Consider a torsionally elastic (GJ = 8000 lb in2) wind-tunnel model of a uniform wing, the
ends of which are rigidly fastened to the wind-tunnel walls. The model has a symmetric
airfoil, a span of 3 ft, and a chord of 6 in. The sectional lift-curve slope is 6 per rad. The
aerodynamic center is located at the quarter-chord, and both the mass centroid and the
elastic axis are at the mid-chord.

This problem involves a torsionally elastic wind-tunnel model of a uniform wing with given geometric and aerodynamic properties. The key details given are:

- **Torsional stiffness**: \( GJ = 8000 \, \text{lb} \cdot \text{in}^2 \)
- **Wing span**: 3 ft (36 inches)
- **Wing chord**: 6 in
- **Symmetric airfoil**: This indicates that the lift distribution is symmetric about the mid-span, and the angle of attack is evenly distributed.
- **Sectional lift-curve slope**: 6 per rad, meaning for every radian of change in the angle of attack, the lift coefficient increases by 6.
- **Aerodynamic center**: Located at the quarter-chord point.
- **Mass centroid and elastic axis**: Both located at the mid-chord point, simplifying any coupling effects between bending and twisting.

The **torsional stiffness** \( GJ \) governs how much the wing twists under an applied torque. The fact that the wing is **rigidly fastened** at both ends to the wind-tunnel walls implies a boundary condition where the wing cannot twist at its ends.

Here are some next steps that could be explored based on the problem description:

1. **Determine the torsional displacement along the span**: You might want to compute the torsional angle distribution along the wing due to the aerodynamic load.
  
2. **Calculate the aerodynamic moment** about the elastic axis using the lift-curve slope: The aerodynamic moment can induce a torsional moment, which would be resisted by the wing's torsional stiffness.

3. **Investigate flutter conditions**: Since this is a wind-tunnel model, an interesting consideration could be whether the wing is at risk of torsional flutter at higher airspeeds.

4. **Define the twist distribution**: Based on the lift distribution and aerodynamic moment, we could explore how much twist the wing experiences along its span.

5. **Potential for aeroelastic instability**: You might also be concerned about how the coupling of aerodynamic loads and wing flexibility affects stability.

Do you want to focus on a specific aspect of this problem (e.g., calculating torsional displacement, lift distribution, or flutter speed)? Let me know how you want to proceed or if you need clarification on any of these aspects.

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Here are five possible questions you could explore from this problem:

1. How do you calculate the torsional deflection along the span of the wing?
2. How does the position of the aerodynamic center affect the twisting moment on the wing?
3. What is the relationship between the sectional lift-curve slope and the total lift generated by the wing?
4. How does the torsional stiffness \( GJ \) impact the wing's resistance to twisting under aerodynamic load?
5. What are the conditions for aeroelastic instability (flutter) in this wind-tunnel setup?

**Tip**: Understanding the relationship between the torsional stiffness and the aerodynamic moment is crucial for analyzing wing torsion and potential flutter behavior.

Consider a torsionally elastic (GJ = 8000 lb in2) wind-tunnel model of a uniform wing, the ends of which are rigidly fastened to the wind-tunnel walls. The model has a symmetric airfoil, a span of 3 ft, and a chord of 6 in. The sectional lift-curve slope is 6 per rad. The aerodynamic center is located at the quarter-chord, and both the mass centroid and the elastic axis are at the mid-chord.

Explore the torsionally elastic wind-tunnel model of a uniform wing, including calculations of torsional stiffness (GJ = 8000 lb in2), aerodynamic moments, and the lift-curve slope. Understand the impact of aerodynamic center placement and aeroelasticity for wind tunnel setups. Suitable for undergraduate aerospace engineering students.

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