Damping

Analysis Preliminaries

7.1 Modeling Assumptions

7.1.7 Damping

In section 4.8 the use of non-classical nonlinear damping was mentioned. Nonlinear inter-story damping is assembled through the use of story dampers. The damping strength,

F/J,

and the cutoff velocity, v§, are chosen to give a reasonable elastic damping in the lowest mode of the building. Figure 4.11 is repeated here to show these values. In each story, the damping strength is chosen to be equivalent to

F

0

---~~--~~---.-x

Figure 7.2 Nonlinear inter-story damping force.

the shear spring strength pgH defined above. This model makes sense physically, since the damping force provided by non-structural elements will level off as those elements get damaged. To make the inter-story damping proportional to the shear

building stiffness, two coefficients a2 and a3 are chosen so that

(7.3)

(7.4) and

Since

F{j

⁼

FgH,

a₃ = 1.0 for each story. The velocity required to achieve a desired level of damping in the first mode can be determined experimentally by applying a small impulse to the building and determining the damping from the observed decay in roof response. This will be the damping for low levels of excitation. For higher levels of excitation, the damping forces will level off when inter-story velocities exceed the cutoff velocity. The impulse loading is thus used to determine the factor

a2

=

vg in an iterative manner.

A half-period step impulse of low excitation is applied to the models. The decay in response at the roof is determined. The excitation is low enough to ensure elastic response. For stiffness-proportional damping, the damping in the first mode can be expressed as

(7.5)

( =

a¹w

2

where a 1 is the coefficient defined in section 4.8 for which C = a1K.

For shear building stiffness-proportional damping, C

=

(a3/a2)F8H must be satisfied for the unknown a2 and a3

=

1. Recalling the values of these parameters, this becomes Cvg

=

F8, which is true by definition. While the damping is actually proportional to the strength of the shear building, the yield shear and damping strengths are equivalent, so the damping "stiffness" is proportional to the shear building stiffness.

The rate of decay of the impulse loading determines the level of damping. For a

single degree of freedom system, the equation of motion without forcing is

(7.6) (7.7)

mx+ci+kx

=

0 or x

+

^2(wn:i:

+

^w~x

=

0

where 2(wn

=

c/m and w~

=

kjm. The response can be determined from

(7.8)

where wd

=

wn.J1- (^{2 •} The response at peak n occurs at timet. The response at the next peak n+ 1 occurs one damped period later, or at time t+27r/wd· The ratio of displacements at these steps can be expressed as

(7.9)

Xn+l This can be rewritten as

(7.10)

ln~

^{=27r ( .}

Xn+l ~

For small (, this can be solved for ( as

(7.11) ^{(~-ln--. 1 Xn}

27r Xn+l

When the response to an impulse is recorded, the first five or six peaks are deter- mined, and an average damping value is determined using equation (7.11).

If the approximated value of a2 is not correct, the next approximation can be determined from the following relationship

(7.12) ^{(desired ~} (a3/a2)desired (current ( a3 / a2) current

( ) ( ) (current

a2 desired ~ a2 current I" •

':.desired

(7.13) or

Equation (7.13) is used to approximate the value of a2 that should be used to achieve

the desired level of damping and another impulse loading is performed to confirm the resulting damping.

The damping for elastic response in the first mode is easily determined from the preceding approximations and trial impulses. Special consideration was made for the 10-story building that has two almost equivalent first modal frequencies.

If the impulse is applied in a direction other than one of the principal directions, both modes are excited and the observed decay in one mode is actually accelerated by the other mode picking up energy from that mode. This can be seen in figure 7.3, which shows an impulse applied in the E-W direction of the 10-story building, and a second impulse applied along the axis of symmetry. The impulse in the E-W direction shows energy increasing at first in the N-S direction. Energy is transferred into both of the two lowest modes and decays in both modes. The impulse along the axis of symmetry decays without energy transferring to another mode. The damping for this building was calculated using the axis of symmetry impulse (figure 7.4).

Once the damping matrix is determined, the damping for higher modes or non- linear levels of response can be assessed. In classical damping, the damping matrix can be diagonalized by the same modal matrix transformation used to diagonalize the mass and stiffness matrices. Classical damping allows the equations of motion to be separated into modal equations. If the damping matrix is not constructed from a Caughey series, or nonlinear response is being modeled, then the damping should be considered non-classical (Chopra 1995). The damping used in this work is considered non-classical because it is not proportional to stiffness and mass as in a Caughey series and also because it is nonlinear. While the damping is pro- portional to the shear building stiffness, this stiffness in turn is not proportional to the global stiffness. Additionally, the stiffness will change for nonlinear response, so the damping will change. There will be off-diagonal terms that cause interaction between modes during damped response, similar to the coupling mentioned above for multiple equal modes in elastic response. This effect exists even for the first mode, but the impulses used produce energy predominantly in the first mode. Since

Half period impulse along axis of symmetry E ¹

0 ...;-

0.5

c: Q)

E Q)

0 0

0. ttl

(/)

~ -0.5

0 0

a: -1

0 2 4 6 8 10 12 14 16

Time, sec.

Half period impulse in E-W direction

E 1 E-Wresponse

0 N-S response

...;- c: 0.5

Q)

E Q) \ I

~ 0 _\

0. I

(/)

~ -0.5

0 0

a: -1

0 2 4 6 8 10 12 14 16

Time, sec.

Figure 7.3 Effect of impulse direction.

E 0

...

_c:

Q)

E Q) 0 as

a. _(/)

'5

-

⁰ 0

a:

1

0.5

oo - E

Q)<O

(/)"it

<0<0

,... ...

t\ic:i

Damped response to impulse along axis of symmetry

oo _E

Q)<O (/),_

<O"it

C'><O

'<ic:i

-1~----~----~---~----~----~---~----~----~

0 2 4 6 8

Time, sec.

10 12

Figure 7.4 10-story impulse damped response.

14 16

stiffness proportional damping increases with frequency, any energy that leaks to higher modes from the first mode is quickly damped out.

The coupling of modes in a damped non-classical system makes it difficult to quantify the level of damping in each mode as a function of excitation level. Free vibrations initiated in one mode may end up in a lower mode even as the vibrations of the original mode are damped out. If instead the system is forced with a forcing function set to a specific modal frequency and shape, the level of damping for that mode can be estimated by determining the amplitude of the steady state response.

Consider a single degree of freedom system as an analogy:

(7.14)

mx +ex+

kx =Po sinwt.

The steady state response is

(7.15) ^{Po (1-} (w/wnf) sinwt- 2(w/wn coswt

Xp

= k

(1-(w/wn)²)2

+

(2(w/wn)²

which can be simplified for the case desired where w

=

^{Wn as}

(7.16) Xp

= -k

^{Po 1}

2( ^COSWnt

=

-apCOSWnt.

The value p0jk is just the static displacement,

as.

Looking at the amplitude of response at steady state, ap, and making the static displacement substitution, the damping ratio can be determined from

(7.17) ⁽⁼

as.

2ap

While this is only true for the single degree of freedom system, the damping esti- mated for multiple dof systems using this approach is accurate because the energy is forced into the desired mode. With free response from a displaced shape equal to one mode, the energy can leak to other modes as it is damped out of the desired mode. While energy can leak from the forced mode, steady state response in that mode was observed for the analyses performed, so the amount of energy leaked to

other modes is small relative to the input energy in the desired mode.

A quantitative example of the nonlinear damping used in this work follows for the 13-story building. Note that this is just an example to show the behavior of the nonlinear damping. The shear building damping is estimated by approximating the 13-story building as a thirteen-degree of freedom spring and nonlinear damping system with 3% damping in the first mode for low levels of excitation. For this system, the tangent damping matrix is approximated as

(7.18) C

=

^a1K

+

^{a3K8 H.}

a2

The overall target damping for the first mode is 3.0%, and the coefficient a1 was chosen so that there would be 0.5% damping due to classical, stiffness proportional damping. Even for the most extreme nonlinear response, there will still be 0.5%

damping in the first mode due to this elastic damping contribution.

The nonlinear equation of motion for the 13 dofs were solved using the same iterative scheme as formulated in the previous chapters. The forcing function is accelerations applied to the masses at the frequencies and shapes of each of the first five modes. The mode shapes are normalized so that the largest amplitude at any story is 1.0. The largest magnitude of the input accelerations range from 0.02g to 0.50g. In this way, the damping as a function of both frequency and amplitude can be approximated. The results are compared to Rayleigh damping, where coefficients were chosen to achieve 3.0% damping in both the first and second modes. The results are shown in figure 7.5 for acceleration amplitudes of 0.02g, 0.05g, 0.10g, 0.15g, 0.25g, and 0.50g, where g is gravitational acceleration. The lines are drawn solely to keep track of the damping across modes for constant levels of excitation. The minimum nonlinear damping (Min NL Damping) will be the 0.05% linear stiffness-proportional damping that was included in the total damping.

0.18 0.16 0.14

).J>

0 0.12

~

g' 0.1

·c. ~ 0.08 0

0.06 0.04 0.02

---

- T -

- e -

Damping for nonlinear model Rayleigh Damping

Min N[ Damping NL Damping

2

/ / /

/

/ /

/

0

3

/ / / / / / /

4 Mode

0.15g

_...0 0.25g

0.50g

5

Figure 7.5 Nonlinear damping for several modes and excitation levels.