- Email: [email protected]
High resolution analysis of axisymmetric wave modes in cylindrical structures
Ultrasonics
34 (1996) 297-306
High resolution analysis of axisymmetric wave modes in cylindrical structures* J. Vollmann *, J. Dual Institute of Mechanics,
ETH Ziirich, Swiss Federal Institute of Technology,
CH-8092 Zurich, Switzerland
Abstract The relation between frequency and complex wavenumber of guided, axisymmetric waves in circular cylindrical structures is investigated theoretically and experimentally. In particular, a theoretical model is derived to calculate the complex dispersion relation of axisymmetric wave modes in a thin-walled elastic shell containing a viscoelastic medium. The influence of the viscoelastic material properties on the dynamic response of the shell is discussed numerically and the results are confirmed by an extended series of experiments. New dynamic metrology is developed to measure complex dispersion relations, i.e. complex phase velocities as a function of frequency, in a range between 1 kHz and 2 MHz for up to 40 travelling wave modes. The phenomenon of ‘backward wave propagation’ is clearly measured. Excellent agreement between numerical and experimental results was found over a wide parameter range. Keywords:
interaction;
Waves in cylindrical structures; Viscoelastic Spectral analysis of structural waves
loaded
1. Introduction Interference of P-, SH- and SV-waves and their reflections in a three-dimensional medium causes so-called ‘wave modes’ of guided waves in structures. In a limited interval of frequency and wavenumber there regularly exists a limited number of propagating modes. The relation between frequency and complex wavenumber or complex phase velocity of such a mode information about material properties, contains homogeneity, geometry and boundary conditions of a structure. This dispersion relation can be measured by analysing arrival times of narrow band pulses or broad band pulse Fourier analysis. However, these methods turn out to be insufficient when many propagating modes with similar group velocities exist and when outgoing pulses interfere with reflected pulses from the ends of the specimen. Therefore, a new method is developed to derive the complex dispersion relation directly from experimental data. In the case of axi-
* Corresponding author. Fax: +41-1/632-l e-mail: [email protected]
l-45;
-his paper was awarded with the R.W.B. Stephens the Ultrasonics International 1995 Conference.
Student
Prize at
0041-624X/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0041-624X(95)00000-0
shell; Backward
wave propagation;
Dynamic
fluid-strugture
symmetric waves in a cylindrical structure it is sufficient to consider one-dimensional wave propagation in the direction of the symmetry axis in order to characterize the three-dimensional structure. The propagating waves can be described as follows: “=p
II=1
where u denotes the displacement of a surface point of the structure and p the number of travelling modes in the x-direction. o indicates the frequency, k the complex wavenumber which can be regarded as ‘frequency’ in space domain. The imaginary part of the wavenumber k indicates a decaying or a growing wave mode depending on its sign, To determine the dispersion relation experimentally, the structure is subjected to a broad-band pulse exciting as many modes as possible in the frequency range of interest. After recording the displacement of the surface in a definite interval of time and for many points along the shell, a spectrum analysis in time and space leads directly to the dispersion relation. In this research project the method described above is used to investigate material properties of anisotropic
298
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
and isotropic structures. It can also be applied to systems of cylindrical structures consisting of more than one material. For example, it can be used to inspect a coating of a cylindrical rod or tube. However, we shall focus here on the dynamic behaviour of a thin-walled elastic cylindrical shell containing a linear viscoelastic medium. By varying the complex moduli, this medium can be modelled as an inviscid fluid, a linear elastic medium or anything between the two extremes. Previous work in this field was done by Plona et al. Cl] and Sinha et al. [2], who analysed waves in a thickwalled tube, surrounded and/or filled with water. There the water is considered inviscid. Alleyne and Cawley [ 31 have used a two-dimensional Fourier transform method to measure velocities of Lamb waves in a plate. The results presented in the paper above, is limited to the plane of real wavenumber and frequency.
R>>h Fig. 1 Differential shell elements and forces involved in axisymmetric wave propagation.
2. Theoretical model 2.1. The shell A theory for thin-walled elastic shells, including the effects of shear deformation and rotary inertia, forms the basis of the dispersion relation. Due to the curvature and lateral contraction, shell wall bending and longitudinal motion are coupled in a cylindrical shell. The ratio of radial and axial motion of the shell surface is frequency dependent. For that reason the dispersion relation of a shell containing a viscoelastic medium is governed by the shell properties as well as by the bulk modulus, the shear modulus and the density of the medium enclosed. Forces acting on a differential shell element are shown in Fig. 1. In the following equations, Eqs. (l)-(3), & denotes the boundary stress between shell and medium. Z indicates the moment of inertia and 5 the slope of the crosssection due to bending. N,,, = /Ou,,tt + &
(1)
Qr,, - 2 = /Oww + dk
(2)
-Mu
+ Q, = PJ~,,, + d;.
aij = A*(W)EkkSij + 2~*(O)&ij.
(5)
A Maxwell model is used for both, J*(w), Eq. (6) and p*(o), Eq. (7), to describe the frequency dependent relation between stress and strain. As illustrated in Fig. 2, this model consists of a spring and damper on the same axis subjected to the same force. For high frequency, compared with l/T, this model is governed by the spring, for low frequency by the damper.
(3) p*(m) =
&Jo io + l/Tn ’ po10
io + l/T,.
The dynamic behaviour of the viscoelastic core is specified by three-dimensional theory. Therefore, the vector field of the displacement is expressed in the potential form, see Eq. (4). x Y.
of Maxwell behaviour for dilatation and shear
Eq. (5) shows the constitutive equation for a viscoelastic medium, where aij denotes the stress tensor, Eijthe strain tensor and 6, the Kronecker symbol.
n*(o) =
2.2. The viscoelastic core
uJ=Pf$?+P
Fig. 2. Illustration deformation.
(4)
The viscoelastic model can easily be modified by altering Eq. (6) and Eq. (7). Approaches for the scalar potential a, and for the vector potential u, containing Bessel functions with complex arguments, satisfy the Helmholtz equations, which are extended to linear viscoelasticity.
299
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
3.1. Experimental
Interferometer
--------
set-up
A heterodyne laser interferometer and a phasedemodulator, developed and manufactured at the Institute of Mechanics, ETH Ziirich, represents the most essential part of the experimental set-up [4]. The interferometer enables us to measure displacements of less than one nanometer for a frequency range and bandwidth between 1 kHz and 8 MHz without touching the specimen. Fig. 3 shows the experimental set-up. The waves are excited by a broad band pulse which is repeated with a frequency of 20 Hz. After measuring the displacement versus time 300 times and averaging them, the laser interferometer is shifted to the next location. Depending on the approximate number of expected modes within the frequency range considered, the displacement array is recorded at 100 to 160 locations. The recorded data are transferred to a cluster of eight Hewlett Packard 9000 workstations, where the spectrum analysis is carried out. 3.2. Signal processing
Fig. 3. Experimental set-up.
2.3. Dispersion relation
Elaborating all boundary and compatibility conditions between the shell and core, and providing a harmonic set up in time and space for the shell displacement, leads to a homogeneous system of linear equations for four remaining constants, which is solvable if the determinant of the coefficients vanishes. Since this condition contains o, and k as well as all geometrical and material parameters of the shell and the core, it represents an implicit dispersion relation. It is solved numerically with a complex root finding algorithm.
3. Experiments Several combinations of shells and ‘fluids’, like water, milk, alcohol, glycerine, silicone oil and polyisobutylen are investigated in the project. However, the results presented in this paper refer to a steel shell with a diameter of 26 mm and a wall thickness of 0.5 mm, filled with a high viscosity silicone oil (Dow Corning 200 fluid (60 000 cs)) and an isotropic aluminium rod with a diameter of 18 mm.
As illustrated in Fig. 4, the recorded signals are processed with FFT in the time domain. To decompose the complex wavenumber dependence in space domain, a complex Total-Least-Squares Algorithm [ 53 is applied. This algorithm, originally developed for the signal processing of NMR signals, demands fewer measurements than FFT, allows a lower signal-to-noise ratio and has no intrinsic resolution limitations. It extracts both the real and imaginary part of the wavenumber from the measured data. 4. Results In a field ‘somewhere’ between acoustics, fluid mechanics and solid mechanics, many effects and phenomena interfere and the borders between these classical disciplines fade with increasing generality of the constitutive equations. In fluid mechanics shear resistance in a boundary layer is assumed to be proportional to the velocity gradient, whereas an acoustic fluid is supposed to behave inviscid. From the measured dispersion curves of a cylindrical shell containing a Maxwell fluid, five material parameters; &, pO, T,, T, and the density need to be determined, when the material constants of the shell are known from a preliminary series of experiments carried out on an empty shell. In the case of silicone oil the projection of the complex dispersion curves to the real o, k-plane is dominated by the velocity of P-waves c1 in the enclosed medium. Fig. 5 shows numerical results of a shell filled with an inviscid fluid in which P-waves propagate with the same velocity as in the silicone oil investigated. The measured dis-
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
300
Piezoelectric Four-dimensional
0 J.VoUmann
Excitation
Array:
1995
Fig. 4. Illustration of the signa processing.
persion relation is represented in Fig. 6. Fig. 7 compares experimental and numerical dispersion relations in one diagram and Fig. 8 shows the projection of the diagram, presented in Fig. 7, into the real plane. Increasing the elastic shear resistance while keeping the value of c1 constant, the dispersion curves leave the real plane periodically as shown in Fig. 9. (Due to numerical problems not all modes in Fig. 9 could be calculated.) Regarding the measured dispersion diagram Fig. 10, the curves occur very clearly in a range starting at k = 1500 rad m-l, frequency = 200 kHz ending at k = 3200 rad m-r, frequency = 1 MHz. In that specific area the imaginary part of the wavenumber becomes very small in the calculated dispersion relation for a shear elastic ‘fluid’, see Fig. 9. This effect could be an explanation for the clear detection of the curves in that range. Most likely the Complex Linear Prediction Method reaches its limits by interpreting the periodically increasing imaginary part of the wavenumber as noise, apart from the area mentioned above. Scaling the diagram of the measured dispersion curves
in the imaginary axis, an alternating imaginary part of the wavenumber, as calculated for a shear elastic ‘fluid’, is observed in Fig. 10. As far as the authors are aware, the mechanical behaviour of fluids subjected to shear forces alternating with high frequency (MHz range), is not very well known yet. To check the validity of the theoretical model especially in the range of strong shear elasticity the dispersion curves of axisymmetric waves in an aluminium rod with circular cross-section is also measured. This dispersion relation is calculated, running the same computer program used for the shell filled with viscoelastic medium. Therefore the relaxation times T’,‘,and T, are set to high values compared with the reciprocal value of the lowest frequency within the range considered. The material properties found in this manner, correspond to those found with other dynamic methods. See Figs. 11 and 12. Since the damping of the waves propagating in an aluminium rod is much lower than in a silicone filled shell, the dispersion curves of waves reflected at the
301
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
Im
600000
i
Fig. 5. Numerical
frqWI
results for a fluid with c1 equal to c, of silicone oil.
Im
Fig. 6. Measured
dispersion
curves of axisymmetric
waves in a cylindrical
shell containing
silicone otl
J. Vollmann, .I Dual/ Ultrasonics 34 (1996) 297-306
302
Im
WI
Fig. 7. Experimental and numerical dispersion curves in one diagram.
end of the rod are detected as well, see Fig. 13. Since waves propagate in both directions, the phenomenon of ‘backward-wave propagation’, mentioned by Onoe, McNiven and Mindlin in 1962 [ 61, where group velocity
and phase velocity have opposite signs, is clearly measured. See Fig. 14. A detailed investigation of the phenomenon of ‘backward wave propagation’ will be published soon.
frq[Hz1 800000
600000
Re 1000
1500
-2000
2500
3000
3500
Fig. 8. Projection of the curves shown in Fig. 7 into the real plane.
4000
j-ad/m]
8
w
304
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
300000
frq
WI 200000 \
Fig. 11. Numerical results of the dispersion relation in an aluminium rod.
frq
Fig. 12. Experimental and numerical dispersion curves of an aluminium rod in one diagram.
J. Vollmann, J. Dual J Ultrasonics 34 (1996) 297-306
305
Im k [radlm]
300000
frqWI
‘3
:
Fig. 13. Measured dispersion relation of an aluminium rod (diameter: 18 mm).
Fig. 14. Area of ‘backward wave propagation’ in a measured dispersion relation of an aluminium rod.
306
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
5. Conclusions Combining modern spectral estimation methods with a high resolution laser interferometer, we have developed a new measurement technique in structural mechanics. We have investigated some aspects of dynamic fluid structure interaction in the high frequency range, where classical models reach their limits. Future work will focus on the extension of spectral estimation methods to more than one dimension.
to Roger Brett, who has implemented and compared several Linear Prediction Algorithms to find the most powerful one. Special thanks also to Markus Hageli for the improvement of our laser interferometer.
References Cl1 T.J. Plona, B.K. Sinha, S. Kostek and S-K.Chang, J. Acoust. Sot. Am. 92 (1992) 1144.
VI B.K. Sinha, T.J. Plona, S. Kostek and S.-K. Chang, J. Acoust.
Acknowledgements Many thanks to Dr Stephan Kaufmann for his kind support in numerically solving the dispersion relation,
Sot. Am. 92 (1992) 1132. c31 D. Alleyne and P. Cawley, J. Acoust. Sot. Am. 89 (1991) 1159. [41 J. Dual, M. Hlgeli, M.R. Pfaffinger and J. Vollmann, Ultrasonics 34 (1996) 291 (this issue). CSI C.F. Tirendi and J.F. Martin, J. Magnetic Resonance 85 (1989) 162. 161 M. Onoe, H.D. McNiven and R.D. Mindlin, J. Appl. Mech. 29 (1962) 61.
34 (1996) 297-306
High resolution analysis of axisymmetric wave modes in cylindrical structures* J. Vollmann *, J. Dual Institute of Mechanics,
ETH Ziirich, Swiss Federal Institute of Technology,
CH-8092 Zurich, Switzerland
Abstract The relation between frequency and complex wavenumber of guided, axisymmetric waves in circular cylindrical structures is investigated theoretically and experimentally. In particular, a theoretical model is derived to calculate the complex dispersion relation of axisymmetric wave modes in a thin-walled elastic shell containing a viscoelastic medium. The influence of the viscoelastic material properties on the dynamic response of the shell is discussed numerically and the results are confirmed by an extended series of experiments. New dynamic metrology is developed to measure complex dispersion relations, i.e. complex phase velocities as a function of frequency, in a range between 1 kHz and 2 MHz for up to 40 travelling wave modes. The phenomenon of ‘backward wave propagation’ is clearly measured. Excellent agreement between numerical and experimental results was found over a wide parameter range. Keywords:
interaction;
Waves in cylindrical structures; Viscoelastic Spectral analysis of structural waves
loaded
1. Introduction Interference of P-, SH- and SV-waves and their reflections in a three-dimensional medium causes so-called ‘wave modes’ of guided waves in structures. In a limited interval of frequency and wavenumber there regularly exists a limited number of propagating modes. The relation between frequency and complex wavenumber or complex phase velocity of such a mode information about material properties, contains homogeneity, geometry and boundary conditions of a structure. This dispersion relation can be measured by analysing arrival times of narrow band pulses or broad band pulse Fourier analysis. However, these methods turn out to be insufficient when many propagating modes with similar group velocities exist and when outgoing pulses interfere with reflected pulses from the ends of the specimen. Therefore, a new method is developed to derive the complex dispersion relation directly from experimental data. In the case of axi-
* Corresponding author. Fax: +41-1/632-l e-mail: [email protected]
l-45;
-his paper was awarded with the R.W.B. Stephens the Ultrasonics International 1995 Conference.
Student
Prize at
0041-624X/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0041-624X(95)00000-0
shell; Backward
wave propagation;
Dynamic
fluid-strugture
symmetric waves in a cylindrical structure it is sufficient to consider one-dimensional wave propagation in the direction of the symmetry axis in order to characterize the three-dimensional structure. The propagating waves can be described as follows: “=p
II=1
where u denotes the displacement of a surface point of the structure and p the number of travelling modes in the x-direction. o indicates the frequency, k the complex wavenumber which can be regarded as ‘frequency’ in space domain. The imaginary part of the wavenumber k indicates a decaying or a growing wave mode depending on its sign, To determine the dispersion relation experimentally, the structure is subjected to a broad-band pulse exciting as many modes as possible in the frequency range of interest. After recording the displacement of the surface in a definite interval of time and for many points along the shell, a spectrum analysis in time and space leads directly to the dispersion relation. In this research project the method described above is used to investigate material properties of anisotropic
298
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
and isotropic structures. It can also be applied to systems of cylindrical structures consisting of more than one material. For example, it can be used to inspect a coating of a cylindrical rod or tube. However, we shall focus here on the dynamic behaviour of a thin-walled elastic cylindrical shell containing a linear viscoelastic medium. By varying the complex moduli, this medium can be modelled as an inviscid fluid, a linear elastic medium or anything between the two extremes. Previous work in this field was done by Plona et al. Cl] and Sinha et al. [2], who analysed waves in a thickwalled tube, surrounded and/or filled with water. There the water is considered inviscid. Alleyne and Cawley [ 31 have used a two-dimensional Fourier transform method to measure velocities of Lamb waves in a plate. The results presented in the paper above, is limited to the plane of real wavenumber and frequency.
R>>h Fig. 1 Differential shell elements and forces involved in axisymmetric wave propagation.
2. Theoretical model 2.1. The shell A theory for thin-walled elastic shells, including the effects of shear deformation and rotary inertia, forms the basis of the dispersion relation. Due to the curvature and lateral contraction, shell wall bending and longitudinal motion are coupled in a cylindrical shell. The ratio of radial and axial motion of the shell surface is frequency dependent. For that reason the dispersion relation of a shell containing a viscoelastic medium is governed by the shell properties as well as by the bulk modulus, the shear modulus and the density of the medium enclosed. Forces acting on a differential shell element are shown in Fig. 1. In the following equations, Eqs. (l)-(3), & denotes the boundary stress between shell and medium. Z indicates the moment of inertia and 5 the slope of the crosssection due to bending. N,,, = /Ou,,tt + &
(1)
Qr,, - 2 = /Oww + dk
(2)
-Mu
+ Q, = PJ~,,, + d;.
aij = A*(W)EkkSij + 2~*(O)&ij.
(5)
A Maxwell model is used for both, J*(w), Eq. (6) and p*(o), Eq. (7), to describe the frequency dependent relation between stress and strain. As illustrated in Fig. 2, this model consists of a spring and damper on the same axis subjected to the same force. For high frequency, compared with l/T, this model is governed by the spring, for low frequency by the damper.
(3) p*(m) =
&Jo io + l/Tn ’ po10
io + l/T,.
The dynamic behaviour of the viscoelastic core is specified by three-dimensional theory. Therefore, the vector field of the displacement is expressed in the potential form, see Eq. (4). x Y.
of Maxwell behaviour for dilatation and shear
Eq. (5) shows the constitutive equation for a viscoelastic medium, where aij denotes the stress tensor, Eijthe strain tensor and 6, the Kronecker symbol.
n*(o) =
2.2. The viscoelastic core
uJ=Pf$?+P
Fig. 2. Illustration deformation.
(4)
The viscoelastic model can easily be modified by altering Eq. (6) and Eq. (7). Approaches for the scalar potential a, and for the vector potential u, containing Bessel functions with complex arguments, satisfy the Helmholtz equations, which are extended to linear viscoelasticity.
299
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
3.1. Experimental
Interferometer
--------
set-up
A heterodyne laser interferometer and a phasedemodulator, developed and manufactured at the Institute of Mechanics, ETH Ziirich, represents the most essential part of the experimental set-up [4]. The interferometer enables us to measure displacements of less than one nanometer for a frequency range and bandwidth between 1 kHz and 8 MHz without touching the specimen. Fig. 3 shows the experimental set-up. The waves are excited by a broad band pulse which is repeated with a frequency of 20 Hz. After measuring the displacement versus time 300 times and averaging them, the laser interferometer is shifted to the next location. Depending on the approximate number of expected modes within the frequency range considered, the displacement array is recorded at 100 to 160 locations. The recorded data are transferred to a cluster of eight Hewlett Packard 9000 workstations, where the spectrum analysis is carried out. 3.2. Signal processing
Fig. 3. Experimental set-up.
2.3. Dispersion relation
Elaborating all boundary and compatibility conditions between the shell and core, and providing a harmonic set up in time and space for the shell displacement, leads to a homogeneous system of linear equations for four remaining constants, which is solvable if the determinant of the coefficients vanishes. Since this condition contains o, and k as well as all geometrical and material parameters of the shell and the core, it represents an implicit dispersion relation. It is solved numerically with a complex root finding algorithm.
3. Experiments Several combinations of shells and ‘fluids’, like water, milk, alcohol, glycerine, silicone oil and polyisobutylen are investigated in the project. However, the results presented in this paper refer to a steel shell with a diameter of 26 mm and a wall thickness of 0.5 mm, filled with a high viscosity silicone oil (Dow Corning 200 fluid (60 000 cs)) and an isotropic aluminium rod with a diameter of 18 mm.
As illustrated in Fig. 4, the recorded signals are processed with FFT in the time domain. To decompose the complex wavenumber dependence in space domain, a complex Total-Least-Squares Algorithm [ 53 is applied. This algorithm, originally developed for the signal processing of NMR signals, demands fewer measurements than FFT, allows a lower signal-to-noise ratio and has no intrinsic resolution limitations. It extracts both the real and imaginary part of the wavenumber from the measured data. 4. Results In a field ‘somewhere’ between acoustics, fluid mechanics and solid mechanics, many effects and phenomena interfere and the borders between these classical disciplines fade with increasing generality of the constitutive equations. In fluid mechanics shear resistance in a boundary layer is assumed to be proportional to the velocity gradient, whereas an acoustic fluid is supposed to behave inviscid. From the measured dispersion curves of a cylindrical shell containing a Maxwell fluid, five material parameters; &, pO, T,, T, and the density need to be determined, when the material constants of the shell are known from a preliminary series of experiments carried out on an empty shell. In the case of silicone oil the projection of the complex dispersion curves to the real o, k-plane is dominated by the velocity of P-waves c1 in the enclosed medium. Fig. 5 shows numerical results of a shell filled with an inviscid fluid in which P-waves propagate with the same velocity as in the silicone oil investigated. The measured dis-
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
300
Piezoelectric Four-dimensional
0 J.VoUmann
Excitation
Array:
1995
Fig. 4. Illustration of the signa processing.
persion relation is represented in Fig. 6. Fig. 7 compares experimental and numerical dispersion relations in one diagram and Fig. 8 shows the projection of the diagram, presented in Fig. 7, into the real plane. Increasing the elastic shear resistance while keeping the value of c1 constant, the dispersion curves leave the real plane periodically as shown in Fig. 9. (Due to numerical problems not all modes in Fig. 9 could be calculated.) Regarding the measured dispersion diagram Fig. 10, the curves occur very clearly in a range starting at k = 1500 rad m-l, frequency = 200 kHz ending at k = 3200 rad m-r, frequency = 1 MHz. In that specific area the imaginary part of the wavenumber becomes very small in the calculated dispersion relation for a shear elastic ‘fluid’, see Fig. 9. This effect could be an explanation for the clear detection of the curves in that range. Most likely the Complex Linear Prediction Method reaches its limits by interpreting the periodically increasing imaginary part of the wavenumber as noise, apart from the area mentioned above. Scaling the diagram of the measured dispersion curves
in the imaginary axis, an alternating imaginary part of the wavenumber, as calculated for a shear elastic ‘fluid’, is observed in Fig. 10. As far as the authors are aware, the mechanical behaviour of fluids subjected to shear forces alternating with high frequency (MHz range), is not very well known yet. To check the validity of the theoretical model especially in the range of strong shear elasticity the dispersion curves of axisymmetric waves in an aluminium rod with circular cross-section is also measured. This dispersion relation is calculated, running the same computer program used for the shell filled with viscoelastic medium. Therefore the relaxation times T’,‘,and T, are set to high values compared with the reciprocal value of the lowest frequency within the range considered. The material properties found in this manner, correspond to those found with other dynamic methods. See Figs. 11 and 12. Since the damping of the waves propagating in an aluminium rod is much lower than in a silicone filled shell, the dispersion curves of waves reflected at the
301
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
Im
600000
i
Fig. 5. Numerical
frqWI
results for a fluid with c1 equal to c, of silicone oil.
Im
Fig. 6. Measured
dispersion
curves of axisymmetric
waves in a cylindrical
shell containing
silicone otl
J. Vollmann, .I Dual/ Ultrasonics 34 (1996) 297-306
302
Im
WI
Fig. 7. Experimental and numerical dispersion curves in one diagram.
end of the rod are detected as well, see Fig. 13. Since waves propagate in both directions, the phenomenon of ‘backward-wave propagation’, mentioned by Onoe, McNiven and Mindlin in 1962 [ 61, where group velocity
and phase velocity have opposite signs, is clearly measured. See Fig. 14. A detailed investigation of the phenomenon of ‘backward wave propagation’ will be published soon.
frq[Hz1 800000
600000
Re 1000
1500
-2000
2500
3000
3500
Fig. 8. Projection of the curves shown in Fig. 7 into the real plane.
4000
j-ad/m]
8
w
304
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
300000
frq
WI 200000 \
Fig. 11. Numerical results of the dispersion relation in an aluminium rod.
frq
Fig. 12. Experimental and numerical dispersion curves of an aluminium rod in one diagram.
J. Vollmann, J. Dual J Ultrasonics 34 (1996) 297-306
305
Im k [radlm]
300000
frqWI
‘3
:
Fig. 13. Measured dispersion relation of an aluminium rod (diameter: 18 mm).
Fig. 14. Area of ‘backward wave propagation’ in a measured dispersion relation of an aluminium rod.
306
J. Vollmann, J. Dual/ Ultrasonics 34 (1996) 297-306
5. Conclusions Combining modern spectral estimation methods with a high resolution laser interferometer, we have developed a new measurement technique in structural mechanics. We have investigated some aspects of dynamic fluid structure interaction in the high frequency range, where classical models reach their limits. Future work will focus on the extension of spectral estimation methods to more than one dimension.
to Roger Brett, who has implemented and compared several Linear Prediction Algorithms to find the most powerful one. Special thanks also to Markus Hageli for the improvement of our laser interferometer.
References Cl1 T.J. Plona, B.K. Sinha, S. Kostek and S-K.Chang, J. Acoust. Sot. Am. 92 (1992) 1144.
VI B.K. Sinha, T.J. Plona, S. Kostek and S.-K. Chang, J. Acoust.
Acknowledgements Many thanks to Dr Stephan Kaufmann for his kind support in numerically solving the dispersion relation,
Sot. Am. 92 (1992) 1132. c31 D. Alleyne and P. Cawley, J. Acoust. Sot. Am. 89 (1991) 1159. [41 J. Dual, M. Hlgeli, M.R. Pfaffinger and J. Vollmann, Ultrasonics 34 (1996) 291 (this issue). CSI C.F. Tirendi and J.F. Martin, J. Magnetic Resonance 85 (1989) 162. 161 M. Onoe, H.D. McNiven and R.D. Mindlin, J. Appl. Mech. 29 (1962) 61.