DETERMINATION OF NATURAL FREQUENCY AND DAMPING RATIO

Transcription

1 Hasa G Pasha DETERMINATION OF NATURAL FREQUENCY AND DAMPING RATIO OBJECTIVE Deterie the atural frequecy ad dapig ratio for a aluiu catilever bea, Calculate the aalytical value of the atural frequecy ad copare with the experietal value APPARATUS 1. Test rig. Frequecy aalyzer. Fuctio/Wavefor geerator THEORY ON VIBRATION Mechaical Vibratio Mechaical Vibratio is defied as the otio of a syste (a particle or a body) which oscillates about its stable equilibriu positio. Mechaical Vibratio geerally results whe a syste is displaced fro a positio of stable equilibriu. The syste teds to retur to its equilibriu positio by virtue of restorig forces. However the syste geerally reaches its origial positio with certai acquired velocity that carries it beyod that positio. Ideally this otio ca repeat idefiitely. Free Vibratio Whe the vibratio otio is aitaied by the restorig forces oly, the vibratio is tered as free vibratio. Natural frequecy Natural frequecy is defied as the lowest iheret rate (cycles per secod or radias per secod) of free vibratio of a vibratig syste. Its uit is or rad s -1 ad it is desigated by ω. Dapig Dapig is dissipatio of eergy i a oscillatig syste. It liits aplitude at resoace. All vibratig syste are daped to soe degree by frictio forces. These forces ca be caused by dry frictio or Coulob frictio, betwee rigid bodies, by fluid frictio whe a rigid body oves i a fluid, or by iteral frictio betwee the olecules of a seeigly elastic body. Viscous dapig ad Coefficiet of viscous dapig Viscous dapig is caused by fluid frictio at low ad oderate speeds. It is characterized by the fact that the frictio force is directly proportioal ad opposite to the velocity of the ovig body. The agitude of the frictio force exerted o the pluger by the surroudig fluid is equal to c x. Where c is kow as the coefficiet of viscous dapig expressed i N s/. It depeds o the physical properties of the fluid ad depeds o the costructio of the dashpot. Critical dapig coefficiet Assuig that the otio of the syste is defied by the followig differetial equatio: 1

2 Hasa G Pasha x + cx + kx 0 The otio is tered as critically daped whe the coefficiet of viscous dapig equals ω ad it is desigated by c c. Dapig ratio Dapig ratio is defied as the ratio of the coefficiet of viscous dapig to critical dapig coefficiet. It is desigated by ζ. Measureet of dapig ratio experietally - Logarithic Decreet A coveiet way to easure the aout of dapig preset i a syste is to easure the rate of decay of free oscillatios. The larger the dapig, the greater is the rate of decay. Rate of decay of the oscillatio Cosiderig a daped vibratio expressed by the geeral equatio: ςω t x Xe si( 1 ς ωt + φ) Logarithic decreet ca be defied as the atural logarith of the ratio of ay two successive aplitudes. x 1 x δ l l x x 1 0 δ ςω τ ς d

3 Hasa G Pasha SCHEMATIC DIAGRAM DESCRIPTION The test rig cosists of a rectagular cross-sectio, aluiu catilever bea. The free ed of the bea is coected with a ferroagetic disc with a pickup device below it. This echais serves to trace the vibratio. It is based o the laws of electro-agetic iductio. Whe the bea vibrates, the gap size chages ad this causes the flux desity to vary which is calibrated ad read fro a volteter ad also fed to a oscilloscope. The gap betwee the ferroagetic disc ad the pickup device is adjusted such that it is ot less tha 5 ties the expected aplitude of vibratio. TABULATION Bea Legth 50 Sl No No of cycles Scaled Aplitude Iitial Fial Scaled Tie Iitial Fial Natural Frequecy Logarithic decreet Dapig ratio

4 Hasa G Pasha Bea Legth 450 Sl No No of cycles Scaled Aplitude Iitial Fial Scaled Tie Iitial Fial Natural Frequecy Logarithic decreet Dapig ratio Copariso of stadard values with experietal values Bea legth Natural Frequecy Sl No Relative error % Stadard Experietal PROCEDURE 1. Set the bea legth to 50. Excite the aluiu catilever bea. Record the output wave 4. Observe ad tabulate the scaled iitial ad fial values (of a set of 10 successive oscillatios) of the aplitude ad tie period 5. Repeat steps through 4 for a bea legth of Calculate the atural frequecy ad dapig ratio 7. Calculate the stadard value of atural frequecy ad copare it with the experietal values FORMULAE δ x 1 1 x 0 l l x x ς δ ω τ π ς d ς 4

5 Hasa G Pasha δ f τ d 1000 τ τ fial iitial f ω.5 EI l I bh 4 1 b h ρ kg -1 δ Logarithic decreet X 0 Aplitude of the first cycle M x Aplitude of the th cycle M N Nuber of cycles ζ Dapig ratio τ d Daped vibratio tie period S ω Natural frequecy rad s -1 f Natural frequecy E Modulus of Elasticity/Youg s odulus Pa I Moet of area about cetral axis parallel to width B Breadth of the bea -4 M H Thickess of the bea ρ Desity of the bea M kg kg - for Aluiu 5

6 Hasa G Pasha SAMPLE CALCULATION δ 1 x l 0 x l ς δ f τ d 1000 τ τ fial iitial 10 * I bh 1 (0.076) (0.0061) x

7 Hasa G Pasha b h ρ ( 0.076)(0.0061)(700) kg -1 f ω.5 l EI.5 (0.5) (7x ) ( x 10 (1.517) -9 ) SOURCES OF ERROR The error calculated by coparig the experietal value of the atural frequecy with the stadard value is as a result of the fact that ay vibratio is daped to soe extet. I this case the Coulob dapig caused due to air was eglected. Error ca also be attributed to the fact that the aterial i the catilever ight ot be uiforly distributed i the aterial cotiuu as assued. RESULT The atural frequecy ad dapig ratio for the aluiu catilever bea were foud experietally. The results are tabulated below: Bea legth Natural Frequecy Dapig Ratio The stadard value of the atural frequecy was calculated ad copared to the experietal value. The % of relative error was calculated as 1.10 % ad 7.91 % bea legths of 0.5 ad