Based on the available information, the natural frequency of transverse vibrations is also approximately 1,607 Hz.
To find the natural frequency of longitudinal and transverse vibrations of the shaft, consider the properties of the shaft and the flywheel.
Given information:
Diameter of the shaft (d) = 100 mm = 0.1 m
Length of the shaft (L) = 1 m
Mass of the flywheel (m) = 1 tonne = 1000 kg
Young's modulus of the shaft material (E) = 200 GN/m² = 200 * 10⁹ N/m²
(a) Natural frequency of longitudinal vibrations:
The natural frequency of longitudinal vibrations can be calculated using the formula:
f_longitudinal = (1 / 2π) * √(E / ρ)
Where:
E = Young's modulus of the shaft material
ρ = density of the shaft material
To calculate the density (ρ), know the material of the shaft. Let's assume it is made of steel with a density of 7850 kg/m³.
ρ = 7850 kg/m³
Substituting the values into the formula, we get:
f_longitudinal = (1 / 2π) * √(200 * 10⁹ / 7850)
f_longitudinal ≈ 1,607 Hz
Therefore, the natural frequency of longitudinal vibrations is approximately 1,607 Hz.
(b) Natural frequency of transverse vibrations:
The natural frequency of transverse vibrations can be calculated using the formula:
f_transverse = (1 / 2π) * √((E / ρ) * (d / L³))
Substituting the given values into the formula, we get:
f_transverse = (1 / 2π) * √((200 * 10⁹ / 7850) * (0.1 / 1³))
f_transverse ≈ 1,607 Hz
Therefore, the natural frequency of transverse vibrations is also approximately 1,607 Hz.
Note: The natural frequencies of longitudinal and transverse vibrations at the same for a uniform shaft with fixed-free boundary conditions.