DAMPING LOSS FACTOR ESTIMATION IN PLATES

Transcription

1 Molecular and Quantum Acoustics vol. 4, (003) 61 DAMPING LOSS FACTOR ESTIMATION IN PLATES Marek IWANIEC The Group of Structural Acoustics and Smart Materials, AGH University of Science and Technology Al. Mickiewicza 30, Cracow, POLAND The paper aimed at experimental determining of Damping Loss Factor coefficient (DLF) in plate elements. Author presented an alternative method of structure exciting by the use of point force applied to a sample through a simple mechanical clutch. Keywords: Damping Loss Factor, experimental modal analysis 1. INTRODUCTION Vibrating structures accumulate the energy of vibration in the form of kinematical energy (stored by mass) and potential energy (stored by elasticity). Each real, vibrating structure dissipates and radiates energy. Internal dissipation leads to conversion of mechanical energy into thermal energy. There are as many damping mechanisms [3] as the ways of transforming the mechanical energy into heat. Mechanisms of dissipation vary with respect to surface friction, liquid viscosity, turbulence, acoustical radiation, rotational flows, magnetic and mechanical hysteresis. While analyzing a damping vibration it is easy to notice that one ore two dissipation mechanisms overweigh and reduce the effects of the others.. DAMPING IN PLATE ELEMENTS If an acoustical pressure ( ) ( ) p0 x, y, t = p x, y cos ω t acts on a plate then a load of this plate can be expressed as a function of a plate mode shape Ψ (x,y): (, ) = (, ) p x y P Ψ x y where: m and n describe a mode shape, P are constants: m, n= 0

2 6 Iwaniec M. P p( x, y) Ψ (, ) A = Ψ ( x, y) da A x y da where: A - plate surface. Velocity distribution ( ) ( ) superposition of mode participation factors: where: V m n is constant. V0 x, y, t = V x, y cosω t on a plate surface is obtained through a (, ) = (, ) υ x y V Ψ x y m, n= 0 Considering transverse vibrations of a homogenous plate [1,] it can be stated that a modal velocity V m n of a resonant mode depends on modal pressure P m n : V P = η µ ω where: η - plate modal damping loss factor (depending on mode m, n); µ = ρ h p - plate mass for a surface unit [kg/m ]; ρ p - plate material density [kg/m 3 ]; h plate thickness [m]; ω m.n - resonant frequency for a mode m, n [rad / s]; Rectangular plate simply supported at the edge, of dimensions a and b defined in directions of coordinates x and y (0 < x < a, 0 < y < b), under homogenous pressure of a distribution (, ) =, has a following mode shape: Ψ = sin ( mπ x a) sin ( nπ y b) p x y p c Then P = 16 p 0 c π if m and n are odd otherwise Natural frequency ω m. n equals: ω π = clh m + 1 a n b The maximal intensity in mode m, n resonance (where m and n are odd) is described by the equation: σ 3EP max ρ η ω = 96 pc 4 cl ph π η h m + a (p c - homogenous effective sound pressure above a plate). n b

3 Molecular and Quantum Acoustics vol. 4, (003) 63 Damping of propagating waves can be analyzed for a free propagating elastic rectilinear { } ( ω ) bending waves for which: υ ( x y t ) υ e j t k b,, = Re x 0 ( k b = π λ b - wave number) If the damping effect is introduced by the complex Young modulus: E = E ( + j ) equation: kbη x 4 ( x, y, t ) = e cos( t k x) υ υ ω 0 b 1 η then the is known as an equation of wave propagation, of an amplitude decreasing exponentially with respect to distance. If a propagation path is increased by a wave length λ amplitude will change by a e η π / and level decay on a wave length will equal: ( λ ) ( η π = log e ) = 10( log e) π η 13. η λ = 6 db (η - loss factor). = π k b then the Free vibration decay (described by t ) is proportional to a loss factor and a middle band frequency (or a mode natural frequency): ( db / sec ) = 7. 3η f t n 3. MEASUREMENT METHODS A rectangular plate is a multimodal system its response can be treated as a superposition of responses of many single degree of freedom systems. For low frequencies near a basic resonance it can be assumed that a single-frequency harmonic excitation results in appearance of one vibration mode only. In higher frequencies each excitation results in appearance of many modes that have an influence on each other. In such a case measurements of resonant amplitude, bandwidth or logarithmic damping decrement are of no use. Decrease method consisting in a reverberation time measurement in convenient frequency bands is much more useful. A plate is excited by a band of a white noise; excitation is interrupted and the slope of a decaying signal envelope is determined on the basis of a decay coefficient t. A loss factor η is calculated on the basis of t and a middle frequency of an excitation excitation band: D π E pot 7,3 f t η = = ( 7 ) max where: D dissipated energy. While carrying out damping measurement it is necessary to eliminate energy loss in constrains as scrupulously as possible. Therefore tested plates are usually hung at long strings in order to create boundary conditions similar to entirely free. Suspension point lies at a plate symmetry axis and should correspond with a node of many modes. Accelerometer should be light so as not to deform system response and an electric filter should have an adequate n

4 64 Iwaniec M. amplitude frequency characteristics. Harmonic and modal analysis are the most frequently used measurement methods. In order to minimize the influence of excitation type on the accuracy of measurements the acoustic wave excitation is applied. However, there is a fundamental disadvantage of the use of such an excitation: it is difficult to excite structure vibrations in the low frequency band. 4. MEASUREMENT SET UP FOR INTERNAL DAMPING LOSS FACTOR MEASUREMENTS WITH THE USE OF A CLUTCH The research aimed at experimental determination of Damping Loss Factor coefficient (DLF) in plate elements. The experiment was carried out in the Laboratory of Structural Acoustics and Biomedical Engineering at AGH. Plate elements were suspended from a carrying construction by the use of elastic strands (Fig. 1). The signal of excitation in a form of a white noise, generated by the use of the computer software, through the amplifier and an inverse filter was passed to a typical magneto-dynamic inductive shaker. The output force was transmitted to a tested element through a special connector disengaging clutch. Elimination of the shaker influence on the plate vibration during the measurement of vibration decay time stood for the essence of the experiment. Damping Loss Factor was investigated in the frequency third-octave bands from 100 Hz to 4000 Hz. Fig. 1. Measurement set up. A test stand for DLF measurement (Fig. 1) was consisted of a piezoelectric accelerometer of the PD-1 type, a low pass filter of a khz cut off frequency and a multi-channel measurement cart PROSIG connected to a PC computer. The circuit of vibration excitation

5 Molecular and Quantum Acoustics vol. 4, (003) 65 consisted of a programmable random noise generator, voltage amplifier, inverse filter correcting exciter inductive characteristic, power amplifier, exciter and a glass connector serving as a clutch. PC computer inductive exciter coupling PROSIG Transduce r Fig.. Block scheme of a measurement circuit. When the plate was excited the connector was being broken. A signal obtained from a piezoelectric accelerometer was amplified and filtered and then written in the operational memory and, when the experiment was over, rewritten to a hard disc file and subjected to further processing. In the Fig. 3 there is presented an example of a sample velocity level spectrum and in the Fig. 4 vibration decay characteristic obtained during an experiment is shown. 150 db /3 OCTAVE No. 1 Hz 16 Hz 15 Hz 1 khz 8 khz Total Fig. Cursor: f[0] = 15 Hz A = 119. db 3. Example of a spectrum obtained during measurements. 150 db Fig. 4. Vibration level decay in time. TIME INTERSECTION for 1/3 oct.: 15 Hz Cursor: t[1] = 0.5 s A = 119. db s

6 66 Iwaniec M. materials. In the Table 1 there are presented the parameters of tested samples made of different Table 1 Physical parameters of tested plates. Material Length Width Thickness Property St 3 steel ρ=780 kg/m 3, E=, Pa, ν=0,3 Aluminium ρ=,7 g/m 3, E=0, MPa, ν=0,3 Brass ρ=8,3 g/m 3, E= MPa, ν=0,3 Plexiglas Plates were suspended from a truss, carrying construction of a high stiffness in two points located 0 [mm] below the plate upper edge. 5. RESULTS OF DAMPING LOSS FACTOR MEASUREMENTS The results of damping loss factor measurements were compared to the results of measurements carried out for the same elements in the anechoic room with the use of an acoustic field excitation generated by the system of broad-band loudspeakers. In the Fig. 9 there are shown the DLF values measured in third-octave bands obtained with the use of the current method (presented in detail in this paper). 0, ,0050 0,0000 0, , ,00050 steel brass plexiglass aluminium 0, frequency [Hz] Fig. 5. Measured values of DLF

7 Molecular and Quantum Acoustics vol. 4, (003) 67 While comparing the results of damping loss factor measurements and the results of previous measurements in which the samples were excited by the use of an acoustic field it can be noticed that general trend and character of damping loss factor frequency characteristics remained unchanged. Yet the relative error with reference to measurements with the use of acoustic field depicted in the Fig. 10 indicates significant differences of measured values relative error, % steel brass plexiglass frequency Hz aluminium Fig. 6. Relative error of DLF measurements. The differences between measurement methods in which the structure was excited by an acoustic field and mechanical force applied through a clutch are significant and even approach to 100 %. The results of both experiments are closest in the low frequency bands for samples made of steel and brass, in the present experiment the materials of the smallest inside damping. The most significant errors were observed for a sample made of plexiglass the material of the highest internal damping. Interpretation of uncertainness of obtained results and determination of quantitive influence of individual factors on measurement accuracy require more detailed research. 6. CONCLUSIONS While carrying out DLF measurements or other dynamic analyses it is extremely important to choose a proper way of structure exciting and a place of the exciting force application. Usually the structure is driven to movements by the use of modal hammer, magneto-dynamic shaker, piezoelectric glued elements and, under certain circumstances, by

8 68 Iwaniec M. the use of an acoustic wave. In the paper the author presented an alternative method of structure exciting by the use of point force applied through a simple mechanical clutch. In the proposed method disadvantages of other DLF coefficient measurement methods were eliminated: - shaker mass has no influence on the measurement results, - the influence of an aerodynamic damping of shaker elements is eliminated, - damping resistance that results from the electromotoric force being induced in a shaker coil is also eliminated, - the main advantage is the easiness of obtaining high levels of vibration amplitude. The bigger difference of measured levels corresponds to the longer vibration decay time, which allows for more accurate determination of a slope of vibration decay characteristic, - in the proposed method, in contrary to the method with an acoustic excitation, the problem of sample exciting after turning the loudspeakers off (by the energy stored in the acoustic cavity around loudspeakers) do not appear, - presented measurement technique is economical and very simple in comparison to other types of disengaging clutches. The proposed way of structure exciting can also be applied to parametric modal analysis and structure dynamic analysis. REFERENCES 1. A. Cacracciolo, C. Valette, Damping mechanisms governing plate vibration Acta acustica 3, pp , (1995).. T. Okada, J. Fujimoto, Radiation noise reduction by applying vibration damping material, Inter noise 94, Yokohama, (1994). 3. Z. Engel, Ochrona środowiska przed drganiami i hałasem Wydawnictwo Naukowe PWN, Warszawa J. Giergiel, Tłumienie drgań mechanicznych. AGH, Kraków Heckl M., Muller H.A.: Taschenbuch der technischen Akustik. Springer-Verlag, F. F.Yap, J. Woodhouse, Investigation of Damping Effects on Statistical Energy Analysis of Coupled Structures, J. of Sound and Vibration, 197 (3), p , (1996).