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Dynamic Stability of Cylindrical Shells under Moving Loads by Applying Advanced Controlling Techniques—Part II: Using Piezo-Stack Control

Dynamic Stability of Cylindrical Shells under Moving Loads by Applying Advanced Controlling... Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 927078, 7 pages doi:10.1155/2009/927078 Research Article Dynamic Stability of Cylindrical Shells under Moving Loads by Applying Advanced Controlling Techniques—Part II: Using Piezo-Stack Control 1 2 Khaled M. Saadeldin Eldalil and Amr M. S. Baz Department of Mechanical Engineering, Tanta University, Sperbay, Tanta, Egypt Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA Correspondence should be addressed to Khaled M. Saadeldin Eldalil, eldalil01@msn.com Received 2 April 2009; Accepted 6 July 2009 Recommended by Mohammad Tawfik The load acting on the actively controlled cylindrical shell under a transient pressure pulse propelling a moving mass (gun case) has been experimentally studied. The concept of using piezoelectric stack and stiffener combination is utilized for damping the tube wall radial and circumferential deforming vibrations, in the correct meeting location timing of the moving mass. The experiment was carried out by using the same stiffened shell tube of the experimental 14 mm gun tube facility which is used in part 1. Using single and double stacks is tried at two pressure levels of low-speed modes, which have response frequencies adapted with the used piezoelectric stacks characteristics. The maximum active damping ratio is occurred at high-pressure level. The radial circumferential strains are measured by using high-frequency strain gage system in phase with laser beam detection system similar to which used in part 1. Time resolved strain measurements of the wall response were obtained, and both precursor and transverse hoop strains have been resolved. A complete comparison had been made between the effect of active controlled and stepped structure cases, which indicate a significant attenuation ratio especially at higher operating pressures. Copyright © 2009 K. M. S. Eldalil and A. M. S. Baz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The integration of viscoelastic damping materials into various structures has been widely theoretical studied. Euler- Bernoulli beam theory has been used to model viscoelas- The production of long slender gun systems to meet increased exit velocity requirements of rounds has subse- tic materials using the Rayleigh Ritz approximation for quently increased the effect of precision and accuracy of damping. Timoshenko beam theory incorporates both shear firing as well as barrel/round interactions during firing. A deformation and rotational deformation in the formulation. lightweight, low-cost method is desired to damp the induced The modal strain energy method [2]makes useofthe vibrations thereby increasing performance of the gun system. relationship between damping factors and modal loss factors Some experimental and theoretical methods are sug- in order to assign damping factors to real elastic modes gested for damping and controlling vibrations in the gun assign approximation to light damping. The Golla-Hughes- tubes. A relatively inexpensive and lightweight method of McTavish (GHM) method [3, 4] uses a finite element damping vibrations in some structures is to apply a surface approach where viscoelastic damping is introduced as a series treatment of a viscoelastic material and a constraining layer, of minioscillator terms and auxiliary dissipation coordinates. as a passive control combination [1], which deals with DiTaranto and Blasingame [5] and Mead and Markus [6] the transverse vibrations in the system resulted in shear derived a sixth-order PDE to model the transverse vibrations deformation of the viscoelastic material, which in turn of a three-layer beam system based on the equations dissipates the energy. This technique is very effective for developed for flexural vibrations of layered plates. In this damping terrain and firing-induced vibrations and hence approach damping of the viscoelastic layer is incorporated increases precision and accuracy. through the use of a complex shear modulus. 2 Advances in Acoustics and Vibration Table 1: The effective material properties for the data of the stack 1 2 n resonator. Property Real Imaginary Z Z Z Z Z Z 1 1 1 1 1 1 2 −11 −13 s (m /N) 2.00 × 10 −3.0 × 10 T −11 −10 ε (F/m) 3.00 × 10 −6.17 × 10 −10 −12 d (C/N) 3.925 × 10 −7.66 × 10 Z Z 2 2 k 0.5 −0.001 Martin’s general solution for the admittance of a piezo- N C N N C C 0 0 electric stack of area A, n layers, and total length nL is [19] nγ 2N Y (ω) = iωnC + tanh ,(1) Z 2 Figure 1: Equivalent circuit representation of a stack with the ST mechanical ports of each equivalent circuit representing a layer where connected in series and n electrical ports in parallel. ε A 33 2 C = 1 − k , 0 33 Ro et al. [7] and Sung [8] developed this technique the- Ad N = , oretically and experimentally by applying active constrained Ls damping layer (ACLD); they modeled the tube/ACLD system 0.5 by using Golla-Hughes-McTavish method in order to predict 1 Z = Z Z 2+ , ST 1 2 the tube response in the time domain. They calculated (2) the transient response using finite element method; the 0.5 predicted response is compared with that of a tube controlled γ = 2 arcsin h , 2Z with passive constrained layer damping treatment. The pre- 2 dictions of the finite element model are validated experimen- ωL tally. The results obtained indicted that ACLD treatment has Z = iρν A tan , 2ν achieved significant attenuation of the structure vibrations. D 2 On the other hand, the methods for damping radial ρν A iN Z = + , and circumferential vibrations of the shell tube walls are i sin(ωL/ν ) ωC completely different. Using periodic stiffeners distributed on the shell tube surface is tried theoretically by Ruzzene and baz D D and ν = 1/ ρs is the acoustic velocity at constant [9, 10], Aldareihem and Baz [11, 12], and others [13–15], and T D electric displacement. The constants ε , s , d are the 33 33 studied experimentally in part 1, Eldalil and Baz [16]. free permittivity, the elastic compliance at constant elec- In this work, the concept of applying a feedforward tric displacement, and the piezoelectric charge coefficient, control by using piezoelectric stack is experimentally tried. respectively. Using (1)to(2), Martin demonstrated that in To the best of our knowledge, this technique is not examined the limit of large n (n> 8), the acoustic wave speed in the before for damping the radial and circumferential deforma- material was determined by the constant field elastic constant tions of the shell walls theoretically or experimentally. The E E s (1/ ρs ). In the limit of n> 8 an analytical equation 33 33 selection of the piezoelectric stacks is based on its dynamic for the admittance was presented which allowed for direct characteristics which are meeting the shell walls radial and determination of material constants from the admittance circumferential deformation dynamic modes. A feedforward data [3]. In this limit the admittance was shown to be control system is suggested and designed in order to achieve the optimum attenuation ratio and satisfy suitable dynamic iAnω (k ) ω 2 33 stability. Y = 1 − (k ) + tan ,(3) ω/4 f 4 f s s 1.1. Piezoelectric Stack Characteristics. Piezoelectric stacks where the series resonance frequency is are used in variety applications that require relatively high force and larger displacement than single element piezo- 1 1 f = . (4) electric transducers can produce. These include microposi- 2πL ρs tioning systems, solid-state pumps/switches, noise isolation The effective material properties are shown in Table 1,for mounts, ultrasonic drills, and stacked ultrasonic transducers. the data of the stack resonator shown in Figure 2. The stack The solution for the zero bond length stacks was derived by Martin [17, 18]. His model was derived from Mason’s length nL = 0.02 m, Area A = (0.01) m , and density equivalent circuit of n layers connected mechanically in series ρ = 7800 kg/m . and electrically in parallel as shown in Figure 1. Formoredetails, referto[19]. Advances in Acoustics and Vibration 3 1e 1 D E 1e 2 _ n = 10 1e 3 1e 4 Figure 3: Basic mechanisms of regulation, from left to right: _ buffering, feedforward, and feedback block diagrams. 1e 5 1e n = 1 d measured disturbance Feedforward control 1e 7 ff 1e 8 032 64 98 130 162 194 226 158 Process Frequency (kHz) Y(s) output de Figure 2: Typical conductance as a function of frequency for n = 1 to 10 layers. Figure 4: Traditional feedforward control structure. 2. Feedforward Control 2.1. Concept of Feedforward Control. Feed-forward control, To eliminate the effect of the measured disturbance, we need which is used herein, will suppress the disturbance defor- only choose q so that: ff mations before it has had the chance to affect the system’s essential variables. This requires the capacity to anticipate P − Pq = 0, d ff the effect of perturbations on the system’s goal. Otherwise (6) ∼−1 ∼ q = P P , the system would not know which external fluctuations to ff consider as perturbations, or how to effectively compensate where a ∼ over a process transfer function indicates that it their influence before it affects the system. This requires that is a model of the process. Even in the above case, where the control system is able to gather early information about the feedforward controller can perfectly compensate the these fluctuations. Figure 3 shows block diagram of typical measured disturbance, Seborg et al. [20]. feedforward control system; the effect of disturbances D on the essential variables E is reduced by an active regulator R. Feedforward control can entirely eliminate the effect 3. Experimental Setup of the measured disturbance on the process output. Even when there are modeling errors, feed-forward control can The experiments were carried out with in-door Pneumatic often reduce the effect of the measured disturbance on 6/12-feet helium Gun Facility in the Vibration and Noise the output better than that achievable by feedback control Control Laboratory at University of Maryland; it has a alone. However, the decision as to whether or not to use stainless steel tube length of 6 feet. The detailed description feedforward control depends on whether the degree of is found in part 1, Eldalil and Baz [16]. improvement in the response to the measured disturbance justifies the added costs of implementation and maintenance. 3.1. Piezoelectric Stack Configuration. The piezoelectric stack The economic benefits of feedforward control can come and shell tube combination is shown in Figure 5. As indicated from lower operating costs and/or increased salability of the in part 1 [16], the stiffeners are composed of three parts, two product due to its more consistent quality. half rings (2, 3), in Figure 5, and one outer ring, surrounding the shell tube (1); consequently, in active control, the outer 2.2. Mathematical Formulation. Figure 4 shows a traditional ring is removed and the stacks (4, 7) are rested on the feedforward control scheme. The transfer function between upper and lower half rings. A stainless steel clamp strip (6) the process output Y and the measured disturbance d as is used to hold the components fixed in position, and the shown in Figure 4 is screw (5) is used to create initial tension in the clamp strip and consequently generates compression force distributed around the shell tube substituting the removed outer ring of Y (s) = P − Pq d. (5) d ff the stiffener. Conductance G(S) 4 Advances in Acoustics and Vibration Tension screw Strip clamp Upper piezoelectric stack Upper half ring Gun tube Lower half ring Figure 7: Piezoelectric stack control view picture. Lower piezoelectric stack S.S. 130 cal, 1600psi, plain (loc.2) 1.5 Figure 5: Double Piezoelectric stack configuration. 0.5 165 mm −0.5 Strain gage no. 2 145 mm Piezo-foil-sensor −1 0.102 0.103 0.104 0.105 0.106 0.107 Shell tube Bullet direction Time (s) s.g.2 Piezoelectric actuators Ph. shift Amplifier (a) s.s130 cal,1600psi, w177ss, one ring Figure 6: Piezoelectric stacks/feedforward control scheme. 1.5 0.5 3.2. Feedforward Control Elements Representation. The con- trol circuit scheme is shown in Figure 6, of type feedforward; −0.5 it is composed of piezoelectric foil pressure sensor located −1 at distance ×1 (165 mm) upstream of the strain gage of 0.104 0.105 0.106 0.107 0.108 0.109 location number 2, and at distance ×2 (145 mm) upstream Time (s) the piezoelectric stack actuator. The output signals are phase s.g.2 w17ss,1ring shifted and amplified, then feed the piezoelectric stacks, which are located at distance ×3 (20 mm) upstream the (b) strain gage of location number 2. The sensor has a time s.g.2w17ss, active.2piezo advance of Δt1 (0.391 milliseconds) and the system (phase 1.5 shifter, amplifier, and piezoelectric stacks) has a time lag of Δt2 (0.372 milliseconds), so for ideal operating case the two 0.5 times should be equal, or Δt2 < Δt1byfew percent. By this way the actuators will have sensible impact on the tube −0.5 vibrations; the phase can be adjusted by using phase shifter. −1 0.111 0.112 0.113 0.114 0.115 0.116 The piezoelectric stack control setup view picture is shown Time (s) in Figure 7. s.g.2w177ss, act.2piezo 3.3. Experimental Measurements and Results. The measure- (c) mentsare carried outattwo operating helium pressures, Figure 8: (a) Output signal at location number 2 of plain tube at 1600 and 2000 psi, and the controller (piezo-stack) is 1600 psi. (b) Output signal at location number 2 of tube with 17 installed very close to location number 2 on the gun tube, as single stiffeners. (c) Output signal of location number 2 with 17 indicated in part 1 [16]. The expected deforming vibration single stiffeners and with active control at 1600 psi. due to these pressures ranges between 7.2 and 25 kHz, Part 1 [16], that is, far enough from the first mode frequency of the piezoelectric stack (64 kHz), as shown in Figure 2. off, which is equivalent to the stiffened tube; is shown in Figure 8(b), and the active controlled time domain signals is 3.3.1. Measurements at Pressure Level of 1600 psi. The time shown in Figure 8(c). A time domain comparison of active resolved measurement of plain tube is shown in Figure 8(a), and stiffened tube, is shown in Figure 9, and the comparison and that of the arrangement when the control is turned analysis of the frequency domains is shown in Figure 10. Strain (V) Strain (V) Strain (V) Advances in Acoustics and Vibration 5 Comparison of active and stiffened tube, loc. no.2, 1600psi S.S 130 cal. 2000psi, plain 1.5 1.5 0.5 0.5 0 −0.5 −1 −0.5 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 Time (s) −1 s.g.2 0.104 0.105 0.106 0.107 0.108 0.109 0.11 piez.2 Time (s) piez.3 Stiffeners (a) Active S.S.130 cal, 2000psi, s.g.2, w17ss Figure 9: Time domain comparison between active and stiffened 1.5 tube at location no. 2 at operating pressure 1600 psi. 0.5 S.S 130 cal-1600psi, w17ss, active-2piezostacks (subcritical) 0.07 0.06 −0.5 −1 0.05 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1 Time (t) 0.04 s.g.2 piezo3 0.03 piezo2 Laser (b) 0.02 Active and with 17ss, s.g.2 1.5 0.01 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ×10 Frequency (Hz) −0.5 −1 Plain 17ss −1.5 17ss and active 0.088 0.089 0.09 0.091 0.092 0.093 0.094 0.095 Time (t) Figure 10: Frequency domain comparison between plain, stiffened tube, and with active piezo-stack control. 2bullet,s.g.2 (c) The attenuation ratio due to using active piezoelectric Figure 11: (a) Output signal at location number 2 of plain tube at 2000 psi. (b) Output signal at location number 2 of tube with 17 stack is attained to 33%, at vibration mode of 7.2 kHz and single stiffeners at 2000 psi. (c) Output signal of location number 2 48% at vibration mode of 9.5 kHz for pressure level of with 17 single stiffeners and with active control at 2000 psi. 1600 psi. The effect of active control is decreased at higher vibration modes. 3.3.2. Measurements at Pressure Level of 2000 psi. The time sensor and amplifier output signals (piezo-stqack loading domain measurement of the plain tube at pressure level of signal or the control effort) is shown in Figure 13,at 2000 psi is shown in Figure 11(a) and that of the stiffened amplifier gain factor of 2. tube with piezo-stack arrangement without control is shown The radial strain vibration is attenuated by using the in Figure 11(b), and the controlled time resolved signal is active piezoelectric control by a ratio of 46% at vibration shown in Figure 11(c). The frequency domain comparison is mode of 7.2 kHz and 65% at 9.5 kHz. The attenuation ratio is shown in Figure 12. A comparison of the piezoelectric foil increased at higher pressure level of 2000 psi than at pressure Amplitude Amplitude (V) Strain (V) Strain (V) Strain (V) 6 Advances in Acoustics and Vibration S.S 130 cal-1600psi, w17ss, active-2piezostacks (subcritical) A feedforward control scheme is designed and con- 0.07 structed by using piezoelectric foil sensor to predict in advance the incoming dynamics vibration and piez-stacks 0.06 actuators. The control system is checked at two pressures levels; 0.05 first pressure level is 1600 psi and the second pressure level is 2000 psi. The attenuation ratio is predicted at two modes 0.04 of low-frequency vibrations. At first mode of vibration, 7.2 kHz, the attenuation ratio 0.03 is found to be about 33% and 46% at pressure levels of 0.02 1600 psi and 2000 psi, respectively. And at second mode of vibration, 9.5 kHz, the attenu- 0.01 ation ratio is found to be about 48% and 65% at pressure levels of 1600 psi and 2000 psi, respectively. At higher modes of vibrations the attenuation effect decreases to lower ratios. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ×10 The frequency is shifted to lower values by about 13%. Frequency (Hz) The frequency domain comparison for the two pressure Plain levels indicates that the stability is satisfied and no spill over 17ss occurred. 17ss and active Figure 12: Frequency domain comparison between plain, stiffened References tube, and with active control at 2000 psi. [1] M. Z. Kiehl and C. P. T. W. Jerzak, “Modeling of passive constrained layer damping as applied to a gun tube,” Shock and Comparison between sensing signal Vibration, vol. 8, no. 3-4, pp. 123–129, 2001. and control signal [2] C. D. Johnson and D. A. Kienholz, “Finite element prediction 0.5 of damping in structures with constrained viscoelastic layers,” 0.3 AAIA Journal, vol. 20, no. 9, pp. 1284–1290, 1982. [3] D. J. McTavish and P. C. Hughes, “Modeling of linear viscoelas- 0.1 tic space structures,” Journal of Vibration and Acoustics, vol. 105, pp. 103–110, 1993. −0.1 [4] D. H. Golla and P. C. Hughes, “Dynamics of viscoelastic structures—a time-domain finite element formulation,” Jour- −0.3 nal of Applied Mechanics, vol. 52, pp. 897–906, 1985. −0.5 [5] R. A. DiTaranto and W. Blasingame, “Composite damping of vibrating sandwich beams,” Journal of Engineering for Industry, −0.7 0.0965 0.1015 0.1065 0.1115 0.1165 [6] D. J. Mead and S. Markus, “The forced vibration of a Time (s) threelayer, damped sandwich beam with arbitrary boundary conditions,” Journal of Sound and Vibrations,vol. 10, no.2,pp. Control signal 163–175, 1969. Sensing signal [7] J. Ro, K. S. El-Din, and A. Baz, “Vibration control of tubes Figure 13: Comparison between piezoelectric sensor and amplified with internally moving loads using active constrained layer signals at amplifier gain factor of 2. damping,” in Proceedings of ASME Annual Meeting, vol. 223, pp. 1–11, Dallas, Tex, USA, November 1997. [8] Y.-G. Sung, “Modeling and control with piezoactuators for a simply supported beam under a moving mass,” Journal of level of 1600 psi, by a value of 40% for the first mode and Sound and vibration, vol. 250, no. 4, pp. 617–626, 2002. by 35% for the second mode of vibrations. The frequency is [9] M. Ruzzene and A. Baz, “Dynamic stability of periodic shells shifted to lower values by about 13%. This due to increasing with moving loads,” Journal of Sound and Vibration, vol. 296, no. 4-5, pp. 830–844, 2006. the control effort nonlinearly with strain deformations which [10] M. Ruzzene and A. Baz, “Response of periodically stiffened are occurred at the two pressure levels of the first and second shells to a moving projectile propelled by an internal pressure experiments; consequently, the control signal is magnified wave,” Mechanics of Advanced Materials and Structures, vol. 13, sharply. no. 3, pp. 267–284, 2006. [11] O. J. Aldraihem and A. Baz, “Moving-loads-induced instability 4. Conclusion in stepped tubes,” Journal of Vibration and Control, vol. 10, no. 1, pp. 3–23, 2004. A cylindrical shell is actively controlled by using piezoelectric [12] O. J. Aldareirhem and A. Baz, “Dynamic stability of periodic stacks for attenuating the radial circumferential vibrations stepped beams under moving loads,” Journal of Sound and due to moving pressure propelling mass. Vibration, vol. 250, no. 2, pp. 835–848, 2002. Amplitude (V) Amplitude Advances in Acoustics and Vibration 7 [13] R. N. Miles and P. G. Reinhall, “An analytical model for the vibration of laminated beams including the effects of both shear and thickness deformation in the adhesive layer,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 108, pp. 56–64, 1986. [14] E. L. Kathe, “MATLAB modeling of non-uniform beams using the finite element method for dynamic design and analysis,” Tech. Rep. ARCCB-TR-960 10, US Amy Armaments Research, Development and Engineering Center; Close Combat Arma- ments Center, Benet ` Laboratories, Watervliet, NY, USA. [15] M. J. Lam, W. R. Saunders, and D. J. Inman, “Modeling active constrained layer damping using Golla-Haughes-McTavish approach,” in Smart Structures and Materials 1995: Passive Damping, vol. 2445 of Proceedings of SPIE, pp. 86–97, San Diego, Calif, USA, March 1995. [16] K. M. S. Eldalil and A. M. S. Baz, “Dynamic stability of cylindrical shells under moving loads by applying advanced controlling techniques Part 1-Using Periodic Stiffeners,” in Advanced Materials for Application in Acoustics and Vibration, Cairo, Egypt, January 2009. [17] G. E. Martin, “Vibrations of coaxially segmented longi- tudinally polarized ferroelectric tubes,” The Journal of the Acoustical Society of America, vol. 36, no. 8, pp. 1496–1506, [18] G. E. Martin, “On the theory of segmented electromechanical systems,” The Journal of the Acoustical Society of America, vol. 36, no. 7, pp. 1366–1370. [19] S. Sherrit, S. P. Leary, Y. Bar-Cohen, B. P. Dolgin, and R. Tasker, “Analysis of the impedance resonance of piezoelectric stacks,” in Proceedings of the IEEE Ultrasonics Symposium, vol. 2, pp. 1037–1040, San Juan, Puerto Rico, October 2002. [20] D. E. Seborg, T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, John Wiley & Sons, New York, NY, USA, 1989. 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Dynamic Stability of Cylindrical Shells under Moving Loads by Applying Advanced Controlling Techniques—Part II: Using Piezo-Stack Control

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Copyright © 2009 Khaled M. Saadeldin Eldalil and Amr M. S. Baz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 927078, 7 pages doi:10.1155/2009/927078 Research Article Dynamic Stability of Cylindrical Shells under Moving Loads by Applying Advanced Controlling Techniques—Part II: Using Piezo-Stack Control 1 2 Khaled M. Saadeldin Eldalil and Amr M. S. Baz Department of Mechanical Engineering, Tanta University, Sperbay, Tanta, Egypt Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA Correspondence should be addressed to Khaled M. Saadeldin Eldalil, eldalil01@msn.com Received 2 April 2009; Accepted 6 July 2009 Recommended by Mohammad Tawfik The load acting on the actively controlled cylindrical shell under a transient pressure pulse propelling a moving mass (gun case) has been experimentally studied. The concept of using piezoelectric stack and stiffener combination is utilized for damping the tube wall radial and circumferential deforming vibrations, in the correct meeting location timing of the moving mass. The experiment was carried out by using the same stiffened shell tube of the experimental 14 mm gun tube facility which is used in part 1. Using single and double stacks is tried at two pressure levels of low-speed modes, which have response frequencies adapted with the used piezoelectric stacks characteristics. The maximum active damping ratio is occurred at high-pressure level. The radial circumferential strains are measured by using high-frequency strain gage system in phase with laser beam detection system similar to which used in part 1. Time resolved strain measurements of the wall response were obtained, and both precursor and transverse hoop strains have been resolved. A complete comparison had been made between the effect of active controlled and stepped structure cases, which indicate a significant attenuation ratio especially at higher operating pressures. Copyright © 2009 K. M. S. Eldalil and A. M. S. Baz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The integration of viscoelastic damping materials into various structures has been widely theoretical studied. Euler- Bernoulli beam theory has been used to model viscoelas- The production of long slender gun systems to meet increased exit velocity requirements of rounds has subse- tic materials using the Rayleigh Ritz approximation for quently increased the effect of precision and accuracy of damping. Timoshenko beam theory incorporates both shear firing as well as barrel/round interactions during firing. A deformation and rotational deformation in the formulation. lightweight, low-cost method is desired to damp the induced The modal strain energy method [2]makes useofthe vibrations thereby increasing performance of the gun system. relationship between damping factors and modal loss factors Some experimental and theoretical methods are sug- in order to assign damping factors to real elastic modes gested for damping and controlling vibrations in the gun assign approximation to light damping. The Golla-Hughes- tubes. A relatively inexpensive and lightweight method of McTavish (GHM) method [3, 4] uses a finite element damping vibrations in some structures is to apply a surface approach where viscoelastic damping is introduced as a series treatment of a viscoelastic material and a constraining layer, of minioscillator terms and auxiliary dissipation coordinates. as a passive control combination [1], which deals with DiTaranto and Blasingame [5] and Mead and Markus [6] the transverse vibrations in the system resulted in shear derived a sixth-order PDE to model the transverse vibrations deformation of the viscoelastic material, which in turn of a three-layer beam system based on the equations dissipates the energy. This technique is very effective for developed for flexural vibrations of layered plates. In this damping terrain and firing-induced vibrations and hence approach damping of the viscoelastic layer is incorporated increases precision and accuracy. through the use of a complex shear modulus. 2 Advances in Acoustics and Vibration Table 1: The effective material properties for the data of the stack 1 2 n resonator. Property Real Imaginary Z Z Z Z Z Z 1 1 1 1 1 1 2 −11 −13 s (m /N) 2.00 × 10 −3.0 × 10 T −11 −10 ε (F/m) 3.00 × 10 −6.17 × 10 −10 −12 d (C/N) 3.925 × 10 −7.66 × 10 Z Z 2 2 k 0.5 −0.001 Martin’s general solution for the admittance of a piezo- N C N N C C 0 0 electric stack of area A, n layers, and total length nL is [19] nγ 2N Y (ω) = iωnC + tanh ,(1) Z 2 Figure 1: Equivalent circuit representation of a stack with the ST mechanical ports of each equivalent circuit representing a layer where connected in series and n electrical ports in parallel. ε A 33 2 C = 1 − k , 0 33 Ro et al. [7] and Sung [8] developed this technique the- Ad N = , oretically and experimentally by applying active constrained Ls damping layer (ACLD); they modeled the tube/ACLD system 0.5 by using Golla-Hughes-McTavish method in order to predict 1 Z = Z Z 2+ , ST 1 2 the tube response in the time domain. They calculated (2) the transient response using finite element method; the 0.5 predicted response is compared with that of a tube controlled γ = 2 arcsin h , 2Z with passive constrained layer damping treatment. The pre- 2 dictions of the finite element model are validated experimen- ωL tally. The results obtained indicted that ACLD treatment has Z = iρν A tan , 2ν achieved significant attenuation of the structure vibrations. D 2 On the other hand, the methods for damping radial ρν A iN Z = + , and circumferential vibrations of the shell tube walls are i sin(ωL/ν ) ωC completely different. Using periodic stiffeners distributed on the shell tube surface is tried theoretically by Ruzzene and baz D D and ν = 1/ ρs is the acoustic velocity at constant [9, 10], Aldareihem and Baz [11, 12], and others [13–15], and T D electric displacement. The constants ε , s , d are the 33 33 studied experimentally in part 1, Eldalil and Baz [16]. free permittivity, the elastic compliance at constant elec- In this work, the concept of applying a feedforward tric displacement, and the piezoelectric charge coefficient, control by using piezoelectric stack is experimentally tried. respectively. Using (1)to(2), Martin demonstrated that in To the best of our knowledge, this technique is not examined the limit of large n (n> 8), the acoustic wave speed in the before for damping the radial and circumferential deforma- material was determined by the constant field elastic constant tions of the shell walls theoretically or experimentally. The E E s (1/ ρs ). In the limit of n> 8 an analytical equation 33 33 selection of the piezoelectric stacks is based on its dynamic for the admittance was presented which allowed for direct characteristics which are meeting the shell walls radial and determination of material constants from the admittance circumferential deformation dynamic modes. A feedforward data [3]. In this limit the admittance was shown to be control system is suggested and designed in order to achieve the optimum attenuation ratio and satisfy suitable dynamic iAnω (k ) ω 2 33 stability. Y = 1 − (k ) + tan ,(3) ω/4 f 4 f s s 1.1. Piezoelectric Stack Characteristics. Piezoelectric stacks where the series resonance frequency is are used in variety applications that require relatively high force and larger displacement than single element piezo- 1 1 f = . (4) electric transducers can produce. These include microposi- 2πL ρs tioning systems, solid-state pumps/switches, noise isolation The effective material properties are shown in Table 1,for mounts, ultrasonic drills, and stacked ultrasonic transducers. the data of the stack resonator shown in Figure 2. The stack The solution for the zero bond length stacks was derived by Martin [17, 18]. His model was derived from Mason’s length nL = 0.02 m, Area A = (0.01) m , and density equivalent circuit of n layers connected mechanically in series ρ = 7800 kg/m . and electrically in parallel as shown in Figure 1. Formoredetails, referto[19]. Advances in Acoustics and Vibration 3 1e 1 D E 1e 2 _ n = 10 1e 3 1e 4 Figure 3: Basic mechanisms of regulation, from left to right: _ buffering, feedforward, and feedback block diagrams. 1e 5 1e n = 1 d measured disturbance Feedforward control 1e 7 ff 1e 8 032 64 98 130 162 194 226 158 Process Frequency (kHz) Y(s) output de Figure 2: Typical conductance as a function of frequency for n = 1 to 10 layers. Figure 4: Traditional feedforward control structure. 2. Feedforward Control 2.1. Concept of Feedforward Control. Feed-forward control, To eliminate the effect of the measured disturbance, we need which is used herein, will suppress the disturbance defor- only choose q so that: ff mations before it has had the chance to affect the system’s essential variables. This requires the capacity to anticipate P − Pq = 0, d ff the effect of perturbations on the system’s goal. Otherwise (6) ∼−1 ∼ q = P P , the system would not know which external fluctuations to ff consider as perturbations, or how to effectively compensate where a ∼ over a process transfer function indicates that it their influence before it affects the system. This requires that is a model of the process. Even in the above case, where the control system is able to gather early information about the feedforward controller can perfectly compensate the these fluctuations. Figure 3 shows block diagram of typical measured disturbance, Seborg et al. [20]. feedforward control system; the effect of disturbances D on the essential variables E is reduced by an active regulator R. Feedforward control can entirely eliminate the effect 3. Experimental Setup of the measured disturbance on the process output. Even when there are modeling errors, feed-forward control can The experiments were carried out with in-door Pneumatic often reduce the effect of the measured disturbance on 6/12-feet helium Gun Facility in the Vibration and Noise the output better than that achievable by feedback control Control Laboratory at University of Maryland; it has a alone. However, the decision as to whether or not to use stainless steel tube length of 6 feet. The detailed description feedforward control depends on whether the degree of is found in part 1, Eldalil and Baz [16]. improvement in the response to the measured disturbance justifies the added costs of implementation and maintenance. 3.1. Piezoelectric Stack Configuration. The piezoelectric stack The economic benefits of feedforward control can come and shell tube combination is shown in Figure 5. As indicated from lower operating costs and/or increased salability of the in part 1 [16], the stiffeners are composed of three parts, two product due to its more consistent quality. half rings (2, 3), in Figure 5, and one outer ring, surrounding the shell tube (1); consequently, in active control, the outer 2.2. Mathematical Formulation. Figure 4 shows a traditional ring is removed and the stacks (4, 7) are rested on the feedforward control scheme. The transfer function between upper and lower half rings. A stainless steel clamp strip (6) the process output Y and the measured disturbance d as is used to hold the components fixed in position, and the shown in Figure 4 is screw (5) is used to create initial tension in the clamp strip and consequently generates compression force distributed around the shell tube substituting the removed outer ring of Y (s) = P − Pq d. (5) d ff the stiffener. Conductance G(S) 4 Advances in Acoustics and Vibration Tension screw Strip clamp Upper piezoelectric stack Upper half ring Gun tube Lower half ring Figure 7: Piezoelectric stack control view picture. Lower piezoelectric stack S.S. 130 cal, 1600psi, plain (loc.2) 1.5 Figure 5: Double Piezoelectric stack configuration. 0.5 165 mm −0.5 Strain gage no. 2 145 mm Piezo-foil-sensor −1 0.102 0.103 0.104 0.105 0.106 0.107 Shell tube Bullet direction Time (s) s.g.2 Piezoelectric actuators Ph. shift Amplifier (a) s.s130 cal,1600psi, w177ss, one ring Figure 6: Piezoelectric stacks/feedforward control scheme. 1.5 0.5 3.2. Feedforward Control Elements Representation. The con- trol circuit scheme is shown in Figure 6, of type feedforward; −0.5 it is composed of piezoelectric foil pressure sensor located −1 at distance ×1 (165 mm) upstream of the strain gage of 0.104 0.105 0.106 0.107 0.108 0.109 location number 2, and at distance ×2 (145 mm) upstream Time (s) the piezoelectric stack actuator. The output signals are phase s.g.2 w17ss,1ring shifted and amplified, then feed the piezoelectric stacks, which are located at distance ×3 (20 mm) upstream the (b) strain gage of location number 2. The sensor has a time s.g.2w17ss, active.2piezo advance of Δt1 (0.391 milliseconds) and the system (phase 1.5 shifter, amplifier, and piezoelectric stacks) has a time lag of Δt2 (0.372 milliseconds), so for ideal operating case the two 0.5 times should be equal, or Δt2 < Δt1byfew percent. By this way the actuators will have sensible impact on the tube −0.5 vibrations; the phase can be adjusted by using phase shifter. −1 0.111 0.112 0.113 0.114 0.115 0.116 The piezoelectric stack control setup view picture is shown Time (s) in Figure 7. s.g.2w177ss, act.2piezo 3.3. Experimental Measurements and Results. The measure- (c) mentsare carried outattwo operating helium pressures, Figure 8: (a) Output signal at location number 2 of plain tube at 1600 and 2000 psi, and the controller (piezo-stack) is 1600 psi. (b) Output signal at location number 2 of tube with 17 installed very close to location number 2 on the gun tube, as single stiffeners. (c) Output signal of location number 2 with 17 indicated in part 1 [16]. The expected deforming vibration single stiffeners and with active control at 1600 psi. due to these pressures ranges between 7.2 and 25 kHz, Part 1 [16], that is, far enough from the first mode frequency of the piezoelectric stack (64 kHz), as shown in Figure 2. off, which is equivalent to the stiffened tube; is shown in Figure 8(b), and the active controlled time domain signals is 3.3.1. Measurements at Pressure Level of 1600 psi. The time shown in Figure 8(c). A time domain comparison of active resolved measurement of plain tube is shown in Figure 8(a), and stiffened tube, is shown in Figure 9, and the comparison and that of the arrangement when the control is turned analysis of the frequency domains is shown in Figure 10. Strain (V) Strain (V) Strain (V) Advances in Acoustics and Vibration 5 Comparison of active and stiffened tube, loc. no.2, 1600psi S.S 130 cal. 2000psi, plain 1.5 1.5 0.5 0.5 0 −0.5 −1 −0.5 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 Time (s) −1 s.g.2 0.104 0.105 0.106 0.107 0.108 0.109 0.11 piez.2 Time (s) piez.3 Stiffeners (a) Active S.S.130 cal, 2000psi, s.g.2, w17ss Figure 9: Time domain comparison between active and stiffened 1.5 tube at location no. 2 at operating pressure 1600 psi. 0.5 S.S 130 cal-1600psi, w17ss, active-2piezostacks (subcritical) 0.07 0.06 −0.5 −1 0.05 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1 Time (t) 0.04 s.g.2 piezo3 0.03 piezo2 Laser (b) 0.02 Active and with 17ss, s.g.2 1.5 0.01 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ×10 Frequency (Hz) −0.5 −1 Plain 17ss −1.5 17ss and active 0.088 0.089 0.09 0.091 0.092 0.093 0.094 0.095 Time (t) Figure 10: Frequency domain comparison between plain, stiffened tube, and with active piezo-stack control. 2bullet,s.g.2 (c) The attenuation ratio due to using active piezoelectric Figure 11: (a) Output signal at location number 2 of plain tube at 2000 psi. (b) Output signal at location number 2 of tube with 17 stack is attained to 33%, at vibration mode of 7.2 kHz and single stiffeners at 2000 psi. (c) Output signal of location number 2 48% at vibration mode of 9.5 kHz for pressure level of with 17 single stiffeners and with active control at 2000 psi. 1600 psi. The effect of active control is decreased at higher vibration modes. 3.3.2. Measurements at Pressure Level of 2000 psi. The time sensor and amplifier output signals (piezo-stqack loading domain measurement of the plain tube at pressure level of signal or the control effort) is shown in Figure 13,at 2000 psi is shown in Figure 11(a) and that of the stiffened amplifier gain factor of 2. tube with piezo-stack arrangement without control is shown The radial strain vibration is attenuated by using the in Figure 11(b), and the controlled time resolved signal is active piezoelectric control by a ratio of 46% at vibration shown in Figure 11(c). The frequency domain comparison is mode of 7.2 kHz and 65% at 9.5 kHz. The attenuation ratio is shown in Figure 12. A comparison of the piezoelectric foil increased at higher pressure level of 2000 psi than at pressure Amplitude Amplitude (V) Strain (V) Strain (V) Strain (V) 6 Advances in Acoustics and Vibration S.S 130 cal-1600psi, w17ss, active-2piezostacks (subcritical) A feedforward control scheme is designed and con- 0.07 structed by using piezoelectric foil sensor to predict in advance the incoming dynamics vibration and piez-stacks 0.06 actuators. The control system is checked at two pressures levels; 0.05 first pressure level is 1600 psi and the second pressure level is 2000 psi. The attenuation ratio is predicted at two modes 0.04 of low-frequency vibrations. At first mode of vibration, 7.2 kHz, the attenuation ratio 0.03 is found to be about 33% and 46% at pressure levels of 0.02 1600 psi and 2000 psi, respectively. And at second mode of vibration, 9.5 kHz, the attenu- 0.01 ation ratio is found to be about 48% and 65% at pressure levels of 1600 psi and 2000 psi, respectively. At higher modes of vibrations the attenuation effect decreases to lower ratios. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ×10 The frequency is shifted to lower values by about 13%. Frequency (Hz) The frequency domain comparison for the two pressure Plain levels indicates that the stability is satisfied and no spill over 17ss occurred. 17ss and active Figure 12: Frequency domain comparison between plain, stiffened References tube, and with active control at 2000 psi. [1] M. Z. Kiehl and C. P. T. W. Jerzak, “Modeling of passive constrained layer damping as applied to a gun tube,” Shock and Comparison between sensing signal Vibration, vol. 8, no. 3-4, pp. 123–129, 2001. and control signal [2] C. D. Johnson and D. A. 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Reinhall, “An analytical model for the vibration of laminated beams including the effects of both shear and thickness deformation in the adhesive layer,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 108, pp. 56–64, 1986. [14] E. L. Kathe, “MATLAB modeling of non-uniform beams using the finite element method for dynamic design and analysis,” Tech. Rep. ARCCB-TR-960 10, US Amy Armaments Research, Development and Engineering Center; Close Combat Arma- ments Center, Benet ` Laboratories, Watervliet, NY, USA. [15] M. J. Lam, W. R. Saunders, and D. J. Inman, “Modeling active constrained layer damping using Golla-Haughes-McTavish approach,” in Smart Structures and Materials 1995: Passive Damping, vol. 2445 of Proceedings of SPIE, pp. 86–97, San Diego, Calif, USA, March 1995. [16] K. M. S. Eldalil and A. M. S. Baz, “Dynamic stability of cylindrical shells under moving loads by applying advanced controlling techniques Part 1-Using Periodic Stiffeners,” in Advanced Materials for Application in Acoustics and Vibration, Cairo, Egypt, January 2009. [17] G. E. Martin, “Vibrations of coaxially segmented longi- tudinally polarized ferroelectric tubes,” The Journal of the Acoustical Society of America, vol. 36, no. 8, pp. 1496–1506, [18] G. E. Martin, “On the theory of segmented electromechanical systems,” The Journal of the Acoustical Society of America, vol. 36, no. 7, pp. 1366–1370. [19] S. Sherrit, S. P. Leary, Y. Bar-Cohen, B. P. Dolgin, and R. Tasker, “Analysis of the impedance resonance of piezoelectric stacks,” in Proceedings of the IEEE Ultrasonics Symposium, vol. 2, pp. 1037–1040, San Juan, Puerto Rico, October 2002. [20] D. E. Seborg, T. F. Edgar, and D. A. Mellichamp, Process Dynamics and Control, John Wiley & Sons, New York, NY, USA, 1989. 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