LINEAR VIBRATION APPERATUS

DAMPED OSCILATION OF A LIQUID COLUMN

INSTRUCTED BY: Mr. L.A.U.S. Liyanarachchi

NAME: L.N. UdayangaINDEX NO: 090529KCOURSE: B.Sc Engineering (ME)

THEORY

Equilibrium level position

Effective mass and stiffness of the liquid columnm = Al k = 2Ag

Assuming the damping to be viscous- The equation of motion can be written as, ... mx + cx + kx = 0As to the standard format + 2 + 2x = 0, where = {k/m}1/2 and = c/2*{mk}1/2 Logarithmic increments of amplitudes of oscillation and period of oscillation ln [ ar / ar+1 ] = 2/{1-2}1/2T = 2/ {1-2}1/2A Area of cross section of tube c Damping coefficient Damping ratio Un-damped natural frequency k Effective stiffness gAcceleration due to gravity ar Oscillation amplitudes T Period of damped oscillation x Displacement l Length of liquid columnm Mass of liquid column Density of the liquid

Length (l) [cm]Consecutive (uninterrupted Amplitudes) [cm]Period (4T) [s]

arar+1ar+2ar+3ar+4

236.713.78.25.74.33.59.06

12.77.14.4439.19

13.785.24.53.69.27

23016.59.56.34.53.58.93

16.5106.7548.77

16.29.86.75.13.78.86

22018.610.475.13.98.61

17.7107.25.148.78

18.910.57548.61

21019.410.46.44.13.58.49

20.110.76.74.63.58.43

20116.74.63.68.42

20021.5117.253.68.27

21.611.274.93.58.32

21.11175.23.68.22

CALCULATIONS

PI3.1415927

Circumference [cm]12.5

radius0.0198944

area0.0012434

ln(ar)ln(ar+1)ln(ar+2)ln(ar+3)ln(ar+4)mBetaBeta (Avg)Period (Avg)

2.6172.1041.7401.4591.253-0.3370.054

2.5421.9601.4821.3861.099-0.3460.055

2.6172.0791.6491.5041.281-0.3250.0520.0532.293

2.8032.2511.8411.5041.253-0.3850.061

2.8032.3031.9021.6091.386-0.3530.056

2.7852.2821.9021.6291.308-0.3610.0570.0582.213

2.9232.3421.9461.6291.361-0.3840.061

2.8742.3031.9741.6291.386-0.3650.058

2.9392.3511.9461.6091.386-0.3850.0610.0602.167

2.9652.3421.8561.4111.253-0.4360.069

3.0012.3701.9021.5261.253-0.4340.069

2.9962.3981.9021.5261.281-0.4300.0680.0692.112

3.0682.3981.9741.6091.281-0.4360.069

3.0732.4161.9461.5891.253-0.4470.071

3.0492.3981.9461.6491.281-0.4290.0680.0692.068

Average0.062

l [cm]Beta (Avg)Period (Avg)Omega (damped)Omega square (damped)1/l [m-1]Omega square1/betasqrt(l)

236.7000.0532.2932.7447.5280.4227.50618.7211.539

230.0000.0582.2132.8448.0860.4358.05917.1931.517

220.0000.0602.1672.9058.4400.4558.41016.6631.483

210.0000.0692.1122.9838.8950.4768.85314.5371.449

200.0000.0692.0683.0469.2800.5009.23614.4071.414

2g (damped) {m}21.555166

g (damped)10.78

2g {m}21.246221

g10.62

2Ap*(2g)^0.5/c {m}35.263753

c (practical)0.312

Graphs

m = -0.36565

m= -0.37783

m=-0.43313

m= -0.43714

2g=m= 21.74937 g= 10.87468

m=10.97355

DiscussionOscillations occur whenever a force exists that pushes an object back towards a stable equilibrium position whenever it is displaced from it.Such forces abound in nature - things are held together in structured formbecausethey are in stable equilibrium positions and when they are disturbed in certain ways, they oscillate.When the displacement from equilibrium issmall, the restoring force is oftenlinearlyrelated to the displacement, at least to a good approximation. In that case the oscillations take on a special character - they are calledharmonicoscillations as they are described by harmonic functions.The damping ratio, , is adimensionlessmeasure describing how oscillations in a system decay after a disturbance. Many systems show oscillatory behavior when they are disturbed from their position ofstatic equilibrium. A liquid column at rest in a U tube is disturbed by displacing its equilibrium position , will traverse up and down from the initial equilibrium position. On each oscillation, the system is trying to return to its equilibrium position, but surpasses it.Frictionallossesdampthe system and cause the oscillations to gradually decay in amplitude towards zero. The damping ratio is a measure of describing how rapidly the oscillations decay from one to the next.

Oscillations we encounter in nature are almost always damped. It means that the body involved in the oscillation is undergoing a force which is opposed to its velocity. We can identify 3 types of damped oscillations;

1. Critically damped ( = 1)

The system returns to equilibrium as quickly as possible without oscillating

2. Over damped (>1)

The system returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio() ,return to equilibrium slower.

3. Under damped (0