALYSIS OF MDOF FORCED DAMPED SYSTEMS 25–6

Lecture 24. The new kid in (25.3) is the generalized damping matrix Cg = ΦT C Φ, Substituting (25.5)into (25.4) we arrive at the modal EOM for the damped system:

η(t) + Cg η(t) + diag(ω2i ) η(t) = f(t). (25.6)

Here we ran into a major difficulty: matrix Cg generally will not be diagonal. If that happens, the modalEOM (25.6) will not decouple. We seem to have gone through a promising path, but hit a dead end.

§25.3.2. Three Ways Out

There are three ways out of the dead end,*

• Diagonalization. Keep working with (25.6), but make Cg diagonal through some artifice.

• Complex Eigensystem. Set up and solve a different eigenproblem that diagonalizes two matricesthat comprise M, C and K as submatrices. The scheme name comes from the fact that it generallyleads to frequencies and mode shapes that are complex.

• Direct Time Integration or DTI. Integrate numerically in time the EOM in physical coordinates.

Each approach has strengths and weaknesses. (Obviously, else we would mention only one.)

Diagonalization allows straightforward reuse of undamped frequencies and mode shapes, which are fairlyeasy to obtain with standard eigensolution software. The uncoupled modal equations often have physicalinterpretation and mesh well with experiments. Only real arithmetic is necessary. The down side is thatwe dont solve (25.2) directly, so some form of approximation is inevitable. This is counteracted bythe fact that structural damping is often difficult to quantify since it can come from so many sources.Thus the approximation in solving (25.2) may be tolerable in view of modeling uncertainties. This isparticularly true if damping is light.

There are problems, however, in which diagonalization cannot properly represent damping effects.Threeof them are: (1) structures with localized damper devices (e.g., shock absorbers), (2) structure-mediainteraction (e.g., building foundations, tunnels, aeroelasticity, surface ships), (3) active control systems.In those situations one of the two remaining approaches must be taken.

The complex eigensystem method is mathematically irreproachable and can solve (25.2) without addi-tional approximations. No assumptions as to light versus heavy damping are needed. But it involves asubstantial amount of preparatory work since the EOM must be transformed to the so-called state spaceform. For a large number of DOF, solving complex eigensystems is unwieldy. Physical interpretationof complex frequencies and modes is less immediate and require substantial expertise in math as well asengineering experience. But in the three scenarios listed above it shoud be preferred to diagonalization.

Direct time integration (DTI) has the advantage of being completely general. Numerical time integrationcan in fact handle not only the linear EOM (25.2), but also nonlinear systems as well as non-viscousdamping mechanisms. No transformations to mode coordinates is necessary, and no complex arithmeticemerges. The main disadvantage is that it requires substantial expertise in computational handling ofODE, which is a hairy topic onto itself. Since DTI can only handle fully numerically-specified models,the approach is not particularly useful during preliminary design stages when free parameters floataround.

Because the last two approaches lie outside the scope of an introductory course (they are taught at themaster or senior-elective levels), our choice is easy: diagonalization it is.

* In the textbook of Craig-Kurdila the first two approaches listed above are called mode superposition through real modes ofthe undamped system, and mode superposition through complex modes of the damped system, respectively. See their Section10.1.10 for details.

25–6

25–7 §25.5 DIAGONALIZATION BY RAYLEIGH DAMPING

§25.4. Diagonalization by Modal Damping

In this approach the generalized damping matrix Cg = Φ CΦ is assumed to be diagonal from the start,using modal damping factors

Cg = diag[ 2ξiωi ], i = 1, 2, . . . n. (25.7)

in which n is the number of DOF and ξi denotes the damping factor for the i th natural mode. As aresult, the modal EOM (25.6) decouple, reducing to n canonical second-order equations in the modalamplitudes

ηi + 2ξiωi ηi + ω2i η = fi (t), i = 1, 2, . . . n. (25.8)

These equations can be solved using the methods described in Lecture 21. The solutions can be super-posed via the mode decomposition assumption u(t) = Φη(t) to get the physical response.

The method is straightforward. Two technical difficulties remain. First, how are modal damping factorspicked? Second, what is the error incurred by the decoupling assumption?

§25.4.1. Guessing Damping Factors

One time honored approach is educated guessing. This is of course the only possibility if (a) little isknown about the damping level and sources and (b) there are no experimental results; for example thestructure exists only on paper. The structural engineer then makes recourse to experience with similarsystems. With an air of authority she says: “Ahem, let assign 1% to modes 1 through 5, 2% to modes 6through 20, and 4% to all higher ones.” Done.

There is some method in this madness. First, damping factors of well constructed structures are typicallysmall compared to unity: 1 to 5% is typical. Second damping generally increases with frequency, thereason being that more hysteresis cycles take place within a fixed time interval.

§25.4.2. Energy Equivalent Damping Factor

This technique is applicable if C is available. It can be shown that the power dissipated by viscousdamping in an unforced system governed by the physical-coordinates EOM (25.2) with p = 0 is

D = 12 u C u. (25.9)

The kinetic and potential energies of the system are

T = 12 u M u, V = 1

2 u K u, (25.10)

Now suppose that the structure is moving in one of the undamped modes, say φi , The damping factorcan be chosen so that the energy dissipated in that mode matches that of the full discrete system. Thisleads to the Rayleigh Quotient rule. (Topic to be developed further, since it is not in C-K.)

25–7

Lecture 25: MODAL ANALYSIS OF MDOF FORCED DAMPED SYSTEMS 25–8

§25.5. Diagonalization by Rayleigh Damping

This diagonalization procedure is widely used for civil structures, especially for seismic response cal-culations. The viscous damping matrix C is directly defined as a linear combination of the mass andstiffness matrix:

C = a0 M + a1 K. (25.11)

in which a0 and a1 are selected constants (with appropriate physical dimensions). The method is alsocalled proportional damping in the literature. Applying the modal matrix congruential transformationto this C results in

Cg = ΦT C Φ = diag[Cr ] = diag[a0 + a1 ω2i ] = diag[2ξi ωi ], (25.12)

whence the effective modal damping factor is

ξi = 1

2

(a0

ωi+ a1ωi

). (25.13)

Choosing the damping factor for two modes of different frequencies and solving (25.13) for a0 and a1

yields C from (25.11). In practice the stiffness proportional term is more physically relevant (because, aspreviously noted, damping usually increases with frequency). Thus engineers tend to adjust the choiceof those frequencies so that ξi is roughly minimized for the lowest frequency mode.

25–8