Chemical Engineering - Apllied Mechanics

Full exam

Exam of APPLIED MECHANICS Study Track in Chemical Engineering September 06, 2016 Exercise n.1 This exercise must be solved only by students who have failed the Intermediate Test or previous exams, or by students who want to improve the mark they got with the Intermediate Test or exams. Students who want to improve their mark about kinematics must only solve the part of kinematics of the exercise n.1. Students who want to improve their mark about dynamics must only solve the part of dynamics of the exercise n.1, by assuming that all the kinematic parameters that are necessary to be defined, in order to solve the problem, are known (in this case, please, draw arbitrary but likely direction and versus of these kinematic parameters). The mechanical system illustrated in Fig. 1 can move in a vertical plane. The bar HO, hinged to ground at point O, has a barycentre G 1, a mass m 1 and a mass moment of inertia J O, evaluated about an axis passing through point O. A slider, having a barycentre C and a mass m 3 can move on the inclined plane AB. This slider is connected to the bar HO by means of the bar CD, whose mass is negligible. A rigid body, having a barycentre G 2, a mass m 2 and a mass moment of inertia J G2, is supported on the bar HO. It is assumed that no slippage occurs between this body and the bar HO, owing to the static friction coefficient f s. The kinetic friction coefficient between the slider and the inclined plane is f k. At the given time the slider is moving with velocity v and acceleration a (see Fig. 1) under the effect of the unknown drive force F d. All the geometrical parameters are assumed to be known (the thickness of the bar HO and the slider are negligible). You are asked to describe the procedure to: 1 evaluate the angular velocity and angular acceleration of the bar HO; 2 evaluate the absolute velocity and acceleration of the barycentre G 2; 3 evaluate the magnitude of the drive force F d; 4 evaluate the reactions at point O, along with the internal forces at point D; 5 evaluate the contact forces between the body of barycentre G 2 and the bar HO; 6 check that no slippage occurs between this body and the bar HO; 7 write the expression of the kinetic energy of the system. N.B.: You are asked to write the equations that allow one to calculate any unknown kinematic and mechanical parameter that is cited in the expressions of your solution procedure. If the kinematic analysis is carried out using a mobile reference frame it is mandatory to indicate the origin of the frame and its type of motion. You are also asked to do a qualitative graphical solution of the vector equations used to evaluate the kinematic parameters (regardless of the method you use). Exercise n.2 The mechanical system illustrated in Fig. 2 can move in a vertical plane. The structure ACDBC is composed of bars whose mass is negligible in comparison to the mass m 2 of the pulley hinged at point D, while the mass moment of inertia of this pulley is J D. An inextensible massless wire is wrapped around the pulley. The centre O of a disk, having a mass m 3 and a mass moment of inertia J O, is connected at one end of the wire. This disk is in contact with an inclined plane at point H. No slippage occurs between disk and supporting plane (f r and f s are the rolling friction coefficient and static friction coefficient, respectively). The opposite stretch of the wire is wrapped around the external surface of a disk, having a mass m 1 and a mass moment of inertia J 1, rigidly mounted on the output shaft of a gearbox, T. Let us denote τ and η the transmission ratio and the efficiency coefficient of the gearbox whose input shaft is the drive shaft of an electrical motor, M. The mass moment of inertia of the motor is J m while the drive torque is M d. At the current time the velocity of the point O is v. The stretch of wire KO is parallel to the inclined plane. All the geometrical parameters are assumed to be known. You are asked to describe the procedure to: 1 evaluate the angular acceleration of the shaft of the electrical motor; 2 evaluate the reaction forces at points A and B, along with the internal forces at point D; 3 evaluate the contact forces at point H. N.B.: You are asked to write the equations that allow one to calculate any unknown kinematic and mechanical parameter that is cited in the expressions of your solution procedure. Exercise n.3 The vibrating system illustrated in Fig. 3 moves in a vertical plane. The disk has a m 1 and a mass moment of inertia J B. An inextensible massless wire is wrapped around the disk. A rigid body of mass m 2, which can only move in vertical direction, is connected to the wire at point K. The centre of the disk is connected to the upper constraint by means of an elastic element of stiffness k and a viscous damper of constant c. The upper constraint moves with the periodic motion: y (t) = Y 1 cos ( Ω 1 t ) + Y 2 cos (3 Ω 1 t ). It is assumed that no slippage occurs between wire and disk. All the geometrical parameters are assumed to be known. You are asked to: 1 write the equation of motion of the system; 2 evaluate the static equilibrium position of the system; 3 evaluate the natural frequency and the damping ratio of the system; 4 write the expression of the system response in the transient and in the steady state of its motion; 5 evaluate the maximum amplitude of the vibration of point K in the steady state; 6 evaluate the force transmitted to the constraint at point A (in the steady state response); 7 explain in which condition the system resonance is excited; 8 show a procedure to evaluate a suitable range of the stiffness k that allows the maximum amplitude of vibration of point K to be mitigated.