Rotating blade (body force in axial loading)
A blade is fixed to a rigid rotor of radius R spinning at omega(w) rad/sec around the vertical z axis, neglect the effect of gravity,
calculate the peak stress in the blade
calculate the blade elongation.
calculate the displacement of the blade mid section.
To calculate the peak stress in the blade, the blade can be considered as a rotating cantilever beam under axial loading. The stress is caused by the centrifugal force acting on the blade.
The centrifugal force (F_c) acting on an elementary mass element (dm) within the blade can be expressed as:
F_c = dm * R * ω²
where R is the radius of the rotor, ω is the angular velocity (w) in rad/sec, and dm is the mass per unit length of the blade.
The stress (σ) in the blade can be calculated by dividing the centrifugal force by the cross-sectional area (A) of the blade:
σ = F_c / A
To calculate the blade elongation, we can use the axial strain formula.
The axial strain (ε) in the blade can be obtained by dividing the change in length (ΔL) by the original length (L) of the blade:
ε = ΔL / L
The axial strain is related to the stress through Hooke's Law:
ε = σ / E
where E is the Young's modulus of the blade material.
Therefore, the blade elongation (ΔL) can be calculated as:
ΔL = σ * L / E
To calculate the displacement of the blade midsection, we can use the equation for the deflection of a cantilever beam under axial loading.
The deflection (δ) of the blade midsection can be calculated using the following formula:
δ = (σ * L³) / (3 * E * I)
where I is the moment of inertia of the blade cross-section.
The moment of inertia (I) can be calculated based on the geometry of the blade cross-section.
Please note that in order to calculate the actual values for these parameters, you will need to know the specific properties of the blade material, such as the mass per unit length (dm), cross-sectional area (A), Young's modulus (E), and the moment of inertia (I) of the blade.
To calculate the peak stress in the blade, we need to consider the centrifugal force acting on the rotating blade due to its mass. The centrifugal force will cause axial loading on the blade.
Step 1: Calculate the centrifugal force acting on the blade.
The centrifugal force can be calculated using the equation:
Fc = m * R * ω²
where Fc is the centrifugal force, m is the mass of the blade, R is the radius of the rotor, and ω is the angular velocity.
Step 2: Calculate the stress in the blade.
The stress in the blade can be calculated using the equation:
σ = Fc / A
where σ is the stress, Fc is the centrifugal force, and A is the cross-sectional area of the blade.
Step 3: Calculate the blade elongation.
The elongation of the blade can be calculated using the equation for axial deformation:
δ = (σ * L) / (E * A)
where δ is the elongation, σ is the stress, L is the length of the blade, E is the Young's modulus of the material, and A is the cross-sectional area of the blade.
Step 4: Calculate the displacement of the blade mid-section.
The displacement of the blade mid-section can be calculated using the equation for axial displacement:
Δz = (δ * L) / 2
where Δz is the displacement, δ is the elongation, and L is the length of the blade.
Please provide the values for the mass of the blade, radius of the rotor, angular velocity, length of the blade, cross-sectional area of the blade, and Young's modulus of the material to proceed with the calculations.