Introduction

Damping is restraining of vibratory motion. We have heard the word damping in the subjects like oscillations, noise, alternating current, RLC circuits etc. Even though we had no idea at that time, when we were playing with the swing its motion fizzled out automatically. It’s because of the damping.

Vibrations have desirable effects and adverse effects. Example for desirable effect is music. But for most of the engineering applications vibrations are undesirable. So damping out vibrations is a desirable effect in engineering applications.

When we are inside an automobile, since roads are not in the same conditions everywhere, we with the automobile body undergo mechanical vibrations. So we should feel some degree of uncomfortable. But in reality we do not. Why? What is done there to reduce the vibrations and prevent uncomfortable?

Let’s take another example. We know that Japan is highly vulnerable for earthquakes. Most of them are slight earthquakes. During an earthquake ground is induced forced vibrations. Then how can they maintain tall buildings without fail? What do they have in there? After doing this practical we’ll try to find out the answers for these.

Notations:

ar - Amplitude of the rth oscillation

T - Period of damped oscillation

x - Displacement from equilibrium position

l - Length of liquid column

m - Mass of liquid column

ρ - Density of the liquid

A - Area of the cross section of the tube

c - Damping coefficient

β - Damping ratio

ω - Un -damped natural frequency

k - Effective stiffness

g - Acceleration due to gravity

Procedure

Length of the u tube water column was set to 250 cm by measuring the height of the liquid column by a meter ruler.

Then we put our mouth to the one end of the u tube and blow air to oscillate the liquid column.

Five consecutive amplitudes and corresponding time is measured. For one length practical was repeated 3 times. Some amount of water was removed from the tab underneath to set the next length

of the liquid column. Repeat the practical for 5 lengths.

Graph of ln(ar) vs r for l = 250cm

0 1 2 3 4 5 62

2.2

2.4

2.6

2.8

3

3.2

r

ln(ar)

Graph of ln(ar) vs r for l = 240cm

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.52

2.2

2.4

2.6

2.8

3

3.2

r

ln(ar)

Graph of ln(ar) vs r for l = 230cm

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.52

2.2

2.4

2.6

2.8

3

3.2

3.4

r

ln(ar)

Graph of ln(ar) vs r for l = 220cm

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.52

2.2

2.4

2.6

2.8

3

3.2

3.4

r

ln(ar)

Graph of ln(ar) vs r for l = 210cm

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.52

2.2

2.4

2.6

2.8

3

3.2

r

ln(ar)

Table for graph ln(ar) vs r for l = 250cm

r ln(ar)

1 2.95

2 2.88

3 2.68

4 2.47

5 2.32

Gradient = M = 3.12−2

7 = 0.17

Table for graph ln(ar) vs r for l = 240cm

r ln(ar)

1 3.12

2 2.91

3 2.72

4 2.51

5 2.32

Gradient = 3.32−2

6.6 = 0.2

Table for graph ln(ar) vs r for l = 230cm

r ln(ar)

1 3.18

2 2.94

3 2.71

4 2.55

5 2.36

Gradient = 3.36−2

6.7 = 0.20

Table for graph ln(ar) vs r for l = 220cm

r ln(ar)

1 3.15

2 2.87

3 2.66

4 2.43

5 2.31

Gradient = 3.32−2

6.3 = 0.21

Table for graph ln(ar) vs r for l = 210cm

r ln(ar)

1 3.12

2 2.88

3 2.67

4 2.47

5 2.36

Gradient = 3.28−2

6.6 = 0.19

Using the gradients (m) we can find,

m= ln (ar )−ln (ar+1)

(r+1 )−r

= ln (ar )−ln (ar+1)

But we have,

ln ( arar+1)= 2πβ

√1−β2

Therefore,

m = 2πβ

√1−β2

β = m

√m2+4π 2

m β

0.16 0.0250.20 0.0320.20 0.0320.21 0.0330.19 0.030

Period of oscillation:

T= 2 π

ω√1−β2

ω= 2π

T √1−β2

β l 1/l T ω2

0.025 250 0.40 2.235 7.9001580.032 240 0.42 2.205 8.1198320.032 230 0.43 2.148333 8.5538360.033 220 0.45 2.070833 9.2066620.030 210 0.48 2.030833 9.571098

l β ω m c

250 0.030 2.810722 3.365 0.57240 0.030 2.849532 3.23 0.55230 0.030 2.924694 3.09 0.54220 0.030 3.034248 2.96 0.53210 0.030 3.093719 2.83 0.52

C = 2β ωmwe assumed that beta as the average of 5 values = 0.03

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

sqrt(l)

c

Table for ω2 vs 1/l graph

β l 1/l T ω2

0.025 250 0.40 2.235 7.9001580.032 240 0.42 2.205 8.1198320.032 230 0.43 2.148333 8.5538360.033 220 0.45 2.070833 9.2066620.030 210 0.48 2.030833 9.571098

Gradient=9.5711−7.9000.48−0.40

= 20.88

Gravitational acceleration = 20.88/2 = 10.44 ms-2

ω2 vs 1/l graph

0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.490

2

4

6

8

10

12

1/l

Discussion

About the practical

Here I’m going to discuss the importance of this experiment. Here we induced vibratory motion to a liquid column. What we could see from our results is the amplitude of the vibration is gradually decreased and fizzles out with the time. Why? Because it acts like a damper. Because of the viscosity of the fluid an energy loss occurs converting the kinetic energy of vibratory motion to other type of energy such as heat. By mathematically modeling this, we could see that the gradient of ln(ar) vs r graph showing this decreasing amplitude. But at what time it decays down faster? That’s what’s important to us as engineers. Because we need to reduce vibrations as soon as possible, to reduce undesirable effects like fatigue. The gradient of the ln(ar) vs r graph is what matters here. Higher the gradient, lesser the time to decrease the vibration. So it’s obvious that the length of the liquid column also matters. And also un-damped natural frequency is inversely proportional to the length of the liquid column. But the damping coefficient is proportional to the square root of the length of the liquid column. So we have a compromise of the length of the liquid column. And at the end result we could see that damping coefficient is independent of the length of the liquid column.

Application Aspect

As I stated in the introduction what happens in the automobile is there is a damper (viscous damper) usually called shock absorber. This is how it reduces uncomfortable for the passengers travelling inside.

And let’s look at the Japan example. How do they safeguard tall buildings?

The need of mitigate wind, ocean wave and earthquake induced vibrations in structures like tall buildings, long span bridges, offshore platforms has led the interest of damping devices. The impact dampers are a very useful way to suppress unwanted high amplitude vibrations in small scale systems. But for large structures it’s difficult. Because advancement of the technology allow tall buildings and other structures to be built from light flexible material, with relatively low intrinsic damping. Vibrations are increased because the stiffness of the building is decreased. Traditionally we increased the stiffness by adding more mass. But it’s not cost effective. So If we don’t do something for it those buildings are highly vulnerable to fail.

The response for this is to dissipate the vibration energy by auxiliary damping devices. A tuned liquid column damper (TLCD) is a special type of dampers relying on a liquid in a u tube like container to counteract the forces acting on the structure. This is how tall buildings in Japan are secured.