Modal Analysis-hw-04 The schematic diagram of a large cannon is shown below. When the gun is fired, high-pressure gases accelerate the projectile inside the barrel to a very high velocity. The reaction force pushes the gun barrel in the opposite direction of the projectile. Since it is desirable to bring the gun barrel to rest in the shortest time without oscillation, it is made to translate backward against a critically damped spring-damper system called the recoil mechanism. In a particular case, the gun barrel and the recoil mechanism have a mass of 500 kg with a recoil spring stiffness of 104 N/m. The gun recoils 0.4 m upon firing. Determine 1) the critical damping coefficient of the damper, 2) the initial recoil velocity of the gun, and 3) the time taken by the gun to return to a position 0.1 m from its initial position. Using Excel, plot the recoil of the gun versus time. Be sure to show how you determined the correct time interval for this plot so that it is a smooth curve without using excessive time increments.

Modal Analysis-hw-04 (cont’d)

Modal Analysis-hw-05 A body supported by an elastic structure is submerged in a medium that produces a damping effect which is proportional to velocity. As the body undergoes damped oscillations it has a period of 1 s. At one instant the amplitude of vibration is seen to be 100 mm, and 10 s later the amplitude has been reduced to 1 mm. Determine the period of free vibration if the damping effect of the medium were negligible.

Modal Analysis-hw-05 (cont’d)

Modal Analysis-hw-06 A bridge is excited by dropping a weight at its center so as to determine the amount of damping exhibited by the structure. By dropping the weight at center span, the fundamental mode was excited and the frequency of vibration was measured to be 1.5 Hz and after 2 s the amplitude of vibration was found to have decreased to 90% of the initial maximum. From a kinetic energy analysis, the effective or equivalent mass of the bridge is known to be 105 kg. Assuming viscous damping and simple harmonic motion, determine the damping coefficient, the logarithmic decrement, and the damping ratio.

Modal Analysis-hw-06 (cont’d)

Modal Analysis-hw-07 The figure below shows a diagrammatic end view of one-half of a swing-axle suspension of an automobile which consists of a horizontal half-axle OA pivoted to the chassis at O, a wheel/tire assembly rotating about the center line of the axle, and a spring of stiffness k and a viscous damper with a damping coefficient c both located vertically between the axle and the chassis. The mass of the half-axle is m1 and its radius of gyration about O is h. The mass of the wheel/tire assembly is m2 and it may be regarded as a thin uniform disk having an external radius r and located at a horizontal distance s from the pivot O. The spring and damper are located at horizontal distances q and p from the pivot O, respectively. Determine the equation for angular movement of the axle-wheel assembly about the pivot O, and obtain from that equation an expression for the frequency of free damped oscillations of the assembly. Express the frequency in terms of the given parameters and the undamped natural frequency of the assembly.

Modal Analysis-hw-07 (cont’d)