Deriving Potential Energy from a Force Law

In summary, when a block is dropped from a height onto the end of a cantilever beam, the maximum deflection is proportional to the static deflection, with a factor of 1 + (1+2h/y)^1/2. The potential energy stored in the beam should also be considered in the calculation, similar to a spring. The potential energy can be derived by considering the force law and using F dot dx.
  • #1
AngelofMusic
58
0
When an elastic beam AB supports a block of weight W at a given point B, the deflection [tex]y_s[/tex] (static deflection) is proportional to W. Show that if the same block is dropped form a height h onto the end B of a cantilever beam, the maximum deflection [tex]y_m = y_s(1 + (1+\frac{2h}{y})^\frac{1}{2}).[/tex] Neglect the weight of the beam and any energy dissipated in the impact.

I have:

[tex]y_s = kW[/tex]

T1 = 0
V1 = mgh
T2 = 0 when deflection is at a max
[tex]V2 = -mgy_m[/tex]

I'm pretty sure that V2 should also include the potential energy stored in the beam, but I don't know how to express that. Would it be similar to a spring? V(beam) = 1/2ky^2 ? That's my guess, but there must be a logical way of proving it.

I think I can do the rest once I find the expression for the potential energy inside the elastic beam.
 
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  • #3
Originally posted by AngelofMusic
I have:

[tex]y_s = kW[/tex]
...
... there must be a logical way of proving it.
Yes, there is. You have identified a force law that should look familiar (if you put the k on the other side). How do you derive the potential energy from that force law (think about F dot dx).
 

1. How is deflection of an elastic beam defined?

The deflection of an elastic beam refers to the amount of bending or displacement that occurs when a load is applied to the beam. It is measured as the distance between the original position of the beam and its new position after the load is applied.

2. What factors affect the deflection of an elastic beam?

The deflection of an elastic beam is affected by several factors, including the material properties of the beam (such as Young's modulus and moment of inertia), the type and magnitude of the applied load, the length and shape of the beam, and the support conditions at each end of the beam.

3. How is the deflection of an elastic beam calculated?

The deflection of an elastic beam can be calculated using various methods, such as the Euler-Bernoulli beam theory, the Timoshenko beam theory, or numerical analysis using finite element methods. These methods take into account the aforementioned factors and use equations and formulas to determine the deflection at different points along the beam.

4. What are the practical applications of studying deflection of an elastic beam?

Understanding the deflection of an elastic beam is crucial in engineering and construction, as it allows for the design of structures that can withstand different types of loads and maintain their stability. It is also important in fields such as aerospace and mechanical engineering, where beams are used in various applications.

5. How can the deflection of an elastic beam be controlled?

The deflection of an elastic beam can be controlled by adjusting the material properties, dimensions, and support conditions of the beam, as well as the type and placement of the applied load. In some cases, additional support, such as beams or columns, may be added to reduce the deflection of a beam and improve its overall strength and stability.

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