A shaft of length 0.95 m supported simply at the ends, carrying a body of mass 84 kg at 0.38 m from one end. Determine the natural frequency of transverse vibration of the shaft. Assume E= 200 GN/m2 and shaft diameter = 65 mm.
To determine the natural frequency of transverse vibration of the shaft, we can use the formula for the natural frequency of a simply supported beam:
f = (1 / (2π)) * √((k / m))
Where:
- f is the natural frequency of vibration,
- k is the stiffness of the beam, and
- m is the mass of the body attached to the beam.
To find the stiffness of the beam, we can use the formula for the stiffness of a circular shaft:
k = (π^2 * E * I) / (L^3)
Where:
- E is the Young's modulus of the material,
- I is the area moment of inertia of the shaft, and
- L is the length of the shaft.
First, let's calculate the area moment of inertia of the shaft:
I = (π * D^4) / 64
Where D is the diameter of the shaft.
Given that the shaft diameter is 65 mm, we need to convert it to meters:
D = 65 / 1000 = 0.065 m
Now we can calculate the area moment of inertia:
I = (π * (0.065^4)) / 64
Next, we can calculate the stiffness of the beam using the formula:
k = (π^2 * E * I) / (L^3)
Given that the Young's modulus is E = 200 GN/m^2 and the length of the shaft is L = 0.95 m, we can substitute them into the formula:
k = (π^2 * 200 * 10^9 * I) / (0.95^3)
Now we have the stiffness of the beam, k.
Finally, we can calculate the natural frequency of vibration using the formula:
f = (1 / (2π)) * √((k / m))
Given that the mass of the body attached to the beam is m = 84 kg, and the body is located at a distance of 0.38 m from one end, we can substitute the values into the formula:
f = (1 / (2π)) * √(k / m)
Now we have the natural frequency of transverse vibration of the shaft.