Question

An annular membrane of inner radius \(a\) and outer radius \(b\) is fixed about its inner edge. A uniform force per unit length of magnitude \(P_{0}\) oriented at an angle \(\beta\) from the plane of the membrane is applied around the periphery of the outer edge. Establish the boundary conditions for the structure if it is modeled as an ideal membrane.

Short answer

Answer: The boundary conditions for the annular membrane are as follows: - For the inner edge (r = a): 1. u(a) = 0 (no displacement in radial direction) 2. w(a) = 0 (no displacement in vertical direction) - For the outer edge (r = b): 1. t_{rr}(b) = P_0 * cos(𝛽) (radial force balance) 2. t_{rz}(b) = P_0 * sin(𝛽) (vertical force balance)

Step-by-step solution

1. Identify given quantities and given boundary conditions

We are given the following quantities and boundary conditions: - Inner radius: a - Outer radius: b - Uniform force per unit length around the outer edge: P_0 - Angle at which the force is applied: \beta

2. Analyze the forces acting on the membrane

In order to establish the boundary conditions for the structure, we must first analyze the forces acting on the membrane. There are two main forces acting on the membrane: 1. Tension in the membrane 2. The force applied around the outer edge, P_0

3. Decompose P_0 into horizontal and vertical components

Since the force P_0 is acting at an angle \beta from the plane of the membrane, we will decompose P_0 into its horizontal and vertical components: - Horizontal component of P_0: P_{0x} = P_0 \cos(\beta) - Vertical component of P_0: P_{0y} = P_0 \sin(\beta)

4. Establish boundary conditions for the inner edge

As the membrane is fixed at the inner edge, we must have no displacement: 1. Displacement in radial direction (u) at the inner edge (r = a): u(a) = 0 2. Displacement in vertical direction (w) at the inner edge (r = a): w(a) = 0

5. Establish boundary conditions for the outer edge

For the outer edge (r = b), we must consider the forces acting on the membrane. Due to the applied force P_0, we can write the boundary conditions as follows: 1. Radial force balance at the outer edge (r = b): t_{rr}(b) = P_{0x} 2. Vertical force balance at the outer edge (r = b): t_{rz}(b) = P_{0y}

6. Conclusion

The boundary conditions for the annular membrane of inner radius a and outer radius b under a uniform force per unit length P_0, applied at an angle \beta, are as follows: - For the inner edge (r = a): 1. u(a) = 0 2. w(a) = 0 - For the outer edge (r = b): 1. t_{rr}(b) = P_0 \cos(\beta) 2. t_{rz}(b) = P_0 \sin(\beta)

Key concepts

Ideal Membrane

An ideal membrane is a theoretical model used in physics and engineering to simplify the study of membranes. It assumes that the membrane is a two-dimensional surface without any thickness. This means that it cannot withstand bending or support compressive forces. Instead, it can only handle tension forces that lie along the plane of the membrane.

Membranes in the real world, like drum skins or soap films, can be approximated as ideal membranes under certain conditions. When the forces acting on the membrane are predominantly tensile, and any bending or secondary effects are negligible, this model is particularly useful. Understanding the behavior of an ideal membrane helps us simplify complex equations, making it easier to analyze the structures they form.

Force Decomposition

Force decomposition is a technique used to break down a force vector into its components along mutually perpendicular axes. In the context of the annular membrane, the force per unit length, denoted as \(P_0\), is applied at an angle \(\beta\) from the plane of the membrane. To fully understand the effect of this force, it must be decomposed into horizontal and vertical components.

The horizontal component is calculated using the cosine of the angle: \(P_{0x} = P_0 \cos(\beta)\). This is the part of the force acting parallel to the plane of the membrane. The vertical component is found using the sine of the angle: \(P_{0y} = P_0 \sin(\beta)\), which acts perpendicular to the membrane plane.

This decomposition allows us to analyze how different parts of a force influence membrane behavior. It is a crucial step in establishing boundary conditions and ensuring structural stability.

Boundary Conditions

Boundary conditions specify the constraints imposed on the membrane at its edges, which are crucial for solving physical equations governing the membrane's behavior. In the context of the annular membrane:

• Inner Edge: This edge is fixed, meaning there is no movement. Thus, the displacement in the radial direction \(u(a) = 0\), and in the vertical direction \(w(a) = 0\).
• Outer Edge: Here, the boundary conditions are defined by the forces experienced. The radial force balance is expressed as \(t_{rr}(b) = P_{0x}\) and the vertical force balance as \(t_{rz}(b) = P_{0y}\), where the components \(P_{0x}\) and \(P_{0y}\) are derived from the decomposition of the total force \(P_0\).

Understanding these conditions is vital for predicting how the membrane will react to external forces, ensuring it remains secure and functional.

Tension in Membranes

Tension is a force that acts along the plane of a membrane, trying to stretch it. In an ideal membrane, only tensile forces are considered because it cannot resist bending. Tension plays a critical role in maintaining the membrane's shape and stability.

In the case of the annular membrane, the tension comes from two main sources:

• The intrinsic tension in the membrane material itself, which tends to keep it taut and flat.
• The external force \(P_0\) applied at the outer edge exerting additional tension. This force componentizes into \(P_{0x}\) and \(P_{0y}\), which influence the overall tension distribution.

Proper analysis of tension ensures that the membrane structure can withstand applied forces without deforming or failing. It helps in designing membranes for specific applications, ensuring their optimal performance and safety.