forced vibration and output-only procedures for estimating modal

8/17/2019 Forced Vibration and Output-Only Procedures for Estimating Modal

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698 EVACES’07

2 MODAL MASS: CONCEPT AND ESTIMATION

The equation of motion of a viscously damped SDOF oscillator representing vibration mode r (of a dynamic system with N degrees of freedom) is given by:

( ) ( ) ( ) ( )r 2

r r r r r r

r

p tq t 2 q t q t

m

+ ζ ω + ω =&& & (1)

where ( )r q t , is the modal (or generalised) displacement which has physical units e.g. [m], r m [kg] is modal mass, r ζ is modal damping ratio and r ω [rad/s] is the circular natural frequency.The modal mass can be calculated as:

{ } [ ]{ }T

r r r m M= ψ ψ (2)

where { }r

ψ is the Nx1 rth arbitrarily scaled mode shape vector whose elements have no units,and [ ]M and is the NxN mass matrix of the dynamic system with N degrees of freedom.

For any kind of dynamic loading, a larger r m leads to lower modal (and hence physical) re-sponses and vice versa. Therefore, reliable estimation of modal mass for the excited mode of vi-

bration is crucial for reliable prediction of the actual dynamic response.

Physical interpretation of modal massTwo methods for scaling (or normalisation) of mode shapes are common: mass-normalisation,where scaling is set so that modal mass has unit value of whatever mass units are in use (typi-cally kilograms), and unity scaling, where the maximum absolute mode shape value over alltranslation DOFs is a non-dimensional unity. The latter form is preferred for vibration service-ability calculations since modal mass obtained via unity scaled mode shapes must always be ofthe same order as, but equal to or lower than, the physical mass of the vibrating structure. Thismeans that modal mass has a tangible physical meaning.

Classical methods for estimating modal mass

Modal mass can be obtained through finite element analysis as well as by modal analysis using

vibration excitation and response measurements made on a full-scale structure. Modal analysistypically employs curve-fitting of frequency response functions (FRFs) or impulse responsefunctions (IRFs). Strictly speaking, it is not possible to obtain modal mass experimentally with-out knowing the exact excitation force and the corresponding vibration response.

FRF-based experimental method for estimating modal mass

A receptance FRF ( ) jk α ω is the ratio, for a constant frequency ω , between harmonic displacement response ( ) jX ω at degree freedom (DOF) j and harmonic force ( )k F ω applied atnode k:

( ) ( )( )

( )

N j,r k, r

j,k 2 2r 1 r r r r

1

m i2=

ψ ψα ω = ⋅

ω − ω + ζ ω ω∑ . (3)

where ( ) j,r ψ is the arbitrarily scaled mode shape element corresponding to the jth DOF in therth mode shape. The quantity

( )( ) j,r k, r r j,k

r

Am

ψ ψ= (4)

is a modal constant or a residue.As a result of curve-fitting of experimental receptances to equation (3) for response at a fixed

DOF k, but with responses at varying DOFs j, a column vector of modal constants can be con-structed for each mode r .


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Experimental Vibration Analysis for Civil Engineering Structures 699

{ } ( )

( )

( )

( )

1,r

r j.k k,r j,r

r

N, r

1A

m

⎧ ⎫ψ⎪ ⎪⎪ ⎪⎪ ⎪

= ψ ψ⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪ψ⎩ ⎭

M

M

(5)

If we scale the mode shapes such that the maximum absolute value is unity and the DOFwhere this maximum amplitude occurs is denoted as jmax, then ( ) jmax,r 1ψ ≡ .

If a point mobility FRF measurement is made at jmax (i.e. j=k=jmax), then it follows that:

{ }

( )

( )

1,r

r j.k

r

N,r

1A 1 1

m

⎧ ⎫ψ⎪ ⎪⎪ ⎪⎪ ⎪

= ⋅ ⋅ ⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪

ψ⎩ ⎭

M

M

(6)

so that unity-scaled modal mass can be recovered from the modal constant corresponding tothe point mobility FRF via:

r

r jmax, jmax

1m

A= (7)

Theoretical background of the proposed method

The method advocated here is based on a single person dynamic excitation of a footbridge andmeasurements of the corresponding dynamic response.

The method involves measurement and post-processing of dynamic responses due to jumpingand/or walking at a frequency which excites footbridge resonance of a mode whose modal massneeds to be determined. The post-processing makes direct or indirect use of an existing databaseof walking and/or jumping forces recorded in the laboratory for various frequencies (Brownjohnet al., 2004).

As footbridges are designed to convey people, a human-induced excitation is the most-logicaland easiest form of dynamic excitation to apply. Compared with directly measured artificial ex-citation of footbridges (e.g. by shaker) it may be as reliable, and is certainly simpler, to recordthe (human-induced) excitation forces separately in a suitably equipped biomechanics labora-tory. Due to the nature of human-induced excitation, a reasonable assumption is made that foreach excitation frequency the forces recorded in the laboratory are changed little compared tothe unmeasured forces actually applied to the structure during human-induced excitation. Themethod has two variants:

1.

A procedure which makes use of the change of velocity per cycle during the initial resonant

build up, and2. A procedure which compares measured build up with simulations using a database of labo-

ratory-recorded human-induced dynamic forces.

Procedure 1: Change of velocity

The idea here is to idealise a footfall due to, say, jumping as an impulsive force. Regardless ofmodal damping and frequency values, in the case of single mode r, if the impulse is applied andthe response measured at the point of maximum response of mode r (i.e. j=jmax), then for unityscaled modal mass,

j

j

r

Ix

mΔ =& (8)


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700 EVACES’07

Equation (8) gives the relationship between impulse and change of velocity, which can bequantified, and the unknown modal mass. If the mode is represented by a single degree of free-dom oscillator which is excited by a sequence of impulses jI applied at times nT , where T isthe natural period of the oscillator and n is a positive integer, the velocity response after the nthimpulse is given approximately by

( ) ( ) j

j

r

n 1 I nx nT 1m 2δ

+ ⎛ ⎞≈ −⎜ ⎟⎝ ⎠& . (9)

The approximation is best for small values of modal damping r ζ and with further approxima-tion and considering only the first few cycles of build up, the n 2δ term can be neglected andequation (8) can be applied to average incremental velocit. If peak velocities in each cycle are

plotted against cycle number, the increment is taken as the tangent to the plot at the origin.Figure 1 shows a resonant build up of response for an SDOF system having mass of

mr =1000 kg and natural frequency of 1.51 Hz. In the plots on the left the system is subjected toreal jumping at 1.51 Hz, and on the right a sequence of idealised sharp, impulses I . Lower left

plots show displacement and velocity build up due to real jumping and lower right plots show build up to ideal impulses.

Figure 1: Comparison of recorded jumping forces (A) and sequence of equivalent impulses (D) and theirrespective simulated displacement and velocity responses (B,C) for recorded jumping; (E,F) for equiva-lent impulses.

The identical ideal impulses are obtained from the average area of the individual jump foot-falls. From basic mechanics, this must be the same as the weight W of the jumper multiplied

by the time interval T between jumps (or divided by the jumping rate f ):

WWT I

f = = (10)

However, as Figure 1 shows, real jumping is not a sequence of perfect impulses. Therefore,in the case of resonance it generates lower response than the perfect impulses applied with a fre-quency identical to the natural frequency of the system. An ‘impulse correction factor’ ( ICF )can be derived using simulations of the kind shown in Figure 1 for a range of jumping frequen-cies. For jumping rates above approximately 1.7 Hz the impulse correction factor is close to 1.0

A)

C)

B)

F)

E)

D)


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which means that for faster jumping, the forcing function more closely resembles perfect impulses, which is to be expected.

Using the ICF, modal mass r m can be calculated by combining equations (8) and (10):

( ) f W

xf ICF

1m

k

r &Δ

= (11)

Looking at equation (11), all that is needed is an estimate of modal frequency of the full-scalestructure which will be taken as the jumping frequency f, the weight of the jumper W and a re-cording of the single-mode response build up due to jumping from which increment of peak ve-locity

k x&Δ between consecutive cycles during early resonant build-up can be obtained.

Procedure 2: Comparison of measured and simulated resonant responses

The authors have collected a database of human-induced excitations from jumping, walking,swaying and bouncing forces for several people from instrumented force plates and a treadmill.As an improvement on Procedure 1, in Procedure 2 the subject jumps, bounces, sways or evenwalks at a rate close to the known or estimated modal frequency of the structure at or near jmax.

The recorded response is compared with response simulated using the laboratory recording ofthe nominally identical force function. This simulation is done for an oscillator having unitmass, the estimated modal natural frequency of the mode investigated (within a few %) and areasonable estimate of the modal damping.

While accurate estimates of modal damping and frequency are always desirable for the simu-lation, precise values are not necessary. Figure 2 shows the percentage errors in simulations ofresponse to jumping (in terms of peak to peak amplitude at the eighth cycle) when using valuesof damping and frequency other than the ‘correct’ values, these being 1% damping and pacingrate 1.51 Hz, as used in Figure 1. An imperfect match of a few percent between structure fre-quency (say, 1.45-1.55Hz) and pacing frequency makes little difference, whereas the error dueto incorrect damping is linear as per equation (9). This means that, provided only the first fewcycles of build-up are used, imperfect timing of jumping (or other activities) is not crucial andthose activities that normally excite the bridge can be used to estimate modal mass values.

Figure 2: Effect of poor imperfectly timed jumping and variation of damping on real or simulated re-sponse to jumping. Datum is jumping at 1.51Hz to match structure frequency with 1% damping.


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702 EVACES’07

3 CASE STUDIES

To demonstrate and verify the proposed methodology, two case studies where the method has been applied are presented.

Western Approach Footbridge, PlymouthThis bridge, shown in Figure 3, is a 27.6 m span simply supported steel framed covered walk-way between a car park and busy shopping centre and was investigated as part of a student pro-

ject on efficient methods for footbridge vibration serviceability assessment (Brownjohn & Mid-dleton, 2005). No drawings were available for the bridge so part of the assessment includedcreating a finite element model from a visual survey. A brief vibration test was conducted withminimal equipment, identifying the first vertical mode, VS1, at 4.5Hz with 1.2% damping as be-ing readily excited by pedestrians.

There were no resources available to obtain reliable FRFs with artificial excitation, and no portable force plate. The bridge was in constant use, but some timed walking and jumping exer-cises were used for estimating modal mass via Procedure 2. Figure 4 shows response build up(and subsequent decay) in mode VS1 due to jumping at 2.25Hz, and Figure 5 compares the peak

values of acceleration during the build up from with those from a simulation using a laboratoryrecording of the force generated by jumping at the same frequency. Using the same frequencyand damping, a good match was found between simulation and measurement using a modalmass of 18,000 kg, almost identical to the modal mass of 18,035 kg obtained from the finiteelement model created in the bridge study.

Figure 3: Western Approach Footbridge, Plymouth.

Figure 4 Response build up due to jumping athalf of the mode VS1.

Figure 5 Fitting simulated peak response for pre-recorded jumping sequence applied at an18,000kg SDOF system to measured peaks of

the bridge response for similar jumping.

48 50 52 54 56

-1

-0.5

0

0.5

1

[ m /

s e c

2 ]

seconds

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

[ m / s e c

2 ]

seconds

18,000kg masswalkway


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Natural Frequency: 4.72 Hz

Walkway at Singapore Polytechnic

This 46m long steel pedestrian bridge (Figure 6) links two teaching blocks at Singapore Poly-technic. It is a steel space truss and frame system with the deck floor comprising timber plankswith a layer of stone tiling. Only one end is fully supported, the other is connected to the adja-cent structure by a narrow elastomeric expansion joint leaving a 15m cantilever.

Ambient vibration measurements identified the first vertical mode (Figure 7) at 5 Hz with1.3% damping, both values having significant amplitude dependence. As with the Western Ap- proach Footbridge, jumping was used to estimate the bridge modal mass experimentally, butthis was a unique occasion when a portable force plate was available for use at the bridge.Hence it was possible to record the vertical force generated by a 785 N student jumping at thefree end of the bridge at 2.5 Hz, half the first mode frequency.

Figure 8 shows the effect of a short sequence of jumping then standing still. The top trace isthe bridge response, second trace is the force plate signal and the bottom trace is the simulatedresponse of a SDOF oscillator having mass mr =1000 kg, frequency 5 Hz and damping

r ζ =1.3%. Using procedure 2 but in this case with the actual in-itue recorded jumping forces, ,comparison of response build-up for top and bottom plots leads to an experimental modal massestimate mr =7380 kg, reasonably close to the modal mass of 8500 kg estimated from the finiteelement model.

Figure 6 Singapore Polytechnic walkway

looking towards the cantilevered end

Figure 7 First vertical mode from FE model and

hammer testing

4

MODAL MASS BY FORCED VIBRATION TESTING

For the Singapore Polytechnic walkway, the modal mass was estimated by two other methods.

Hammer testing was used first to estimate modal mass for lateral and vertical modes. Figure 9shows the inertance frequency response function (FRF) for hammer force one side of the walk-way and response at the other, both at the free end. The two modes show are the first verticaland first torsional modes and the modal mass estimate for the first vertical modes. The dataquality were too poor to be useful for recovering modal mass estimates.

With the benefit of synchronous recordings of force and response available with the portableforce plate, classical curve fitting was applied to the experimental FRF. Figure 10 shows thecurve fit which is clearly high enough quality to allow recovery of a modal mass estimate,which as shown, is 8134kg.

Force plates are seldom used but when they are they are effective as demonstrated by Blake- borough and Williams (2003) and the measurements reported here.

mode: 1 f=5.033Hz


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704 EVACES’07

-1

0

1

w a l k w

a y a c c [ m / s e c 2 ]

0

1000

2000

j u m p f o r c e [ N ]

57 58 59 60 61 62 63 64 65-10

0

10

seconds 1 0 0 0 k g a c c [ m / s e c

2 ]

Figure 8 785 N student jumping at 2.5Hz on Singapore Polytechnic walkway. Top: measured accelera-

tion response. Middle: measured contact force. Bottom: simulated acceleration response for 1000 kgmass.

Figure 9 FRF for hammer testing; input and output at opposite sides of walkway. Units are m/sec2/kN.

0 5 10 15-3

-2

-1

0

1

2

F/Hz

Real : ch3 vs ch1

0 5 10 15-6

-4

-2

0

2

4

F/Hz

Imag : ch3 vs ch1

-4 -2 0 2

-4

-2

0

2

Nyquist: ch3 vs ch1


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-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

I m a g i n a r y

Real

n5 ch1/ch3 1/mod=8134 φ=175.4° f=4.929Hz ζ= 1.22%

Mod(RE)=ψi.ψ j/ψT[m]ψ Mod(TF)=ψi.[m].ψ/ψ

T[m]ψ = ψiL / M

Figure 10 FRF and modal parameters estimates from force plate and accelerometer data

5

CONCLUSION

The method described in this paper is used by the authors as part of the initial vibration service-

ability assessment of existing footbridges and has also been used on lively low frequency floors.Reasonable estimates are obtained if only the body mass of the tester is known and because ver-tical footfall forces are more repeatable, modal mass estimates from modes excited by verticalexcitation are likely to be more accurate. The modal mass estimates obtained are values forunity scaling at the point of excitation which is at midspan for symmetric modes, while correc-tions can be applied for excitation at other locations if the mode shape has been measured or canreasonably well be assumed. With the addition of a force plate even more reliable modal massvalues can be obtained, either using the recorded force data in the procedure described here or

by application of classical modal analysis procedures.

REFERENCES

Blakeborough, A. and Williams, M. S. (2003). Measurement of floor vibrations using a heel-drop tests.Structures and Buildings. Vol. 156, No. 4, 367-371. Proceedings of the Institution of Civil Engineers.Thomas Telford. ISSN 0965-0911.

Brownjohn, J. M. W., Pavic, A., and Omenzetter, P. (2004b). A Spectral Density Approach for ModellingContinuous Vertical Forces on Pedestrian Structures Due to Walking. Canadian Journal of Civil Engi-neering. National Research Council of Canada, Vol. 31, No. 1, 65-77. National Research CouncilCanada Research Press. ISSN 1208-6029.

Brownjohn, J.M. and Middleton, C. (2005). Efficient dynamic performance assessment of footbridge.Proceedings of ICE, Structures and Buildings , 158 (4), pp. 185-192. ISSN 1478-4629.

Zivanovic, S., Pavic, A. and Reynolds, P. (2005) Vibration Serviceability of Footbridges under Human-Induced Excitation: A Literature Review . Journal of Sound and Vibration, 279 (1), January, pp. 1-74.ISSN 0022-460X.


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forced vibration and output-only procedures for estimating modal

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