Active Vibration Control of Finite Thin-Walled Cylindrical Shells

Transcription

1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MECHANICAL AND MANUFACTURING ENGINEERING Active Vibration Control of Finite Thin-Walled Cylindrical Shells Timothy McGann Bachelor of Engineering (Mechanical) October 2008 Supervisor: Dr N. J. Kessissoglou

2 Abstract Cylindrical shell-like structures exist in pipelines, pressure vessels, aircraft fuselages, ship hulls and submarine hulls. Improved understanding of the dynamic behaviour and control of vibration in these applications can reduce the associated problems of unwanted fatigue stresses, component misalignment, increased wear, energy loss, sonar detectable acoustic signatures of submarines and passenger discomfort due to both noise and vibrations in aircraft. Active control is a technique that involves using a feedforward control loop to inject energy of the right magnitude and phase so as to cancel out any unwanted oscillatory energy in the system. Very few experiments have been documented regarding the use of this technique to control vibrations within thin-walled cylindrical shells. The purpose of this thesis is to begin filling this existing gap in research. An experimental investigation was conducted into the use of active control to attenuate vibrations generated by single frequency excitations. A thin-walled mild steel cylindrical tube with two thick circular plates fixed at each end was used as the test structure in this thesis. The modal characteristics of the system were experimentally determined and validated by comparison with literature. The cylinder was then excited in the axial direction by an inertial shaker driven at one of three selected system natural frequencies. Active control was applied by using a secondary inertial shaker mounted at the opposing cylinder end for each of the chosen frequencies. The use of multiple error sensor control has also been investigated. i

3 The application of active vibration control effectively produced global attenuation of cylinder vibrations for two out of the three selected natural frequencies. Further validation and improvements in resolution of the system natural frequencies are expected to yield higher performance control. The use of multiple error sensors with a single control actuator was found to deteriorate active control performance in comparison to single sensor/single actuator control. ii

4 Statement of Originality The thesis presented herein contains no material or subject that has been accepted previously for the award of any other degree or diploma in any education institution. To the best of my knowledge and belief, all the material presented in this thesis, except where stated and otherwise referenced, is my own original work. I consent that this thesis be made available for loan and photocopying. Tim McGann October 2008 iii

5 Acknowledgements I would like to extend a special thankyou to my thesis supervisor Dr Nicole Kessissoglou for her encouragement and belief in my abilities at all points of this challenging project. Despite her family commitments with her newborn son, I am very grateful that she could still to find the time and energy to share as much relevant guidance and experience as possible. Without such wisdom this project would not have been possible. I would also like to thank Russell Overhall whose extensive experience in the use of acoustics and vibrations technologies was invaluable. Many issues were encountered during experimentation, and Russell s passion and willingness to assist aside from his busy schedule was always greatly appreciated. Finally I wish to thank all of my family and friends for their support throughout the duration of this thesis and for the continued encouragement I have received during my years of university study. Thank you. iv

6 Table of Contents Abstract...i Statement of Originality...iii Acknowledgements...iv Table of Contents...v List of Figures...viii List of Tables...xiii List of Symbols...xv Chapter 1 Introduction and Literature Review Introduction Thesis objectives Literature review Dynamic response of cylindrical shells Active control Thesis Layout...16 Chapter 2 Vibrations Theory General structural dynamics Vibration fundamentals Frequency response function Coherence Modal analysis Theory of cylinder vibration...24 v

7 Chapter 3 Active Control Theory Basic principles Adaptive feedforward control Active control system design...33 Chapter 4 Experimental Testing and Results Introduction Experimental arrangement Experimental rig Description of equipment Determination of natural frequencies Free response experimental procedure Free response experimental results Forced response experimental procedure Forced response experimental results Natural frequencies Determination of mode shapes Mode shape mapping procedure Mode shape results...54 Chapter 5 Active Control SISO control method SISO control mode 1 results SISO control mode 2 results SISO control mode 3 results Dual error sensor control method...69 vi

8 5.2.1 Dual error sensor control results...70 Chapter 6 Discussion Mode 227Hz Mode shape Single error sensor control Dual error sensor control Mode 478Hz Mode shape Single error sensor control Comparison to theory Mode 546Hz Mode shape Failed control...81 Chapter 7 Conclusions and Future Work Conclusions Future Work...85 References...87 Appendix A Tabulated Experimental Data...91 Appendix B Engineering Drawings vii

9 List of Figures Figure 1.1 Figure 1.2 Figure 1.3 Stringer stiffened cylinder...8 Ring stiffened cylinder...8 Mode shapes of ring stiffened circular cylindrical shell support by shear diaphragm [4]...10 Figure 2.1 a) SDOF system b) and free body diagram...17 Figure 2.2 Frequency response of a forced SDOF system Figure 2.3 Example of a time and frequency domain transformation for a vibrating beam Figure 2.4 First mode of vibration in a tensioned string Figure 2.5 Mode separation of frequency response function [28]...24 Figure 2.6 Cylindrical shell co-ordinate system [1] Figure 2.7 First three longitudinal mode shapes of a cylinder [4]...27 Figure 2.8 First 4 circumferential mode shapes of a cylinder [4]...27 Figure 2.9 Combined longitudinal and circumferential modes [1]...28 Figure 2.10 Natural Frequencies of unstiffened cylinder, a = 2m, L = 6m, h = 0.02m...29 Figure 2.11 Natural Frequencies of unstiffened cylinder, a = 2m, L = 3m, h = 0.02m...29 Figure 3.1 Basic principle of superposition...30 Figure 3.2 First patented active noise control concept...31 Figure 3.3 Adaptive feedforward vibration control system...32 Figure 3.4 Adaptive feedforward control system functional diagram [30] Figure 4.1 Cylinder end-plate and shaker assembly sketch...37 viii

10 Figure 4.2 A-frame assembly...38 Figure 4.3 (a) Original surface of inertial shaker. (b) Inertial shaker with adapterplate attachment...39 Figure 4.4 Modified assembly with force transducer...39 Figure 4.5 Equipment Configuration for free response testing Figure 4.6 Hammer impact locations and accelerometer location Figure 4.7 Frequency response functions of the cylinder from an impulse excitation at different locations...45 Figure 4.8 Coherence functions of the cylinder from an impulse excitation at different locations Figure 4.9 Frequency response function of the cylinder from an impulse excitation at point Figure 4.11 Equipment configuration for forced response testing Figure 4.12 Frequency response functions of the cylinder at multiple accelerometer locations using a forced broadband excitation...48 Figure 4.13 Coherence functions of the cylinder at multiple accelerometer locations using a forced broadband excitation Figure 4.14 Frequency response function of the cylinder at point 19 using a broadband excitation Figure 4.15 Coherence function of the cylinder at point 19 using a broadband excitation...49 Figure 4.16 Figure 4.17 Figure 4.18 Experimental mesh definition for 11 x 16 point mesh D cylinder mesh plot of uncontrolled 227Hz mode shape...55 Longitudinal mode shape at 227Hz measured along b = 4 using 33 point mesh ix

11 Figure 4.19 Circumferential mode shape at 227Hz measured about x = 16 using 32 point mesh Figure 4.20 Figure D cylinder mesh plot of uncontrolled 478Hz mode shape...57 Longitudinal mode shape at 478Hz measured along b = 8 using 33 point mesh Figure 4.22 (a) Circumferential mode shape at 478Hz measured about x = 8 using 32 point mesh. (b) Circumferential mode shape at 478Hz measured about x = 24 using 32 point mesh Figure D cylinder mesh plot of uncontrolled 546Hz mode shape...59 Figure 4.24 Longitudinal mode shape at 546Hz measured along b = 16.5 using 33 point mesh Figure 4.25 Circumferential mode shape at 546Hz measured about x = 16 using 32 point mesh Figure 4.26 (a) Circumferential mode shape at 546Hz measured about x = 8 using 32 point mesh (b) Circumferential mode shape at 546Hz measured about x = 24 using 32 point mesh Figure 5.1 SISO Active control hardware configuration...62 Figure 5.2 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 227 Hz. (b) 3- D cylinder mesh plot of controlled magnitudes at 227Hz Figure 5.3 Controlled and uncontrolled magnitudes at 227Hz measured along the cylinder length through b = 4 using a 33 point mesh...65 Figure 5.4 Controlled and uncontrolled response at 227Hz measured around the circumference through x = 16 using a 32 point mesh Figure 5.5 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 478 Hz. (b) 3- D cylinder mesh plot of controlled magnitudes at 478Hz x

12 Figure 5.6 Controlled and uncontrolled magnitudes at 478Hz measured along cylinder length through b = 8 using a 33 point mesh...67 Figure 5.7 (a) Controlled and uncontrolled response at 478Hz measured around the circumference through x = 8 using a 32 point mesh. (b) Controlled and uncontrolled response around measured around the circumference through x = 24 using a 32 point mesh...67 Figure 5.8 Location of error sensor in 546Hz active control attempt...68 Figure 5.9 Dual sensor active control configuration Figure 5.10 Controlled and uncontrolled responses at 227Hz using two error sensors at (16, 4) and (16, 12) and measured along cylinder length through b = 4 using a 33 point mesh...70 Figure 5.11 Controlled and uncontrolled response at 227Hz using two error sensors at (16, 4) and (16, 12) and measured around the circumference through x = 16 using a 32 point mesh Figure 6.1 The (m, n) = (1, 2) mode shape found by magnitude measurements compared with the measurement of imaginary components in Goodwin [1] Figure 6.2 Comparison of lengthwise control results between the use of 2 error sensors and a single error sensor...75 Figure 6.3 Control results about the circumference comparing the use of 2 error sensors and a single error sensor...76 Figure 6.4 Cylinder beginning, middle and end circumferential mode shapes at 478Hz...78 Figure 6.5 n =2 circumferential mode shapes measured at ¼ and ¾ along the cylinder length at 478Hz...78 xi

13 Figure 6.6 Expected control results for the m = 2 lengthwise mode shape [22]...80 Figure 6.7 Active control results comparison to expected theoretical control at 478Hz...81 xii

14 List of Tables Table 4.1: List of components used during experimentation...40 Table 4.2: Natural frequency comparison from impact testing...50 Table 4.3: Natural frequency comparison from forced testing...50 Table A1: Global uncontrolled mode shape data at primary frequency of 227Hz..91 Table A2: Global controlled mode shape data at primary frequency of 227Hz...91 Table A3: Lengthwise mode shape data for both controlled and uncontrolled responses at primary frequency of 227Hz during active control...92 Table A4: Circumferential mode shape data for both controlled and uncontrolled responses at primary frequency of 227Hz during active control...93 Table A5: Global uncontrolled mode shape data at primary frequency of 478Hz..94 Table A6: Global controlled mode shape data at primary frequency of 478Hz...94 Table A7: Lengthwise mode shape data for both controlled and uncontrolled responses at primary frequency of 478Hz during active control...95 Table A8: Circumferential mode shape data for both controlled and uncontrolled responses at primary frequency of 478Hz during active control with error sensor at point (8,8) and accelerometer about circumference x = Table A9: Circumferential mode shape data for both controlled and uncontrolled responses at primary frequency of 478Hz during active control with error sensor at point (8,8) and accelerometer about circumference x = Table A10: Global uncontrolled mode shape data at primary frequency of 546Hz..98 Table A11: Lengthwise mode shape data for uncontrolled response at 546Hz...98 Table A12: Circumferential mode shape data for uncontrolled response at 546Hz..99 xiii

15 Table A13: Lengthwise mode shape data for both controlled and uncontrolled responses at primary frequency of 227Hz under active control using two error sensors Table A14: Circumferential mode shape data for both controlled and uncontrolled responses at a primary frequency of 227Hz under active control using two error sensors xiv

16 List of Symbols M Elemental mass of a single degree of freedom model [kg] C Damping coefficient [kg/s] K Spring constant [N/m] ω Frequency of input forcing [rad/s] ω n Undamped natural frequency [rad/s] ω d Damped natural frequency [rad/s] ζ Damping ratio φ Phase angle of forced response [rad] F 0 Magnitude of general harmonic forcing [N] γ xy( ω) H(ω) X(ω) Y(ω) G xx G yy G xy Coherence function Frequency response function Input autospectrum of frequency response function Output autospectrum of frequency response function Input auto-spectral density Output auto-spectral density Cross spectral density x (t) Time domain input function y (t) Time domain output response u Cylinder axial displacement [m] v Cylinder circumferential displacement [m] w Cylinder radial displacement [m] [L] Donnell Mushtari differential matrix operator {u i } Displacement vector xv

17 x θ z Axial coordinate reference Tangential coordinate reference Radial coordinate reference a Cylinder radius [m] ρ Material density [kg/m 3 ] E Young s modulus [N/m 2 ] v s Poisson s ratio Non-dimensional cylinder length h Shell thickness [m] β 2 n m b Non-dimensional thickness parameter Circumferential mode number Longitudinal mode number Circumferential mesh point coordinate xvi

18 Chapter 1 Introduction and Literature Review 1.1 Introduction Vibrations are inherently present in all aspects of everyday life. Examples of industries where knowledge in the area of vibrations is deemed important include the transport, construction, aerospace, naval, manufacturing, military and music industries to name a few. These applications all contain mechanical systems, which can be viewed upon as comprising of distributed elements with characteristics of mass, stiffness and damping. A vibrating response in these systems occurs when an external or internal force excites the system. Such a force is generally either periodic or random in nature. Periodic loadings are most often a result of mass imbalances in machinery such as motors and propellers or cyclic impacts from reciprocating compressors and punching machines. The system responses from such harmonic forcings are generally steady state motion whilst the response from a single random excitation is expected to be a decaying oscillation. In all cases where the structure is surrounded by a fluid, it is possible for noise generation to occur due to the fluctuating pressure disturbance that arises from vibrating motion. The specific area of vibrations in thin cylindrical shells is applicable to understanding and controlling the dynamic behaviour of aircraft fuselages, submarine hulls, ship hulls, satellite launches, pipelines and pressure vessels where vibrations and the associated noise are considered an issue. Excitations caused by the operation of propellers, motors and other machinery in these applications can generate potentially damaging fatigue stresses, component misalignment, increased wear, energy loss, passenger stress and discomfort from both noise and vibration and finally sonar detectable acoustic 1

19 signatures in submarines. In order to reduce these undesired effects it is necessary to have a knowledge base of the dynamic behaviour of cylindrical systems and of strategies that can be employed to attenuate the vibration and noise levels. Each cylindrical system, like all other mechanical systems, has a series of natural vibration frequencies and mode shapes determined by the system geometry, size, material properties and boundary conditions. It is important to note that structural discontinuities such as shell stiffeners, bulkheads, junctions, changes in diameter and end closures and other complicating factors such as fluid loading and fluid dynamic effects should be considered if a more realistic cylindrical shell vibration analysis is desired. Studies have shown that these factors can play a significant part in determining the free response of the system. Once the free response characteristics such as resonant frequencies of a system have been understood, active and passive control methods can be implemented to reduce the undesired effects of vibration. Passive control involves modifying the mass, stiffness and damper properties to more effectively absorb radiated energy resulting from system disturbances. Active control involves the use of feedback and feedforward control loops to detect the unwanted disturbance and apply a secondary force to minimise the resulting structural response. 2

20 1.2 Thesis objectives This thesis is an extension of the experimental work conducted by Goodwin [1] on the active control of low frequency vibrations in a thin cylindrical shell under an applied harmonic axial excitation. An area of significant research in which this knowledge base can be applied is the global control of low frequency vibration modes in submarines resulting from fluctuating disturbances transmitted through the propeller shafting system. Attenuation of such responses is highly important in military applications where stealth is of the essence and the radiated acoustic signature due to shell vibrations is undesired. The experiment in Goodwin [1] investigated the control of cylindrical vibrations for a single mode shape and concluded that global attenuation of this mode was effective by use of a single error sensor and single actuator. Goodwin [1] based this conclusion on measurements taken along a single axial and a single circumferential antinodal line with the assumption that attenuation also occurred in the unmeasured regions of the cylinder. The objectives of this thesis are to: Verify and improve the results obtained by Wayne Goodwin on the existing experimental rig by taking measurements over the entire cylinder to confirm global control. Investigate the effectiveness of the active control of higher order mode shapes within the frequency limitations of the available EZ-ANCII controller. Compare the performance of multiple error sensor control against single error sensor control as applied to cylinder vibrations. 3

21 1.3 Literature review Dynamic response of cylindrical shells The dynamic behaviour of vibrating cylindrical shells has been an area of research interest since the 1960s and 70s [2]. Knowledge in this field is useful in many engineering applications including aircraft fuselages, submarines, ship hulls, pipelines and pressure vessels. To effectively attenuate undesired vibrations in these applications it is highly necessary to determine the natural frequencies and mode shapes of the cylindrical shell structures involved. Acoustic radiation and structural deflection is strongest at these resonant frequencies and hence they are more appropriate to control [3]. Shell structures are complex forms of plate structures, having all the same characteristics as plates but with the addition of curvature. Unlike beams and plates, the equations of motion and the effects of boundary conditions in thin cylindrical shells are much more complex. This is due to strong interrelation between angular, axial and circumferential displacements within cylindrical shapes. A coupling effect takes place, whereby an axial force or excitation can generate a displacement in radial and tangential directions as well as the expected axial direction. Common agreement has been met over the classical bending theory in plates, which utilize fourth order equations, however literature still remains divided over the most appropriate theory for cylindrical shells, which generally use eighth order equations. Many theories have been developed over the years with differences usually attributed to the point during their derivation at which simplifying assumptions are made, and or the choice of assumptions themselves [4]. These simplifications may include, but are not limited to: neglecting tangential terms and inertial terms, and the use linear approximations in the characteristic 4

22 equations. Despite the popularised use of some of the existing cylindrical shell theories, it is still necessary to specify the theory used when performing analytical work with cylindrical shells. Armenàkas, Gazis and Herrmann [2] detail the theoretical eigen-value solutions to the three dimensional linear theory of elasticity equations for stress free cylindrical surfaces. Numerical computations were performed in Fortran to find the first six natural frequencies for a wide range of geometric parameters. In performing the analysis, the cylinder was assumed to be hollow, isotropic and infinitely long. Their results tabulate and plot the normalised natural frequencies and corresponding radial, tangential and axial mode shapes for varied thickness to radius ratios and thickness to length ratios. It was found that the normalized natural frequencies vary considerably for high thickness to length ratios and vary much less as the thickness to length ratio decreases. As it stands, the results offer a satisfactory check on the validity and applicability of a range of other future and existing simplified shell theories for determining free dynamic behaviour of cylinders. Leissa [4] is referred to in many pieces of literature regarding the theory vibration of cylindrical shells. The publication summarizes a large range of the existing approximate cylinder theories of the era, including the: Donnell-Mushtari, Love-Timoshenko, Reissner-Naghdi-Berry, Vlasov, Sanders, Flügge, and Houghton-Johns. Leissa outlines the principles from which these theories were derived, including strain displacement shell theory, force and moment resultants and the fundamental equations of motion. The assumptions made during the derivation of each cylinder theory are also specified. Comparative tables are provided showing the numerical variations in frequency 5

23 solutions that exist between the approximate methods and the exact solutions to the three-dimensional elasticity theory as seen in Armenàkas et al [2]. It was found that a close agreement was met between theories for shells that were very thin, of moderate length and with small numbers of circumferential waves. In addition to plane isotropic cylinders, complicating effects such as rings and stringers, initial stresses, variable cylinder thickness, large non-linear deflections, shear deformation, rotary inertia, composite layered materials and the effects of surrounding fluids are discussed. In each case, the effects of cylinder end boundary conditions such as free, shear diaphragm and clamped in various combinations are considered. Numerical analysis and justification of the theories outlined in Leissa [4] has continued over the years for many different boundary conditions and shell configurations. Buchanan and Chua [5] recognized the absence of published vibration results for finite length cylinders under the fixed-free and fixed-fixed boundary conditions. They performed a finite element analysis involving both a standard isotropic material and Beryllium to tabulate the non-dimensional natural frequencies and corresponding mode shapes for various length/radius ratios. It was found that as this ratio increased, the effect of material characteristics tended to have more influence on the order of natural frequencies than did the cylinder geometry. It is stated in Buchanan [5] that with enough information, the effects caused by geometry and those caused by material properties can be observed in separation. Various comparative studies regarding cylindrical shell theories have been performed. One such study was that of El-Mously [6] which compared three explicit formulae that are used for predicting the natural frequencies and mode shapes of thin cylindrical 6

24 shells. The formulae considered include: the Weingarten-Soedel, Calladine-Koga and the Timoshenko-beam-on-Pasternak-foundation. All of these approximations contain limitations and restrictions for use depending on the certain geometric ratios. The formulae were numerically compared with the analytical solutions to Flügge s equations and with finite element results. Saijyou and Yoshikowa [7] presents experimental validation for the use of a modified bending stiffness approach to estimate the modes of vibration that are actually excited in a simply supported cylindrical shell. They found that different modes present different levels of modified bending stiffness, which in turn effect the magnitude of the driving force necessary to excite a particular mode. If two modes have similar natural frequencies, the more flexible mode is likely to be excited. An approximate approach using superposition of the axial and circumferential standing waves determined by the wave numbers n, the number of full circumferential waves, and m, the number of longitudinal waves, is presented in Zhang [8]. The axial wave number is approximated from an equivalent beam with similar boundary conditions to represent the cylindrical shell. Finite element modelling was used to validate the method and found the natural frequencies to be within 2% of each other. These results were expressed in Zhang to be reasonably accurate. Wang and Lai [9] claimed that this theory was flawed because it neglects the coupling that exists between axial and circumferential vibration and is only reasonable for relatively long cylindrical shells. Zhang [10] confirmed that the theory was appropriate as it was validated with other methods in literature for simply supported/simply supported, clamped/clamped and clamped/simply supported boundary conditions. A large advantage of this method is 7

25 that it can be easily extended to include more complex boundary and loading conditions without the need for intensive computations. While it is clear that a vast range of theories and approximations exist for idealized cylinders undergoing free vibration, the application of such theories to structures such as submarines and aircraft fuselages requires further effort to account for the existence of complicating effects. This includes studies into the effects of stringer and stiffeners, illustrated in figures 1.1 and 1.2 respectively, on the natural properties and radiated acoustic signatures of cylinders. Figure 1.1 Stringer stiffened cylinder Figure 1.2 Ring stiffened cylinder 8

26 Ruotolo [11] outlines that there are two primary ways for determining the effects of stiffeners on cylindrical shells, which include: treatment of the stiffeners as discrete elements or by averaging the properties over the shell surface as a smeared approach. Figure 1.3 demonstrates the effect that ring stiffeners can have on the lengthwise mode shape of a shear diaphragm - shear diaphragm cylindrical shell for varied circumferential mode numbers. The net mode shape in this example is obvious which therefore justifies the use of a smeared approach for low circumferential mode numbers when an adequate number of ring stiffeners are used. The work undertaken in Ruotolo [11] compared the use of Love s, Donnell s, Sander s and Flügge s shell theories under the smeared stiffener condition for different stiffness scenarios including: only rings, only stringers and rings plus stiffeners. Analytical results were compared with a finite element model and it was found that all theories were in close agreement except for Donnell s, which gave errors of up to 40% for the natural frequencies of the structure. Ruotolo [12] goes on to analytically address the influence of structural stiffness theories on interior noise generation. 9

27 Figure 1.3 Mode shapes of ring stiffened circular cylindrical shell support by shear diaphragm [4] Norwood [13] has written a detailed literature review in regards to cylinders and the effect of boundary conditions, end closures, stiffeners and external and internal fluid loadings, as applies to submarine structures. The review summarises the knowledge that has been developed such as modelling the reduction in modal frequency under external pressure and water loading and the effects of ring stiffeners causing an increase in system natural frequencies. The conclusions from this study were that further work is needed to improve the fluid/structural interaction in finite element analysis and more 10

28 specific study into the behaviour of internal bulkheads and the deep frame stiffeners of submarines is required. Ruzzene and Baz [14] have performed a finite element analysis of the effects of stiffeners, damped stiffening and water loading on the associated acoustic pressure field. The results show that stiffening and damping are suitable methods for passively reducing the vibration and sound radiation from submerged shells. However, low frequencies are much more difficult to attenuate by passive methods and it stands that active control is expected to be a much more effect means of vibration in stiffened cylinders Active control Attenuation of unwanted noise and vibration in mechanical systems is an area of particular interest to engineers. Occupant discomfort due to the internal noise generation within cylindrical structures such as submarines and aircraft fuselages, and the advances in sonar detection of external acoustic signatures in military marine vessels has led to an increasing need for developments in noise and vibration control. Passive control and active control are the two most common attenuation techniques. Passive control has been an area of wide research and is a procedure involving the modification of the physical parameters of a system such as mass, stiffness and damping. However, this technique is limited at low frequencies due to the larger energy absorptive mass requirements creating conflict with lightweight structural necessity and issues with system stability. As a consequence, active control methods have proven more successful in the low frequency range. 11

29 Active vibration control involves actively adding more energy to a system such that when it is superimposed with the original response, the total combined response is reduced. In single frequency noise and vibration control this can be achieved by introducing a secondary disturbance that is 180 degrees out of phase with the original. Although the basic concept of active control has been known for many decades, recent advances in control theory and solid-state transducer and microprocessor technology have allowed the method to become more practically feasible. In order to achieve anti-phase noise and vibration attenuation, it is important to use an appropriate control system to ensure that destructive rather than constructive interference is sustained in a stable manner. Sievers and Andreas [15] outline the control theory behind many systems that can be applied to reducing narrowband and single frequency disturbances. They discuss various methods of control including: adaptive, discrete, analogue, frequency domain and time domain approaches. The discussions indicate the wide variety of options available for control, the choice of which depends on the type of disturbance and level of performance desired. Multiple input multiple output adaptive feedforward systems are discussed and a new compensator design was put forward to increase the robustness of this control algorithm. An experimental analysis of active control of vibratory power transmission in a cylindrical shell is discussed in Pan and Hansen [16]. This analysis involved extending the Flügge equation to account for the linear inertia of cylinder walls as applies to a cylinder of semi infinite length, simply supported at one end and anechoically terminated at the other. The experiment investigated the influence on attenuation control of variables including; error sensor type and location, control force type and location, 12

30 cylinder radius and thickness and the excitation frequency of harmonic radial forces arranged circumferentially around the cylinder rig. The results found that an attenuation of 30dB in transmitted vibration power could be achieved by 3 or more control forces and was more effective in the radial direction than the axial. Extensional wave transmission gives a good approximation to the total power transmission while acceleration and power transmission cost functions are effective if chosen for just a single direction due to the wave coupling present in cylinders. Thomas et al [17,18] conducted research on the active control of sound transmission through a thin cylindrical shell as applied to aircraft with high-speed turbo props. Their model was based on previous works of Bullmore et al [19,20] which looked at producing reductions in low frequency cabin noise related to harmonic propeller blade tones by using primary and secondary sound sources and comparing the results with computational modelling. Thomas et al [17,18] used a theoretical expression giving the total kinetic energy of the cylinder walls to form a control cost function. The optimal configuration of secondary forces for minimizing the radial component of energy was set as the control criteria. The results were found to show that large reductions in vibration energy were difficult to due to the high number of structural modes occurring as a response to the primary forcing. Control was found to be most effective when there were fewer modes governing the response at the low frequency modal density regions. However, Nelson and Elliot outline in numerous sources [17,18,21] that reducing the vibration energy of a distributed structural system does not necessarily imply a reduction in radiated sound levels. 13

31 The behaviour of structural acoustic coupling within cylindrical shells is an area that has only recently received literary coverage. Kessissoglou [22] describes the effect of active vibration control of a submerged cylinder on its radiated sound pressure levels. An analytical cylinder model of length L was set up containing two mass balanced end plates and two internal dummy bulkheads. A primary axial input excitation was applied at one end with a secondary axial control force at the opposing end. Both axial and radial displacements were considered when investigating the radiated sound pressure levels. To control axial displacements, an error sensor was located at each end of the cylinder. For control of radial displacements a ring of error sensors was located around the circumference at selected axial positions along the cylinder. The first two axial resonance modes were observed while maintaining constant axisymmetric circumferential mode. The results indicated that axial attenuation achieved much higher reduction in acoustic pressure level than radial attenuation at all tested resonant frequencies. Active control system design requires decisions regarding the optimal location for control actuators and error sensors within the structure to be controlled. Kessissoglou, Ragnarsson and Lofgren [23] performed an analytical and experimental study into this mater on an L-shaped plate with simply supported conditions along the parallel L- shaped edges and free conditions at the two ends. They found that the level of control is much more dependent on actuator and error sensor position rather than the quantity or type of actuators present. Optimised control occurred when the control force was in line with the primary force and the error sensor midway between in a symmetrical configuration. 14

32 In large shell and plate structures, the use of a single error sensor and single actuator to achieve global control may not produce the best level of attenuation possible. For this reason it was necessary to enhance the available literature with a study on the use of multiple actuators and error sensors. Keir, Kessissoglou and Norwood [24] performed a theoretical and experimental analysis on a T-shaped plate with simply supported conditions along the parallel T-shaped edges and free conditions elsewhere. The results showed that multiple error sensors for a single actuator caused deterioration in control performance. Two error sensors with two dependently driven actuators produced greater attenuation than the single sensor single actuator arrangement with less importance on the choice of error sensor location. Use of three sensors and three actuators showed only slight improvement to the latter. Independently driven actuators were shown to produce better attenuation levels than dependently driven actuators for arbitrary sensor locations. Symmetrical arrangements were found to be the most effective for single actuator and single error sensor control. Whilst active control of vibrations under single frequency or multiple known frequency excitations has received considerable literature coverage, very little has been reported on the application of feedforward broadband structural control. Vipperman, Burdisso and Fuller [25] discuss the use of adaptive least mean square and recursive least mean square algorithms in controlling broadband vibration of a simply supported beam. Various impulse filters were applied and it was found that a Finite Impulse Response filter gave significant improvements in control performance with power reductions of up to 20dB at resonant. 15

33 The combining of both passive and active control strategies is an area of recent development expressed in Baz and Chen [26]. This work outlines the modelling of active constrained layer damping using energy principles to describe the vibratory behaviour of simply supported cylindrical shells with composite fabric walls. The cylinder construction uses a combination of passive constrained layer damping materials with embedded piezo-electric actuators to apply control. Control is achieved by controlling the strain experienced by the material in the constraining layer. Optimal balance between the simplicity of passive techniques and the efficiency of active methods is sought. The advantage of using such a system is that vibrations of large structures can be controlled without the need for large actuation voltages. 1.4 Thesis Layout Chapter 2 presents a summary of the background mathematical theories that are most commonly used to describe vibrations in general and the dynamic behaviour of thin cylindrical shells. Chapter 3 outlines the basic theory behind active control using feedforward systems as applicable to this thesis. Chapter 4 presents the experimental testing and results for the free and forced vibration modal characteristics of the cylinder. Chapter 5 presents the active control experimental procedure and results for attenuating single frequency oscillatory responses in the cylinder. 16

34 Chapter 6 contains a detailed discussion of the trends observed in the experimental results of this thesis. Chapter 7 summarizes the results and concludes the theoretical developments achieved in this thesis. An outline for necessary future work is also given. 17

35 Chapter 2 Vibrations Theory 2.1 General structural dynamics Vibration fundamentals A vibration or oscillation is any repeated motion of a physical system. Every mechanical system can be understood to consist of a continuous distribution of elements each displaying the characteristics of mass, elasticity and damping. A single-degree-offreedom (SDOF) model as shown in figure 2.1 is the most basic unit from which more complex multi-degree-of-freedom systems can be constructed for vibrations analysis. The number of degrees of freedom of a system equals the number of independent coordinates necessary to completely specify the motion of that system. Ideally, mechanical systems such as thin cylindrical shells would be modelled as continuous systems with an infinite number of degrees of freedom. However, obtaining the exact solutions to these systems is often very complicated and sometimes not possible so it is best to use lumped parameter models to approximate the continuous behaviour. In general, results of greater accuracy are obtained by increasing the number of degrees of freedom, however, this comes with the downside of requiring more computations. Figure 2.1 a) SDOF system b) and free body diagram. 17

36 On applying force equilibrium to the free body diagram of figure 2.1b for the system in free vibration, the following homogeneous differential equation is obtained. M & x + Cx& + Kx = 0 (2-1) Where M is the elemental mass, C is the damping coefficient, K is the spring constant and x is the displacement of the mass from its equilibrium position. A single dot above the x denotes the first derivative of displacement with respect to time, known as velocity. The double dot above the x denotes the second derivative of displacement with respect to time, known as acceleration. The general solution to a SDOF system in free vibration is given by an exponentially decaying sine function as follows: x(t) = Ae ω n ζt sin( ω d t + φ) (2-2) Where A is the amplitude, t is the time, φ is the phase angle, ζ is the damping ratio, ω n 2 is the natural frequency and ωd = ωn 1 ζ is the damped natural frequency. The SDOF system shown in figure 2.1 can also be excited by a persistent disturbance instead of an initial excitation as in the free response case. If a harmonic force or displacement excitation is applied then the homogeneous equation in equation (2-1) is modified to include the disturbance and is written as follows: M& x + Cx& + Kx = F0 sinωt (2-3) 18

37 Where F 0 is the amplitude of the forcing and ω is the frequency of the applied harmonic forcing. The general steady state solution to equation (2-3) is given by: F0 x(t) = K sin( ωt φ) 2 ω 1 ωn 2 ω + 2ζ ωn 2 (2-4) The non-dimensional frequency response amplitude is shown in figure 2.2. The important feature to note from this graph is the very high amplitude that occurs when the driving frequency (ω) is somewhat close to the natural frequency (ω n ). Under this condition, the system is described as being driven at resonance. The increased amplitudes due to resonance can lead to increased displacements, increased noise generation and higher stress levels that can accelerate fatigue failure. In order to reduce these undesired resonant effects, it is important to have the ability to change a systems natural frequency, adjust the driving frequency or destructively interfere with the driving signal by wave superposition. 19

38 Figure 2.2 Frequency response of a forced SDOF system Frequency response function Complex oscillatory behaviour is often very difficult to analyse within the time domain. It is much simpler to deal with vibrations data in the frequency domain by performing a Fast-Fourier-Transform (FFT) manipulation. In frequency domain analysis of linear systems, a frequency response function (FRF) represents the transfer function H(ω) of the system and is the mathematical relationship between the input X(ω) and output Y(ω) frequency autospectrums given for a single input/single output set-up as follows [27]: Y( ω) H( ω ) = (2-5) X( ω) 20

39 The transformation between time domain and frequency domain is shown in figure 2.3 where the top three boxes represent the time and spatial domain, whilst the bottom three represent the frequency domain for a vibrating cantilever beam. Figure 2.3 Example of a time and frequency domain transformation for a vibrating beam Coherence The functions X(ω), Y(ω) and H(ω) apply to ideal linear systems which contain no noise. In reality the degree of correlation between measured input and measured output 2 must be checked. This is performed by the coherence function, γ ( ω) which is defined as follows [27]: xy G xy ( ω) 2 γ xy ( ω) = (2-6) G ( ω)g ( ω) xx yy 2 21

40 * * * Where G ( ω ) = X( ω)x ( ω), G ( ω) = Y( ω)y ( ω) and G ( ω) = Y( ω)x ( ω) are the xx yy input auto-spectral density, output auto-spectral density and cross-spectral density respectively. X * (ω) and Y * (ω) are the complex conjugates of the input X(ω) and output Y(ω) respectively. xy The coherence function has an upper bound of 1 indicating a system with no extraneous noise and a lower bound of 0 indicating absolutely no correlation between input and 2 output measurements. The condition 0 < γ ( ω) 1 generally occurs due to: extraneous xy < noise, resolution bias errors, system non-linearity or y(t) caused by additional inputs apart from x(t) Modal analysis A multi degree of freedom (MDOF) system consisting of N degrees of freedom requires N co-ordinates to completely specify its motion and has N natural frequencies. Corresponding to each of these natural frequencies is a mode shape, which describes the expected curvature pattern of system when oscillating at that frequency. An example of the first mode shape of a vibrating string under tension and its resultant when combined with its second harmonic is illustrated in figure

41 Figure 2.4 First mode of vibration in a tensioned string. The collective term for the natural frequency and its corresponding mode shape is called a mode of vibration. A continuous system can be described as having an infinite number of degrees of freedom. This implies an infinite number of modes whereby the superposition of each simple mode shape will result in the total wave motion of the structure under vibration. Gade et al [28] explains that it is possible to break down the FRF of a continuous system into its constituent modes which each have a characteristic resonant frequency, damping and mode shape. This break down is represented in figure 2.5. In general if there is reasonable separation between resonance points and the structure is lightly damped, then coupling between mode shapes is minimal. Under this condition, a system driven at resonance can be considered to behave primarily as a SDOF system. Thus for a thin cylindrical shell, each peak on the FRF is expected to have a unique associated mode, assuming that the natural frequencies are reasonably separated. 23

42 Figure 2.5 Mode separation of frequency response function [28]. 2.2 Theory of cylinder vibration Cylindrical shell theory is most commonly understood by using the cylindrical coordinates shown in figure 2.5. The theory in this report is presented in terms of axial displacement, u, circumferential displacement, v, and radial displacement, w. Figure 2.6 Cylindrical shell co-ordinate system [1]. The simplest theory used to describe the motion of a cylinder is the Donnell-Mushtari set of equations. These equations were developed for uniform unstiffened cylindrical shells of homogenous isotropic linearly elastic material properties undergoing relatively small displacements. 24

43 25 With appropriate boundary conditions, the equations can be solved to obtain the eigen values, which in turn give the natural frequencies of the system. Leissa [4] outlines the Donnell-Mushtari equations of cylindrical motion in matrix form as follows: [ ] [ ] 0 } u { i = L (2-7) Where {u i } is the displacement vector given by = w v u } u { i (2-8) u, v, and w are the components of displacement in the x, θ, z directions respectively as shown in figure 1. [L] is a differential matrix operator. The Donnell Mushtari matrix system is given by = w v u t E ) v (1 s a a 1 R 1 x a v a 1 t E ) v (1 a 1 x 2 v) (1 x 2a v) (1 x a v x 2a v) (1 t E ) v (1 2a v) (1 x ρ θ β θ θ ρ θ θ θ ρ θ (2-9) Where a is the cylinder radius, ρ is the density, E is the Young s modulus, ν is Poisson s ratio, s = x/a is the non-dimensional length, and β 2 = h 2 /12a 2 is the nondimensional thickness parameter, h is the shell thickness.

44 The boundary conditions applied to the system in equation (2-9) are an adapted form of the simply supported condition from beam and plate theory at both ends of the finite length cylinder. This condition is generally parametrically described in the following manner: u 0 v = w = 0 x = 0,L w = w / z 0 (2-10) Leissa [4] adopts the term shear diaphragm boundary conditions to describe the simply supported situation that exists for a cylindrical shell. These conditions are used because the cylinder considered in this thesis is enclosed at both ends by flat, thin plates that behave in a very similar way to the conditions described in (2-10). For harmonic motion, the following general solutions for axial, tangential and radial displacements in terms of circumferential and longitudinal mode numbers (n) and (m) respectively are given as follows: mπx jωt u = U nm cos(nθ)cos e (2-11) n= 0 m= 1 L mπx jωt v = Vnm sin(nθ)sin e (2-12) n= 0 m= 1 L jωt w = Wnm cos(nθ)sin e (2-13) n= 0 m= 1 L mπx On substituting the general solutions in (2-11), (2-12) and (2-13) into the equations of motion and finding the determinant of the coefficient matrix, the resulting characteristic equation can be solved to yield the natural frequencies and corresponding modes of vibration. 26

45 The mode shapes of a cylindrical shell are described by two integers; the longitudinal mode number, m, and the circumferential mode number, n. The longitudinal mode number m represents the number of half sine waves that fit along a cylinders length whilst the circumferential wave number n represents the number of full sine waves around the circumference of the cylinder. Figure 2.4 shows the first three cylinder shapes corresponding to the longitudinal wave numbers m = 1, 2, 3. Figure 2.7 First three longitudinal mode shapes of a cylinder [4] Figure 2.8 shows the first four circumferential shapes corresponding to the circumferential mode numbers n = 0, 1, 2, 3. Figure 2.8 First 4 circumferential mode shapes of a cylinder [4] 27

46 Circumferential and longitudinal modes generally occur simultaneously and various configurations exist for different natural frequencies. Figure 2.9 gives an illustration of the types of lower order mode shape combinations that can occur in thin cylindrical shells. Figure 2.9 Combined longitudinal and circumferential modes [1] Unlike many other structures, the simplest modes of vibration in cylindrical shells do not necessarily have the lowest natural frequencies. Solutions to the Donnell - Mushtari equations yield 3 frequency roots for every set of fixed mode numbers. The lowest frequency is the most generally reported as it is the strongest sound radiator. If the natural frequencies are known, the associated mode shape can be classified by the dominant vibrational form, whether radial, axial or circumferential. The lowest frequency is usually associated primarily with radial motion [2]. Generally the lowest 28

47 natural frequency will occur for a circumferential mode greater than 1 and is dependent on the geometry of the cylinder as recognized by comparison of figures 2.10 and Figure 2.10 Natural Frequencies of unstiffened cylinder, a = 2m, L = 6m, h = 0.02m Figure 2.11 Natural Frequencies of unstiffened cylinder, a = 2m, L = 3m, h = 0.02m 29

48 Chapter 3 Active Control Theory 3.1 Basic principles Active control is a technique encompassing the principle of destructive interference to actively cancel out any unwanted acoustic or vibration disturbances within a system. Take the simple case of a single frequency, fixed amplitude, sinusoidal disturbance as represented in figure 3.1 by the blue line. The principle of superposition suggests that a secondary control signal, shown by the red line, 180 degrees out of phase with disturbance will provide complete cancellation, provided that the frequency and magnitudes are equal at the point interest. Figure 3.1 Basic principle of superposition Paul Lueg established the concept of active noise control in his 1936 patent [29]. He considered the use of a microphone and loudspeaker to apply noise cancellation to a one-dimensional propagating sound wave in a duct. This is shown in figure 3.2, whereby a microphone detects an upstream pressure disturbance of sinusoidal form and a control circuit measures then sends the same noise in anti-phase through a 30

49 loudspeaker. While Lueg s concept was valid, the available technology at the time used for detection, processing and generation of sound was not available. Figure 3.2 First patented active noise control concept Over the past 70 years there has been substantial advancement in microprocessor technology, which has lead to an increased academic interest in the use of active control systems. The control of vibrations in structures can take either passive or active forms. The former uses spring/damper systems to isolate the system from its vibratory source. Although this system is proven to work effectively for a large frequency bandwidth, the general rule of thumb is that the lower the frequency, the more spring and damper material is required for energy dissipation. This presents an issue in areas such as aircraft design and submarine operations, where system weight and size consideration is of high importance. Active vibration control provides a more suitable lightweight solution for the attenuation of low frequency vibrations than the existing passive control methods. 31

50 3.2 Adaptive feedforward control The method for actively controlling vibrations throughout the experiments in this thesis involves the use of an adaptive feedforward control loop illustrated in figure 3.3. This system incorporates a slight addition to the basic system envisaged by Lueg in figure 3.2. A reference signal of the incoming disturbance is measured and the controller predicts an output that is expected to attenuate the disturbance signal. In addition to the feedforward set-up seen in Lueg s system, an error sensor is used to monitor the output at a specified location, allowing for the control system to adapt and improve its cancellation in an iterative process. Figure 3.3 Adaptive feedforward vibration control system An adaptive feedforward control loop has many advantages over a feedback loop in that it can offer prevention of a disturbance by producing a cancellation signal prior to the disturbance taking effect. In a feedback set-up, the disturbance will have already passed through the system and this is often very undesired, especially in applications such as submarine sonar stealth. Adaptive feedforward control, when stably converging, also has superior attenuation performance over feedback control, rendering it more suitable. 32

51 The main disadvantage of adaptive feedforward control is the requirement of a reference measurement to accurately predict an impending disturbance. While this might be quite simple for a tonal disturbance, a random broadband excitation is much more difficult to predict. Suitable techniques may need to be implemented in practice to rapidly determine the incoming disturbance before it propagates through the system prior to application of control. 3.3 Active control system design The three basic hardware components required for an active vibration control system are a sensor, controller and an actuator. Various sensors can be used including accelerometers, proximity detectors, fibre optics and piezo-electric materials. Actuators generally consist of inertial shakers or piezo-electric crystals. Control systems can come as pre-packaged multi-channel units with single input/single output (SISO) and multiple input/multiple output (MIMO) capabilities, single channel controllers, or derived from scratch depending on the users preference. When designing an active control system there are a few preliminary tasks that should be performed to achieve the best and most stable level of vibration control. The first task is to select the location for the error sensor and control actuator within the system. While there are algorithms and functions that can be used to calculate the optimum location for these components, commonsense was found to be appropriate during this thesis. The error sensor location was generally chosen to be at the anti-nodal points of the uncontrolled mode shape. All system disturbances requiring control in this thesis were generated using single frequency sine wave input signals. The second task is to set 33

52 the input and output signal gains to ensure that there is adequate energy to achieve control and to also ensure that the system was not driven beyond its linear behaviour. The controller consists of two fundamental parts. The first is the digital control filter, which is responsible for generating a control output signal in real time based on filter weight parameters. The second part is the adaptive algorithm that plays the role of tuning the filter weights based on three inputs. These inputs are the reference signal and the error signal, both of which update the algorithm in real time, and the cancellation path identification transfer function. Prior to performing control, it is necessary to create a model to define the transfer path between the control actuator and the corresponding error sensor. This is because the control excitation signal takes time to transmit through the physical system and the predicted outputs would vary for each sensor/actuator configuration. The function of the error signal is to indicate any residual vibration disturbance that may still exist while control is being applied. The adaptive algorithm applies a least mean square function to these residual vibration magnitudes in order to continually update the control filter weights. Figure 3.4 provides a block diagram to assist the understanding of the adaptive control systems internal functions. 34

53 Figure 3.4 Adaptive feedforward control system functional diagram [30]. One of the most influential parameters in determining control system stability is the convergence coefficient. Analogous to finite element and numerical methods, a smaller convergence coefficient corresponds to a more refined mesh size which is often less likely to diverge and become unstable in the iterative loop. However, the adaptive process becomes drastically slowed and often stagnant if the selected convergence coefficient is too small. So a compromise is to be found for the quickest and most effective control solution. Finding such a compromise generally requires a large amount of time spent performing trial and error convergence tests. It can be concluded from this chapter that whilst active control appears simple in theory, a significant amount of time and preliminary thought must be spent in order to configure a system to achieve optimal control. 35

54 Chapter 4 Experimental Testing and Results 4.1 Introduction This chapter provides details on the cylindrical rig and experimental instrumentation used during the determination of system natural frequencies and the plotting of mode shapes. The testing procedures used and their corresponding results are also included. 4.2 Experimental arrangement Experimental rig The testing for this thesis was conducted on a suspended cylindrical tube arrangement as constructed by Goodwin [1]. The cylinder is a 1100mm length mild steel tube of mean diameter 148.8mm and shell thickness of 1.6mm providing reasonable dynamic flexibility. Welded to each end of the cylinder is a 6mm thick circular end-plate to provide a rigid support through which the disturbance and control vibration signals can be transmitted. The configuration seen in figure 4.1 shows the assembly arrangement consisting of removable fasteners and adapter-plates to allow for easy removal of components if necessary. The configuration shown is repeated at both ends of the cylinder. 36

55 Figure 4.1 Cylinder end-plate and shaker assembly sketch The cylinder was suspended from an A-frame by elastic ockie straps, as shown in figure 4.2, to simulate free-free boundary conditions. The low stiffness of these straps ensured that rigid body motion in any of the six degrees of freedom was of very low frequency, causing negligible interference with the responses generated by higher experimental frequency excitations. Such rigid body motions that occurred throughout testing include: translation in the axial and transverse directions by swing, translation in the vertical direction by bounce and some rotational motion. 37

56 Figure 4.2 A-frame assembly Design Modifications to Existing Rig A decision was made to introduce a force transducer between the inertial shaker and the square adapter plate at one end of the cylinder to monitor the coherence between the primary forcing input and accelerometer output. Threaded studs on both ends of the force transducer were chosen as the preferred option for attachment. These allow for easy disassembly of the system should a new configuration be required in future work. The studs required unified course thread (UNC) tapped holes in the components on either side of the transducer. However, one of these components was the inertial shaker, which could not be modified. Consequently a small circular adapter-plate machined from a piece of standard aluminium bar stock was designed to accommodate the tapping on the shaker side as shown in figure 4.3(b). A tapping was also added to the centre of 38

57 the existing square adapter plate. The new cylinder, force transducer and shaker assembly is shown in figure 4.4. Figure 4.3 (a) Original surface of inertial shaker. (b) Inertial shaker with adapterplate attachment Figure 4.4 Modified assembly with force transducer 39

58 4.2.3 Description of equipment A large selection of common electronic equipment was used to obtain vibration measurements throughout the testing procedures. For each testing stage including: impact testing, broadband excitation, mode shape plotting and active control there were different set-up configurations and instrumentation requirements. A list of the common equipment used to perform the experiments is given in table 4.1. Table 4.1: List of components used during experimentation Equipment Name Type of Model Personal Computer Laptop/PC DELL Latitude D800 Signal analyser and generator Pulse Active Controller EZ-ANC Active Controller EZ-ANCII Bruel & Kjær Pulse Front- End Type 3560C Shaker Inertial Actuator Ultra Electronics D/L2 Amplifier Charge Amplifier Bruel & Kjær Type 2635 Accelerometer Piezoelectric Accelerometer Bruel & Kjær Type 4393 Hammer Impact Hammer Bruel & Kjær Type 8202 Force Transducer Dynamic Force Transducer Bruel & Kjær Type 8200 Power Pack Laboratory Power Supply Powertech MP-3084 Inertial Shaker Signal Conditioner Signal Conditioner Ultra Electronics (D/L2) The Pulse Front-End is a multi-channel device capable of generating and analysing noise and vibrations signals. The device provides an interface between transducer and actuator instrumentation and the Pulse software program. The Pulse software contains a database of calibration data for common Bruel & Kjær instrumentation from which the components in use can be selected. In taking measurements an appropriate trigger level was set such that excessive loading was not required to initiate the measuring process. A 40

59 number of functions are available in Pulse for data analysis and comparison including frequency response spectra, coherence plots and time domain functions. The EZ-ANCII active noise controller is a multi-channel device capable of generating a primary disturbance signal from which a reference and error signal can be used in determining a control output. The system can be used as either a SISO or MIMO control loop depending on the requirements. The adaptive feedfoward control algorithm used in the EZ-ANCII is based on a filtered least mean square algorithm applied to the error signal. Optimal control convergence can be achieved by modifying the appropriate algorithm parameters through a software interface. Piezoelectric accelerometers were used throughout experimentation to convert mechanical movements into charge pulses. A mass bonded to a piezoelectric crystal inside the accelerometer casing generates a compressive force on the crystal when accelerations are experienced, which produces a charge. The transducers are lightweight and can be easily moved and reattached to measurement locations by a magnetic mount. The component has a high natural frequency that is well above the range seen in this thesis. The impact hammer was used to generate an impulse force on the test structure. The hammer tip was chosen from a range of materials with different stiffness depending on the frequency band chosen for analysis. Softer tips such as nylon are easier to control and more suited to low frequency range, while steel tips are better suited to higher frequency ranges. A force transducer is fixed within the hammer to transform the impulse into a charge signal by the same means as the piezoelectric accelerometer. 41

60 Inertial shakers utilise the oscillation of a permanent magnet inside an energised AC wire coil. The magnet shakes according to the solenoid coils input electric signal. The supports constraining the magnet have a significant influence on the resonant frequency of the actuator. Driving at this frequency should be avoided to prevent unexpected dynamic behaviour. The charge amplifier is required to enhance the charge signals received from accelerometer and force transducer equipment to be more easily read by the analysers. The amplifiers can be used to adjust transducer sensitivity, voltage gain and upper and lower band pass frequency levels. 4.3 Determination of natural frequencies Free response experimental procedure When a system is harmonically forced at one of its natural frequencies, instabilities can arise causing significant noise and vibration amplitudes. For this reason the natural frequencies of the cylinder were chosen as the driving frequencies for the primary forcing during active control. This is based on the assumption that if large amplitude responses can be controlled then the smaller amplitudes at other frequencies will also be reduced. In order to determine the free response natural frequencies of the cylindrical shell it was desired to generate a frequency domain response function of the cylinder resulting from an initial impulse excitation. This was achieved by using an impact hammer to strike the rig in the radial direction and an accelerometer to map the output response. The equipment configuration for this procedure is shown in figure 4.5. Unlike Goodwin [1] who carried out free response testing with only a single control shaker mounted to the rig, the free response testing in this thesis had both primary and 42

61 secondary shakers attached. This minimised any effects caused by adding or removing lumped masses from the system that may have shifted the natural frequencies between free response testing and active control testing. Figure 4.5 Equipment Configuration for free response testing. Before the test was conducted, it was necessary to choose an appropriate hammer tip for exciting a consistent energy level across the impulse frequency spectrum. A hard plastic tip was chosen as the impact surface for the hammer. Other tip choices included rubber, nylon and hardened steel. The tip hardness is an important factor in determining the input frequency band. A harder tip will excite higher frequencies with greater energy than a soft tip, which provides better energy to the low-end frequencies. The desired frequency band for this thesis was chosen to be up to 800Hz. Autospectrum testing showed that the hard plastic tip had a 3dB roll off over this frequency range indicating a suitable energy level at all frequencies. 43

62 Trigger settings in the Pulse unit were configured to initiate data acquisition when the rig was struck. An exponential window was introduced to the accelerometer response time spectrum to prevent leakage to be sure that the signal finished decaying to zero within the sample time. Prior to taking measurements, correct use of the hammer was practiced to ensure unwanted occurrences, such as double impacts due to rebound, did not take place during measurement Free response experimental results Figure 4.6 Hammer impact locations and accelerometer location. Free response results were collected for different locations of the hammer impact and a fixed accelerometer location. The locations of these impact points are indicated by the red points shown in figure 4.6 along the length of the cylinder. The 33 nodal points were previously marked out by Goodwin [1] and were used as a reference in this thesis. The accelerometer was fixed at point 4, indicated by the blue dot, during each impact test. A linear average of ten impact measurements per hammer impact location was set in the Pulse FFT analyser settings. For each impact location, a frequency response function and corresponding coherence function were collected. As discussed in chapter 2, the coherence function is a measure of linearity of the data between input and output responses of the system. A coherence of 1 indicates a high correlation between input and output measurements. Figures 4.7 and 4.8 show the results collected from all 44

63 impact test points whilst figures 4.9 and 4.10 indicate a single result from the point 13 for greater clarity. FRF Hammer Test 40 Acceleration (db) point7 point10 point13 point16 point19 point22 point25 point28 point Frequency (Hz) Figure 4.7 Frequency response functions of the cylinder from an impulse excitation at different locations. Hammer Coherence Test 1.2 Coherence point7 point10 point13 point16 point19 point22 point25 point28 point Frequency (Hz) Figure 4.8 Coherence functions of the cylinder from an impulse excitation at different locations. 45

64 FRF - Point Acceleration (db) Frequency (Hz) Figure 4.9 Frequency response function of the cylinder from an impulse excitation at point 13. Coherence - Point Coherence Frequency (Hz) Figure 4.10 Coherence function of the cylinder from an impulse excitation at point

65 4.3.3 Forced response experimental procedure A second test was carried out to determine the natural frequencies of the cylinder. A pseudo-random noise signal was generated by the Pulse system to drive the primary shaker. The input broadband signal spanned from 0 to 800Hz. Figure 4.11 shows the experimental set-up for this test. A force transducer measured the input signal while an accelerometer obtained the dynamic response of the cylindrical rig. Multiple accelerometer points were measured to account for the effects of various mode shape nodal and anti-nodal points that may exist within the frequency range of interest. This allowed all natural frequencies to be identified and then compared with those obtained from the free response hammer test. Figure 4.11 Equipment configuration for forced response testing Forced response experimental results The frequency response function and coherence function were generated by the Pulse front-end and are displayed in figure 4.12 through to

66 Forced Broadband FRF 120 Acceleration (db) p7 p10 p13 p16 p19 p22 p25 p28 p Frequency (Hz) Figure 4.12 Frequency response functions of the cylinder at multiple accelerometer locations using a forced broadband excitation. Forced Broadband Coherence 1.2 Coherence p7 p10 p13 p16 p19 p22 p25 p28 p Frequency (Hz) Figure 4.13 Coherence functions of the cylinder at multiple accelerometer locations using a forced broadband excitation. 48

67 Forced Broadband FRF - Point Acceleration (db) Frequency (Hz) Figure 4.14 Frequency response function of the cylinder at point 19 using a broadband excitation. Forced Broadband Coherence - Point Coherence Frequency (Hz) Figure 4.15 Coherence function of the cylinder at point 19 using a broadband excitation. 49

68 4.3.5 Natural frequencies The system natural frequencies were identified from the peaks of both the free response FRF and the forced FRF. These results have been compared in tables 4.2 and 4.3 and with those obtained by Goodwin [1] for validation. Table 4.2: Natural frequency comparison from impact testing Free Response Natural Frequencies Goodwin [1] - Free Response (Hz) Natural Frequencies (Hz) N/A Table 4.3: Natural frequency comparison from forced testing Forced Response Natural Goodwin [1] Forced Response Frequencies (Hz) Natural Frequencies N/A N/A In general, all natural frequencies obtained from the experiments in this thesis are very similar to those obtained in Goodwin [1]. Any minor differences can be attributed to the slight modification in the set-up of the experimental rig, whereby both shaker masses were fixed to the rig during testing. In contrast, Goodwin only had the primary shaker attached. The similarity between the results of this thesis and Goodwin s results provide 50

69 adequate validation for the data. It was decided that any slight variation between the results obtained between the forced and free response testing can be neglected and the nearest whole number frequency chosen for use during mode shape plotting and active control. The selected natural frequencies for control in this thesis include 227Hz, 478Hz and 546Hz. 4.3 Determination of mode shapes A mode shape is a natural property of an oscillatory system which describes the pattern of vibration amplitude across the geometry of the system. Each natural frequency has its own mode shape. The mode shapes of the cylinder were measured to obtain a greater understanding of the vibration levels and to determine the nodal and anti-nodal regions for the specified resonant frequencies of 227Hz, 478Hz and 546Hz. This information was used for choosing the optimum error sensor location during active control. For example, an error sensor placed on a nodal line often produced poor attenuation and created difficulties in achieving a stable control algorithm. In theory, axial excitation should excite only the breathing modes of a cylinder. Hence, both the circumferential and longitudinal mode shapes were measured and a three dimensional map of the vibration levels across the cylinder was obtained Mode shape mapping procedure To map the acceleration magnitudes over the entire cylinder, a mesh was marked out on the outer surface using a marker pen. As shown in figure 4.16, the mesh consisted of 11 equally spaced points along the cylinder length and 16 around the circumference, to give a total of 176 points. The longitudinal mesh was selected to be relatively coarse (element size = 110mm) because pre-testing evaluations confirmed that longitudinal 51

70 mode numbers m of up to only m = 3 existed for the chosen driving frequencies. The circumferential plot required a slightly more refined mesh for better mode definition. A coordinate system was also introduced to track the mesh node positions. The x-axis in figure 4.16 refers to the longitudinal location starting from 0 at the primary shaker end of the cylinder. The b-axis refers to the circumferential location starting from 1 at the datum line. Figure 4.16 Experimental mesh definition for 11 x 16 point mesh. 52

71 The following procedure was followed to experimentally determine the mode shapes for each of the chosen resonant frequencies of 227Hz, 478Hz and 564Hz: 1. The instrumentation was set up as shown in figure A signal was generated by the Pulse system to drive the primary shaker using a sinusoidal disturbance. 3. An accelerometer was used to measure the magnitude of radial acceleration at each of the 176 node points by traversing lengthwise along the cylinder for each circumferential co-ordinate b. At each location, the Pulse system generated a FRF based on the linear average of 10 data measurements from which the magnitude at the driven frequency was displayed. A period of 30seconds was allowed between each mesh point measurement. This was to allow for the decay of any unwanted transients created by the magnetic snapping force between the accelerometer and the steel cylinder. 4. The procedure was run for each of the frequencies and the results collected as a matrix of data (see Appendix A) to be plotted in a three-dimensional mesh surface cylinder format in MatLab. 5. To obtain a clearer view of each of the mode shapes, and to check for effects such as symmetry that are not necessarily obvious from the 3-dimensional plots a second procedure was run. The lengthwise mode shape was measured and plotted using a 33 point mesh (element size = 34mm) passing through an antinodal point of the circumferential mode. The circumferential mode shape was measured and plotted using a 32 point mesh passing through the anti-nodal point/s of the longitudinal mode. 53

72 6. The separate longitudinal and circumferential mode plots provided a more accurate indication of where on the cylinder control error sensors could be placed for optimum attenuation Mode shape results When a 227Hz sine wave signal was sent axially into the cylinder rig, the three dimensional response was measured as shown in figure 4.17 as a series of acceleration magnitudes. The results show a half sine wave along the cylinder length and two full sine waves (magnitude) around the circumference indicating the (n, m) = (2, 1) mode shape. The lengthwise mode shape shown in figure 4.18 was measured along the antinodal line b = 4 (see figure 4.16). The circumferential mode shape shown in figure 4.19 was measured at the centre of the cylinder about the line x =

73 Figure D cylinder mesh plot of uncontrolled 227Hz mode shape. 55

74 227Hz Lengthwise mode shape Acceleration (db) Distance along cylinder (mm) Un-controlled Figure 4.18 Longitudinal mode shape at 227Hz measured along b = 4 using 33 point mesh. 227Hz Circumferential Mode shape Figure 4.19 Circumferential mode shape at 227Hz measured about x = 16 using 32 point mesh. 56

75 When a 478Hz sine wave signal was sent axially into the cylinder rig, the threedimensional response was measured as shown in figure 4.20 as a set of acceleration magnitudes. The results show two half sine waves along the cylinder length and two full sine waves (magnitude) around the circumference indicating the (n, m) = (2,2) mode shape. The lengthwise mode shape shown in figure 4.21 was measured along the antinodal line b = 8 (see figure 4.16). The circumferential mode shapes shown in figure 4.22 (a) and figure 4.22 (b) were measured about the lines x = 8 and x = 24 which correspond to locations ¼ and ¾ along the cylinder length respectively. Figure D cylinder mesh plot of uncontrolled 478Hz mode shape. 57

76 478Hz Lengthwise mode shape 0-5 Aceleration (db) Distance along cylinder (mm) Figure 4.21 Longitudinal mode shape at 478Hz measured along b = 8 using 33 point mesh. Figure 4.22 (a) Circumferential mode shape at 478Hz measured about x = 8 using 32 point mesh. (b) Circumferential mode shape at 478Hz measured about x = 24 using 32 point mesh. 58

77 When a 546Hz sine wave signal was sent axially into the cylinder rig, the threedimensional response was measured as shown in figure 4.23 as a series of acceleration magnitudes. The lengthwise mode shape shown in figure 4.24 was measured along the anti-nodal line b = 16.5 (see figure 4.16). The circumferential mode shapes were measured about the lines x = 8 and x = 16 and x = 24 as these showed varied circumferential behaviour. The longitudinal mode number m is identified as m = 3 [3] whilst the circumferential mode is undecided as figures 4.25 and 4.26 both indicate characteristics of n = 1 and n = 3. Figure D cylinder mesh plot of uncontrolled 546Hz mode shape. 59

78 546Hz Lengthwise mode shape Acceleration (db) Distance along cylinder length (mm) Figure 4.24 Longitudinal mode shape at 546Hz measured along b = 16.5 using 33 point mesh. Figure 4.25 Circumferential mode shape at 546Hz measured about x = 16 using 32 point mesh. 60

79 Figure 4.26 (a) Circumferential mode shape at 546Hz measured about x = 8 using 32 point mesh (b) Circumferential mode shape at 546Hz measured about x = 24 using 32 point mesh. 61

80 Chapter 5 Active Control 5.1 SISO control method A discussion on how active control works and the basic system parameter set-up is given in chapter 3.3. Figure 5.1 shows the hardware configuration used to apply single input/single output (SISO) control to the cylindrical rig. Four channels out of the twenty available in the EZ-ANCII controller were used for generator output, control signal output, reference signal input and error signal input. Figure 5.1 SISO Active control hardware configuration. 62

81 The following procedure was used for single error sensor control at the chosen frequencies of 227Hz, 478Hz and 546Hz: 1. The error sensor location was selected based on the point of maximum acceleration amplitude obtained from the plotted mode shape results. 2. Once the error sensor was attached to the cylinder it s charge amplifier was set to produce an amplitude between 0.5 and 0.75 on the EZ-ANCII software interface display. Input gain settings were also adjusted to achieve this. 3. The EZ-ANCII signal generator was set to produce a sine wave signal output of the desired frequency. 4. Filtering, adaptive algorithm and system cancellation path identification variables were adjusted until stable control was achieved in the system. 5. While the generator was left running, the Pulse front-end system was used to obtain the magnitude of vibration of the uncontrolled signal prior to control. 6. The active control mode was then switched on and left until the error signal had converged to a stable value. This was generally close to 10 % of the original amplitude. The pulse unit was again used to obtain the controlled magnitude of vibration. The active control mode was then switched off. 7. Steps 5 and 6 were repeated for each nodal point in the selected 176 point cylindrical mesh and for each of the selected driving frequencies. 8. After the three-dimensional control data was obtained, a more refined set of data was collected by repeating steps 5 and 6 along a 33 point longitudinal mesh and around a 32 point circumferential mesh. 63

82 5.1.1 SISO control mode 1 results The error sensor was placed at the point (16,4) corresponding to halfway along the cylinder length and a quarter of the way around the circumference from the reference line (see figure 4.16). Figure 5.2(a) and 5.2(b) display the uncontrolled and controlled vibration magnitude plots at a primary 227Hz sine wave shaker excitation. The lengthwise and circumferential control plots are shown in figure 5.3 and figure 5.4 respectively. Figure 5.2 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 227 Hz. (b) 3- D cylinder mesh plot of controlled magnitudes at 227Hz. 64

83 Figure 5.3 Controlled and uncontrolled magnitudes at 227Hz measured along the cylinder length through b = 4 using a 33 point mesh. Figure 5.4 Controlled and uncontrolled response at 227Hz measured around the circumference through x = 16 using a 32 point mesh. 65

84 5.1.2 SISO control mode 2 results The error sensor was placed at the point (8,8) corresponding to a quarter of the way along the cylinder length and halfway around the circumference from the reference line (see figure 4.16). Figure 5.5(a) and 5.5(b) display the uncontrolled and controlled vibration magnitude plots at a primary 478Hz sine wave shaker excitation. The lengthwise and circumferential control plots are shown in figure 5.6 and figure 5.7 respectively. Figure 5.5 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 478 Hz. (b) 3- D cylinder mesh plot of controlled magnitudes at 478Hz. 66

85 Figure 5.6 Controlled and uncontrolled magnitudes at 478Hz measured along cylinder length through b = 8 using a 33 point mesh. Figure 5.7 (a) Controlled and uncontrolled response at 478Hz measured around the circumference through x = 8 using a 32 point mesh. (b) Controlled and uncontrolled response around measured around the circumference through x = 24 using a 32 point mesh. 67

86 5.1.3 SISO control mode 3 results Application of active control to the 546Hz mode shape was unsuccessful. The error sensor was located at the point of maximum acceleration as shown in figure 5.8. However, convergence of the adaptive control algorithm was never achieved. The system cancellation path identification was defined on multiple occasions and a large range of convergence coefficient values were tried. Reasons for the lack of success in controlling this mode shape are discussed in chapter 6. Figure 5.8 Location of error sensor in 546Hz active control attempt. 68

87 5.2 Dual error sensor control method The single input/single output control results all show that the maximum attenuation levels occur in and around the error sensor location. This leads to the assumption that the use of multiple error sensors throughout the cylinder will produce improved levels of attenuation. However, Kessissoglou et al [23] states that while performance can improve if a second error sensor is placed on the same anti-nodal line as an optimally located single error sensor, arbitrarily locating the second sensor will in fact deteriorate the performance. This statement was based on results obtained from the control of rectangular plate mode shapes. It was decided to test this theory for cylindrical shells. The active control unit and instrumentation were therefore set-up as shown in figure 5.9. The same procedure as per single error sensor control was used. Figure 5.9 Dual sensor active control configuration. 69