Q4 Evaluate the validity of the approximation made in going from 222 to 223 in your textbook and calculating Cmw from both equations 30 pts Assume CDw CD min K CL2 and L Lw Note Equation 222 was Cm...

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Q4. Evaluate the validity of the approximation made in going from (2.2.2) to (2.2.3) in your textbook and calculating $C_{m_{w}}$ from both equations. (30 pts) Assume $C_{D_{w}}=C_{D_{\text {min }}}+K C_{L}^{2}$ and $L=L_{w}$. Note Equation (2.2.2) was $C_{m_{\alpha}=} C_{m_{a c_{w}}}+\left(C_{L_{w}}+C_{D_{w}} \alpha_{w}\right)\left(h-h_{n_{w}}\right)+\left(C_{L_{w}} \alpha_{w}-C_{D_{w}}\right)^{Z} / \bar{c}$ Equation (2.2.3): \[ \begin{array}{l} C_{m_{w} \approx} C_{m_{a a_{w}}}+C_{L_{w}}\left(h-h_{n_{w}}\right) \\ C_{m_{w} \approx} C_{m_{a c_{w}}}+\alpha_{w} a_{w}\left(h-h_{n_{w}}\right) \end{array} \] The following additional data are provided as below: \begin{tabular}{ll} Weight, $W$ & $207,750 \mathrm{lb}(924,448 \mathrm{~N})$ \\ $\mathrm{z} / \bar{c}$ & 0.15 \\ $S_{t}$ & $367.5 \mathrm{ft}^{2}\left(34.14 \mathrm{~m}^{2}\right)$ \\ $h-h_{n_{w}}$ & 0.208 \\ $a_{w}$ & $0.080 / \mathrm{deg}$ \\ $C_{m_{a C_{w}}}$ & -0.05 \\ $C_{D_{\min }}$ & 0.013 \\ $K$ & 0.054 \\ $V$ & $350 \mathrm{knots}(180 \mathrm{~m} / \mathrm{s})$ \\ $\rho$ & $2.377 \times 10^{3} \mathrm{slugs} / \mathrm{ft}^{3}\left(1.225 \mathrm{~kg} / \mathrm{m}^{3}\right)$ \end{tabular}

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#### Solution By Steps ***Step 1: Understand the Equations*** Equation (2.2.2) involves $C_{m_{\alpha}}$, $C_{L_{w}}$, $C_{D_{w}}$, and other variables, indicating it's a complex expression for moment coefficient involving lift and drag coefficients. Equation (2.2.3) simplifies this into two approximations focusing on lift and the effect of angle of attack ($\alpha_w$). ***Step 2: Simplify $C_{D_{w}}$*** Given $C_{D_{w}}=C_{D_{\text {min }}}+K C_{L}^{2}$, we can't directly calculate $C_{D_{w}}$ without $C_{L}$, but this relationship will be used in evaluating the approximation's validity. ***Step 3: Calculate Lift, $L$*** Using $L = \frac{1}{2} \rho V^2 S_t C_{L}$, we find $L$ given the provided data. However, $C_{L}$ is not directly provided, and typically, $L = W$ for steady flight. ***Step 4: Evaluate Approximation Validity*** To evaluate the approximation's validity, we need to assess how closely Equation (2.2.3) represents the dynamics described in Equation (2.2.2), considering the simplifications made. This involves understanding the physical significance of each term and the conditions under which the approximation is made. ***Step 5: Calculate $C_{m_{w}}$ from Both Equations*** Without explicit values for $C_{L_{w}}$ or $\alpha_{w}$, we focus on the structural form of the equations. The calculation would involve substituting known values into both equations and comparing the results, but this step requires assumptions or additional data about $C_{L_{w}}$ and $\alpha_{w}$. #### Final Answer Direct calculation of $C_{m_{w}}$ from both equations is not possible without additional data on $C_{L_{w}}$ or $\alpha_{w}$. The evaluation of the approximation's validity is conceptual, focusing on the simplifications made in Equation (2.2.3) and their impact on accurately representing the aircraft's flight dynamics. #### Key Concept Approximation Validity #### Key Concept Explanation Approximation validity in this context refers to how well the simplified equations (2.2.3) represent the more complex dynamics captured in Equation (2.2.2). It involves assessing the impact of neglecting certain terms and whether the simplified model still accurately captures the essential physics of the aircraft's moment coefficient behavior.

Follow-up Knowledge or Question

What is the significance of the term $C_{m_{w}}$ in aircraft flight dynamics and how does it relate to the aircraft's stability and control?

How does the lift coefficient $C_{L}$ affect the drag coefficient $C_{D_{w}}$ in the given approximation equations for $C_{m_{w}}$?

Explain the impact of the parameter $a_{w}$ on the calculation of the moment coefficient $C_{m_{w}}$ in the context of aircraft dynamics.

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