A free vibration test is conducted on an empty elevated water tank. A cable attached to the tankapplies a lateral (horizontal) force of 72 kN and pulls the tank horizontally by 50 mm. The cable issuddenly cut, and the resulting free vibration is recorded. At the end of five complete cycles, thetime is 2.0 s and the amplitude is 25 mm.(a) From these data compute the following: (i) Damping ratio
Question
A free vibration test is conducted on an empty elevated water tank. A cable attached to the tankapplies a lateral (horizontal) force of 72 kN and pulls the tank horizontally by 50 mm. The cable issuddenly cut, and the resulting free vibration is recorded. At the end of five complete cycles, thetime is 2.0 s and the amplitude is 25 mm.(a) From these data compute the following: (i) Damping ratio
Solution
To calculate the damping ratio, we first need to find the logarithmic decrement. The logarithmic decrement (δ) is a measure of the rate of decay of the amplitude of the free vibration. It can be calculated using the formula:
δ = 1/n * ln(A1/A2)
where:
- n is the number of cycles (5 in this case)
- A1 is the initial amplitude (50 mm in this case)
- A2 is the final amplitude after n cycles (25 mm in this case)
Substituting the given values into the formula, we get:
δ = 1/5 * ln(50/25) = 0.18
Next, we calculate the damping ratio (ζ) using the formula:
ζ = δ / (2πsqrt(1 - δ^2))
Substituting the calculated value of δ into this formula, we get:
ζ = 0.18 / (2πsqrt(1 - 0.18^2)) = 0.028
So, the damping ratio is 0.028.
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