Molecular ad Quatum Acoustics vol. 7, (6) 79 ANALYSIS OF DAMPING EFFECT ON BEAM VIBRATION Jerzy FILIPIAK 1, Lech SOLARZ, Korad ZUBKO 1 Istitute of Electroic ad Cotrol Systems, Techical Uiversity of Czestochowa, 17 Armii Krajowej Str., - Czestochowa, Polad Departmet of Civil Egieerig, Military Uiversity of Techology, Kaliskiego Str., -98 Warsaw, Polad, lechsolarz@iteria.pl Istitute of Applied Physics, Military Uiversity of Techology, Kaliskiego Str.,-98 Warsaw, Polad, kzubko@wat.edu.pl The paper presets aalysis of dampig effect o beam vibratio forced by impact. The air-related value of dampig is researched. All the cosideratios are based o the assumptio that the particular mout does ot produce dampig. Furthermore, they are true oly for the researched rage of the variability of the beam s geometry ad the resultat resoace frequecies. Two applied methods of modellig produce results, which differ by oe order of magitude, but both of them are sufficietly small to be regarded as egligible. Keywords: acceleratio sesor, acceleratio meters, air dampig, vibratio dampig 1. INTRODUCTION The catilever beams are sigificat elemets of the SAW (Surface Acoustic Wave) acceleratio sesors.
8 Filipiak J., Solarz L., Zubko K. Rys.1. SAW acceleratio sesors produced i Physical Acoustic Research Uit of MUT (Warsaw) They are made of aisotropic materials, such as quartz. I the calculatios [1], the effective material costats are used. Rys.. Quartz plates with SAW delay lies ad resoators. These plates are mouted i the SAW acceleratio sesors see i Fig. 1. The measured resoace frequecies ad vibratio dampig decremets eable computatio of the costats. To aalyze the vibratios of the catilever beam i the frequecy domai below the secod eigefrequecy, the followig approximatio ca be used, e.g.:
Molecular ad Quatum Acoustics vol. 7, (6) 81 Krylov model, where the Youg modulus ad the vibratio dampig decremet are described by the followig formula: ( ωτ ) ωl l E = 1ρ 1 + (1.1) k h ωτ δ = πτω 1+ (1.) where: ω - measured frequecy of damped vibratios ρ - desity of the medium ad l - legth of the beam b - width of the beam h - height of the beam m - aggregated mass τ - dampig characteristic (relaxatio) time k = 1,875,9689m 1+ ρbhl Raleigh model, where the Youg modulus is described by the formulae: (1.) ( ωτ ) ωl E =,966ρ 1 + (1.) h These two approaches produce similar results of dampig characteristic time (τ ) ad Youg modulus (E) calculatios for specific cuts, whereas the Raleigh method ivolves a lesser workload. O the basis of the measured parameters, the chages i the eigefrequecy ca be modelled, takig ito accout chages of the beam geometry or the aggregated mass at the ed of the beam. This eables the descriptio of dyamics of the catilever beams i meters or sesors, which are usually made of mechaically aisotropic materials. The experimets described i [] have demostrated a small impact of dampig o the beam s vibratios isofar as the measured parameters are cocered. Experimets were performed for the beams with the followig dimesios: width: -5 mm, height:.5-1 mm, legth -7 mm. The sesors operatig frequecy is up to some khz. Exemplary deflectios, velocities ad acceleratios of the sesor plate ed are see i table 1.
8 Filipiak J., Solarz L., Zubko K. Table 1. Frequecy [ Hz] Deflectio [ mm] Velocity [m/s] Acceleratio [m/s ] 1 1 1 1,1 1 5 We use simple models of iteractio betwee air ad plates because it is well grouded for the rage of velocity metioed above. We try to discuss a model of the dumpig effect caused by air surroudig beam see i the Fig.. y x aggregated mass - m h - height b - width w(x,t) l- legth casig Fig.. The schematic view of catilever beam.. DAMPING EFFECT OF WAVES RADIATED INTO THE AIR..1. EIGENFUNCTIONS We kow the eigefuctios (ormal, modal fuctios) of the beam cylidrically bet. The eigefuctios W ( X ) are solutios to the boudary problem for the equatio: W ( X ) k W ( X ) = '''', X l x = (.1) where k is liked with the -th circular frequecy of free vibratios, which are ot damped by the relatio: ω h l = k, l a ( 1 ) 1 ν E a =. (.) ρ where a is for the velocity of the logitudial wave i the rod made of the same material.
Molecular ad Quatum Acoustics vol. 7, (6) 8 We are lookig for the W (x) fuctios, which fulfil the boudary coditios: dw for x = W =, =, (.) d x for d W d W x = l = ; =, (.) d x d x The eigefuctios (.1), (.), (.) form the orthogoal, ormalized set. 1 W k ( X ) Wl ( X ) dx = δ k l The eigefuctios are described by the formula: ( k ) ( k ). (.5) S W ( X ) = F U ( k X ) V ( k X ) T k =1. 1 875 (.6) where S( ), T( ), U( ), V( ) are Krylov fuctios [].. FORMULATION OF THE SIMPLIFIED MATHEMATICAL PROBLEM OF AIRWAVES DAMPING EFFECT. We examie a flat problem, i.e. =. The equatio describig the vibratios of the beam y with air ifluece take ito accout has the form: Fuctio ( x t ) w E h b w ρ hb + + b p( x, z =, t ) = (.7) t 1 x ( 1 ν ) w, fulfils the boudary coditios equivalet to (.), (.). I the gas, acoustic pheomea occur which ca be described usig the liear wave equatio: φ z ( x, z, t ) φ ( x, z, t ) 1 φ ( x, z, t ) + x a t = z > (.8) Where a = ad p( x, z, t ) = ρ κ p ( x, z, t ) φ ρ (.9) t The kiematical compatibility coditio has the form: ( x, z =, t ) w( x t ) φ, = z t. (.1)
8 Filipiak J., Solarz L., Zubko K.. SOLUTION TO THE SIMPLIFIED PROBLEM We seek the potetial φ fulfillig the coditio of emissio or decreasig. We will try to solve a simplified problem: with istallig ot mutually affectig layers of gas which ca be portrayed as pistos with walls havig a zero thickess, placed directly above the beam. As a result, i equatio (.8) the part φ ( x, z, t ) x is omitted. We are lookig for a solutio, which is a sum with regard to the eigefuctios of the udamped system: w x N N ( x, t ) S ( t ) W, ( x, z, t ) φ ( z, t ) W = = 1 l From the simplified equatio (.8), we get: ( z, t ) = Φ t a x φ =. (.11) = 1 l z φ. (.1) Cosiderig the kiematical coditio (.1), we get: ( t ) = a S ( t ) Φ, (.1) therefore, the pressure is equal to: p z x = ρ. (.1) = 1 a l N ( x, z, t ) a S t W Whe we take ito accout (.1) ad (.5) i (.8), we get ( t ) + χ S ( t ) + ω S ( t ) = S (.15) where χ is equal to ρ a = (.16) ρ h χ i additio, ω is determied by (.). Whe we defie the logarithmic decremet as a atural logarithm from the ratio betwee the amplitude of the former half-period to the amplitude of the curret half-period we wil get χ ρ a = χ Th π = π 1 1 ω ρ a l h ( ν ) k δ (.17) For small dampig values, the followig approximatio is possible:
Molecular ad Quatum Acoustics vol. 7, (6) 85 τ δ ρa (.18) πω ρhω air = δ ρ a l τ air = (.18a) πω k Eh. VISCOUS INTERACTION MODELLING. The air iteractio is modelled by force distributed over the plate. The force is proportioal to the velocity of the plate i the poit x. The costat of proportioality depeds o the coefficiet of air viscosity η 1 ad parameters of the plate. Fig. Qualitative view of air-dampig force distributio o the plate The iteral dampig is described by Voight model []. The deductio of the equivalet coefficiet of dampig by the Rayleigh method [] gives us the formula E b d 1l 1l τ + it.56η1 (.1). E b h c eq =. Costitutive relatio ε σ = E ε + τ it E. (.) t defies material costat τ it. Total characteristic dampig time τ tot is the sum of material costat τ it ad the part caused by viscous iteractio of the air τ air
86 Filipiak J., Solarz L., Zubko K. τ = τ it + (.) tot τ air Where.96η l Eh 1 τ air = (.) b. RESULTS OF AIR DAMPING MODELLING If we do ot apply the simplificatio (.18), we will oly be able to obtai umerical solutios. Below (Table ), we preset two cases of modellig of a quartz beam based o (.18). Table. Case I Data: E=7 GPa, ρ=65 kg/m, ν =., χ = 1., ρ = 1.5 kg/m, p =11.15 kpa, h=.8 mm, l=5 mm, Calculated: k 1 =1.8751, a=58.5 m/s, a =6.5 m/s Results: Resoace frequecy f =7.7 Hz Period T=.611e- s Logarithmic decremet of dampig δ=.71669 Case II Data: E=7 GPa, ρ=65 kg/m, ν =., χ = 1., ρ = 1.5 kg/m, p =11.15 kpa, h=5. mm, l=5 mm, Calculated: k 1 =1.8751, a=58.5 m/s, a = 6.5 m/s Results: Resoace frequecy f =17Hz Period T=5.7781e- s Logarithmic decremet of dampig δ=.186 The quartz beams differ i thickess (h): the first (I) is.8 mm thick, ad the secod (II) 5 mm. Their legths (l) are the same. I the above-metioed modellig, dampig is characterized by logarithmic decremet δ. This parameter is defied i [] e.g. However, the compariso of dampig betwee these two cases ca be more coveietly doe based o the parameterτ. Its relatio with the parameter δ is show below: δ δ τ = 1 (.1) πω π
Molecular ad Quatum Acoustics vol. 7, (6) 87 Now we ca compare the results of the dampig modellig for the above examples usig the relatio (.18) ad (.). The compariso is show i the Table below. The experimets [], [5] coducted for the beam with a thickess of.8 mm allow us to determie aggregated dampig time (material, moutig, air) τ = µs. Table. Parameters of the beam Air dampig (.18) Air dampig (.) Height [mm].8 5..8 5. Frequecy [Hz] 7.7 17 7.7 17 τ [s] 1.6 8. 5. CONCLUSIONS The modellig cofirmed a miimal ifluece of the air o the researched beams dampig time. The air-related values of dampig time are idetified usig the method described i paper. These values are several times smaller tha aggregated dampig. A sigificat chage i the thickess of the beam does ot affect i a measurable way the air-related volume of dampig characteristic time. Two methods of modellig produce results, which differ by oe order of magitude but both of them, are sufficietly small to be regarded as egligible. This is cosistet with the results of the experimets described i [], [5] where it was assumed that the impact of the beam s dampig related to the air is egligible. It is ot possible to cofirm the value of the dampig experimetally, with the apparatus available to the authors. The theoretical cosideratios are based o the assumptio that the particular mout does ot produce dampig. Furthermore, they are true for the researched rage of the variability of the beam s geometry ad the resultat resoace frequecies. REFERENCES 1. J. Filipiak, L. Solarz, K. Zubko, Aalysis of Acceleratio Sesor by the discrete model Molecular & Quatum Acoustics 5, 89-99, ().. J. Filipiak, K. Zubko, Determiatio of dampig i piezoelectric crystals, Molecular&Quatum Acoustics 6, 75-8, (5).. S. Kaliski, ad all, Vibratios ad Waves, PWN Elsevier, NY, Amsterdam, Warsaw 199.. Z. Dżygadło, S. Kaliski, L. Solarz, E. Włodarczyk, Vibratio ad Waves i Solids, Library of Applied Mechaics, PWN, Warsaw, 1966 (i Polish)
88 Filipiak J., Solarz L., Zubko K. 5. K. Zubko, Determiatio of elastic ad viscoelastic parameters of piezoelectric crystals by Rayleigh method, PhD thesis (i Polish), Military Uiversity of Techology, Warsaw 6.