THEORETICAL AND EXPERIMENTAL STUDY OF MODAL STRAIN ANALYSIS

Journal of Sound and Vibration (1996) 191(2), 251–260

THEORETICAL AND EXPERIMENTAL STUDY OF MODAL STRAIN ANALYSIS L. H Y  T. P. L Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

 D. B. L  K. Z. X Department of Engineering Mechanics, Tsinghua University, Beijing, China (Received 16 September 1993, and in final form 20 April 1995) The relationship between the strain mode and displacement mode for vibrating elastic structures is derived. It is based on the idea that when a structure is subjected to dynamic loading, its strain response can be expressed by the superposition of contributions from the ‘‘natural strain modes’’. Both the natural strain and natural displacement modes represent the dynamic characteristics of a vibrating structure. By using the finite element method, it has been possible to relate the Strain Frequency Response Function (SFRF) to the Displacement Frequency Response Function (DFRF). The technique used in the modal strain test and the method used for parameter identification are described. An experimental study of the modal strain analysis of cantilever rectangular thin plates has been performed. This shows that the strain mode is more sensitive to local changes of the structure than the displacement mode. 7 1996 Academic Press Limited

1. INTRODUCTION

Nowadays, in the design of moving machines and dynamically loaded structures, it has become normal practice to consider the dynamic and fatigue strength of the systems. These two problems are related to the dynamic strain and the dynamic stress of the structure. In order to make the analysis more accurate and reliable, there has been a tendency in the past decade to study the establishment of structural strain/stress response prediction models directly from experimental measurements and the analysis of the load-strain transfer function [1–5]. This has been followed by further studies of modal strain analysis leading to practical engineering applications [3, 4, 6]. The strain response of an elastic vibrating system can be obtained by the linear superposition of the contributions of ‘‘eigenmodal strain fields’’ which are also called ‘‘strain modes’’ [3, 6]. The strain mode and displacement mode are both intrinsic dynamic characteristics of a structure and they correspond to each other; but the latter is not so sensitive to local changes in a structure, such as holes, grooves and cracks. Therefore, structural modal strain analysis is more useful in the dynamic design of structures with such features. In this paper, the basic formulation of modal strain analysis, including the matrix for the strain frequency response function and the model for strain response prediction, is given. The relationship between modal strain analysis and modal displacement analysis, as well as the method of parameter identification for the modal 251 0022–460X/96/120251 + 10 $18.00/0

7 1996 Academic Press Limited

. .    .

252

strain analysis are discussed. Experiments for the modal strain analysis of cantilever plates, with and without holes, are performed to show the advantage of using modal strain analysis. This makes it possible to use modal strain analysis in the detection of structural damage or changes in the characteristics of a machine during its operational life. 2. RELATIONSHIP BETWEEN STRAIN MODE AND DISPLACEMENT MODE

The relationship between modal strain and nodal displacement of a finite element is to be derived. The expression for strain response is also to be derived, and thus a relationship between the strain and displacement modes is obtained. 2.1.       Let {de }i represent the vector of all nodal displacements in element i, and {d}i represent the displacement vector of a certain point in element i. Then {d}i = [P]{a}i ,

(1)

where [P] is the matrix of the displacement function which is generally a polynomial function of the space variables, and {a}i is the coefficient vector of this polynomial; its components are constants to be determined (a list of principal nomenclature is given in the Appendix). In order to determine {a}i uniquely, the number of unknown constants should equal the total number of nodal displacements in an element. At the nodes of an element the space variables have given values, which means, from equation (1), that {de }i = [A]i {a}i ,

(2)

and the elements in [A]i have known values. The coefficient vector can be obtained from {a}i = [A]−1 i {de }i .

(3)

Substituting equation (3) into equation (1) gives {d}i = [P][A]−1 i {de }i . The strain {o}i at any point in element i is then {o}i = [D]{d}i = [D][P][A]−1 i {de }i = [B]i {de }i ,

(4)

where [B]i = [D][P][A]−1 i , i = 1, 2, . . . , n, [D] is a differential operator and n is the total number of elements. The n equations (4) can be integrated together as

K {o}1L K [B]1 G {o} G G [B]2 G 2G G G * G =G G {o}iG G G *G G G G G k {o}nl k

... [B]i ...

L G G G G G G [B]nl

K {de }iL G {d } G G e 2G G * G, G {de }iG G * G G G k {de }nl

(5)

or, in a more compact form, {o} = [B]{d},

(5')

  

253

where {o} and {d} are the strain vector and the nodal displacement vector for n elements, respectively. When the continuity conditions on the displacements at the joint nodes of elements are considered, equation (5) can be transformed to a global co-ordinate system through the co-ordinate transform {d} = [b]{ds }

(6)

where {ds } is the nodal displacement vector in the global co-ordinate system and [b] is the co-ordinate transform matrix. Then equation (5') can be expressed as {o} = [B][b]{ds }.

(7)

2.2.          The equations of motion for a finite element model can be written down as [Ms ]{d s } + [Cs ]{d s } + [Ks ]{ds } = { fs }.

(8)

{ds } = {Us } e jvt .

(9)

If { fs } = {Fs } e jvt then

If proportional damping is assumed, equation (8) can be rewritten as (−v 2[Ms ] + jv[Cs ] + [Ks ]){Us } = {Fs }.

(8')

By using the superposition of modal contributions to express the displacement response, the solution of equation (8') can be expressed as {Us } = [F][ yr ][F]T{Fs } = [H]{Fs },

(9')

where [F] = [{81 }{82 } · · · {8k } · · · {8m }], yk = (kk − v 2mk + jvck )−1 ,

[ yr ] − diag[ y1 , y2 , . . . , yk , . . . , ym ], m

[H] = [F][ yr ][F]T = s yr {8r }{8r }T.

(10)

r=1

kk , mk and ck are the kth modal stifiness, modal mass and modal damping, respectively, v is the frequency of excitation, [H] is called the matrix of the displacement frequency response function (DFRF), [F] is the mode matrix, {8r } is the rth real-value eigenmode vector, m is the total number of modes considered, and j = z−1. From equations (7), (9) and (9') it can be shown that {o} e jvt = [B][b][F][ yr ][F]T{Fs } e jvt = [C o][ yr ][F]T{Fs } e jvt m

m

r=1

r=1

= s yr {cro }{8r }T{Fs } e jvt = [C o]{q} = s qr {Cro },

(11)

where [C o] = [{c1o } {c2o } · · · {cko } · · · {cmo }] = [B][b][F], {cko } = [B][b]{8k },

qk = yk {8k }T{Fs } e jvt .

(12)

{cko } is called the kth strain mode corresponding to {8k }. From the energy point of view, {cko } corresponds to the kth natural energy equilibrium state; therefore, it is also a characteristic parameter of a vibrating structure. Equation (11) is the model for the strain response prediction and qk is the modal co-ordinate. The physical meaning of yk is that

. .    .

254

it is the frequency response function of qk , when considered as a single-degree-of- freedom system.

3. EXPERIMENTAL MODAL STRAIN ANALYSIS AND PARAMETER IDENTIFICATION

When the above-mentioned strain analysis method is adopted for a finite element calculation, the following problems are inevitable. First, at the joint nodes between elements only the displacement is continuous, and the strain is generally not continuous; second, the interpolation function in each element is an assumed static shape function which is different from the dynamic modal function; finally, it is generally difficult to determine the actual damping characteristics of any structure. Experimental determination of the various parameters in equation (10) is an alternative method for solving this problem. Since the strain mode and its corresponding parameters are obtained from the experiment, in principle the above-mentioned problems do not exist. 3.1.       (SFRF)    From equation (11), m

{o} = [C o][ yr ][F]T{Fs } = [H o]{Fs } = s yr {cro }{8r }T{Fs },

(11')

r=1

where m

[H o] = [C o][ yr ][F]T = s yr {cro }{8r }T

(13)

r=1

is called the matrix of the strain frequency response function (SFRF). The elements of this matrix can be expressed as m

Hijo = s yr ciro 8jr .

(14)

r=1

This is the strain response at point i induced by a unit force applied at point j. It can be seen from equation (14) that Hijo $ Hjio : i.e., the SFRF matrix is not symmetrical. Indeed, the SFRF matrix is generally not a square matrix. Equation (13) can be expanded to the following form:

KH11o GH21o G. . . o kHN1

o H12 o H22 ... o HN2

... ... ...

o H1n L o H2n G . . .G o HNn l

KC1ro 81r C1ro 82r . . . C1ro 8nr L GC2ro 81r C2ro 82r . . . C2ro 8nr G . (15) = s yr G ... ... ... G r=1 o o o kCNr 81r CNr 82r . . . CNr 8nr l N × n N×n m

It can be seen that the SFRF matrix possesses the following properties: (i) any element Hijo of this matrix contains the information about kr , mr and cr for all the modes; (ii) any row of this matrix contains the information about the displacement mode {8r } for all the modes; (iii) any column of this matrix contains the information about the strain mode {C ro } for all the modes. Therefore, in order to obtain all the modal parameters, any one row

  

255

o

together with any one column of [H ] should be measured. It is not enough to measure only one row or one column. 3.2.      It can be seen by a comparison between equations (10) and (13) that the DFRF matrix [H] and the SFRF matrix [H o] contain entirely the same modal parameters of kr , mr and cr . However, the expression for [H] contains only the displacement mode {8r }, while [H o] contains both {8r } and the strain mode {cro }. Therefore, if the modal displacement analysis of a structure has been made, only one column of the SFRF matrix needs to be measured to obtain all the strain modes {cro }. Thus, it is only necessary to excite one selected point and acquire the strain response at all the measurement points, which enables {cro } to be obtained after data processing. The method for experimental modal strain analysis can be summarized as follows. (i) Perform a modal displacement analysis of the structure to be tested, to obtain modal parameters kr , mr , cr and {8r } (r = 1, 2, . . . , m); any method developed for modal displacement testing and parameter identification can be adopted at this step. (ii) Excite the structure at only one selected point (e.g., point number t) and measure the strain response at all the N measurement points; N curves of strain transfer function of the tth column in [H o] can be obtained. (iii) Suppose that the real modes are considered; the following formula then can be used for curve fitting [6]:

K H1to L G H2to G G G G * G m 1 G oG = s H 2j(jv − Sr ) tt G G r=1 G G G * G kHNto l

K rR1to L G rR o G G G G * G m 1 G oG − s 2j(jv − S* r Rtt r ) G G r=1 G G G * G k rRNto l 2t

K rR1to L G rR2to G G G G * G G oG , G r Rtt G G G G * G krRNto l

o T {cro } + (mr nr /8tr )[r R1to , rR2to , . . . , rRNt ],

(16)

(17)

where sr = −sr + jnr , nr = zVr2 − sr2 ,

sr = cr /2mr ,

s* r = −sr − jnr , Vr = zkr /mr ,

r

Rito = ciro 8tr /(mr nr ).

(18)

The residue vector can be identified by using equation (16). Because the 8tr have been obtained from the displacement mode identification, the strain mode can be obtained by using equation (17).

4. EXPERIMENTAL MODAL STRAIN ANALYSIS OF CANTILEVER PLATES

As a practical example, the modal strain analysis of a cantilever plate with and without holes has been performed experimentally. The background of this example is related to engineering applications in construction of bridges, ships, etc., since the thin plate components in these structures often have holes. Therefore, the study of the dynamic stress concentration of a plate with holes is of general significance.

256

. .    .

Figure 1. A specimen for experimental modal strain analysis: an aluminium rectangular thin cantilever plate without a hole. Dimensions in mm. Plate thickness 3·3 mm.

4.1.  The specimen of a 252 mm × 193 mm × 3.3 mm aluminium rectangular thin cantilever plate without holes is shown in Figure 1. Its built-in edge is clamped by a jaw vice. Each edge of the plate is divided into six equal parts. There are a total of 49 measurement points. Since the structure is symmetrical, strain gauges are arranged along x or y or both directions at the measurement points on half of the plate. On the other half-plate strain gauges are arranged only at points 31, 33, 38 and 40 for the purpose of checking. For the experimental modal strain analysis of the plate with holes, two equal small circular holes each of radius 8 mm were drilled. Their centres were at points 17 and 19 respectively, as shown in Figure 1, respectively. The other conditions of the plate are the same as the plate without holes. In order to measure the strain at the circumference of the holes, four strain gauges are arranged symmetrically for each hole (see Figure 2).

4.2.      The schematic diagram for the set-up of this experimental modal strain analysis is shown in Figure 3.

Figure 2. A specimen for experimental modal strain analysis: an aluminium rectangular thin cantilever plate with holes (only the part near the holes is shown).

  

257

Figure 3. A schematic diagram of the set-up for Strain Frequency Response Function measurement.

The method of testing and parameter identification adopted in this study is as follows. (i) The displacement response at point 41 shown in Figure 1 is measured by using an accelerometer and the excitations are made by using a hammer at all the 49 measurement points including point 41. One row in the DFRF matrix [H] is measured to identify the modal parameters kr , mr , cr and the displacement mode {8r } (r = 1, 2, . . . , m). (ii) The hammer was used to excite the structure at point 41 and measurements of the strain response were taken at various points by using strain gauges. The tth (t = 41) column in [H o] is measured and {rRto } (r = 1, 2, . . . , m) in the residue matrix corresponding to this column can be obtained by using equation (16). (iii) By using equation (17), the strain mode {cro } is then calculated. 4.3.     The displacement modal frequencies for plates with and without holes obtained from experimental tests and finite element calculations are listed in Table 1. Reasonable agreement is shown between the experiment and the calculations. The first four displacement modes and their corresponding eigen frequencies for a plate without holes are shown in Figures 4(a)–(d), respectively, where the solid and the dotted lines correspond to the deformed and undeformed configurations of the plate, respectively (the same key is adopted for Figures 5–7). The first four displacement modes and their corresponding eigenfrequencies for the plate with holes are shown in Figures 5(a)–(d), respectively. Comparison of Figures 4 and 5 shows that for plates with and without holes the differences between their eigenfrequencies are quite small. Their displacement modes of the same order are nearly the same even at the circumferences of the holes; i.e., the small holes have very little influence on the displacement modes. Thus, the strain/stress concentration at the circumferences of the holes cannot be detected correctly from the displacement modes. T 1 Experimental and calculated results for displacement modal frequencies for plates with and without holes Method of analysis Experiment Calculation Experiment Calculation

Specimen of analysis Plate Plate Plate Plate

without hole without hole with holes with holes

Mode 1 (Hz)

Mode 2 (Hz)

Mode 3 (Hz)

Mode 4 (Hz)

44·273 135·775 43·888 (0·86%)† 127·90 (5·8%) 43·80 134·21 43·129 (1·5%) 128·04 (4·6%)

277·881 266·78 (3·9%) 278·25 264·15 (5·1%)

471·458 433·39 (8·07%) 471·25 436·22 (7·4%)

†The figures in parentheses are the relative differences between calculated results and experimental results.

258

. .    .

Figure 4. The displacement modes of the plate without a hole: (a) mode 1, v1 = 44·27 Hz; (b) mode 2, v2 = 135·77 Hz; (c) mode 3, v3 = 277·88 Hz; (d) mode 4, v4 = 471·46 Hz.

Modal strains along the x- and y-directions have been obtained. Here only those significant events occurring in the modes along the x-direction are discussed. The first four strain modes and their corresponding eigenfrequencies for plates without and with holes are shown in Figures 6 and 7, respectively. Although the form of deformation as a whole is quite similar in Figures 6 and 7, the strain obviously changes at the circumferences of the holes in Figure 7. In Figure 7(a) the change of strain at point 17 is more obvious than at point 19, since point 17 is nearer to the built-in edge. In Figure 7(b) ox at the circumferences of the two holes is comparatively small due to the mode being one of torsional displacement. In Figure 7(c) the deformation curvatures at the circumferences of the two holes and thus the ox are comparatively large due to the mode being one of bending displacement. In Figure 7(d) the change of curvature of the plate is large for this torsional displacement mode, and therefore the change of ox at the circumference of the holes is again very evident.

Figure 5. The displacement modes of the plate with holes; (a) mode 1, v1 = 43·80 Hz; (b) mode 2, v2 = 134·21 Hz; (c) mode 3, v3 = 278·25 Hz; (d) mode 4, v4 = 471·25 Hz.

  

259

Figure 6. The strain modes along the x-direction of the plate without a hole: (a) mode 1, v1 = 43·75 Hz; (b) mode 2, v2 = 134·31 Hz; (c) mode 3, v3 = 276·32 Hz; (d) mode 4, v4 = 470·24 Hz.

Figure 7. The strain modes along the x-direction of the plate with holes: (a) mode 1, v1 = 47·35 Hz; (b) mode 2, v2 = 134·31 Hz; (c) mode 3, v3 = 276·32 Hz; (d) mode 4, v4 = 470·24 Hz.

5. CONCLUSIONS

The relationship between the strain and displacement modes of a vibrating structure has been discussed. The theoretical background, fundamental formulae and experimental technique for modal strain analysis as well as a method for parameter identification have been introduced. The model for strain and stress response prediction established by using modal superposition can be applied to dynamic strength analysis and fatigue life estimation effectively. In order to reveal the difference between the modal strain analysis and the modal displacement analysis, experiments on plates with and without holes have been carried out. The strain frequency response functions were measured directly by using strain gauges, and the strain mode shapes were obtained. It was found that for plates with and without holes the modal frequencies obtained from modal displacement and modal strain analysis are nearly the same. This is as expected, because modal frequencies represent the dynamic characteristics of the whole structure. The strain mode, however, is more sensitive to

260

. .    .

structural local changes than the displacement mode. The extent of the sensitivity depends on the position of local structural change and the deformation form of each mode. Experimental modal strain analysis has some special problems which do not exist in modal displacement analysis. First, more experimental work needs to be done in modal strain analysis than in modal displacement analysis; second, the strain responses at higher modes are not as strong as the corresponding displacement responses under general testing conditions; therefore, the requirement for the sensitivity and accuracy of the instruments is higher. ACKNOWLEDGMENTS

This investigation has been supported by both the Research Committee of the Hong Kong Polytechnic University and research funds from the Open Research Laboratory for Structural Engineering and Vibration in Tsinghua University, Beijing, China. The authors are greatly indebted to these two institutions. REFERENCES 1. B. H and D. J. E 1984 Proceedings of 2nd IMAC, 627–634. The use of strain gauges in force determination and frequency response function. 2. T. X. S, P. Q. Z, W. Q. F and T. C. H 1986 Proceedings of 4th IMAC, 31–37. The application of time domain method in strain modal analysis. 3. D. B. L 1989 Proceedings of 7th IMAC, 1285–1289. The principle and techniques of experimental strain modal analysis. 4. O. B and D. J. E 1989 Journal of Modal Analysis, 68–76. Modal strain/stress fields. 5. W. F. T 1990 Proceedings of 8th IMAC, 1246–1289. Use of dynamic strain measurement for the modelling of structures. 6. D. B. L 1989 Modal Analysis with Applications. Beijing: Astronautics Book Company; first edition. See pp. 237–244.

APPENDIX: NOMENCLATURE Hij Hijo j kr m mr n N qr [r R o] sr {cro } {8r } vr zr

displacement frequency response function (DFRF) strain frequency response function (SFRF) =z−1, imaginary unit modal stiffness total number of modes considered modal mass number of measured points of displacements number of strains to be measured modal co-ordinate residue matrix of SFRF complex frequency strain mode displacement mode modal frequency modal damping ratio