When an airplane (see Fig. 6.37 (a)) undergoes symmetric vibrations, the fuselage can be idealized as a concentrated central mass M0 and the wings can be modeled as rigid bars carrying end masses M, as shown in Fig. 6.37( b). The flexibility between the wings and the fuselage can be represented by two torsional springs of stiffness kt each. (a) Derive the equations of motion of the airplane, using Lagrange's equations with x and θas generalized coordinates. (b) Find the natural frequencies and m