The vibration is the vertical direction of an airplane and its wings can be modeled as a three-degree-of-freedom system with one mass corresponding to the right wing, one mass for the left wing, and one mass for the fuselage. The stiffness connecting the three masses corresponds to that of the wing and is a function of the modulus E of the wing.
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a) The model is given in Figure P4. Calculate the natural frequencies and mode shapes. Plot the mode shapes and interpret them according to the airplane's deflection.
b) Consider the airplane of Figure P4 with modal damping of with 0.1 in each mode. Suppose that the airplane hits a gust of wind, which applies an impulse of 3?(t) at the end of the left wing and ?(t) at the end of the right wing. Calculate the resulting vibration of the cabin [??2(??)].
c) Consider again the airplane of Figure P4 with 10% modal damping in each mode. Suppose that this is a propeller-driven airplane with an internal combustion engine mounted in the nose. At a cruising speed the engine mounts transmit an applied force to the cabin mass (4m at ??2) which is harmonic of the form 50??????10??. Calculate the effect of this harmonic disturbance at the nose and on the wind tips after subtracting out the translational or rigid motion.

Frequencies refer to the number of occurrences of a repeating event per unit time. It is commonly used when discussing sound and electromagnetic radiation, such as radio and light waves. In these cases, frequency describes the number of wave cycles per second, measured in hertz (Hz). It is also used to describe the number of occurrences of a particular event within a given time frame. For example, a frequency chart may show the number of times a certain type of crime occurred in a given month or year.

The natural frequencies and mode shapes of the three-degree-of-freedom system with one mass corresponding to the right wing, one mass for the left wing, and one mass for the fuselage can be calculated by solving the eigenvalue equation for the system. The mode shapes are the eigenvectors of the system and represent the relative displacement of the three masses for each mode. The mode shapes can be plotted to show the deflection of the wings and fuselage in response to the applied force or displacement.
To calculate the resulting vibration of the cabin, the impulse response of the system needs to be calculated with modal damping of 0.1 in each mode. This can be done by solving the differential equation describing the system and applying the impulse force at the ends of the left and right wings. The resulting vibration of the cabin can be found by solving the resulting equation.
To calculate the effect of the harmonic disturbance on the cabin and wing tips, the response of the system needs to be calculated with 10% modal damping in each mode. This can be done by solving the differential equation describing the system and applying the harmonic force at the cabin mass. The resulting vibration of the cabin and wing tips can be found by solving the resulting equation. After subtracting out the translational or rigid motion, the effect of the harmonic disturbance can be determined.

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