Rotating blade (body force in axial loading)
blade is fixed to a rigid rotor of radius R spinning at ω rad/sec around the vertical z-axis. Neglect the effects of gravity.
Calculate the peak stress in the blade: σmaxn
Calculate the blade elongation: δ
Calculate the displacement of the blade mid-section: ux(L/2)
1. rho*omega^2/2*(L^2 + 2*R*L)
2. (rho*omega^2)/(2*E)*(2/3*L^3 + R*L^2)
3. (rho*omega^2)/(2*E)*(11/24*L^3 + 3/4*R*L^2)
Simply obtained by using definitions and proper integrations.
To calculate the peak stress in the blade (σmaxn), the blade can be considered as a cantilever beam under axial loading due to rotation. The equation for the stress in a rotating cantilever beam can be given as:
σ = ρ * ω^2 * r
where:
ρ is the density of the material,
ω is the angular velocity of the rotor,
r is the radial distance from the axis of rotation.
In this case, the blade is fixed to a rigid rotor of radius R spinning at ω rad/sec around the vertical z-axis. Since the blade is fixed to the rotor, the radial distance r is equal to the radius R. Assuming the density of the material is known, you can substitute these values into the equation to calculate the peak stress σmaxn.
To calculate the blade elongation (δ), we can use Hooke's law, which states that the elongation of a material is directly proportional to the applied stress and the material's elastic modulus (E). The formula for elongation δ is given by:
δ = (σmaxn * L) / (E * A)
where:
L is the length of the blade,
E is the elastic modulus of the material,
A is the cross-sectional area of the blade.
In this case, the applied stress σmaxn is the peak stress calculated earlier, and the length of the blade L can be obtained from the given information. The elastic modulus E and the cross-sectional area A should be known or provided.
To calculate the displacement of the blade mid-section (ux(L/2)), we can use the formula for displacement in a cantilever beam under axial loading:
ux(x) = (σmaxn * x^2) / (2 * E * I)
where:
ux(x) is the displacement of the beam at a distance x from the fixed end,
I is the moment of inertia of the beam's cross-section.
In this case, x = L/2 corresponds to the mid-section of the blade, and the applied stress σmaxn is the peak stress calculated earlier. The elastic modulus E and the moment of inertia I should be known or provided.
By substituting the known values into the equations, you can calculate the peak stress in the blade (σmaxn), the blade elongation (δ), and the displacement of the blade mid-section (ux(L/2)).