- Email: [email protected]
On elastic beams parameter identification using eigenfrequencies changes and the method of added mass
Computational Materials Science 25 (2002) 207–217 www.elsevier.com/locate/commatsci
On elastic beams parameter identification using eigenfrequencies changes and the method of added mass Matjaz Skrinar
*
Faculty of Civil Engineering, University of Maribor, Smetanova 17, SI 2000 Maribor, Slovenia
Abstract Recently much attention has been focused on the problems of parameter identification based on the measured dynamic response of the structure. Inverse identification can be used either for the reconstruction of missing parameters of the computational analytical model, or as a tool for monitoring the structure during its utilisation. Inverse identification is usually performed with the implementation of the dynamic response of the structure. The method presented in this paper uses measurements of flexural vibration of a beam and of the same beam carrying a known additional concentrated mass to identify the beam’s structural parameters, such as mass and stiffness (i.e. flexural rigidity). From the eigenfrequencies of these two structures and from the analytical solutions of the dynamic response of both structures, the original and the modified, the structural parameters can be determined with satisfactory accuracy. The method is nondestructive and can be applied to many one-dimensional structures for which an analytical description is available. The method only requires measurements of the structural response at a single point, because it does not need information about eigenshapes but only about eigenfrequencies, and measurements performed at several points offer correlated and more reliable results. It is not iterative and therefore demands a modest computational effort, in some special cases it even offers simple expressions. Thus it can be described as a simple and easy-to-handle method having the advantage of performing measurements in situ with a rather simple equipment. One of its main advantages is that after the identification procedure is finished, the added mass is simply removed and the structure remains without negative effects. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 03.20; 46.10; 06.30.F Keywords: Parameter identification; Dynamic response; Flexural vibration; Beam; Additional mass; Mass; Flexural rigidity; Measurements; Eigenfrequencies
1. Introduction and objectives The motion of a distributed parameter system is generally described by partial differential equa-
*
Tel.: +386-222-94-358; fax: +386-225-24-179. E-mail addresses: [email protected], [email protected] (M. Skrinar).
tions. The parameters contained in the equations of motion such as mass and stiffness (and damping) are in general continuous functions of the spatial variables. When the response of the structure has to be determined the mass and stiffness are assumed to be known parameters. Due to safety reasons the actual response of the structure is usually measured in some time intervals and the validity of the original computational
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 6 5 - 3
208
M. Skrinar / Computational Materials Science 25 (2002) 207–217
model, particularly its parameters, should be investigated. In cases when all the parameters appearing in the equations of motion are not known, they can be obtained from the structure response and from an appropriate computational model of the structure. A frequently used approach is first to elicit and measure the system response and then to repeat the procedures on the structure where a controlled change was carried out. The missing system parameters are afterward identified from the comparison of the results obtained from the original structure with the results of a slightly modified structure. This is the essence of the inverse parameter identification that must be nondestructive. Gladwell [1] discussed an identification method in which he altered the boundary conditions and determined the resulting changes in response spectra. In this way he was able to obtain the mass and the stiffness matrix coefficients of the discrete mathematical model of the cantilever beam. Gladwell’s result is a pure theoretical one, showing that the discrete model can be reconstructed uniquely from two (three) suitably chosen spectra. Gladwell further considers continuous non-uniform models (mass distribution m and flexural rigidity EI are variable). Following the idea of Gladwell, Skrinar and Umek [2] introduced an approach where they applied a controlled modification of the mass (instead of altering the boundary conditions) of the structure in an appropriate point (not necessary at free end). The method was applied to identify the parameters of a simple supported beam where the masses were supposed to be concentrated. The method was experimentally verified with the identification of the parameters of a simply supported beam, where the masses were assumed to be concentrated in discrete points. Encouraged by the experimental results, the theoretical extension of the previously presented method requiring a smaller number of input data, particularly suitable for cantilevers, was introduced later [3]. Beams carrying a concentrated mass were also investigated by Low [4] and Chai and Low [5], not in the sense of inverse identification, but from the eigenfrequency determination point of view. Two procedures for the prediction of changes in eigenfrequencies caused
by a mass added on simply supported beams were presented by Skrinar and Umek [6]. Both procedures are based on Taylor’s expansion of equations resulting from the governing differential equation for the structure with an added mass. They required the eigenfrequencies of the original structure (i.e. without added mass) to be known. An analytical solution was presented for small added masses (in comparison with the total mass of the original structure) and a numerical procedure was shown for an arbitrary value of the additional mass. This article presents a study of the inverse identification by the added mass method of moderate structural systems with uniformly distributed mass m and flexural rigidity EI. The method requires that the structure possesses an analytical description or can be modelled by means of other convenient discretisation methods. The method requires some eigenfrequencies of the original structure and some of the modified structure with a known added mass. Before a single element is built in a structure it is relatively easy to determine its physical properties such a mass and flexural rigidity, but when the structure is completed this task is not so easy any more. Therefore, problems of this kind can be addressed as a serviceability/maintenance problem. Thus the objective of the presented study is to demonstrate the development of relatively simple formulas for a quick in situ determination of physical properties, as for example stiffness and mass distribution that can be implemented for beams with various types of boundary conditions. I believe that the presented study of the inverse identification method can be expanded to moderate structures for which an analytical description of eigenmotion can be obtained, or to structures described by other convenient discretisation methods. For successful identification of the structure some eigenfrequencies of the considered structure and some eigenfrequencies of the slightly modified structure are required. The modification of the structure is performed by adding a known supplementary mass to a selected location. To accomplish the derived theoretical solutions of the studies, experimental investigations were completed. The dynamic response of both types of
M. Skrinar / Computational Materials Science 25 (2002) 207–217
beams (original structure and the beam with an added mass) was measured. From the measured eigenfrequencies flexural rigidity and mass per unit of length were determined within engineering accuracy. This represents the basis of methods for the identification of the properties of an elasticallyclamped beam, where mass, flexural rigidity and rotational stiffness at both ends are to be determined. It is also believed that this approach can be utilised for inverse identifications of bridges.
o4 wðxÞ Y ðtÞ ¼ 0 ox2 and can be rewritten as follows:
wðxÞmY€ ðtÞ þ EI
o4 wðxÞ EI € Y ðtÞ ox2 ¼ 0 þ Y ðtÞ wðxÞm o4 wðxÞ EI Y€ ðtÞ ox2 ¼ Y ðtÞ wðxÞm
209
ð4Þ
ð5Þ
ð6Þ
The solution requires that both sides are equal to a constant, for example x2 : 2. Elastic behaviour of a slender elastic beam A slender beam is considered to be a simple but also representative and very frequently used engineering structure. The eigen frequencies of a beam with uniform material properties can be obtained through the closed analytical solution from the equation of motion. Considering an elementary case that neglects shear, damping, and axial-force effects, the solution for free bending transverse vibration of a beam is obtained by solving the differential equation of motion of a Bernoulli– Euler beam that can be written as follows (that can be found in many references dealing with dynamics of structures, see for example [7]): 2 o2 v o2 ov m 2 þ 2 EI ¼0 ð1Þ ot ox2 ox where m is the mass distribution of the unit of length, EI is the flexural rigidity, the product of the elasticity modulus with the moment of inertia of the cross-section, v is the transverse displacement, which is a function of the spatial coordinate x and time t. The solution of Eq. (1) is assumed in the form of a product of two functions: vðx; tÞ ¼ wðxÞY ðtÞ
ð2Þ
The introduction of Eq. (2) into Eq. (1) results in o2 Y ðtÞ o2 o2 wðxÞ wðxÞm þ 2 EI Y ðtÞ ¼ 0 ð3Þ ot2 ox2 ox For a uniform homogeneous beam the above equation is simplified
Y€ ðtÞ ¼ x2 Y ðtÞ o4 wðxÞ ox2 ¼ x2 wðxÞm
ð7Þ
EI
ð8Þ
The partial differential Eq. (1) is thus transformed into two ordinary differential equations (spatial and temporal) (7) and (8) with two unknown functions wðxÞ and Y ðtÞ Y€ ðtÞ þ x2 ¼ Y ðtÞ o4 wðxÞ x2 m wðxÞ ¼ 0 ox2 EI
ð9Þ ð10Þ
Introducing x2 m ð11Þ EI the differential Eq. (10) can be rewritten as: b4 ¼
o4 wðxÞ b4 wðxÞ ¼ 0 ox2 The solution is:
ð12Þ
wðxÞ ¼ A1 sinðbxÞ þ A2 cosðbxÞ þ A3 sinhðbxÞ þ A4 coshðbxÞ
ð13Þ
Four actual boundary conditions of the structure under consideration must be introduced into Eq. (13) to construct the eigenvalue or the eigenfrequency problem. To determine the eigenfrequencies the problem generally reduces into a solution of a system of four homogeneous linear equations. The system has a nontrivial solution if the determinant of the system, which represents a
210
M. Skrinar / Computational Materials Science 25 (2002) 207–217
characteristic equation, is equal to zero. A general solution or the nth circular or natural frequency x0n of the considered uniform beam is obtained in the form of: rffiffiffiffiffiffi vn p2 EI 0 xn ¼ 2 n ¼ 1; 2; . . . ; 1 ð14Þ m L where vn are constants that depend on actual boundary conditions and L is the length or span of the beam. The eigenfrequencies therefore depend on L, EI and m. Let us now suppose that some of the natural frequencies of a particular beam are known (for example from the measurements) and that the properties EI and m are to be identified. Supposing further that the length L can be easily and accurately measured on the actual structure, only the ratio between EI and m can be obtained from Eq. (14): 0 2 2 EI xn L ¼ ð15Þ m vn p2 whereas the separate actual values for EI and m cannot be obtained. If only the natural frequencies (no matter how many of them) are known, the correct values EI and m remain unknown. To solve the problem an additional equation is required that can be obtained from a structure with a controlled change. This required equation can for example be obtained from the governing equation of motion of the considered structure with an added known mass.
Fig. 1. A beam carrying an additional mass.
w1 ðxÞ ¼ A1 sinðbxÞ þ A2 cosðbxÞ þ A3 sinhðbxÞ þ A4 coshðbxÞ
ð16Þ
w2 ðxÞ ¼ B1 sinðbxÞ þ B2 cosðbxÞ þ B3 sinhðbxÞ þ B4 coshðbxÞ ð17Þ where the constant b was already given by Eq. (11) and x now represents the circular eigenfrequency of the structure with added mass. To evaluate the eight coefficients Ai and Bi (i ¼ 1–4) four boundary conditions and four compatibility conditions at the mass location must be implemented. To determine the eigenfrequencies a system of eight homogeneous equations must be solved. The equations possess again a nontrivial solution if the determinant of the system is equal to zero. The determinant of the system represents a characteristic equation. Due to the complexity of the problem the analytical solutions do not exist and the problem is solved numerically.
4. The idea of the identification process 3. The equation of motion for a flexural beam with supplementary mass Let us consider an elastic, homogeneous beam with an attached mass with the weight M and the moment of inertia Jc located at a known distance, for example L1 from the left support (Fig. 1). The process of the analysis is similar as in the case of the original system with the difference that the mass separates the beam into two parts. This results in two ordinary spatial differential equations with two spatial functions w1 ðxÞ and w2 ðxÞ for the part to the left and the to the right of the added mass:
On the considered structure, length L is measured, and the structure is afterwards excited to vibrate freely. The dynamic response is measured in time domain in one (or more) arbitrarily chosen points by the attached accelerometers. The recorded data are further transformed in the frequency domain (Fourier domain) and analysed to detect the eigenfrequencies (as many as can possibly be detected from the analysed signal). From the known analytical expressions that exist for standard boundary conditions for a single beam it is possible to identify only the ratio between the flexural rigidity EI and the mass per unit length m,
M. Skrinar / Computational Materials Science 25 (2002) 207–217
but the actual values remain unknown. All values computed from various eigenfrequencies are then averaged to obtain a more reliable result. To obtain the required additional equation to separate both unknown values, a known mass M with the moment of inertia Jc is attached to a chosen point of the structure and the process of dynamic response measurements and eigenfrequencies identification is repeated. For the modified structure a corresponding mathematical model is established which yields the characteristic equation from the homogeneous equations. The ratio EI/m that appears in the characteristic equation of motion is replaced by the value identified on the original structure. The coefficient b that appears in Eqs. (16) and (17), and consequently in the characteristic equation, is defined by Eq. (11) and can thus be computed using the length L measured on the actual structure, the ratio EI/m identified from the original structure and the eigenfrequencies x measured on the modified structure. With the measured eigenfrequencies x and the known value of coefficient b it is now possible to evaluate directly either m or EI. The remaining parameter is afterwards obtained from the previously identified ratio EI/m. The algorithm needs no iteration. To increase the reliability and the accuracy, various added masses can be used at several locations. The proper selection of the weight of the added mass can be a very delicate task since the original mass of the structure is unknown. A relatively small mass will yield small eigenfrequency changes that might not be distinguished from the measurements’ error. On the other hand, a very big added mass can significantly change the behaviour of the system that might put under question the chosen computational model. The simplest solution is to perform identification with several added masses. For simply supported boundary conditions the identification formulas are given in the following subsection. They were also verified by experimental identification. The development of the corresponding equations for other types of boundary conditions follows the same steps and similar equations can be obtained for inverse identification.
211
5. Dynamic behaviour of an elastic, simply supported slender beam with an added mass A simply supported beam with an attached mass M with the moment of inertia Jc located at the distance L1 from the left support is considered. There are several approaches to the solution of the motion of the system with an added mass that could also be implemented for inverse identification. One of the simplest is clearly the Rayleigh principle. The efficiency of this approach depends on the right selection of shape functions, and static deflection functions are considered to be the simplest shape functions. For this type of structure the most natural place for the added mass is probably the centre of the span. For this case the mass distribution m of the structure can be obtained from the equation for the computation of the first eigenfrequency of a simply supported beam with an added mass at the midspan given by Low [4]: 35Mm21 p2 m¼ ð18Þ 420 EI 2 2 17m p 1 L3 m where m1 (Hz) represents the first eigenfrequency of the simply supported beam with an added mass. An alternative form of shape functions is the trigonometric-series’ representation. The use of the shape function v ¼ sinðpx=LÞ leads to the following expression: 8L3 Mm21 m¼ ð19Þ EI p2 4L4 m21 m In Eqs. (18) and (19) the moment of inertia Jc of the added mass is not presented as it does not influence the response due to the symmetry of the structure. The second equation yields better results due to more appropriate shape functions. Although several shape function offering similar solutions can be easily found it is a more prosperous approach to perform the analysis of the governing equation because it allows an easier introduction of higher eigenfrequencies. As previously mentioned, the process of the analysis is the same as for the original system, with the exception of the mass separating the beam into two parts, which leads to two functions w1 ðxÞ and w2 ðxÞ for
212
M. Skrinar / Computational Materials Science 25 (2002) 207–217
the left and the right part of the added mass (Eq. (16)). The eight unknown coefficients Ai and Bi (i ¼ 1– 4) can be determined from four boundary conditions describing the displacements and bending moments at both ends: w002 ðLÞ ¼ 0
w2 ðLÞ ¼ 0
ð20Þ
and the four compatibility conditions at the mass location describing compatibility conditions for displacements, rotations, bending moments and shear forces: w1 ðxÞjx¼L1 ¼ w2 ðxÞjx¼L1 ; w01 ðxÞx¼L ¼ w02 ðxÞx¼L ; 1 1 00 00 0 2 EI w1 ðxÞ x¼L w2 ðxÞ x¼L Jc w1 ðxÞ x¼L x ¼ 0; 1 1 1 000 000 2 EIðw1 ðxÞ w2 ðxÞÞ x¼L þ Mx w1 ðxÞjx¼L1 ¼ 0 1
ð21Þ The first two boundary conditions yield simple solutions A2 ¼ A4 ¼ 0
ð22Þ
The remaining boundary and compatibility conditions result in a system of homogeneous linear equations. Such a system has always a trivial solution with no engineering meaning, and a nontrivial solution when the determinant of the system is equal to zero. Therefore the determinant of the following matrix must vanish: 2
0 6 0 6 6 s 1 6 6 c1 6 4 b1 s1 Jc x2 c1 Mx2 s1 b3 c1
0 0 sh1 ch1 b1 sh1 Jc x2 ch1 b3 ch1 þ Mx2 sh1
sL sL s1 c1 b1 s 1 b3 ch1
cL cL c1 s1 b1 c 1 b3 sh1
with symbols s1 ¼ sinðbL1 Þ
sh1 ¼ sinhðbL1 Þ
c1 ¼ cosðbL1 Þ ch1 ¼ coshðbL1 Þ sL ¼ sinðbLÞ
shL ¼ sinhðbLÞ
cL ¼ cosðbLÞ chL ¼ coshðbLÞ b1 ¼ bEI
b3 ¼ b3 EI
EI2 D1 þ EID2 þ D3 0
ð25Þ
with the coefficients D1 ¼ 8b4 sL shL
w001 ð0Þ ¼ 0;
w1 ð0Þ ¼ 0;
The determinant of matrix (23) can be rewritten in a general form as:
ð24Þ
D2 ¼ 2bMx2 ðchL sL cosh ðbð L 2L1 ÞÞsL þ ðcL cos ðbð L 2L1 ÞÞÞshL Þ þ 2b3 Jc x2 ð chL sL cosh ðbð L 2L1 ÞÞsL þ ðcL þ cos ðbð L 2L1 ÞÞÞshL Þ D3 ¼ MJc x4 ð cos ðbð L 2L1 ÞÞchL cL cosh ðbð L 2L1 ÞÞÞ MJc x4 ðsL shL þ sin ðbð L 2L1 ÞÞ sinh ðbð L 2L1 ÞÞÞ ð26Þ It is obvious that if the added mass is not presented, Eq. (25) reduces to D1 0, which is the standard equation for eigenvalue problems of simply supported beams. Eq. (25) is in fact a simple quadratic equation with the following solutions:
EI ¼ EI ¼
shL shL sh1 ch1 b1 sh1 b3 ch1
D2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D22 4D1 D3
2D1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi D2 þ D22 4D1 D3
ð27Þ
2D1 3 chL chL 7 7 ch1 7 7 sh1 7 7 b1 ch1 5 b3 sh1
ð23Þ
If the solutions from Eq. (27) have the opposite sign, it is clear that only the solution with the positive sign has an engineering meaning. If both solutions are positive, we need to know which is the correct one. The required indication about the correct solution can be obtained from the equation derived from the characteristic equation if the moment of
M. Skrinar / Computational Materials Science 25 (2002) 207–217
inertia Jc is negligible. Eq. (25) reduces into a linear equation EID1 þ D2 0
ð28Þ
where D1 ¼ 8b4 sL shL D2 ¼ 2bMx2 ðchL sL coshðbðL 2L1 ÞÞsL þ ðcL cosðbðL 2L1 ÞÞÞshL Þ
ð29Þ
Eq. (28) has a single solution
EI ¼
made of steel with the modulus of elasticity E ¼ 2:1 1011 Pa (value was not measured but taken from the references). The mass distribution was m ¼ 1:72 kg=m0 per unit of length (measured data). The profile was a rectangular structural tubing with a nominal size of 40 20 mm with the thickness of 2 mm. The moment of inertia of the cross-section was I ¼ 1:4216 108 m4 . Attached accelerometers were used in vibration measurements. Using an accelerometer to detect the vibration motion of the structure has some
D2 Mx2 ðchL sL coshðbðL 2L1 ÞÞsL þ ðcL cosðbðL 2L1 ÞÞÞshL Þ ¼ 4b3 sL shL D1
Although simple in comparison to Eq. (26), the importance of Eq. (28) and its solution (30) lies in the fact that it indicates the correct solution among both solutions (27). Even much simpler expressions can be found for some particular locations. If the supplementary mass is located in the middle of the span, the governing equation, due to the symmetry of the structure, reduces to a form of two products, which further yield two solutions: bL tanh bL 2 tan 2 2 EI ¼ Mx ð31Þ 4b3 bL cot bL 2 coth 2 2 EI ¼ Jc x ð32Þ 4b The first equation is the function of the weight of the added mass only, and belongs to symmetrical modes and related eigenfrequencies. The second equation is influenced by the moment of inertia of the added mass and belongs to the antimmetric mode shapes and related eigenfrequencies. Eqs. (31) and (32) represent solutions similar in form to the solutions (18) and (19), but with a much higher accuracy. 6. Experimental investigation To test the efficacy of the proposed procedures a 1780 mm long beam was considered. The beam was
213
ð30Þ
advantages, such as ease of installation and higher sensitivity. However, the disadvantage is that the accelerometer adds its weight, which may lead to lower natural frequencies. Comparing the weight of the accelerometers used (54.2 g) it is obvious that this mass can be usually neglected in comparison with the mass of the beam. The measuring device was a Br€ uel & Kjær accelerometer, Model 4370, and the accelerometer sensitivity was 81.8 mV/g with a useable frequency range to 5000 Hz. The signal from the accelerometer was further amplified with a Br€ uel & Kjær charge Amplifier 2635 and then transmitted to the analogue–digital unit Mowlem Microsystems MM700. Afterwards the data were transmitted to a PC and analysed with the program Matlab. To obtain the natural frequencies either the ambient vibrations were measured or the beam was excited by an impact load. The excitation of the specimen by quick release of a suspended weight was not carried out since satisfactory results were obtained either without an excitation or with a shock impact excitation. The sampling rates were from 20 up to 500 Hz due to technical limitation of the used equipment, with altogether up to 3200 points (due to limited free memory of the analogue–digital data acquisition unit). Table 1 presents the first three detected natural frequencies of the original beam with corresponding evaluated ratios EI=m, computed via Eq. (15).
214
M. Skrinar / Computational Materials Science 25 (2002) 207–217
Table 1 The first three measured eigenfrequencies and related ratios EI=m 2 xn L2 n2 p 2
n
mn (Hz)
EI ¼ m
1 2 3
20.30 83.98 184.14
1676.611 1793.235 1703.008
Average
1724.28
It is obvious that the identified results vary within a range of 4% around the average value and therefore the mean value 1724.28 was used in further identification. Compared with the analytical value (1758.97) the mean value differs for less than 2.0%. The tests on the modified structure were performed with two different additional masses (discs with masses 5 and 10 kg with Jc ¼ 0:010125 kg m2 and Jc ¼ 0:049 kg m2 , respectively). To obtain
more reliable results each mass was added at two locations (quarter and midspan of the length). The inverse procedure was performed twice for each pair of the eigenfrequencies: including the moment of inertia in the analysis (Eq. (27)) and neglecting it (Eq. (30)). As expected, the eigenfrequencies of the system with an added mass were lower than the corresponding eigenfrequencies of the intact beams. It is evident from Tables 2–5 (Figs. 2 and 3) that all four solutions obtained with the first natural frequencies lie within a certain satisfactory range. The solutions obtained with higher frequencies are not reliable. The reason for such errors can be found in the fact that due to their weights the added masses change the structure and make it very similar to a single degree of freedom system. Consequently, it is impossible to correctly detect higher eigenfrequencies from the power spectra. As the highest sampling frequency was also relatively slow (500 Hz) for accurate measurements of
Table 2 Comparison of identified mass distributions for M ¼ 5 kg and L1 ¼ 0:445 m n
1 2 3
mn (Hz)
12.21 55.61 168.0
b
1.35921 2.90073 5.04180
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
6.72 3229.40 1840.51
2888.92 27.26 10007.42
2874.33 3136.19 6157.26
Mass (kg=m0 )
E (N/m2 )
1.67534 1.87279 5.80349
2:00917 1011 2:24597 1011 6:95991 1011
Mass (kg=m0 )
E (N/m2 )
1.78797 1.62465
2:14424 1011 1:94838 1011
Mass (kg=m0 )
E (N/m2 )
1.64730 0.53144 13.69432
1:97554 1011 6:37336 1010 1:64230 1012
Table 3 Comparison of identified mass distributions for M ¼ 10 kg and L1 ¼ 0:445 m n
1 2
mn (Hz)
9.668 51.195
b
1.20948 2.78320
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
20.11 2801.51
3083.13 863.12
3047.07 1701.36
Table 4 Comparison of identified mass distributions for M ¼ 5 kg and L1 ¼ 0:89 m n
1 2 3
mn (Hz)
9.760 72.00 162.30
b
1.21522 3.30063 4.95554
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
5.69 916.40 365.88
2840.57 8546.58 23614.19
2840.57 8546.58 23614.19
M. Skrinar / Computational Materials Science 25 (2002) 207–217
215
Table 5 Comparison of identified mass distributions for M ¼ 10 kg and L1 ¼ 0:89 m n
1 2 3
mn (Hz)
7.52 72.22 128.4
b
1.06669 3.30568 4.40772
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
16.311 4540.45 140.95
3021.18 17053.39 13.53984
3021.18 17053.39 140.95
Mass (kg=m0 )
E (N/m2 )
1.75204 2.63309 0.00785
2:10115 1011 3:15777 1011
Fig. 2. The beam with the added mass M ¼ 5 kg, L1 ¼ 0:445 m.
higher eigenfrequencies, the detected higher eigenfrequencies were spurious. As a result, the inverse identification was based on the data obtained from the first eigenfrequencies only. It is obvious that the values identified using the first natural frequencies of the structure with the added mass give acceptable results compared to the real value 1.72 kg/m0 . Using again the mean value of the identified mass from all four results obtained with first eigenfrequencies only e.g. 1.73 kg/m0 , it is clear that the accuracy is excellent. Assuming that the identified value 1.73 of the mass distribution is precise enough, the value obtained for the flexural rigidity is the EI ¼ 2900:5 N/m2 . If we compare the identified value with the theoretical one (2985.36 N/m2 ) we can see that the identified and the exact value differ for less than 3%. For the first eigenfrequency the solutions of the ‘‘exact’’ (Eq. (27)) and the ‘‘simplified’’ (Eq. (30)) produce equal results in the case when the added mass is located in the middle of the span where,
Fig. 3. The beam with the added mass M ¼ 5 kg, L1 ¼ 0:89 m.
due to the symmetry of the modal eigenshape, the moment of inertia has no influence. In the case of the second eigenfrequency the added mass at the midspan is actually located in the node of the second eigenshape and thus only the moment of inertia influences the motion. Furthermore, it can be observed that the two solutions (Eq. (27)) from the quadratic equation both offered positive values and the simplified
216
M. Skrinar / Computational Materials Science 25 (2002) 207–217
Eq. (30) had to be implemented to indicate the correct value. For the added mass at the midspan the identification was also performed with both equations obtained by Rayleigh’s principle. For the added mass of 5 kg Eq. (18) yields the result m ¼ 1:43 kg=m0 and Eq. (20) yields the result m ¼ 1:63 kg=m0 . For the added mass of 10 kg the results are m ¼ 1:64 kg/m0 and m ¼ 1:73 kg/m, for Eqs. (18) and (20), respectively. In both cases Eq. (20) obviously produces more accurate results, which results from a better shape function. Eqs. (31) and (32) produced the same results as Eq. (27), where the solutions obtained with Eq. (31) were the correct ones. The results obtained from Eq. (32) should actually be zero and the obtained values could serve as an indicator of the achieved accuracy.
7. Comparison of the results and comments Equations for inverse identification were practically verified with experiments. The suggested approach with the averaging of the results has clearly demonstrated its benefits. Although implemented with first eigenfrequencies it can be assumed that the result would show similar correctness by the implementation of precisely measured higher eigenfrequencies in combination with the increased number of reasonably smaller added masses. It should be noted that all masses implemented in the experiments were greater than the mass of the beam. The task of selecting the added mass is very delicate since the mass of the beam should be the first result of identification. It can be assumed that for too small masses the method will practically not be useful because the difference between the eigenfrequencies of the original and the modified structure will be too small (due to limited sensitivity of the measurements). The problem of the weight of the added mass can be easily avoided by implementing several different added masses. The experiments have shown that the method was successfully tested with an added mass that was greater than the mass of the original structure.
8. Conclusions The presented identification method is based on the hypothesis that a concentrated mass can be attached to the original structure. Such an approach is easier to realise than the variation of boundary conditions. The method requires satisfactory measurements of the eigenfrequencies of the original structure and of the structure with the added mass. To perform the identification no iteration is required and the results can be easily obtained in situ, since after obtaining the requested eigenfrequencies, the mathematical effort is reduced to the solution of the quadratic equation, or in some cases even linear equation. The method is superior to some known methods in two views. First, the added mass is easily removable after identification, and the original structure remains unchanged. Second, only natural frequencies are measured, which is especially important when performing measurements on large structures with a relatively moderate equipment. In addition, it was shown with the experiments that solely the first eigenfrequencies are practically enough for the identification, if measured with sufficient precision. With the increasing number of higher natural frequencies of the structure a large number of parameter estimations can be made with different masses, which can yield correlated and more reliable results. Although presented by a simply supported beam, the method is applicable to structural systems that have analytical descriptions or can be modelled by means of other convenient discretisation method. For all four types of boundary conditions for a single beam simple formulae can be easily developed. It is believed that the proposed method can be developed into a simple, non-destructive technique and serve as a useful tool even for more complex structures. In conjunction with satisfactory equipment (exitator) the method could play an important role in monitoring the safety of bridges as a detected discrepancy found between eigenfrequencies of several successive measurements can be attributed to the change in the material properties of the structure.
M. Skrinar / Computational Materials Science 25 (2002) 207–217
References [1] G.M.L. Gladwell, Inverse problems in vibration, Martinus Nijhoff Publisher, The Hague, 1986. [2] M. Skrinar, A. Umek, in: Identification of beams by the method of added mass, Computational methods and experimental measurements VI, vol. 2, Computational Mechanics Publications, Southampton, London, 1993, pp. 365–377. [3] M. Skrinar, A. Umek, Extension of identification methods with boundary changes, Proceedings of the 16th Meeting of the Slovenian Civil Engineers, pp. 191–198 (in Slovenian).
217
[4] K.H. Low, An equivalent-center method for quick frequency analysis of beams carrying a concentrated mass, Comput. Struct 50 (3) (1994) 409–419. [5] G.B. Chai, K.H. Low, On the natural frequencies of beams carrying a concentrated mass, J Sound Vibration 160 (1) (1993) 161–166. [6] M. Skrinar, U.A. Skrinar, The influence of an added mass to a change of eigenfrequencies, J Mech Eng 11–12 (1994) 443–450. [7] M. Skrinar, A. Umek, Structural Dynamics, Chapman and Hall, New York, 1997.
On elastic beams parameter identification using eigenfrequencies changes and the method of added mass Matjaz Skrinar
*
Faculty of Civil Engineering, University of Maribor, Smetanova 17, SI 2000 Maribor, Slovenia
Abstract Recently much attention has been focused on the problems of parameter identification based on the measured dynamic response of the structure. Inverse identification can be used either for the reconstruction of missing parameters of the computational analytical model, or as a tool for monitoring the structure during its utilisation. Inverse identification is usually performed with the implementation of the dynamic response of the structure. The method presented in this paper uses measurements of flexural vibration of a beam and of the same beam carrying a known additional concentrated mass to identify the beam’s structural parameters, such as mass and stiffness (i.e. flexural rigidity). From the eigenfrequencies of these two structures and from the analytical solutions of the dynamic response of both structures, the original and the modified, the structural parameters can be determined with satisfactory accuracy. The method is nondestructive and can be applied to many one-dimensional structures for which an analytical description is available. The method only requires measurements of the structural response at a single point, because it does not need information about eigenshapes but only about eigenfrequencies, and measurements performed at several points offer correlated and more reliable results. It is not iterative and therefore demands a modest computational effort, in some special cases it even offers simple expressions. Thus it can be described as a simple and easy-to-handle method having the advantage of performing measurements in situ with a rather simple equipment. One of its main advantages is that after the identification procedure is finished, the added mass is simply removed and the structure remains without negative effects. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 03.20; 46.10; 06.30.F Keywords: Parameter identification; Dynamic response; Flexural vibration; Beam; Additional mass; Mass; Flexural rigidity; Measurements; Eigenfrequencies
1. Introduction and objectives The motion of a distributed parameter system is generally described by partial differential equa-
*
Tel.: +386-222-94-358; fax: +386-225-24-179. E-mail addresses: [email protected], [email protected] (M. Skrinar).
tions. The parameters contained in the equations of motion such as mass and stiffness (and damping) are in general continuous functions of the spatial variables. When the response of the structure has to be determined the mass and stiffness are assumed to be known parameters. Due to safety reasons the actual response of the structure is usually measured in some time intervals and the validity of the original computational
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 2 6 5 - 3
208
M. Skrinar / Computational Materials Science 25 (2002) 207–217
model, particularly its parameters, should be investigated. In cases when all the parameters appearing in the equations of motion are not known, they can be obtained from the structure response and from an appropriate computational model of the structure. A frequently used approach is first to elicit and measure the system response and then to repeat the procedures on the structure where a controlled change was carried out. The missing system parameters are afterward identified from the comparison of the results obtained from the original structure with the results of a slightly modified structure. This is the essence of the inverse parameter identification that must be nondestructive. Gladwell [1] discussed an identification method in which he altered the boundary conditions and determined the resulting changes in response spectra. In this way he was able to obtain the mass and the stiffness matrix coefficients of the discrete mathematical model of the cantilever beam. Gladwell’s result is a pure theoretical one, showing that the discrete model can be reconstructed uniquely from two (three) suitably chosen spectra. Gladwell further considers continuous non-uniform models (mass distribution m and flexural rigidity EI are variable). Following the idea of Gladwell, Skrinar and Umek [2] introduced an approach where they applied a controlled modification of the mass (instead of altering the boundary conditions) of the structure in an appropriate point (not necessary at free end). The method was applied to identify the parameters of a simple supported beam where the masses were supposed to be concentrated. The method was experimentally verified with the identification of the parameters of a simply supported beam, where the masses were assumed to be concentrated in discrete points. Encouraged by the experimental results, the theoretical extension of the previously presented method requiring a smaller number of input data, particularly suitable for cantilevers, was introduced later [3]. Beams carrying a concentrated mass were also investigated by Low [4] and Chai and Low [5], not in the sense of inverse identification, but from the eigenfrequency determination point of view. Two procedures for the prediction of changes in eigenfrequencies caused
by a mass added on simply supported beams were presented by Skrinar and Umek [6]. Both procedures are based on Taylor’s expansion of equations resulting from the governing differential equation for the structure with an added mass. They required the eigenfrequencies of the original structure (i.e. without added mass) to be known. An analytical solution was presented for small added masses (in comparison with the total mass of the original structure) and a numerical procedure was shown for an arbitrary value of the additional mass. This article presents a study of the inverse identification by the added mass method of moderate structural systems with uniformly distributed mass m and flexural rigidity EI. The method requires that the structure possesses an analytical description or can be modelled by means of other convenient discretisation methods. The method requires some eigenfrequencies of the original structure and some of the modified structure with a known added mass. Before a single element is built in a structure it is relatively easy to determine its physical properties such a mass and flexural rigidity, but when the structure is completed this task is not so easy any more. Therefore, problems of this kind can be addressed as a serviceability/maintenance problem. Thus the objective of the presented study is to demonstrate the development of relatively simple formulas for a quick in situ determination of physical properties, as for example stiffness and mass distribution that can be implemented for beams with various types of boundary conditions. I believe that the presented study of the inverse identification method can be expanded to moderate structures for which an analytical description of eigenmotion can be obtained, or to structures described by other convenient discretisation methods. For successful identification of the structure some eigenfrequencies of the considered structure and some eigenfrequencies of the slightly modified structure are required. The modification of the structure is performed by adding a known supplementary mass to a selected location. To accomplish the derived theoretical solutions of the studies, experimental investigations were completed. The dynamic response of both types of
M. Skrinar / Computational Materials Science 25 (2002) 207–217
beams (original structure and the beam with an added mass) was measured. From the measured eigenfrequencies flexural rigidity and mass per unit of length were determined within engineering accuracy. This represents the basis of methods for the identification of the properties of an elasticallyclamped beam, where mass, flexural rigidity and rotational stiffness at both ends are to be determined. It is also believed that this approach can be utilised for inverse identifications of bridges.
o4 wðxÞ Y ðtÞ ¼ 0 ox2 and can be rewritten as follows:
wðxÞmY€ ðtÞ þ EI
o4 wðxÞ EI € Y ðtÞ ox2 ¼ 0 þ Y ðtÞ wðxÞm o4 wðxÞ EI Y€ ðtÞ ox2 ¼ Y ðtÞ wðxÞm
209
ð4Þ
ð5Þ
ð6Þ
The solution requires that both sides are equal to a constant, for example x2 : 2. Elastic behaviour of a slender elastic beam A slender beam is considered to be a simple but also representative and very frequently used engineering structure. The eigen frequencies of a beam with uniform material properties can be obtained through the closed analytical solution from the equation of motion. Considering an elementary case that neglects shear, damping, and axial-force effects, the solution for free bending transverse vibration of a beam is obtained by solving the differential equation of motion of a Bernoulli– Euler beam that can be written as follows (that can be found in many references dealing with dynamics of structures, see for example [7]): 2 o2 v o2 ov m 2 þ 2 EI ¼0 ð1Þ ot ox2 ox where m is the mass distribution of the unit of length, EI is the flexural rigidity, the product of the elasticity modulus with the moment of inertia of the cross-section, v is the transverse displacement, which is a function of the spatial coordinate x and time t. The solution of Eq. (1) is assumed in the form of a product of two functions: vðx; tÞ ¼ wðxÞY ðtÞ
ð2Þ
The introduction of Eq. (2) into Eq. (1) results in o2 Y ðtÞ o2 o2 wðxÞ wðxÞm þ 2 EI Y ðtÞ ¼ 0 ð3Þ ot2 ox2 ox For a uniform homogeneous beam the above equation is simplified
Y€ ðtÞ ¼ x2 Y ðtÞ o4 wðxÞ ox2 ¼ x2 wðxÞm
ð7Þ
EI
ð8Þ
The partial differential Eq. (1) is thus transformed into two ordinary differential equations (spatial and temporal) (7) and (8) with two unknown functions wðxÞ and Y ðtÞ Y€ ðtÞ þ x2 ¼ Y ðtÞ o4 wðxÞ x2 m wðxÞ ¼ 0 ox2 EI
ð9Þ ð10Þ
Introducing x2 m ð11Þ EI the differential Eq. (10) can be rewritten as: b4 ¼
o4 wðxÞ b4 wðxÞ ¼ 0 ox2 The solution is:
ð12Þ
wðxÞ ¼ A1 sinðbxÞ þ A2 cosðbxÞ þ A3 sinhðbxÞ þ A4 coshðbxÞ
ð13Þ
Four actual boundary conditions of the structure under consideration must be introduced into Eq. (13) to construct the eigenvalue or the eigenfrequency problem. To determine the eigenfrequencies the problem generally reduces into a solution of a system of four homogeneous linear equations. The system has a nontrivial solution if the determinant of the system, which represents a
210
M. Skrinar / Computational Materials Science 25 (2002) 207–217
characteristic equation, is equal to zero. A general solution or the nth circular or natural frequency x0n of the considered uniform beam is obtained in the form of: rffiffiffiffiffiffi vn p2 EI 0 xn ¼ 2 n ¼ 1; 2; . . . ; 1 ð14Þ m L where vn are constants that depend on actual boundary conditions and L is the length or span of the beam. The eigenfrequencies therefore depend on L, EI and m. Let us now suppose that some of the natural frequencies of a particular beam are known (for example from the measurements) and that the properties EI and m are to be identified. Supposing further that the length L can be easily and accurately measured on the actual structure, only the ratio between EI and m can be obtained from Eq. (14): 0 2 2 EI xn L ¼ ð15Þ m vn p2 whereas the separate actual values for EI and m cannot be obtained. If only the natural frequencies (no matter how many of them) are known, the correct values EI and m remain unknown. To solve the problem an additional equation is required that can be obtained from a structure with a controlled change. This required equation can for example be obtained from the governing equation of motion of the considered structure with an added known mass.
Fig. 1. A beam carrying an additional mass.
w1 ðxÞ ¼ A1 sinðbxÞ þ A2 cosðbxÞ þ A3 sinhðbxÞ þ A4 coshðbxÞ
ð16Þ
w2 ðxÞ ¼ B1 sinðbxÞ þ B2 cosðbxÞ þ B3 sinhðbxÞ þ B4 coshðbxÞ ð17Þ where the constant b was already given by Eq. (11) and x now represents the circular eigenfrequency of the structure with added mass. To evaluate the eight coefficients Ai and Bi (i ¼ 1–4) four boundary conditions and four compatibility conditions at the mass location must be implemented. To determine the eigenfrequencies a system of eight homogeneous equations must be solved. The equations possess again a nontrivial solution if the determinant of the system is equal to zero. The determinant of the system represents a characteristic equation. Due to the complexity of the problem the analytical solutions do not exist and the problem is solved numerically.
4. The idea of the identification process 3. The equation of motion for a flexural beam with supplementary mass Let us consider an elastic, homogeneous beam with an attached mass with the weight M and the moment of inertia Jc located at a known distance, for example L1 from the left support (Fig. 1). The process of the analysis is similar as in the case of the original system with the difference that the mass separates the beam into two parts. This results in two ordinary spatial differential equations with two spatial functions w1 ðxÞ and w2 ðxÞ for the part to the left and the to the right of the added mass:
On the considered structure, length L is measured, and the structure is afterwards excited to vibrate freely. The dynamic response is measured in time domain in one (or more) arbitrarily chosen points by the attached accelerometers. The recorded data are further transformed in the frequency domain (Fourier domain) and analysed to detect the eigenfrequencies (as many as can possibly be detected from the analysed signal). From the known analytical expressions that exist for standard boundary conditions for a single beam it is possible to identify only the ratio between the flexural rigidity EI and the mass per unit length m,
M. Skrinar / Computational Materials Science 25 (2002) 207–217
but the actual values remain unknown. All values computed from various eigenfrequencies are then averaged to obtain a more reliable result. To obtain the required additional equation to separate both unknown values, a known mass M with the moment of inertia Jc is attached to a chosen point of the structure and the process of dynamic response measurements and eigenfrequencies identification is repeated. For the modified structure a corresponding mathematical model is established which yields the characteristic equation from the homogeneous equations. The ratio EI/m that appears in the characteristic equation of motion is replaced by the value identified on the original structure. The coefficient b that appears in Eqs. (16) and (17), and consequently in the characteristic equation, is defined by Eq. (11) and can thus be computed using the length L measured on the actual structure, the ratio EI/m identified from the original structure and the eigenfrequencies x measured on the modified structure. With the measured eigenfrequencies x and the known value of coefficient b it is now possible to evaluate directly either m or EI. The remaining parameter is afterwards obtained from the previously identified ratio EI/m. The algorithm needs no iteration. To increase the reliability and the accuracy, various added masses can be used at several locations. The proper selection of the weight of the added mass can be a very delicate task since the original mass of the structure is unknown. A relatively small mass will yield small eigenfrequency changes that might not be distinguished from the measurements’ error. On the other hand, a very big added mass can significantly change the behaviour of the system that might put under question the chosen computational model. The simplest solution is to perform identification with several added masses. For simply supported boundary conditions the identification formulas are given in the following subsection. They were also verified by experimental identification. The development of the corresponding equations for other types of boundary conditions follows the same steps and similar equations can be obtained for inverse identification.
211
5. Dynamic behaviour of an elastic, simply supported slender beam with an added mass A simply supported beam with an attached mass M with the moment of inertia Jc located at the distance L1 from the left support is considered. There are several approaches to the solution of the motion of the system with an added mass that could also be implemented for inverse identification. One of the simplest is clearly the Rayleigh principle. The efficiency of this approach depends on the right selection of shape functions, and static deflection functions are considered to be the simplest shape functions. For this type of structure the most natural place for the added mass is probably the centre of the span. For this case the mass distribution m of the structure can be obtained from the equation for the computation of the first eigenfrequency of a simply supported beam with an added mass at the midspan given by Low [4]: 35Mm21 p2 m¼ ð18Þ 420 EI 2 2 17m p 1 L3 m where m1 (Hz) represents the first eigenfrequency of the simply supported beam with an added mass. An alternative form of shape functions is the trigonometric-series’ representation. The use of the shape function v ¼ sinðpx=LÞ leads to the following expression: 8L3 Mm21 m¼ ð19Þ EI p2 4L4 m21 m In Eqs. (18) and (19) the moment of inertia Jc of the added mass is not presented as it does not influence the response due to the symmetry of the structure. The second equation yields better results due to more appropriate shape functions. Although several shape function offering similar solutions can be easily found it is a more prosperous approach to perform the analysis of the governing equation because it allows an easier introduction of higher eigenfrequencies. As previously mentioned, the process of the analysis is the same as for the original system, with the exception of the mass separating the beam into two parts, which leads to two functions w1 ðxÞ and w2 ðxÞ for
212
M. Skrinar / Computational Materials Science 25 (2002) 207–217
the left and the right part of the added mass (Eq. (16)). The eight unknown coefficients Ai and Bi (i ¼ 1– 4) can be determined from four boundary conditions describing the displacements and bending moments at both ends: w002 ðLÞ ¼ 0
w2 ðLÞ ¼ 0
ð20Þ
and the four compatibility conditions at the mass location describing compatibility conditions for displacements, rotations, bending moments and shear forces: w1 ðxÞjx¼L1 ¼ w2 ðxÞjx¼L1 ; w01 ðxÞx¼L ¼ w02 ðxÞx¼L ; 1 1 00 00 0 2 EI w1 ðxÞ x¼L w2 ðxÞ x¼L Jc w1 ðxÞ x¼L x ¼ 0; 1 1 1 000 000 2 EIðw1 ðxÞ w2 ðxÞÞ x¼L þ Mx w1 ðxÞjx¼L1 ¼ 0 1
ð21Þ The first two boundary conditions yield simple solutions A2 ¼ A4 ¼ 0
ð22Þ
The remaining boundary and compatibility conditions result in a system of homogeneous linear equations. Such a system has always a trivial solution with no engineering meaning, and a nontrivial solution when the determinant of the system is equal to zero. Therefore the determinant of the following matrix must vanish: 2
0 6 0 6 6 s 1 6 6 c1 6 4 b1 s1 Jc x2 c1 Mx2 s1 b3 c1
0 0 sh1 ch1 b1 sh1 Jc x2 ch1 b3 ch1 þ Mx2 sh1
sL sL s1 c1 b1 s 1 b3 ch1
cL cL c1 s1 b1 c 1 b3 sh1
with symbols s1 ¼ sinðbL1 Þ
sh1 ¼ sinhðbL1 Þ
c1 ¼ cosðbL1 Þ ch1 ¼ coshðbL1 Þ sL ¼ sinðbLÞ
shL ¼ sinhðbLÞ
cL ¼ cosðbLÞ chL ¼ coshðbLÞ b1 ¼ bEI
b3 ¼ b3 EI
EI2 D1 þ EID2 þ D3 0
ð25Þ
with the coefficients D1 ¼ 8b4 sL shL
w001 ð0Þ ¼ 0;
w1 ð0Þ ¼ 0;
The determinant of matrix (23) can be rewritten in a general form as:
ð24Þ
D2 ¼ 2bMx2 ðchL sL cosh ðbð L 2L1 ÞÞsL þ ðcL cos ðbð L 2L1 ÞÞÞshL Þ þ 2b3 Jc x2 ð chL sL cosh ðbð L 2L1 ÞÞsL þ ðcL þ cos ðbð L 2L1 ÞÞÞshL Þ D3 ¼ MJc x4 ð cos ðbð L 2L1 ÞÞchL cL cosh ðbð L 2L1 ÞÞÞ MJc x4 ðsL shL þ sin ðbð L 2L1 ÞÞ sinh ðbð L 2L1 ÞÞÞ ð26Þ It is obvious that if the added mass is not presented, Eq. (25) reduces to D1 0, which is the standard equation for eigenvalue problems of simply supported beams. Eq. (25) is in fact a simple quadratic equation with the following solutions:
EI ¼ EI ¼
shL shL sh1 ch1 b1 sh1 b3 ch1
D2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D22 4D1 D3
2D1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi D2 þ D22 4D1 D3
ð27Þ
2D1 3 chL chL 7 7 ch1 7 7 sh1 7 7 b1 ch1 5 b3 sh1
ð23Þ
If the solutions from Eq. (27) have the opposite sign, it is clear that only the solution with the positive sign has an engineering meaning. If both solutions are positive, we need to know which is the correct one. The required indication about the correct solution can be obtained from the equation derived from the characteristic equation if the moment of
M. Skrinar / Computational Materials Science 25 (2002) 207–217
inertia Jc is negligible. Eq. (25) reduces into a linear equation EID1 þ D2 0
ð28Þ
where D1 ¼ 8b4 sL shL D2 ¼ 2bMx2 ðchL sL coshðbðL 2L1 ÞÞsL þ ðcL cosðbðL 2L1 ÞÞÞshL Þ
ð29Þ
Eq. (28) has a single solution
EI ¼
made of steel with the modulus of elasticity E ¼ 2:1 1011 Pa (value was not measured but taken from the references). The mass distribution was m ¼ 1:72 kg=m0 per unit of length (measured data). The profile was a rectangular structural tubing with a nominal size of 40 20 mm with the thickness of 2 mm. The moment of inertia of the cross-section was I ¼ 1:4216 108 m4 . Attached accelerometers were used in vibration measurements. Using an accelerometer to detect the vibration motion of the structure has some
D2 Mx2 ðchL sL coshðbðL 2L1 ÞÞsL þ ðcL cosðbðL 2L1 ÞÞÞshL Þ ¼ 4b3 sL shL D1
Although simple in comparison to Eq. (26), the importance of Eq. (28) and its solution (30) lies in the fact that it indicates the correct solution among both solutions (27). Even much simpler expressions can be found for some particular locations. If the supplementary mass is located in the middle of the span, the governing equation, due to the symmetry of the structure, reduces to a form of two products, which further yield two solutions: bL tanh bL 2 tan 2 2 EI ¼ Mx ð31Þ 4b3 bL cot bL 2 coth 2 2 EI ¼ Jc x ð32Þ 4b The first equation is the function of the weight of the added mass only, and belongs to symmetrical modes and related eigenfrequencies. The second equation is influenced by the moment of inertia of the added mass and belongs to the antimmetric mode shapes and related eigenfrequencies. Eqs. (31) and (32) represent solutions similar in form to the solutions (18) and (19), but with a much higher accuracy. 6. Experimental investigation To test the efficacy of the proposed procedures a 1780 mm long beam was considered. The beam was
213
ð30Þ
advantages, such as ease of installation and higher sensitivity. However, the disadvantage is that the accelerometer adds its weight, which may lead to lower natural frequencies. Comparing the weight of the accelerometers used (54.2 g) it is obvious that this mass can be usually neglected in comparison with the mass of the beam. The measuring device was a Br€ uel & Kjær accelerometer, Model 4370, and the accelerometer sensitivity was 81.8 mV/g with a useable frequency range to 5000 Hz. The signal from the accelerometer was further amplified with a Br€ uel & Kjær charge Amplifier 2635 and then transmitted to the analogue–digital unit Mowlem Microsystems MM700. Afterwards the data were transmitted to a PC and analysed with the program Matlab. To obtain the natural frequencies either the ambient vibrations were measured or the beam was excited by an impact load. The excitation of the specimen by quick release of a suspended weight was not carried out since satisfactory results were obtained either without an excitation or with a shock impact excitation. The sampling rates were from 20 up to 500 Hz due to technical limitation of the used equipment, with altogether up to 3200 points (due to limited free memory of the analogue–digital data acquisition unit). Table 1 presents the first three detected natural frequencies of the original beam with corresponding evaluated ratios EI=m, computed via Eq. (15).
214
M. Skrinar / Computational Materials Science 25 (2002) 207–217
Table 1 The first three measured eigenfrequencies and related ratios EI=m 2 xn L2 n2 p 2
n
mn (Hz)
EI ¼ m
1 2 3
20.30 83.98 184.14
1676.611 1793.235 1703.008
Average
1724.28
It is obvious that the identified results vary within a range of 4% around the average value and therefore the mean value 1724.28 was used in further identification. Compared with the analytical value (1758.97) the mean value differs for less than 2.0%. The tests on the modified structure were performed with two different additional masses (discs with masses 5 and 10 kg with Jc ¼ 0:010125 kg m2 and Jc ¼ 0:049 kg m2 , respectively). To obtain
more reliable results each mass was added at two locations (quarter and midspan of the length). The inverse procedure was performed twice for each pair of the eigenfrequencies: including the moment of inertia in the analysis (Eq. (27)) and neglecting it (Eq. (30)). As expected, the eigenfrequencies of the system with an added mass were lower than the corresponding eigenfrequencies of the intact beams. It is evident from Tables 2–5 (Figs. 2 and 3) that all four solutions obtained with the first natural frequencies lie within a certain satisfactory range. The solutions obtained with higher frequencies are not reliable. The reason for such errors can be found in the fact that due to their weights the added masses change the structure and make it very similar to a single degree of freedom system. Consequently, it is impossible to correctly detect higher eigenfrequencies from the power spectra. As the highest sampling frequency was also relatively slow (500 Hz) for accurate measurements of
Table 2 Comparison of identified mass distributions for M ¼ 5 kg and L1 ¼ 0:445 m n
1 2 3
mn (Hz)
12.21 55.61 168.0
b
1.35921 2.90073 5.04180
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
6.72 3229.40 1840.51
2888.92 27.26 10007.42
2874.33 3136.19 6157.26
Mass (kg=m0 )
E (N/m2 )
1.67534 1.87279 5.80349
2:00917 1011 2:24597 1011 6:95991 1011
Mass (kg=m0 )
E (N/m2 )
1.78797 1.62465
2:14424 1011 1:94838 1011
Mass (kg=m0 )
E (N/m2 )
1.64730 0.53144 13.69432
1:97554 1011 6:37336 1010 1:64230 1012
Table 3 Comparison of identified mass distributions for M ¼ 10 kg and L1 ¼ 0:445 m n
1 2
mn (Hz)
9.668 51.195
b
1.20948 2.78320
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
20.11 2801.51
3083.13 863.12
3047.07 1701.36
Table 4 Comparison of identified mass distributions for M ¼ 5 kg and L1 ¼ 0:89 m n
1 2 3
mn (Hz)
9.760 72.00 162.30
b
1.21522 3.30063 4.95554
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
5.69 916.40 365.88
2840.57 8546.58 23614.19
2840.57 8546.58 23614.19
M. Skrinar / Computational Materials Science 25 (2002) 207–217
215
Table 5 Comparison of identified mass distributions for M ¼ 10 kg and L1 ¼ 0:89 m n
1 2 3
mn (Hz)
7.52 72.22 128.4
b
1.06669 3.30568 4.40772
EI (Nm2 ) Eq. (27)
Eq. (27)
Eq. (30)
16.311 4540.45 140.95
3021.18 17053.39 13.53984
3021.18 17053.39 140.95
Mass (kg=m0 )
E (N/m2 )
1.75204 2.63309 0.00785
2:10115 1011 3:15777 1011
Fig. 2. The beam with the added mass M ¼ 5 kg, L1 ¼ 0:445 m.
higher eigenfrequencies, the detected higher eigenfrequencies were spurious. As a result, the inverse identification was based on the data obtained from the first eigenfrequencies only. It is obvious that the values identified using the first natural frequencies of the structure with the added mass give acceptable results compared to the real value 1.72 kg/m0 . Using again the mean value of the identified mass from all four results obtained with first eigenfrequencies only e.g. 1.73 kg/m0 , it is clear that the accuracy is excellent. Assuming that the identified value 1.73 of the mass distribution is precise enough, the value obtained for the flexural rigidity is the EI ¼ 2900:5 N/m2 . If we compare the identified value with the theoretical one (2985.36 N/m2 ) we can see that the identified and the exact value differ for less than 3%. For the first eigenfrequency the solutions of the ‘‘exact’’ (Eq. (27)) and the ‘‘simplified’’ (Eq. (30)) produce equal results in the case when the added mass is located in the middle of the span where,
Fig. 3. The beam with the added mass M ¼ 5 kg, L1 ¼ 0:89 m.
due to the symmetry of the modal eigenshape, the moment of inertia has no influence. In the case of the second eigenfrequency the added mass at the midspan is actually located in the node of the second eigenshape and thus only the moment of inertia influences the motion. Furthermore, it can be observed that the two solutions (Eq. (27)) from the quadratic equation both offered positive values and the simplified
216
M. Skrinar / Computational Materials Science 25 (2002) 207–217
Eq. (30) had to be implemented to indicate the correct value. For the added mass at the midspan the identification was also performed with both equations obtained by Rayleigh’s principle. For the added mass of 5 kg Eq. (18) yields the result m ¼ 1:43 kg=m0 and Eq. (20) yields the result m ¼ 1:63 kg=m0 . For the added mass of 10 kg the results are m ¼ 1:64 kg/m0 and m ¼ 1:73 kg/m, for Eqs. (18) and (20), respectively. In both cases Eq. (20) obviously produces more accurate results, which results from a better shape function. Eqs. (31) and (32) produced the same results as Eq. (27), where the solutions obtained with Eq. (31) were the correct ones. The results obtained from Eq. (32) should actually be zero and the obtained values could serve as an indicator of the achieved accuracy.
7. Comparison of the results and comments Equations for inverse identification were practically verified with experiments. The suggested approach with the averaging of the results has clearly demonstrated its benefits. Although implemented with first eigenfrequencies it can be assumed that the result would show similar correctness by the implementation of precisely measured higher eigenfrequencies in combination with the increased number of reasonably smaller added masses. It should be noted that all masses implemented in the experiments were greater than the mass of the beam. The task of selecting the added mass is very delicate since the mass of the beam should be the first result of identification. It can be assumed that for too small masses the method will practically not be useful because the difference between the eigenfrequencies of the original and the modified structure will be too small (due to limited sensitivity of the measurements). The problem of the weight of the added mass can be easily avoided by implementing several different added masses. The experiments have shown that the method was successfully tested with an added mass that was greater than the mass of the original structure.
8. Conclusions The presented identification method is based on the hypothesis that a concentrated mass can be attached to the original structure. Such an approach is easier to realise than the variation of boundary conditions. The method requires satisfactory measurements of the eigenfrequencies of the original structure and of the structure with the added mass. To perform the identification no iteration is required and the results can be easily obtained in situ, since after obtaining the requested eigenfrequencies, the mathematical effort is reduced to the solution of the quadratic equation, or in some cases even linear equation. The method is superior to some known methods in two views. First, the added mass is easily removable after identification, and the original structure remains unchanged. Second, only natural frequencies are measured, which is especially important when performing measurements on large structures with a relatively moderate equipment. In addition, it was shown with the experiments that solely the first eigenfrequencies are practically enough for the identification, if measured with sufficient precision. With the increasing number of higher natural frequencies of the structure a large number of parameter estimations can be made with different masses, which can yield correlated and more reliable results. Although presented by a simply supported beam, the method is applicable to structural systems that have analytical descriptions or can be modelled by means of other convenient discretisation method. For all four types of boundary conditions for a single beam simple formulae can be easily developed. It is believed that the proposed method can be developed into a simple, non-destructive technique and serve as a useful tool even for more complex structures. In conjunction with satisfactory equipment (exitator) the method could play an important role in monitoring the safety of bridges as a detected discrepancy found between eigenfrequencies of several successive measurements can be attributed to the change in the material properties of the structure.
M. Skrinar / Computational Materials Science 25 (2002) 207–217
References [1] G.M.L. Gladwell, Inverse problems in vibration, Martinus Nijhoff Publisher, The Hague, 1986. [2] M. Skrinar, A. Umek, in: Identification of beams by the method of added mass, Computational methods and experimental measurements VI, vol. 2, Computational Mechanics Publications, Southampton, London, 1993, pp. 365–377. [3] M. Skrinar, A. Umek, Extension of identification methods with boundary changes, Proceedings of the 16th Meeting of the Slovenian Civil Engineers, pp. 191–198 (in Slovenian).
217
[4] K.H. Low, An equivalent-center method for quick frequency analysis of beams carrying a concentrated mass, Comput. Struct 50 (3) (1994) 409–419. [5] G.B. Chai, K.H. Low, On the natural frequencies of beams carrying a concentrated mass, J Sound Vibration 160 (1) (1993) 161–166. [6] M. Skrinar, U.A. Skrinar, The influence of an added mass to a change of eigenfrequencies, J Mech Eng 11–12 (1994) 443–450. [7] M. Skrinar, A. Umek, Structural Dynamics, Chapman and Hall, New York, 1997.