Paper Example

7/29/2019 Paper Example

1/12

Experimental evaluation of the damping propertiesof beams and thin-walled structures made of polymeric materials

Giuseppe Catania, Silvio SorrentinoDIEM, Department of Mechanics,

University of Bologna, Viale del Risorgimento 2, 40136 Bologna, Italy

ABSTRACT

In the field of structural vibrations, an appropriate rheological model should be accurate in fitting the experimental

data on a wide interval of frequencies by means of a minimum number of parameters and, in particular, it should

be able to reproduce the experimentally found behavior of the damping ratio n as a function of the natural angular

frequency n. In this study a non-integer order differential linear rheological model is considered, discussing its

effectiveness in solving the above mentioned problem. This model, refferred to as the Fractional Double Kelvin

model, combines the properties of both the Fractional Kelvin and Fractional Zener models, which are considered

to be very effective in describing the viscoelastic dynamic behavior of mechanical structures made of polymers.

An identification method of general validity for viscoelastic models is adopted, based on the concept of equivalent

damping ratio and on the circle-fit technique. It is applied to the analysis of vibrating beams and plates of different

sizes, made of polymeric materials such as Polyethylene, Polyvinyl-chloride and Delrin.

Nomenclature

b width of a beam or a plate x, x spatial coordinateC damping coefficient fractional derivative order

E complex Youngs modulus angleE0 static Youngs modulus strainh thickness of a beam or a plate Poissons ratio

L length of a beam or a plate densityn modal index timeQ modal coordinate angular frequencys Laplace variable n modal natural frequencyW, w displacement n modal damping factor

1 Introduction

When dealing with structural dynamic problems, it would be useful to obtain a viscoelastic model identification

from vibration measurement data only. In this case, however, the direct identification of an optimal set of

parameters from time or frequency domain measurements is a difficult task, especially if the structural dissipative

contributions are slight.

In this paper, an indirect approach is adopted, based on the concept of damping ratio n, focusing the attention on

the behavior of n as a function of the modal frequency n. As is well known, a modal parameter n can beanalytically defined and experimentally estimated by considering a linear viscous dissipative model (based on a

single Newton element). However this theoretical parameter shows a dependency on the modal frequency that in

most cases dramatically fails in fitting the experimental data.

On the contrary, it was shown that a better agreement between theory and experiments can be achieved by

means of non-integer order differential models, obtained by replacing the first derivative (Newton element) with a

fractional derivative (Scott-Blair element) [1]. Extensive literature exists on the application of fractional calculus to

viscoelasticity, since it yields to physically consistent stress-strain constitutive relations with a few parameters,

Proceedings of the IMAC-XXVIIFebruary 9-12, 2009 Orlando, Florida USA

2009 Society for Experimental Mechanics Inc.


2/12

good curve fitting properties and causal behavior [2-3-4].

In particular, regarding the n = n(n) behavior, it was experimentally observed that it may not be monotonic, and

consequently the Single Kelvin and Zener models (either integer or fractional order) may in some cases not be

suitable for fitting the experimental data [5]. In this paper a more refined rheological model is thus considered,

referred to as the Fractional Double Kelvin model, applied to the analysis of vibrating beams and plates of

different sizes, made of polymeric materials such as Polyethylene, Polyvinyl-chloride and Delrin. Structural

damping laws are not included in the analysis, since they may lead to non-causal behavior.

Since in the case of fractional derivative models analytical expressions forn are usually difficult to find, a methodof general validity for viscoelastic models was developed, introducing the concept of equivalent damping ratio

applied to the circle-fit technique [6]. This identification method is based on the assumption that the Nyquist plot of

the mobility for any mode n can be approximated by a circumference, which is still acceptable when considering

fractional derivative models [7-8-9].

2 Linear viscoelastic model and identification technique

According to the circle-fit technique [6], the circular Nyquist plot of the Mobility for each vibration mode allows the

experimental estimate of the related modal damping (Fig.1).As a consequence, the modal damping ratio n forthe classical Integer Single Kelvin model can be evaluated by means of the expression:

=

+

2 2

2 1

2 2 1 1

1

2 tan( ) tan( )n

n

(1)

where n is the natural angular frequency and the other symbols refer to Fig.1.

r0

n

1

2

Im

Re

1

2

n

r

-2 0 2 4 6 8 10 12x 10

-3

-10

-8

-6

-4

-2

0

2x 10

-3

Real [Mobility]

Imaginary[Mobility]

Fig. 1 Nyquist plot of Mobility. General scheme for mode n (left) and experimental plots (right) [5].

The experimental estimates ofnas a function of the natural frequency n usually show a behavior which is veryfar from the straight increasing line passing through the origin predicted by the Integer Single Kelvin model [8-9].

As a consequence, in order to fit such experimental curves, more refined models are needed.

In the present study the Fractional Double Kelvin model is adopted, which can be considered as a generalizationof both the Single Kelvin and the Zener models. Its analogical representation is shown in Fig.2, and its analyticalexpression in the frequency domain is given by:

+ + =+ + +

1 2

1 2

1 1 2 2

1 2 1 2

(i ) (i )( )

(i ) (i )

E C E C E

E E C C (2)

where 1 and 2 are non-integer or fractional derivative orders (values between 0 and 1).


3/12

Expression 2 can be rewritten highlighting the static Youngs modulus E0 as well as the characteristic times (related to the creep behavior) and (related to the relaxation behavior), according to the definitions reported inTab.1:

1 2

1 2

1 2

0 0

1 2

1 (i ) 1 (i ) ( )( )

( )1 (i ) (i )

NE E E

D

+ + = =+ +

(3)

which for1 = 2 = reduces to:

+ + =+

1 2

0

1 (i ) 1 (i )( )

1 (i )E E (4)

The relations among the viscoelastic parameters in the Fractional Double Kelvin model, as well as in theFractional Single Kelvin and Zener models, are reported in Tab.1. Clearly for = 1 these fractional modelscorrespond to the more conventional integer order ones.

Table 1: Parameter relations for the viscoelastic models.

Model0E

1

1

2

2

1

1

2

2

Single Kelvin

= = 2 20,C E 1E

1

1

C

E

0 0 0 0

Zener

=2 0C +1 2

1 2

E E

E E 1

1

C

E 0

+1

1 2

C

E E

+1

1 2

C

E E 0

Double Kelvin+1 2

1 2

E E

E E 1

1

C

E 2

2

C

E

+

+1 2

1 2

C C

E E

+1

1 2

C

E E

+2

1 2

C

E E

E1

E2

C1

C2

Fig. 2 Viscoelastic Double Kelvin analogic model.

At this stage a definition of general validity is needed forn, in order to create a link between the experimentaldamping estimates of Eq.1 and the viscoelastic parameters reported in Tab.1.

For this purpose let us consider, for example, a homogeneous Kirchhoff plate. Its (homogeneous) equilibriumequation in the Laplace domain can be written as:

3 4 4 42 4 4

2 4 2 2 4( , ) ( , ) 0, ( ) ( ) 2 ( ) ( )

12(1 )h

Ehs w s w s

x x y y

+ = = + +

x x (5)

where x denotes the spatial coordinates, h the mass per unit area of the plate and the Poissons ratio [10].


4/12

The expression of the Youngs modulus for the Fractional Double Kelvin model (Eq.3) can be rewritten in theLaplace domain, and then introduced in Eq.5 in place ofE. Multiplying both sides of Eq.5 forD(s) (Eq.3) yields:

1 1 2 2 1 2 1 2 1 1 2 2

1 2

3

2 40

1 2 1 221 ( , ) 1 ( , ) 0

12(1 )h

E hs s s w s s s s w s

+ + + + + + + = x x (6)

By separating the variables, i.e. introducing w(x,s) = W(x)Q(s), Eq.6 can be easily decoupled [11-12-13]. More indetail, it can be rewritten separating the x-dependent part from the s-dependent part in the form:

3 4 20

2

( ) ( )constant

( ) ( )12(1 ) h

E h W s D s

W N s

= = =

x

x(7)

Introducing the modal natural frequency 2n = , equation (7) yields the homogeneous characteristic equationassociated with the n-th mode:

+ + + + + + + =

1 1 2 2 1 2 1 2 1 1 2 2

1 2

2 2

1 2 1 21 1 0ns s s s s s (8)

Clearly, Eq.9 is valid not only for plates, the geometry of the vibrating structure being influential on n only. In thecases of practical interest this equation gives a couple of complex conjugate roots and a real negative one.

Since the real negative part of the complex conjugate roots is responsible for the oscillation decay, a general

definition for an equivalent n can be given as the absolute value of the real part of the complex conjugate roots ofthe characteristic equation, divided by n, i.e.:

2,3

Re( )n

n

s(9)

It should be noted that definition 9 is generally coherent with the equivalent n evaluation via circle-fit technique(with the assumption E1 < E2, C2 < C1, 2


5/12

0 500 1000 1500 2000 2500 30000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Natural frequency [Hz]

Equivalentdampingratio

Single Kelvin

Double Kelvin

Zener

0 500 1000 1500 2000 2500 30000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04



Fractional Single Kelvin

Fractional Double Kelvin

Fractional Zener

Fig. 3 Behavior ofn as a function ofn. Integer order models (left) and fractional order models (right).

0 500 1000 1500 2000 2500 30000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04



Effect of1

variation

1

= 0.5

2

= 0.5

1 = 1.0

2= 0.5

0 500 1000 1500 2000 2500 30000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04



1

= 0.5 2

= 1.0

1

= 0.5 2

= 0.5

Effect of2 variation

Fig. 4 Fractional Double Kelvin model. Variation of1 from 0.5 to 1.0 (left, steps 1 = 0.1, 2 = 0.5) and

variation of2 from 0.5 to 1.0 (right, steps 2 = 0.1, 1 = 0.5).

Table 2: Materials.

Material Density [Kgm3

]

Polyethylene (PE) 964

Polyvinyl chloride (PVC) 1425

Delrin 1437

Tests were made on specimens of different size, as reported in Tab.3 (plates L b and beams with thickness h1and h2, cut at 5 different lengths L), using the experimental setup shown in Fig.5. The system was excited by asuspended shaker with random excitation, and acceleration responses were evaluated by means of 3 ICPpiezoelectric accelerometers placed as shown in Fig. 6. Frequency response functions (inertances) were

estimated on the interval 0-4000 Hz with resolution f= 0.3 Hz, H1 technique [9], 50 averages, Hanning window


6/12

(beams and plates) and on the interval 0-1250 Hz with resolution f= 0.1 Hz, H1 technique, 20 averages,Hanning window (plates only). During the tests, the average temperature was 28C 1C.

Figures 7-9 show the plots of the experimentally estimated damping ratios with respect to the natural frequency.The analytical curves due to the identified Fractional Double Kelvin model (black solid lines) are superimposed onthe experimental data plots. Thick black lines refer to present identification, while thin lines refer to previousidentification [9], based on beam data only. The figures on the right side are enlargements of the low-mediumfrequency range of those on the left side. Table 4 refers to the conventions adopted in Figs.7-9, related tospecimen sizes and accelerometer positions.

Fig. 5 Experimental testing setup: beam (left) and plate (right).

Accelerometer 1

Accelerometer 2

Accelerometer 3

Force sensor

Suspended shaker

3 L8

5L

9

L

54 mm

20 mm10 mm

3L

8

5L

9

L

54 mm

20 mm

Accelerometer 2

Accelerometer 3

3

8

5

9

Accelerometer 1

Suspended shaker

Fig. 6 Sensor position: beam (left) and plate (right).


7/12

The identified parameters of the Fractional Double Kelvin model are reported in Tab.5 (bold red), where they arecompared with the previous identification based on beam data only [9] (green). The data provided by the analysisof vibrating plates made it possible to improve the indentification in the low frequency range (in Figs.7-9 the thinsolid lines refer to the previous identification, while the thick solid lines to the improved one). As a consequence,the viscoelastic parameters identified using beam data only were slightly modified (Tab.5), with the exception ofPVC (in this case the previously identified parameters fit also the new data). The numerical values plotted inFigs.7-9 are reported in the Appendix table. The symbols A i reported in Tab.4 and in the Appendix refer to thefrequency response functions computed from accelerometeri data (Fig.6).

Table 3: Size of specimens.

BEAMS

Material: PE

Width b = 100 mm

Thickness h1 = 25.0 mm h2 = 10.5 mm

Length L [mm]L1

952L2

852L3

748L4

400L5

200L1

952L2

852L3

746L4

400L5

200

Material: PVC

Width b = 100 mm


Length L [mm]L1

971L2

871L3

744L4

400L5

200L1

968L2

868L3

769L4

400L5

200

Material: Delrin

Width b = 100 mm


Length L [mm]L1

951L2

851L3

752L4

400L5

200L1

951L2

851L3

755L4

400L5

200

PLATES

Material PE PVC Delrin

Thickness h [mm] 10 10 10

Width b [mm] 1000 1000 1000

Length L [mm] 946 969 951

Table 4: Meaning of symbols and colors in Figs.7-9.

BEAM (color = length of specimen) PLATE

length L1 L2 L3 L4 L5 size L b

color red magenta cyan blue green color black

symbol = thickness of specimen + position of accelerometer symbol = position of accelerometer

thickness h1 h2 thickness h

Frf A1 A2 A3 A1 A2 A3 Frf A1 A2 A3

symbol O + symbol O +


8/12

Table 5: Material estimated parameters.

Material 1 2 1 [s] 2 [s] [s] 1 [s] 2 [s] Figure

PE0.5

0.45

0.5

0.45

10

102.0 10-8

2.0 10-8

1.25

1.23

6

6

PVC 0.4 0.4 35 0.6 10-8 7.00 7Delrin

0.3

0.5

0.3

0.3

30

301.0 10-8

0.6 10-8

10.0

8.33

4.33 10-98

8

4 Discussion

Examination of the experimentally estimated damping ratios in Figs.7-9 suggests the following remarks:

different specimen shapes and measurement points lead to n values converging on a single curve,depending on the material properties only (and not on the geometry). The dispersion, mainly due tomeasurement errors, may also be partially due to the temperature effect (varying from 27 to 29C);

the curves related to different materials show different behaviors in magnitude and shape;

in the medium-high frequency region the data show a moderately increasing trend with respect to frequency,which in the frequency range examined is linear (or almost linear) for PE and PVC, but it is clearly non-linearfor Delrin (with decreasing slope);

in the low frequency range the data show a rather sharp decreasing trend, leading to a minimum;

the proposed fractional double Kelvin model was able to accurately fit the experimentally found behavior ofnfor all of the different materials under analysis;

in the case of PE and PVC a single value for the fractional derivative order (1 = 2 = ) seems to besufficient for fitting the data. On the contrary, Delrin required the adoption of two different orders 1, 2 to beable to reproduce the data behavior.

It is important to point out that the Fractional Double Kelvin model was adopted for fitting the presentedexperimental data after having recognized the inefficiency of simpler models. The Single Kelvin and Zener models

(both integer or fractional order) do not seem to be able to reproduce the experimentally found dependency ofnto n, giving a monotonic trend (Fig.3), while the Integer Double Kelvin model lacks in flexibility [9].

5 Conclusions and future work

In this paper, the circle-fit technique was applied for the experimental evaluation of an equivalent modal dampingratio, adopting a method valid for any linear viscoelastic model. Plane flexural vibrations of clamped-free beamsand vibrations of plates clamped at one edge were considered, with specimens made of viscoelastic materialssuch as Polyethylene, Polyvinyl Chloride and Delrin. To fit the experimentally found equivalent modal dampingratio, the Fractional Double Kelvin model was adopted. The accuracy of the proposed model was discussed andproved in comparison with simpler viscoelastic models, namely the Simple Kelvin and Zener models.

The presented procedure and rheological model are suitable for possible future application to the analysis ofvibrating viscoelastic structures with more complicated shapes than beams and plates, being possible theirimplementation in the finite element method [11-12-13].

Future work will deal with comparison of the results presented herein with the viscoelastic parameters identifiedthrough creep and relaxation tests on the same materials.


9/12

0 500 1000 1500 2000 2500 3000 35000

0.005

0.01

0.015

0.02

0.025

0.03

0.035


Equivalen

tdampingratio

0 20 40 60 80 100 120 140 160 180 2000

0.005

0.01

0.015

0.02

0.025

0.03

0.035


Equivalen

tdampingratio

Fig. 7 Equivalent damping ratio n as a function ofn. Material: PE.

0 500 1000 1500 2000 2500 3000 35000

0.005

0.01

0.015

0.02

0.025

0.03



0 20 40 60 80 100 120 140 160 180 2000

0.005

0.01

0.015

0.02

0.025

0.03



Fig. 8 Equivalent damping ratio n as a function ofn. Material: PVC.

0 500 1000 1500 2000 2500 3000 35000

0.005

0.01

0.015

0.02

0.025

0.03



0 20 40 60 80 100 120 140 160 180 2000

0.005

0.01

0.015

0.02

0.025

0.03


Equ

ivalentdampingratio

Fig. 9 Equivalent damping ratio n as a function ofn. Material: Delrin.


10/12

Acknowledgments

This study was developed within the laboratory INTERMECH with the contribution of the Regione EmiliaRomagna - Assessorato Attivita Produttive, Sviluppo Economico, Piano telematico, PRRIITT misura 3.4 azione AObiettivo 2.

References

[1] Mainardi F., Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractalsand fractional calculus in continuum mechanics, Springer, 1997.

[2] Nutting P.G., A new general law of deformation, Journal of the Franklin Institute 191, pp. 679-685, 1921.

[3] Jones D.G., Handbook of viscoelastic vibration damping, Wiley, 2001.

[4] Gaul L., The influence of damping on waves and vibrations, Mechanical Systems and Signal Processing13,pp. 1-30, 1999.

[5] Catania G., Sorrentino S., Experimental validation of non-conventional viscoelastic models via equivalentdamping estimates, proceedings of IMECE 2008, Boston, Massachussetts, USA, 2008.

[6] Ewins, D.J., Modal Testing: theory, practice and application, 2nd ed., Research Studies Press, 2000.

[7] Catania G., Sorrentino S., Experimental identification of a fractional derivative linear model for viscoelastic

materials, proceedings of IDETC/CIE 2005, Long Beach, California, USA, 2005.

[8] Catania, G., Sorrentino, S., Fractional derivative linear models for describing the viscoelastic dynamicbehaviour of polymeric beams, proceedings of IMAC 2006, Saint Louis, USA, 2006.

[9] Catania, G., Sorrentino, S., Analytical modeling and experimental identification of viscoelastic mechanicalsystems, in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics andEngineering, Springer, pp. 403-416, 2007.

[10] Blevins R.D., Formulas for natural frequency and mode shape, Krieger, USA, 1979.

[11] Sorrentino S., Fasana A., Finite element analysis of linear systems with fractional derivative dampingmodels, Journal of Sound and Vibration 299 (4-5), pp. 839-853, 2007.

[12] Catania G., Sorrentino S., Discrete spectral modelling of continuous structures with fractional derivativeviscoelastic behaviour, proceedings of IDETC/CIE 2007, Las Vegas, Nevada, USA, 2007.

[13] Catania G., Fasana A., Sorrentino S., A condensation technique for the FE dynamic analysis with fractionalderivative viscoelastic models, Journal of Vibration and Control 2008 14, pp. 1573-1586, 2008.


11/12

Appendix: Damping ratio estimates.

Polyethylene (PE) Polyvinyl chloride (PVC) Delrin

n [%] n [%] n [%]n

[Hz] A1 A2 A3

n

[Hz] A1 A2 A3

n

[Hz] A1 A2 A3

40.8 1.39 1.40 1.41 39.8 1.14 1.14 1.14 42.0 1.29 1.29 1.29114.8 1.09 1.09 1.09 112.7 0.84 0.84 0.84 118.6 1.22 1.22 1.22

224.8 1.06 1.07 1.05 220.2 0.83 0.83 0.82 230.5 1.23 1.24 1.20

503.9 1.20 1.23 1.23 562.3 1.02 1.02 1.02 377.7 1.39 1.47 1.43

595.5 1.19 1.19 1.18 755.5 1.03 1.03 1.03 600.2 1.49 1.48 1.50

789.2 1.15 1.18 1.18 990.2 1.02 1.00 1.03 800.5 1.57 1.61 1.63

1025.8 1.18 1.16 1.22 1851.4 1.24 1.29 1.32 1051.4 1.72 1.85 1.79

BeamL1 - h1

1303.9 1.13 1.16 1.19 2174.8 1.18 1.20 1.19 1330.5 1.63 1.68

16.7 2.57 2.59 2.59 48.3 1.08 1.09 1.10

47.3 1.37 1.35 1.37 94.8 1.24 1.25 1.25

92.3 1.17 1.18 1.16 156.7 1.28 1.29 1.30

151.1 1.18 1.18 1.20 231.1 1.42 1.42 1.43356.4 1.05 1.07 1.08 308.3 1.32 1.31 1.29

416.1 1.22 1.23 1.22 427.3 1.35 1.47

485.2 1.05 1.05 487.9 1.37 1.42

602.7 1.02 1.05 603.6 1.48 1.54

738.6 0.99 1.00 1.12 744.8 1.55 1.47 1.63

898.9 1.12 1.15 1.16 904.2 1.61 1.75 1.81

Beam

L1 - h2

1068.6 1.05 1.06 1.05

51.1 1.46 1.46 1.46 49.8 1.07 1.07 1.07 147.0 1.30 1.30 1.30

142.6 1.06 1.06 1.06 139.5 0.97 0.98 0.97 286.7 1.48 1.50 1.49

276.7 1.08 1.08 1.06 272.9 0.96 0.96 0.96 463.6 1.58 1.61 1.62

457.3 1.06 1.06 1.07 444.5 0.94 0.94 0.94 723.9 1.73 1.80 1.86716.4 1.27 1.33 1.32 993.0 1.70 1.75 1.79

Beam

L2 - h1

970.8 1.20 1.24 1.25 2012.0 1.99 2.09 2.18

20.5 1.55 1.56 1.56 58.6 0.78 0.78 0.78 21.1 1.28 1.31 1.30

58.6 1.32 1.39 1.34 115.5 0.77 0.77 0.77 60.5 1.21 1.21 1.23

113.6 1.16 1.18 1.17 188.9 0.82 0.83 0.83 118.3 1.27 1.28 1.26

183.3 1.20 1.22 1.22 273.9 0.88 0.89 0.89 192.3 1.40 1.41 1.41

257.3 1.13 1.13 576.1 0.86 0.88 0.87 273.0 1.52 1.52

338.0 1.07 1.06 723.3 0.97 0.95 585.2 1.54 1.58

425.2 1.00 1.02 1.01 894.2 0.86 0.87 0.89 752.0 1.58 1.63

588.9 1.06 1.08 1.08 1086.1 0.86 0.87 0.88 929.8 1.68 1.59 1.73

740.5 1.09 1.12 1133.0 1.50 1.55 1.59

914.8 1.08 1.10 1.12

Beam

L2 - h2

1108.3 1.21 1.24 1.24

64.2 0.87 0.87 0.87 67.0 1.35 1.35 1.35

345.5 0.92 0.92 0.92 186.4 1.38 1.38 1.38

546.1 0.93 0.93 0.94 360.1 1.50 1.51 1.50

883.3 1.04 1.02 1.03

Beam

L3 h1

1193.9 1.03 1.04 1.04


12/12

26.4 1.66 1.67 1.68 26.4 0.89 0.89 0.89 27.0 1.25 1.25 1.26

73.6 1.53 1.53 1.53 75.5 0.85 0.85 0.86 75.8 1.25 1.25 1.25

142.0 1.29 1.29 1.26 146.7 0.84 0.84 0.85 147.3 1.24 1.25 1.23

225.1 1.25 1.28 1.29 235.5 0.82 0.83 0.83 237.0 1.48 1.50 1.50

411.1 1.14 1.14 1.13 556.7 0.89 0.89 0.90 333.6 1.45 1.46

572.0 1.19 1.21 1.21 722.7 0.89 0.89 514.5 1.64 1.65 1.63

736.7 1.12 1.14 916.4 0.85 0.87 746.4 1.38 1.45 1.42

930.2 1.19 1.21 1134.2 0.95 0.95 0.97 938.3 1.67 1.72

Beam

L3 - h2

1150.5 1.22 1159.2 1.49 1.62 1.62

33.3 1.74 1.74 1.74 38.0 1.39 1.37 1.35 71.7 1.74 1.73 1.73

204.8 1.35 1.36 1.36 723.6 0.89 0.88 0.90 1257.0 1.79 1.81 1.85

Beam

L4 h1

628.9 1.24 1.24 1.24 2007.0 1.10 1.11 1.11

13.9 1.70 1.70 1.70 94.5 0.98 0.98 0.98 239.8 1.26 1.26 1.28

86.4 1.50 1.50 1.50 248.6 0.89 0.89 0.89 610.2 1.44 1.45 1.47

566.7 1.12 1.12 1.13 599.8 0.78 0.78 0.79 2794.2 2.24 2.23

1301.7 1.11 1.13 1.12 885.5 0.78 0.80 0.82

2288.6 1.47 1.54 1364.2 0.91 0.90 0.91

2640.5 1.57 1.57 2109.5 1.01 1.04

2426.7 1.20 1.18 1.25

Beam

L4 h2

2810.2 1.30 1.35

126.7 1.21 1.22 1.23 142.0 1.45 1.45 1.46Beam

L5 h1 3347.7 1.78 1.80 1.69 2387.7 1.94 1.9859.2 1.20 1.20 1.20 279.8 1.35 1.36 1.36

533.6 0.93 0.93 0.93 548.9 1.36 1.36 1.36

1160.5 1.08 1.10 1.10 2188.3 2.20 2.23

2178.3 1.33 1.38 1.39 3004.2 2.25 2.34

Beam

L5 h2

2998.0 1.33 1.37

2.6 3.41 3.35 3.32 6.2 1.69 1.63 1.77 6.1 1.40 1.52 1.47

6.3 1.99 2.00 2.11 41.6 0.92 0.93 0.92 16.4 0.89 0.90 0.90

23.6 1.38 100.2 0.68 0.70 0.73 21.7 0.99 0.91 0.90

42.8 1.19 1.20 1.20 42.0 0.90 0.92 0.91

50.8 1.29 1.34 1.27 99.6 0.94 0.95 1.05

72.8 1.10 1.02

Plate

f

0.1 Hz

100.6 1.01

50.8 1.27 73.3 1.15 48.9 1.16

110.2 0.97 181.7 0.82 111.4 1.02

265.2 1.16 270.2 0.99 272.0 1.26

422.3 0.87 1209.2 0.93 423.9 1.49

532.7 0.86

Plate

f

0.3 Hz

1152.3 1.13

Paper Example

Documents

Transcript of Paper Example