a uniform cantilever beam of the total mass ml has a concentrated mass M at its free end. determine the effective mass of the beam to be added to M assuming the deflection to be that of a massless beam with a concentrated force at the end, and write the equation for its fundamental frequency

To determine the effective mass of the beam to be added to the concentrated mass M, we first need to understand the concept of the effective mass. In structural engineering, the effective mass refers to the additional mass that could be added to a structure to represent the dynamic behavior of the system accurately.

In this case, we have a uniform cantilever beam with a total mass ml, and a concentrated mass M at its free end. The deflection of the beam is assumed to be the same as that of a massless beam with a concentrated force at the end.

To find the effective mass, we can use the concept of an equivalent system. We need to consider the deflection caused by the concentrated mass M at the free end of the beam and find the additional mass that would result in the same deflection when it is concentrated at the fixed end of the beam.

The equation for the fundamental frequency of a cantilever beam is given by:

f = (1/2π) * √(EI / (ρA*L^4))

Where:
f = Fundamental frequency
E = Modulus of elasticity of the material
I = Moment of inertia of the beam cross-section
ρ = Density of the material
A = Cross-sectional area of the beam
L = Length of the beam

However, since we are assuming the deflection of a massless beam with a concentrated force at the end, the equation for the effective mass can be simplified.

The deflection of a cantilever beam with a concentrated force at the end is given by:
δ = (F * L^3) / (3 * E * I)

Where:
δ = Deflection
F = Concentrated force at the end of the beam

We assume this deflection is equal to the deflection caused by the added effective mass.

Now, let's find the effective mass of the beam. We want to find the additional mass that, when multiplied by the gravitational acceleration (g), will result in the same deflection as the concentrated force M.

Hence, we can write:
(Effective mass) * g = (M * L^3) / (3 * E * I)

To find the equation for the fundamental frequency, we need the effective mass in terms of the concentrated mass M. Rearranging the equation above:

Effective mass = (M * L^3) / (3 * g * E * I)

Substituting this value for the effective mass into the equation for the fundamental frequency gives us the desired equation.

f = (1/2π) * √(EI / (ρA*L^4 + (M * L^3) / (3 * g * E * I)))

In this equation, we have taken into account the additional effective mass (M * L^3) / (3 * g * E * I) that represents the dynamic behavior of the system accurately.

To determine the effective mass of the cantilever beam, we need to consider the equivalent mass that produces the same deflection as a massless beam with a concentrated force at the end.

The equation for the effective mass (meff) of the cantilever beam can be calculated using the following formula:

meff = ml + [(3/4) × (M × L^2 /Lc^2)]

Where:
- ml is the total mass of the cantilever beam
- M is the mass at the free end of the beam
- L is the length of the beam
- Lc is the deflection of the beam

The fundamental frequency (f) of the cantilever beam can be determined using the equation:

f = (1/2π) × √(k / meff)

Where:
- k is the stiffness of the beam

Please note that the calculation of stiffness requires additional information about the beam's material properties and geometry.