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The measurements on a mechanical vibrating system show that it has a mass of 10 kg and that the springs can be combined to give an equivalent stiffness of the springs 6 N/mm. The dashpot is attached to the system which exerts a force of 40 N when the mass has a velocity of 1 m/s
Q. Determine the logarithmic decrement
  • a)
     1.03
  • b)
     0.515
  • c)
     0.258
  • d)
     0.772
Correct answer is option 'B'. Can you explain this answer?
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The measurements on a mechanical vibrating system show that it has a m...
Logarithmic decrement 
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The measurements on a mechanical vibrating system show that it has a m...
Given:
Mass of the system (m) = 10 kg
Equivalent stiffness of the springs (k) = 6 N/mm = 6 * 10^6 N/m
Force exerted by the dashpot (F) = 40 N
Velocity of the mass (v) = 1 m/s

To find:
Logarithmic decrement

Solution:
The equation of motion for the mechanical vibrating system is given by:
m * d^2x/dt^2 + c * dx/dt + k * x = 0

Where,
m = mass of the system
c = damping coefficient (dashpot)
k = stiffness of the springs
x = displacement of the mass

Step 1: Calculate the damping coefficient (c)
The damping coefficient (c) can be calculated using the force exerted by the dashpot and the velocity of the mass:
F = c * v
c = F / v
c = 40 N / 1 m/s
c = 40 Ns/m

Step 2: Calculate the natural frequency (ωn)
The natural frequency (ωn) can be calculated using the formula:
ωn = sqrt(k / m)
ωn = sqrt(6 * 10^6 N/m / 10 kg)
ωn = sqrt(6 * 10^5 rad/s)

Step 3: Calculate the logarithmic decrement (δ)
The logarithmic decrement (δ) can be calculated using the formula:
δ = (1 / N) * ln(x1 / x2)

Where,
N = number of cycles
x1 = initial displacement
x2 = displacement after N cycles

In this case, since the initial displacement and displacement after N cycles are not given, we can assume x1 = 1 m and x2 = 0.5 m (half the initial displacement).

Step 4: Calculate the number of cycles (N)
The number of cycles (N) can be calculated using the formula:
N = ln(x1 / x2) / (2 * π * δ)

Substituting the values:
N = ln(1 m / 0.5 m) / (2 * π * δ)
N = ln(2) / (2 * π * δ)

Step 5: Calculate the logarithmic decrement (δ)
Substituting the value of N in the logarithmic decrement formula, we get:
δ = (1 / N) * ln(x1 / x2)
δ = (1 / (ln(2) / (2 * π * δ))) * ln(1 m / 0.5 m)
δ = (2 * π * δ / ln(2)) * ln(2)
δ = 2 * π * δ

Simplifying the equation:
δ^2 - 2 * π * δ = 0
δ * (δ - 2 * π) = 0

Since δ cannot be zero, we have:
δ - 2 * π = 0
δ = 2 * π
δ = 6.28

Step 6: Convert the logarithmic decrement to the natural logarithm (ln) scale
The logarithmic decrement (δ) can be converted to the natural
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The measurements on a mechanical vibrating system show that it has a mass of 10 kg and that the springs can be combined to give an equivalent stiffness of the springs 6 N/mm. The dashpot is attached to the system which exerts a force of 40 N when the mass has a velocity of 1 m/sQ. Determine the logarithmic decrementa)1.03b)0.515c)0.258d)0.772Correct answer is option 'B'. Can you explain this answer?
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