Problem 22 - Calculus Of Variations, With Applications To Physics And Engineering

Question

Write down the boundary conditions (134) as they read in the following special cageg: (a) Circular plate \((r \leqq a)\) clamped. ANswER: \(\phi=0,(\partial \phi / \partial r)=0\) at \(r=a\). (b) Rectangular plate \((0 \leqq x \leqq a ; 0 \leqq y \leqq b)\) simply supported. ANgwER: \(\phi-0,\left(\partial^{2} \phi / \partial x^{2}\right)-0\) on \(x-0\) and \(x-a ; \phi-0,\left(\partial^{2} \phi / \partial y^{2}\right)=0\) on \(y-0\) and \(y=b .\) Note also that \(\left(\partial^{2} \phi / \partial y^{2}\right)=0\) on \(x=0, x=a ;\) and \(\left(\partial^{2} \phi / \partial x^{2}\right)=0\) on \(y=0, y=b\). Why? (c) Semicircular plate (r \(\leqq a ; 0 \leqq \theta \leqq \pi)\) simply supported. ANswER: Since \(\begin{aligned} K=(1 / a) \text { on } r=a & \\\ \phi & \phi 0, \quad \frac{\sigma}{a} \frac{\partial \phi}{\partial r}+\frac{\partial^{2} \phi}{\partial r^{2}}=0 \text { on } r=a(0 \leqq \theta \leqq \pi) \\ & \phi=0, \quad \frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}=0 \text { on } \theta=0, \theta=\pi . \end{aligned}\) (d) Circular plate \((r \leq a)\) free. ANswER: $$ \left.\begin{array}{r} \frac{\partial}{\partial r}\left(\nabla^{2} \phi\right)+(1-\sigma) \frac{1}{a^{2}} \frac{\partial}{\partial \theta}\left(\frac{\partial^{2} \phi}{\partial \theta \partial r}-\frac{1}{a} \frac{\partial \phi}{\partial \theta}\right)=0 \\ (1-\sigma)\left(\frac{1}{a^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{1}{a} \frac{\partial \phi}{\partial r}\right)-\nabla^{2} \phi=0 \end{array}\right\\} \text { on } r=a $$

Step-by-Step Solution

Answer

The boundary conditions for each case are given by: (a) \( \phi=0, \frac{\partial \phi}{\partial r}=0 \) at \( r=a \). (b) \( \phi=0, \frac{\partial^2 \phi }{ \partial x^{2} }=0 \) on \( x=0, x=a \) and \( \phi=0, \frac{\partial^2 \phi }{ \partial y^{2} }=0 \) on \( y=0, y=b \). (c) \( K = \frac{1}{a} \) on \( r = a \), \( \phi = 0 , \frac{ \sigma }{ a } \frac{ \partial \phi }{ \partial r } + \frac{ \partial^2 \phi }{ \partial r^{2} } = 0 \) on \( r = a \) where \( 0 \leqq \theta \leqq \pi \) and \( \phi = 0 , \frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}} = 0 \) on \( \theta = 0 , \theta = \pi \). (d) \( \frac{\partial}{\partial r}(\nabla^{2} \phi) + (1-\sigma) \frac{1}{a^{2}} \frac{\partial}{\partial \theta}(\frac{\partial^{2} \phi}{\partial \theta \partial r}-\frac{1}{a} \frac{\partial \phi}{\partial \theta}) = 0 \) and \( (1-\sigma)(\frac{1}{a^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{1}{a} \frac{\partial \phi}{\partial r}) - \nabla^{2} \phi = 0 \) on \( r = a \).

Step 1: Circular Plate - Clamped

In this case, the boundary condition says that the function \( \phi \) and its derivative with respect to \( r \), \( \frac{\partial \phi}{\partial r} \), are both zero at the edge of the plate: \( r=a \). That is: \( \phi=0, \frac{\partial \phi}{\partial r}=0 \) at \( r=a \).

Step 2: Rectangular Plate - Simply Supported

For this case, the boundary conditions imply that \( \phi \) and its second derivative (laplacian), with respect to \( x \) and \( y \), are zero on the edges of the plate, i.e. \( x=0, x=a \) and \( y=0, y=b \). That is: \( \phi=0 , \frac{\partial^2 \phi }{ \partial x^{2} } = 0 \) at \( x=0 , x=a \) and \( \phi=0 , \frac{\partial^2 \phi }{ \partial y^{2} } = 0 \) at \( y=0, y=b \).

Step 3: Semincircular Plate - Simply Supported

In this scenario, the function \( \phi \) and its derivative with respect to \( r \) and \( \theta \) are defined as zero at the edge of the plate and at \( \theta = 0, \theta = \pi \). That is: \( K = \frac{1}{a} \) on \( r = a \), \( \phi = 0 , \frac{\sigma}{a} \frac{ \partial \phi }{ \partial r } + \frac{ \partial^2 \phi }{ \partial r^{2} } = 0 \) on \( r = a \) where \( 0 \leqq \theta \leqq \pi \) and \( \phi = 0 , \frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}} = 0 \) on \( \theta = 0 , \theta = \pi \).

Step 4: Circular Plate - Free

In this case, the boundary condition is given by two equations at the edge of the plate \( r = a \). The equations are: \( \frac{\partial}{\partial r}(\nabla^{2} \phi) + (1-\sigma) \frac{1}{a^{2}} \frac{\partial}{\partial \theta}(\frac{\partial^{2} \phi}{\partial \theta \partial r}-\frac{1}{a} \frac{\partial \phi}{\partial \theta}) = 0 \) and \( (1-\sigma)(\frac{1}{a^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}}+\frac{1}{a} \frac{\partial \phi}{\partial r}) - \nabla^{2} \phi = 0 \) on \( r = a \).

Key Concepts

Circular Plate Boundary Conditions

When analyzing a circular plate clamped at its edges, two key boundary conditions are imposed to ensure the stability and physical realism of the model. A clamped condition indicates that there should be no displacement or rotation at the edge of the plate. Mathematically, this is expressed as the function \( \phi \), which represents the deflection or potential, and its radial derivative \( \frac{\partial \phi}{\partial r} \) being zero at \( r=a \).

In essence, \( \phi=0 \) means no displacement occurs at the boundary, while \( \frac{\partial \phi}{\partial r} = 0 \) ensures the plate's edge does not slope, i.e., the tangential plane to the plate's boundary remains horizontal. Understanding this helps comprehend why the plate’s form remains unaltered at the clamped edge, crucial for problems involving vibrations, thermal stresses, or electrical potentials, where continuity and stability are key.

Rectangular Plate Boundary Conditions

In the context of a simply supported rectangular plate, the boundary conditions are particularly interesting because they involve the second derivative of \( \phi \) with respect to both \( x \) and \( y \) coordinates, known as the Laplacian. Similar to the previous case, \( \phi=0 \) on the edges, indicating no displacement at the boundary. However, the condition \( \frac{\partial^2 \phi }{ \partial x^{2} } = 0 \) along \( x=0 \) and \( x=a \) and an analogous condition for \( y \) implies that the plate is not experiencing any bending moment along those edges.

This characteristic accurately represents a simply supported edge on which the plate can freely rotate and isn't subjected to bending stresses. These boundary conditions are integral when solving partial differential equations tied to structural mechanics, heat conduction, or electromagnetism in rectangular domains.

Semicircular Plate Boundary Conditions

Semicircular plates introduce a blend of boundary conditions along straight and curved edges. On the curved boundary, \( r=a \) and \( 0 \leqq \theta \leqq \pi \) for a simply supported edge, we see similar conditions as the circular plate with \( \phi=0 \) and a combination of its derivatives equaling zero. Here, \( \frac{\sigma}{a} \frac{ \partial \phi }{ \partial r } + \frac{ \partial^2 \phi }{ \partial r^{2} } = 0 \) encapsulates both the effect of the radial stress constant \( \sigma \) and the behavior under radial loads.

Additionally, for the straight edges along \( \theta=0 \) and \( \theta=\pi \) the conditions \( \phi=0 \) and \( \frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}} = 0 \) suggest that there is no deflection and no angular component of bending moment at the ends. Comprehending these conditions is crucial for problems concerning mechanics of materials, as well as in fluid dynamics, where semicircular boundaries are common.

Laplace's Equation in Polar Coordinates

Laplace's equation, a fundamental partial differential equation, finds its unique formulation in polar coordinates when interpreting phenomena like electrostatics, gravitation, or heat transfer in circular or annular domains. The equation in polar coordinates takes the form \( abla^2 \phi = \frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2}\frac{\partial^2 \phi}{\partial \theta^2} = 0 \) where \( \phi \) represents a scalar function such as potential or temperature.

It is a classical example illustrating how coordinate systems must adapt to problem symmetry, reducing to a simpler form when searching for radially symmetric solutions. Analyzing this equation under the aforementioned boundary conditions, like those for circular or semicircular plates, allows students to investigate complex physical systems within a manageable mathematical framework.