Question 3.1.2: Suppose a 1-kg bird flies into the 3-kg security camera of E......
Suppose a 1-kg bird flies into the 3-kg security camera of Example 2.1.3, repeated in Figure 3.2. If the bird is flying at 72 kmph, compute the maximum deflection the impact causes based on the design given in Example 2.1.3. Does the maximum deflection violate the design constraint? Ignore damping.
From the design solution of Example 2.1.3, the stiffness of the camera’s mounting bracket is \left(I=b h^3 / 12\right) :
k=\frac{3 E b h^3}{12 l^3}=\frac{\left(7.1 \times 10^{10}\ N / m ^2\right)(0.02\ m )(0.02\ m )^3}{4(0.55\ m )^3}=1.707 \times 10^4\ N / m
The mass of the camera is m_c=3\ kg, so the natural frequency is 75.43\ rad / s. Combining equations (3.7) and (3.8) for \zeta=0, the response is
x(t)=\hat{F} h(t) (3.7)
h(t)=\frac{1}{m \omega_d} e^{-\zeta \omega_n t} \sin \omega_d t (3.8)
x(t)=\frac{F \Delta t}{m_c \omega_n} \sin \omega_n t=\frac{m_b v}{m_c \omega_n} \sin \omega_n t
where m_b v is the linear momentum of the bird, which imparts the impact force F. The impulse is thus
m_b v=1\ kg \cdot 72 \frac{ km }{\text { hour }} \cdot \frac{1000\ m }{ km } \cdot \frac{\text { hour }}{3600\ s }=20\ kg \cdot m / s
This has maximum amplitude of
X=\left|\frac{F \Delta t}{m_c \omega_n}\right|=\left|\frac{m_b v}{m_c \omega_n}\right|=\left|\frac{20\ kg \cdot m / s }{3\ kg \cdot 75.43\ rad / s }\right|=0.088\ m
Thus, the design constraint of holding the vibration of the camera within 0.01 m required in Example 2.1.3 is violated under a bird strike.