### Our Team

### Kellen Andrew

Student Researcher

Hi! My name’s Kellen and I am a third year Aerospace Engineering with a concentration in Astronautics. When I’m not in class, I’m either riding bikes, repairing bikes, or building up an entirely new frame in the Hangar.

### Matthew Schoenau

Student Researcher

Hi! My name is Matthew and I am a 4^{th} year Aerospace Engineering Major with a concentration inn Aeronautics. I enjoy flying, diving and backpacking, but my wallet doesn’t.

### Acknowledgements

We would like to thank Dr. Deffo for his consistent support throughout the project and his dedication to helping us become more capable engineers.

### Digital Poster

## INTRODUCTION/ MOTIVATION

- Modeling an aircraft wing as a cantilevered beam can be a useful tool in conducting structural analysis. [1]
- As the lift distribution on aircraft wings is not uniform, the wings usually undergo both bending and torsional forces.
- The angular displacement and traverse deflection are coupled for composite beams.
- In this project we studied the free vibration of such beams. Finding the natural frequency as an exact solution is very challenging so we used the Rayleigh-Ritz Method instead. [1]
- With the natural frequencies known, one can then optimize a damper for the wing/fuselage system.

## SIGNIFICANT EXPRESSIONS

## Coupled Equations of Motion

$$\left.\overline{\rho I_{p}} \frac{\partial^{2} \theta}{\partial t^{2}}+m d \frac{\partial^{2} v}{\partial t^{2}}+\frac{\partial}{\partial x}\left(\overline{G J} \frac{\partial \theta}{\partial x}-K \frac{\partial^{2} v}{\partial x^{2}}\right)\right)=r(x, t)$$

$$\left.m\left(\frac{\partial^{2} v}{\partial t^{2}}+d \frac{\partial^{2} \theta}{\partial t^{2}}\right)+\frac{\partial^{2}}{\partial x^{2}}\left(\overline{E I} \frac{\partial^{2} v}{\partial t^{2}}-K \frac{\partial \theta}{\partial x}\right)\right)=f(x, t)$$

## Assumed Mode Shapes

**Bending:**

$$\psi_j=\cosh (\gamma_j x)-\cos (\gamma_j x)-\beta_j(\sinh (\gamma_j x)-\sin (\gamma_j x))$$

where,

$$\beta_j=\frac{\cosh (\gamma_j l)+\cos (\gamma_j l)}{\sinh (\gamma_j l)+\sin (\gamma_j l)}$$

**Torsion:**

$$\phi_i=\sin (\alpha_i x)$$

where,

$$\alpha_i=\frac{(2 i-1)\pi}{2}$$

## Coupled Strain Energy

$$\boldsymbol{U}=\frac{1}{2}\int_{0}^{l}\left[\begin{array}{l}\frac{\partial \theta}{\partial x} \\\frac{\partial^{2} v}{\partial x^{2}}\end{array}\right]^{T}\left[\begin{array}{ll}\overline{G J} & -K \\-K & \overline{E I}\end{array}\right]\left[\begin{array}{l}\frac{\partial \theta}{\partial x} \\\frac{\partial^{2} v}{\partial x^{2}}\end{array}\right] dx$$

## Coupled Kinetic Energy

$$K =\frac{1}{2}\int_{0}^{l}\left[m\left(\frac{\partial v}{\partial t}\right)^{2}+2 m d \frac{\partial \theta}{\partial t} \frac{\partial v}{\partial t}+\overline{\rho I_{p}}\left(\frac{\partial \theta}{\partial t}\right)^{2}\right] d x$$

## Lagrange's Equation[2]

$$\frac{\partial}{\partial t}\left(\frac{\partial K}{\partial \dot{q}_{i}}\right)+\frac{\partial U}{\partial q_{i}}=0$$

# Free Vibration of a Wing under Coupled Bending and Torsion

Kellen Andrew & Matthew Schoenau

Dr. Deffo, Aerospace Engineering

## UNCOUPLED MODE SHAPES

**References:**

[1] Dewey H. Hodges, G. Alvin Pierce, *Introduction to structural Dynamics and Aeroelasticity, *second edition, Cambridge University Pres, 2011

[2] Professor J. Kim Vandiver, *An Introduction to Lagrange Equations, *MIT OpenCourse Ware Fall 2011

## PROCEDURE

**Rayleigh Ritz Method**

Relies on solving Lagrange’s Equation using “admissible functions” *(f _{i} , g_{j})* that satisfy the following conditions:

*f*_{i}, g_{j}must satisfy the boundary conditions of the beam.*f*_{i}, g_{j}must be differentiable enough to compute Lagrange’s Equation without issue.*f*_{i}, g_{j}must each form a complete set of functions.*f*_{i}andf_{j}(i≠j) must be linearly independent & g_{i}and g_{j}(i≠j) must be linearly independent

- Uncoupled bending/torsion modes as assumed mode shapes
- Lagrange’s Equation to obtain homogeneous system of equations
- Determinevalues of ω (oscillation frequency) for nontrivial solutions to the system.
- Implementation in terms of the dimensionless parameter, $\omega^2=\frac{\lambda^2 E I}{m L^4}$

## RESULTS

- Rayleigh Ritz Method can be altered to achieve approximations for higher modes as well as different levels of accuracy.
- To find nth mode natural frequency, use assumed mode shapes i = 1,2,…,n in approximation
- Approximation converges to the actual value as more assumed mode shapes are used i > n.
- Approximation is monotonic decreasing, i.e., higher number of modes provide better approximate of lower frequencies.

#### Table of Natural Frequency Coefficients †

† Natural Frequency is obtained by the following:

$$\omega=Coefficient\cdot\sqrt{\frac{E I}{m L^4 }}$$

## DISCUSSION

- Notice that the apparent limit of the approximation depends on whether k and d have the same or opposite signs
- Opposite signs result in slightly higher frequencies for Modes 1,2
- Additionally, the coupled behavior significantly alters the natural frequencies from the uncoupled cases
- With the results above, one could now design a damper system to deaden vibrations near these frequencies and ultimately avoid the potentially catastrophic phenomenon that is resonance.