simple modal analysis lab report

Mechanical Engineering 4P03 (2014-’15): Modal Analysis LabTeaching Assistant: Emma BadowskiMcMaster University

MODAL ANALYSIS OF A SIMPLE STRUCTURE

Umer Javed0942243—[email protected]—C01/G3

March 4, 2015

Contents1 Introduction 1

2 Experimental-procedure 1

3 Results & discussion 23.1 Static-tests . . . . . . . . . . . . . . . . . . 23.2 Dynamic-tests . . . . . . . . . . . . . . . . 23.3 Effect of test-conditions on the damping-

ratio (ζ) . . . . . . . . . . . . . . . . . . . 23.4 Effect of damping on the natural frequency

(fn) . . . . . . . . . . . . . . . . . . . . . 33.5 Beating-phenomenon . . . . . . . . . . . . 33.6 Effect of stiffness on the deflection-ratio

(X/δST ) . . . . . . . . . . . . . . . . . . . 43.7 Notes on selecting damping & stiffness for

structure-design . . . . . . . . . . . . . . . 43.8 Experimental limitations & sources of errors 4

4 Conclusion 5

Appendix A Sample calculations 5A.1 Static-compliance (SC) . . . . . . . . . . . 5A.2 Logarithmic-decrement (δ) . . . . . . . . . 5A.3 Damping-ratio (ζ) . . . . . . . . . . . . . . 5A.4 Amplitude-ratio (X/δST ) . . . . . . . . . . 6A.5 Compliance-ratio (CR) . . . . . . . . . . . 6A.6 Steady-state deflection at 10N fluctuating

load at fn . . . . . . . . . . . . . . . . . . 6

1 Introduction

Modal analysis is a fundamental process for de-signing and creating structures whose real-life be-haviour can be reliably modelled based on their

static and dynamic properties. This lab studies theseproperties of a simple structure through a series oftests with the system’s mass, damping and stiffness[1].

The results of these experiments are presented,below, and interpretations are made to connectthese results with real-life behaviour and expecta-tions.

2 Experimental-procedure

There are seven different tests carried out in thislab:

1. Static and dynamic tests of the standard simple-structure.

2. A dynamic test with added mass.3. A dynamic test with added damping (sandbag).4. A dynamic test with external load applied.5. Static and dynamic tests with gussets added.6. Static and dynamic tests with gussets and 2

cross-braces added.7. Static and dynamic tests with gussets and 4

cross-braces added.

Data for the static tests is recorded from a dialdisplacement-gauge, as the displacement of the sys-tem, in response to the added mass. The recordingsfor the dynamic tests are made via two accelerom-eters (one on the experimental structure, and theother built-in the impact-hammer). This data is pro-cessed in an adjacent workstation, using LabView.Each dynamic-test is carried out four times.

1

ME4P03, 2014-’15 Modal Analysis Lab

Ten-seconds of response data is recorded, for thedynamic tests, however, a discrepancy was noted inthis particular lab’s data. There were missing val-ues, in all seven dynamic tests. The normal proce-dure requires developing an average-spectrum fromthe four trials, for each test. This was not possible,given the inconsistent data, therefore, for property-derivations, only the highest-quality (least-noisy)data was used. Acquired data is plotted in Excelfor further analysis and the derivation of the rele-vant static and dynamic properties of the structure,under various experimental conditions.

3 Results & discussion

3.1 Static-tests

The results from all four static-tests are illustratedin figure-1. The gradient of each test-case is equalto the static-compliance of that particular configu-ration.

3.2 Dynamic-tests

As listed, earlier, the four-trials for each of the seventests did not have an equal number of data, andwas unsuitable for developing a reliable average-spectrum of the these tests. Based on the least-noisydata, table-1 lists the system’s properties, for eachimpact-test.

3.3 Effect of test-conditions on thedamping-ratio (ζ)

Figure-2 illustrates the damping ratio for the firstfour test-conditions. The most significant peak is attest-3, where a sandbag was placed on top of thesystem. As expected, the system’s equivalent damp-ing increased—roughly twice as high as the otherfour tests.

50 100 150 200

0

200

400

Applied load, L (N)

Defl

ecti

on,D

(µm

)

Simple-structure D/L = 2.35

Structure with gussets-only D/L = 0.593

With gussets and two cross-braces D/L = 0.151

With gussets and four cross-braces D/L = 0.112

Figure 1: Plot of deflection at various loads, for the structurewith four different configurations. The static-compliance (gradi-ent) decreases as the equivalent-stiffness of the system increases.These values are listed in the legend, by the respective configu-ration, and in the

1 2 3 4

0.015

0.020

0.025

0.030

Test #

Dam

ping

-rat

io,ζ

Figure 2: Variation of the damping ratio for the first four tests.Test-3 introduced the sandbag.

Mechanical Engineering 2 McMaster University

ME4P03, 2014-’15 Modal Analysis LabTa

ble

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Test

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st-4

Test

-5Te

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Test

-7N

atu

ralf

requ

ency

,fn

(Hz)

24.8

21.6

22.3

24.4

49.1

90.8

86.4

Acc

eler

atio

n/F

orce

atf n

4.52

×10−5

2.14

×10−5

1.22

×10−5

7.39

×10−6

2.91

×10−6

2.57

×10−5

1.65

×10−5

Dyn

amic

-com

plia

nce

atf n

4.03

×10−3

1.16

×10−3

6.52

×10−4

3.66

×10−4

6.08

×10−5

7.24

×10−5

4.03

×10−3

Stat

ic-c

ompl

ian

ce2.

352.

352.

352.

350.

593

0.15

10.

112

Log-

decr

emen

t,δ

9.90

×10−2

9.35

×10−2

1.93

×10−1

1.04

×10−1

3.31

×10−1

1.33

×10−1

5.37

×10−2

Dam

pin

g-ra

tio,ζ

1.58

×10−2

1.49

×10−2

3.07

×10−2

1.65

×10−2

5.27

×10−2

2.11

×10−2

8.54

×10−3

Am

plit

ude

-rat

io,X

/δ S

T31

.733

.616

.330

.29.

4923

.758

.5C

ompl

ian

ce-r

atio

1.72

×10−3

4.94

×10−4

2.78

×10−4

1.56

×10−4

1.02

×10−4

4.81

×10−4

3.58

×10−2

Defl

ecti

onat

10N

load

(m)

4.03

×10−2

1.16

×10−2

6.52

×10−3

3.66

×10−3

6.08

×10−4

7.24

×10−4

4.03

×10−2

3.4 Effect of damping on the naturalfrequency (fn)

In this experiment, the damping values are sig-nificantly low; even with the added sandbag, thedamping-ratio is still three-times lower than 0.1,the lower threshold of the damping-ratio (under-damped). Therefore, the damping has an insignif-icant effect on fn. For considerable damping, thedamped natural frequency (in radians/second) isdetermined by:

ωd = ωn

√1− ζ2

For the given precision of the data, in table-1,there is no change in the natural frequency, with therespective damping.

3.5 Beating-phenomenon

Beat, or beating, phenomenon is the presence of amodulated-decay, instead of a simple exponentialdecay, in the time-response of a structure [2]. Thisis visible in figure-3, which is the response of thesystem’s acceleration over-time, for test-5. This testhad the most prominent beating pattern, comparedto tests 6 and 7 (which were the only three wherethis phenomenon was noted).

Prima facie, this is triggered in the three tests withthe gussets added to the corners of the frame. Asthe amount of bracing is increased, from tests 6 to7, the beating’s effect on the acceleration amplitudedecreases but it’s frequency increases. The addedbracing increases the equivalent stiffness of the sys-tem and it may contribute to the system’s resilienceagainst modulation.

Beating in this apparatus may be related to thenature of the ‘coupling’ [2]. Without the gussets,the structure is simply supported by the solid legs.These solid legs have a rigid connection with thebase. The addition of the gussets increases the com-plexity of this connection by adding another cou-pling that is not as rigid as the solid-legs, on theirown.



0.1 0.2 0.3 0.4 0.5 0.6

−2

−1

0

1

2

Time, t (s)

Acc

eler

atio

n,a

(m/s

2)

Figure 3: Acceleration of the structure, after the impact, fortest-5. The sinusoidal, modulated-decay, instead of a pureexponential-decay, is known as the beat phenomenon.

3.6 Effect of stiffness on the deflection-ratio (X/δST )

Experimentally, only tests 1 and 4 are ideal forcomparing the effect of increasing stiffness, on theamplitude-ratio. Only that pair has comparabledamping-ratios (tests 5, 6 and 7 would also havebeen good candiates for studying the effects of in-creasing stiffness, from the braces, but their damp-ing ratios are significantly different, in each test).

As stiffness was increased, through the externalload, in test-4, it’s amplitude-ratio decreased byabout 5%.

3.7 Notes on selecting damping & stiff-ness for structure-design

An increase in the stiffness will increase the nat-ural frequency of the system. In structure-design,this may appear to increase the safety of the struc-ture, however, it only shifts the frequency. A poorly-analyzed frequency-shift may even bring fn closerto other excitation frequencies on the system, exac-erbating the resonance on the structure.

Therefore, a safer procedure would involve im-proving the damping of the system. An overdampedsystem may not be ideal, from operational and bud-

getary perspectives. A detailed analysis of the sys-tem and its anticipated excitation frequencies wouldallow for a suitable damper to avoid resonance.

3.8 Experimental limitations & sourcesof errors

The experimental apparatus is seldom perfect andimmune from errors and discrepancies. This is clearwith the initial observation of incorrect number ofentries for the measurements that are collected (seesection-2). In addition to that, the following mayplay a role in decreasing the accuracy and precisionof this experiment:

• Improper orientation of the impact hammer: Im-pact is required to be reasonably perpendicularto the side of the top-plate, to only excite thebending mode. A strike that is not perpendicu-lar would also excite the structure, torsionally,decreasing the accuracy of the measurementson the response.

• Insufficient settling of the apparatus: The baseof the apparatus is struck with a soft-mallet,multiple-times, to settle any stresses and in-crease the overall rigidity of the structure. Lackof proper settling introduces unwanted damp-ing into the system, which would decrease theaccuracy of the measurements on the response.

• Electromagnetic-noise in the data-logging circuit:Significant noise is detected in the initial partof most of the test-results. Since the data is col-lected for a few seconds, the overall impact ofthis is diminished, as ‘cleaner’ data is availablefor review. If, however, data is collected formuch shorter time-periods, this noise reducesthe usability of the measurements.

• Inaccurate data retreival from the measure-ments: Table-1 shows a decrease in the naturalfrequency between test 6 and 7, while it shouldincrease, as the system’s overall stiffness is in-creased with the added brace. It is possible thisis because of improper determination of the fnvalue, from data (which was post-processed inExcel). The fn value may have been for a pointadjacent to the fn peak.



4 Conclusion

A structure’s static and dynamic properties and be-havior can be studied by analyzing its response toimpact loads, under varying operating conditionsand configurations. Altering the equivalent-mass,stiffness, and damping of the system leads to dif-ferent responses and changes the properties of thestructure, namely the natural-frequency.

Real structures can be designed and adequatelytested in this form to increase their safety and sta-bility. While this apparatus is made simple, to aidthe short analysis, in this lab, improvements in thestructure and the underlying assumptions could bemade to increase the comparability to a larger, morecomplex structure.

The results, tabulated in table-1, offer a quickrundown of how the system’s dynamics change, un-der different test configurations. The changes in thesystem’s damping-ratio and natural-frequency areobvious, as test-conditions are changed.

There are some errors, noted in this lab, which arelisted as well, to assist in improving future iterationsof this experiment.

References[1] McMaster University, Mech Eng 4P03—Composite Labora-

tory, Experiment (M.A.): Modal Analysis, 2015.

[2] S. K. Yalla and A. Kareem, “Beat phenomenon in com-bined structure-liquid damper systems,” Engineering Struc-tures, no. 23, p. 622.

A Sample calculations

From table-1, only fn, the acceleration/force atfn, and the dynamic-compliance at fn, were deter-mined from the experimental measurements. All re-maining properties were derived from those initialmeasurements, based on the theory outlined in thelab handout.

A.1 Static-compliance (SC)

The following equation calculates the SC for thesimple-structure with a 44.448N (10 lb) load.

SC =DeflectionStatic-load

=100µm

44.488N= 2.25

For the experimental calculations, the gradient ofthe best-fit line was used, to determine the static-complianace; these two derivations are about 3%different.

A.2 Logarithmic-decrement (δ)

The logarithmic-decrement is determined using theacceleration values at two points (x1 and xn+1) ofknown spacing (n). The following equation is usedto determine δ for test-1:

δ =1

nln

(x1xn+1

)=

1

6ln

(2.5658

1.4166

)= 9.9× 10−2

A.3 Damping-ratio (ζ)

The damping ratio (for test-1) is derived using thevalues of δ:



ζ ∼=δ

2π

∼=9.9× 10−2

2π∼= 1.58× 10−2

A.4 Amplitude-ratio (X/δST )

The amplitude ratio (for test-1) is derived fromthe following expression, for systems with low-damping:

X

δST=

1

2ζ

=1

2(1.58× 10−2)

= 31.7

A.5 Compliance-ratio (CR)

The compliance ratio is the ratio of the dynamic andstatic-compliance values. For test-1:

CR =Dynamic-compliance at fn

Static-compliance

=4.03× 10−3

2.35

= 1.72× 10−3

A.6 Steady-state deflection at 10N fluc-tuating load at fn

Experimental measurements included the Fouriertransform of the displacement signal (displace-ment/force, of units m/N). To determine the de-flection at 10N fluctuating load at the system’s fn,the displacement/force value, from the signal-graph(also listed in table-1, was multiplied by 10N. Fortest-1:

Deflection = (Deflection at fn) (Load)

= (4.03× 10−3)(10)

= 4.03× 10−2 m


simple modal analysis lab report

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