Question

For studying wing vibrations, a wing of mass \(M\) and finite dimensions has been idealized by assuming it to be supported using a linear spring of equivalent stiffness \(k\) and a torsional spring of equivalent stiffness \(k_\theta\) as shown in the figure. The centre of gravity (CG) of the wing (idealized as an airfoil) is marked. The number of degree(s) of freedom for this idealized wing vibration model is \underline{\hspace{1cm}}. \;(Answer in integer) \begin{center} \includegraphics[width=0.5\textwidth]{08.jpeg} \end{center}

Hint:

For 2-D wing-section (aeroelastic) models, ask: which motions are allowed (but resisted) by springs? Each allowed motion is a DOF. The classic minimal model keeps plunge \(h\) and pitch \(\theta\) \(\Rightarrow\) 2 DOF.

Answer 1

Step 1: Start from the unconstrained rigid–airfoil in a plane. \\ A rigid body moving in a plane has \(3\) mechanical DOFs: two translations \((x,y)\) of a reference point (e.g., CG) and one in–plane rotation \((\theta)\) about an axis normal to the plane. For an airfoil section used in typical 2-D aeroelastic models, we adopt the standard coordinates: - \(h\): vertical translation (plunge) of the elastic axis/CG, positive downward; - \(x\): streamwise translation (surge); - \(\theta\): pitch (rotation about the elastic axis/CG), positive nose-up.

Step 2: Identify what the support/springs allow. \\ From the figure and description: - A linear spring of stiffness \(k\) is attached in the vertical direction. This resists (but does not kinematically prevent) plunge \(h\). - A torsional spring of stiffness \(k_\theta\) is attached at/near the elastic axis. This resists (but does not prevent) pitch \(\theta\). - There is no spring or guide permitting streamwise motion \(x\); the mounting implies the chordwise translation is constrained by the support (the section is held in place horizontally). Hence \(x\) is not a generalized coordinate. Thus the only admissible small motions are \(h\) and \(\theta\).

Step 3: Count independent generalized coordinates. \\ Each independent permissible motion adds one DOF: \[ q_1 = h \text{(plunge)}, q_2 = \theta \text{(pitch)}. \] Therefore, the idealized system has \[ \boxed{\text{DOF} = 2}. \]

Step 4: (Insight) Why not 1 or 3 DOF? \\ - \(\mathbf{1}\) DOF would require either \(h\) or \(\theta\) to be kinematically fixed. The presence of both springs explicitly allows both motions. - \(\mathbf{3}\) DOF would require free surge \(x\) as well, which is not allowed by the depicted support (no axial slide or spring in the \(x\)-direction). (Optional) Governing form (to see the two coordinates). \\ A small-motion 2-DOF rigid-airfoil model leads to \[ \begin{bmatrix} m & m\,x_\theta \\ m\,x_\theta & I_\theta \end{bmatrix} \!\begin{bmatrix}\ddot h \\ \ddot\theta\end{bmatrix} + \begin{bmatrix} c_h & 0 \\ 0 & c_\theta \end{bmatrix} \!\begin{bmatrix}\dot h \\ \dot\theta\end{bmatrix} + \begin{bmatrix} k & 0 \\ 0 & k_\theta \end{bmatrix} \!\begin{bmatrix}h \\ \theta\end{bmatrix} = \begin{bmatrix}F_a(h,\theta,\dot h,\dot\theta) \\ M_a(h,\theta,\dot h,\dot\theta)\end{bmatrix}, \] confirming two generalized coordinates \(h\) and \(\theta\) are sufficient to describe the dynamics.

Final Answer: \\ \[ \boxed{2} \]