The deflection of the plate can be noticed by going to a co-...

The deflection of the plate can be noticed by going to a co- rotating frame. In this frame each element of the plate experiences a pseudo force proportional to its mass. These forces have a moment which constitutes the bending moment of the problem. To calculate this moment we note that the acceleration of an element at a distance $\xi$ from the axis is $a=\xi \beta$ and the moment of the forces exerted by the section between $x$ and $l$ is $$ N=\rho l h \beta \int_{x}^{l} \xi^{2} d \xi=\frac{1}{3} \rho \ln \beta\left(l^{3}-x^{3}\right) $$ From the fundamental equation $$ E I \frac{d^{2} y}{d x^{2}}=\frac{1}{3} \rho l h \beta\left(l^{3}-x^{3}\right) $$ The moment of inertia $I=\int_{-h / 2}^{+h / 2} z^{2} l d z=\frac{l h^{3}}{12 .}$ Note that the neutral surface (ie. the surface which contains lines which are neither stretched nor compressed) is a vertical plane here and $z$ is perpendicular to it. $$ \begin{gathered} \frac{d^{2} y}{d x^{2}}=\frac{4 \rho \beta}{E h^{2}}\left(l^{3}-x^{3}\right) . \text { Integrating } \\ \frac{d y}{d x}=\frac{4 \rho \beta}{E h^{2}}\left(l^{3} x-\frac{x^{4}}{4}\right)+c_{1} \end{gathered} $$ Since $\frac{d y}{d x}=0$, for $x=0, c_{1}=0 .$ Integrating again, $$ \begin{gathered} y=\frac{4 \rho \beta}{E h^{2}}\left(\frac{l^{3} x^{2}}{2}-\frac{x^{5}}{20}\right)+c_{2} \\ c_{2}=0 \text { because } y=0 \text { for } x=0 \end{gathered} $$ Thus $\lambda=y(x=l)=\frac{9 \rho \beta l^{5}}{5 E h^{2}}$

The deflection of the plate can be noticed by going to a co- rotating frame. In this frame each element of the plate experiences a pseudo force proportional to its mass. These forces have a moment which constitutes the bending moment of the problem. To calculate this moment we note that the acceleration of an element at a distance $\xi$ from the axis is $a=\xi \beta$ and the moment of the forces exerted by the section between $x$ and $l$ is
$$
N=\rho l h \beta \int_{x}^{l} \xi^{2} d \xi=\frac{1}{3} \rho \ln \beta\left(l^{3}-x^{3}\right)
$$
From the fundamental equation
$$
E I \frac{d^{2} y}{d x^{2}}=\frac{1}{3} \rho l h \beta\left(l^{3}-x^{3}\right)
$$
The moment of inertia $I=\int_{-h / 2}^{+h / 2} z^{2} l d z=\frac{l h^{3}}{12 .}$
Note that the neutral surface (ie. the surface which contains lines which are neither
stretched nor compressed) is a vertical plane here and $z$ is perpendicular to it.
$$
\begin{gathered}
\frac{d^{2} y}{d x^{2}}=\frac{4 \rho \beta}{E h^{2}}\left(l^{3}-x^{3}\right) . \text { Integrating } \\
\frac{d y}{d x}=\frac{4 \rho \beta}{E h^{2}}\left(l^{3} x-\frac{x^{4}}{4}\right)+c_{1}
\end{gathered}
$$
Since $\frac{d y}{d x}=0$, for $x=0, c_{1}=0 .$ Integrating again,
$$
\begin{gathered}
y=\frac{4 \rho \beta}{E h^{2}}\left(\frac{l^{3} x^{2}}{2}-\frac{x^{5}}{20}\right)+c_{2} \\
c_{2}=0 \text { because } y=0 \text { for } x=0
\end{gathered}
$$
Thus $\lambda=y(x=l)=\frac{9 \rho \beta l^{5}}{5 E h^{2}}$

Key Concepts

Co-rotating Frame and Pseudo Forces

In a co-rotating frame, additional fictitious forces, known as pseudo or inertial forces, appear because the frame is accelerating. These forces are proportional to the mass of the element and affect the apparent dynamics of the system, leading to observable phenomena like the deflection of the plate.

Bending Moment

The bending moment represents the internal moment that causes a structure like a beam or plate to bend under load. It is calculated as the sum of the moments produced by distributed forces acting on the structure, and it directly influences the curvature and deflection of the element according to its mechanical properties.

Moment of Inertia

The moment of inertia, particularly the second moment of area, quantifies the distribution of a structure's cross-sectional area relative to a reference axis. It plays a crucial role in determining a beam or plate's resistance to bending, as it enters directly into the relationship between bending moment and the resulting curvature.

Euler-Bernoulli Beam Theory

This theory provides a fundamental relationship between the bending moment applied to a beam, its flexural stiffness (a product of the material's Young's modulus and moment of inertia), and the beam’s curvature. It is the cornerstone of classical beam theory, used to derive the differential equation governing the deflection of beams under various loading conditions.

Integration of Differential Equations with Boundary Conditions

In structural mechanics, solving for the deflection of beams or plates typically involves integrating the governing second-order differential equation. The integration introduces constants, which are then determined by applying physical boundary conditions such as known deflections or slopes at the supports, ensuring that the solution accurately reflects the behavior of the actual structure.

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