Hybrid Viscous-Structural DampingIdentification Method

Vikas Arora

Abstract Damping still remains one of the least well-understood aspects of generalvibration analysis. The effects of damping are clear, but the characterization ofdamping is a puzzle waiting to be solved. A major reason for this is that, in contrastwith inertia and stiffness forces, it is not clear which state variables are relevant todetermine the damping forces. In this paper, a new hybrid viscous-structuraldamping identification method is proposed. The proposed method is a directmethod and gives explicit structural and viscous damping matrices. The effec-tiveness of the proposed structural damping identification method is demonstratedby two numerical examples. First, numerical study of lumped mass system ispresented which is followed by a numerical study of fixed-fixed beam. The effectsof coordinate incompleteness and different level of damping are investigated. Theresults have shown that the proposed method is able to identify accurately thedamping of the system.

Keywords Damping identification � Hybrid viscous-structural dynamic system �Direct method

1 Introduction

The modeling of damping is a very complex and still considered somewhat anunknown or grey area. A major reason for this is that, in contrast with inertia andstiffness forces, it is not in general clear which state variables are relevant todetermine the damping forces. A commonly used model originated by Rayleigh [1]assumes that instantaneous generalized velocities are the only variables. However,it is important to avoid the misconception that, this is the only model of vibration

V. Arora (&)Institute of Technology and Innovation, University of Southern Denmark, Campusvej 55,5230 Odense, Denmarke-mail: viar@iti.sdu.dk

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_18

209

damping. Any model, which guarantees that the energy dissipation rate is non-negative, can be a potential candidate to represent the damping of a given structure.In broad terms, damping mechanisms can be divided into three categories namelymaterial damping, boundary damping and viscous damping. Material damping canarise from variety of micro structural mechanisms [2] but for small strains it is oftenadequate to represent it through an equivalent linear, visco-elastic continuum modelof the material. Heckl [3] found that linear theory produce acceptable responsepredictions for panels whose damping mechanism arose from a bolted joint onbeam. Oliveto and Greco [4] conducted a study on how the modal damping ratioschange with different boundary conditions and they found that Rayleigh-typedamping is actually independent of the boundary conditions and modal dampingratios can be easily converted from one boundary condition to another. Maia et al.[5] discussed the use of fractional damping concept for the modeling the dynamicbehavior of the linear systems and showed how this concept allows for clearerinterpretation and explanation of the behavior displayed by common viscous andhysteric damping models. Agrawal and Yuan [6] modeled the damping forcesproportional to the fractional derivative of displacements and the fractional differ-ential equations governing the dynamics of a system. Adhikari and Woodhouse [7]developed four indices to quantify non-viscous damping in discrete linear system.Chen et al. [8] presented a method for getting the spatial model from complexfrequency response function. Unfortunately, it is unrealistic to assume that allpertinent information is given to solve for damping matrix. Actually, data fromtesting is neither complete nor error free. Minas and Inman [9] proposed a methodwhich assumes that analytical mass and stiffness matrices are determined a priorifrom a finite element model. The identified damping matrix is assumed to be real,symmetric and positive definite. The structure must exhibit complex modes for thisprocedure and the solution is limited to real symmetric positive definite dampingmatrices. Beliveau [10] uses natural frequencies, damping ratios, mode shapes andphase angles to identify parameters of viscous damping matrix. The identification isperformed iteratively. Lancaster [11] proposed a method of identifying the mass,stiffness and damping matrices of a system directly given only the eigenvalues andeigenvectors. This method is only for calculating the viscous damping. The shortfallof this method comes in normalizing the eigenvectors, which requires knowledge ofthe very same damping matrix which we wish to, find in the end. Pilkey [12, 13]proposed two methods for computing the damping matrix using modal data. Thefirst method is an iterative method which requires prior knowledge of the massmatrix and eigenvalues and eigenvectors. The second method requires moreinformation but less computationally intensive. Both the methods developed fromthe Lancaster [11] concept. Oho et al. [14] proposed a method of identifyingexperimental set of spatial matrices valid only for the hysterical damping for theentire frequency range of interest using FRFs. The limitation of this method is that itis unable to predict correctly in the modal domain. Lee and Kim [15] proposed analgorithm for the identification of the damping matrices which identifies the viscousand structural damping matrices using frequency response matrix. The accuracy ofthe identified damping matrices depends almost entirely on the accuracy of the

210 V. Arora

measured FRFs, especially their phase angles. Adhikari and Woodhouse [16]identified the damping of the system as viscous damping. However, viscousdamping is by no means the only damping model. Adhikari and Woodhouse [17]identified non-viscous damping model using an exponentially decaying relaxationfunction. Phani and Woodhouse [18] proposed that complex modes arising out ofnon-proportional dissipative matrix hold the key to successful modeling andidentification of correct physical damping mechanisms in the vibrating systems butthese identified complex modes are very sensitive to experimental errors and errorsarising out from fitting algorithms. Some research efforts have also been made toupdate the damping matrices. Imregun et al. [19] extended the response functionmethod (RFM) given by Lin and Ewins [20] to update proportional viscous andstructural damping matrices by updating the coefficients of viscous and structuraldamping matrices. Arora et al. [21] identified the structural damping matrix usingcomplex frequency response functions (FRFs) of the structure. Arora et al. [22]proposed a viscous damping identification method in which viscous damping isidentified explicitly. This procedure is a two steps procedure. In the first step, massand stiffness matrices are updated and in the second step, viscous damping isidentified using updated mass and stiffness matrices obtained in the previous step.Pradhan and Modak [23] used the normalized FRFs for updating damping matrices.The normalized FRFs are estimated from the complex FRFs. Arora [24] proposed astructural damping identification method in which normalized FRFs are used. Theproposed method is a direct method and gives explicit structural damping matrix.

The proposed hybrid viscous-structural damping identification method requiresprior knowledge of accurate mass and stiffness matrices. The identified viscous-structural damping matrices are both symmetric and positive definite. The effec-tiveness of the proposed structure damping identification method is demonstratedby two numerical simulated examples. Firstly, a study is performed using a lumpedmass system. The lumped mass system study is followed by case involvingnumerical simulation of fixed-fixed beam. The effect of coordinate incompletenessand different levels of damping in a structure are investigated. The results haveshown that the proposed method able to identify the damping matrices accuratelywith all levels of damping.

2 Hybrid Viscous-Structural Damping IdentificationMethod

In the following section, the theory and procedure of proposed direct structuraldamping identification method is developed. The governing equations of motion ofa multi degree of freedom (DOF) structure with structural damping can be written inmatrix form as:

Hybrid Viscous-Structural Damping Identification Method 211

M½ � x:: tð ÞCþ K½ �x tð ÞCþ C½ � x: tð ÞCþ D½ �x tð ÞC¼ f tð Þ ð1Þ

where [M],[K] and [D] are n × n DOF mass, stiffness and structural dampingmatrices respectively. All these are real matrices. j = √−1. x(t) and f(t) are n × 1vectors of time-varying displacements and forces and ‘N’ is the total number ofdegrees of freedom in the finite element model. The superscript C indicates complexvalue corresponding to damped system. For harmonic excitation, f tð Þ ¼ FðxÞejxtand x tð Þ ¼ XðxÞejxt. The Eq. (1) becomes:

K½ � � x2 M½ �� �X xð ÞCþ j C½ �xþ D½ �ð ÞX xð ÞC¼ F xð Þ ð2Þ

Real or normal dynamic stiffness matrix (DSM) is given by:

ZðxÞN� � ¼ K½ � � x2 M½ � ¼ aðxÞN� ��1 ð3Þ

where aðxÞN� �represents real or normal FRF matrix. Using above equation, Eq. (1)

becomes:

ZðxÞN� �X xð ÞCþ j C½ �xþ D½ �ð ÞX xð ÞC¼ F xð Þ ð4Þ

Multiply both sides by real or normal FRF matrix aðxÞN� �. The Eq. (4)

becomes:

X xð ÞCþ j a xð ÞN� �C½ �xþ D½ �ð ÞX xð ÞC¼ a xð ÞN� �

F xð Þ ð5Þ

G(x) is a transformation matrix:

GðxÞ ¼ a xð ÞN� �C½ �xþ D½ �ð Þ ð6Þ

The Eq. (5) becomes:

X xð ÞCþ jGðxÞX xð ÞC¼ a xð ÞN� �F xð Þ ð7Þ

Equation (7) can be rewritten as:

IDþ jGðxÞð ÞX xð ÞC¼ a xð ÞN� �F xð Þ ð8Þ

where ID represents identity matrix. The displacement vector X xð ÞC is related to

input force vector and also to complex FRF matrix a xð ÞCh i

as:

xðxÞ ¼ a xð ÞCh i

f ðxÞ ð9Þ

212 V. Arora

complex FRF matrix aðxÞCh i

consist of real and imaginary parts as:

aðxÞCh i

¼ aðxÞCRh i

þ j aðxÞCIh i

ð10Þ

where aðxÞCRh i

and aðxÞCIh i

represents real and imaginary part of complex FRF

matrix aðxÞCh i

. The subscripts R and I represents real and imaginary values

respectively. The complex FRF aðxÞCh i

is available from experimentation.

Substituting Eq. (10) in Eq. (8), we get:

IDþ jGðxÞð Þ aðxÞCRh i

þ j aðxÞCIh i� �

¼ a xð ÞN� � ð11Þ

aðxÞCRh i

� GðxÞ aðxÞCIh i� �

þ j GðxÞ aðxÞCRh i

þ aðxÞCIh i� �

¼ a xð ÞN� � ð12Þ

since the right-hand side of the above equation is real so the imaginary part of theleft-hand side must be equal to a zero matrix for all frequencies, hence:

GðxÞ aðxÞCRh i

þ aðxÞCIh i

¼ 0 ð13Þ

GðxÞ ¼ � aðxÞCRh i�1

aðxÞCIh i

ð14Þ

using Eq. (6). The above equation becomes:

a xð ÞN� �C½ �xþ D½ �ð Þ ¼ � aðxÞCR

h i�1aðxÞCIh i

ð15Þ

C½ �xþ D½ �ð Þ ¼ � aðxÞCRh i�1

aðxÞCIh i

a xð ÞN� ��1 ð16Þ

In the matrix form Eq. (16) can be written as:

x 1½ � C½ �D½ �

� ¼ � aðxÞCR

h i�1aðxÞCIh i

a xð ÞN� ��1 ð17Þ

since the normal dynamic stiffness matrix [Z(x)N] is inverse of normal FRF

aðxÞN� �matrix ZðxÞN� � ¼ aðxÞN� ��1

� �given in Eq. (3). Using Eq. (3), Eq. (17)

becomes:

Hybrid Viscous-Structural Damping Identification Method 213

C½ �D½ �

� ¼ � x 1½ ��1 aðxÞCR

h i�1aðxÞCIh i

Z xð ÞN� � ð18Þ

The above equation is the basic equation for the hybrid viscous-structuraldamping identification method. The identified viscous and structural dampingmatrices are both symmetric and positive definite. The Eq. (18) is applied to fre-quency range of interest. This makes Eq. (18) over determined. If nf is the numberof frequency points used to identify the structural damping. The Eq. (15) becomes:

C½ �D½ �

� n�n

¼ �

x1 1x2 1... ..

.

..

. ...

..

. ...

xnf 1

2666666664

3777777775

þ aðx1ÞCRh iaðx2ÞCRh i...

..

.

..

.

aðxnf ÞCRh i

2666666666664

3777777777775

þ

ðn�nf Þ�n

�

aðx1ÞCIh i

Z x1ð ÞN� �aðx2ÞCIh i

Z x2ð ÞN� �...

..

.

..

.

aðxnf ÞCIh i

Z xnf� �Nh i

2666666666664

3777777777775ðn�nf Þ�n

ð19Þ

superscript ‘+’ denotes pseudo-inverse of a matrix. Equation (19) is used to identify

structural damping matrix [D] of the structure. aðxÞCRh i

and aðxÞCIh i

are the real

and imaginary part of the FRF matrix, which are obtained from the experimenta-tion. The accurate normal dynamic stiffness matrix (DSM) Z xð ÞN� �

depends uponaccurate knowledge of mass and stiffness matrices. For the proposed structuraldamping identification method mass and stiffness matrices are reduced to thedegrees of freedom measured using iterated IRS method [25] considering themeasurement points.

3 Case Study of Lumped Mass System

2 degree- of freedom lumped mass system shown in the Fig. 1 is described bylumped masses M1, M2 of 5 kg each, spring stiffnesses K1, K2 of 2 × 106 N/m each,the structural damping D1, D2 of 6 × 104, 8 × 104 N/m respectively and viscousdamping C1, C2 of 200, 400 Ns/m respectively. Because this is a contrivedexample, the described result is known ahead of time. As the system has 2 DOF, 4complex FRFs are calculated for each frequency forming a 2 × 2 complex FRFmatrix of a function of frequency. The dynamic stiffness matrix is also calculatedfrom the analytical mass and stiffness matrices for each frequency. As all the 2natural frequencies are in the frequency range of 0–200 Hz.

214 V. Arora

The frequency range from 0 to 200 is used to identify the viscous and structuraldamping using the proposed procedure. In this case study, it is assumed thataccurate mass and stiffness matrices are known. Figure 2a shows the overlay ofanalytical FRF, which is undamped, and simulated experimental FRF obtainedconsidering structural damping in the experimental data. It can be observed that theanalytical FRF and simulated experimental FRF do not match with each other atresonance regions. Figure 2b shows the overlay of simulated experimental FRF andidentified damped FRF. It can be observed from Fig. 2b that after damping iden-tification, simulated experimental FRF and identified damped FRF match exactlyand the two are virtual indistinguishable.

4 Case Study of Fixed-Fixed Beam Structure

A simulated study on a fixed-fixed beam is conducted to evaluate the effectivenessof the proposed method. The dimensions of the beam are 910� 50� 5 mm. Thebeam is modeled using thirty as shown in Fig. 3. The displacements in y-direction

M1 M2

D1

K1

C1

D2

K2

C2

Fig. 1 2 degree of freedomlumped mass system

0 20 40 60 80 100 120 140 160 180 200-150

-140

-130

-120

-110

-100

-90

-80

Frequency in Hz

Rec

epta

nce

MA

G in

dB

Simulated Experimental FRFAnalytical (undamped) FRF

0 20 40 60 80 100 120 140 160 180 200-145

-140

-135

-130

-125

-120

-115

-110

-105

-100

-95

Frequency in Hz

Rec

epta

nce

MA

G in

dB

Simulated Experimental FRFIdentified damped FRF

(a) (b)

Fig. 2 Overlay of simulated experimental FRF and damped identified FRF of lumped masssystem. a Before damping identification. b After damping identification

Hybrid Viscous-Structural Damping Identification Method 215

and the rotation about the z-axis are taken as two degrees of freedom at each node.It is assumed that prior knowledge of accurate and stiffness is known. Dampingconsidered for all the cases is proportional damping being proportional to both mass[M] and stiffness [K] matrices [21]. The proposed method has been evaluated for thecases of different levels of damping in the numerical data. The real life measureddata is always incomplete. Incompleteness is considered by assuming that onlylateral degrees of freedom, at all the 29 nodes are measured.

The performance of the proposed method is judged on the basis of accuracy withwhich the identified damped FRFs obtained by proposed method with the simulatedexperimental FRFs. The case of simulated experimental data obtained using highlevel of structural damping and low level of viscous damping is considered. Fig-ure 4a shows the overlay of analytical FRF, which is undamped, and simulatedexperimental FRF obtained considering high level of structural damping and lowlevel of viscous damping in the experimental data. It can be observed that theanalytical FRF and simulated experimental FRF do not match with each other atresonance and anti-resonance regions. After damping identification using proposedmethod, the identified and simulated FRFs match completely as shown in Fig. 4b.

Fig. 3 Beam structure and its FE mesh

0 100 200 300 400 500 600 700 800 900 1000-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

Frequency in Hz

Rec

epta

nce

MA

G in

dB

Analytical (Udamped) FRFSimulated Experimental FRF

0 100 200 300 400 500 600 700 800 900 1000-200

-180

-160

-140

-120

-100

-80

-60

-40

-20

Frequency in Hz

Rec

epta

nce

MA

G in

dB

Identified damped FRFSimulated Experimental FRF

(a) (b)

Fig. 4 Overlay of simulated experimental FRF and damped identified FRF for high structural andlow viscous damping. a Before damping identification. b After damping identification

216 V. Arora

5 Conclusions

In this paper, a new method of hybrid viscous-structural damping identificationmethod is proposed. The proposed structural damping identification methodrequires prior knowledge of accurate mass and stiffness matrices. The proposedmethod doesn’t require initial damping estimates. The identified viscous structuraldamping matrices are both symmetric and positive definite. The proposed methodaddresses the difficulties for updating the damping matrices. The proposed methodis working successfully for the cases of simulated numerical data. Simulatednumerical case studies based on a lumped mass system and fixed-fixed beamstructure with both viscous and structural damping models are carried out to assessthe effectiveness of the proposed method. The success of these cases has proven thefeasibility of the proposed method. The results have shown that the proposedmethod able to identify the structural damping matrices accurately in all levels ofdamping in the structure.

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