Chemical Engineering - Apllied Mechanics

Full exam

APPLIED MECHANICS Study Track on Chemical Engineering July 22 nd, 2015 Exercise n.1 (only for students who have not got a positive score about kinematic and dynamic yet) The mechanical system illustrated in Fig. 1 moves in a vertical plane. A hydraulic actuator (represented by means of: the slotted link hinged at point O to a fixed frame, the piston having barycentre A and the massless link AB) moves the link CD, hinged at point C to the ground. The slotted link, whose barycentre is G 1, has a mass m 1 and a mass moment of inertia J G1. The piston has a mass m 2 and a mass moment of inertia J A. The link CD, whose barycentre is G 3, has a mass m 3 and a mass moment of inertia J D, evaluated about an axis passing through point D. The absolute velocity and acceleration of point G 3, at the current time, are v and a respectively (see Fig. 1). An unknown horizontal resistance force, F, is applied to point D. The kinetic friction coefficient between piston and slotted link BC is f k. The pressure of the fluid contained in the hydraulic actuator is p. All the geometrical parameters are assumed to be known. You are asked to describe the procedure to: 1 evaluate the velocity and acceleration of the barycentre G 1; 2 evaluate the magnitude of the resistance force F applied to point D; 3 evaluate the external reactions at points O and C; 4 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. Students who only passed a previous test about kinematics have only to answer questions from n.2 to n.4 of the exercise n.1 by assuming arbitrary but realistic values of the kinematic parameters that must be taken into account in the dynamic analysis. Students who only passed a previous test about dynamics have only to answer question n.1. Pay attention ! Exercises n.2 and 3 on the back page. Exercise n.2 The mechanical system illustrated in Fig. 2 moves in a vertical plane. A force F is applied to the end of a wire wrapped around a pulley (n.1) mounted on the input shaft of a gearbox whose efficiency coefficient and transmission ratio are η and τ, respectively. The end A of the wire wrapped around a further pulley (n.2), mounted on the output shaft of the gearbox, is connected to a rigid body of mass m 4 that is supported by a slanted plane whose kinetic friction coefficient is f k. The velocity of the mass m 4 is v while the acceleration a is unknown. The pulley grounded at point O has a mass m 3 and a mass moment of inertia J O. The mass moments of inertia of pulley n.1 and n.2 are J 1 and J 2, respectively. The wire AB is parallel to the slanted plane. All the wires are assumed to be massless and inextensible. Besides, no slippage is assumed to occur between wires and pulleys. All the geometrical parameters are assumed to be known. You are asked to describe the procedure to: 1 evaluate the acceleration of the mass m 4; 2 evaluate the reaction forces at point O. Exercise n.3 The underdamped mechanical system illustrated in Fig. 3 moves in a vertical plane. The “L” shaped frame vibrates with a harmonic law: y(t) = Y o cos ( Ω t ). The centre O of the disk is connected to the above mentioned frame, at point A, by means of an elastic element of stiffness k and a viscous damper of constant c. Let us denote as m 1 and J O the mass and mass moment of inertia of the disk, respectively. No slippage is assumed to occur between disk and supporting frame. The rolling friction, which can be assumed to be null, and the static friction coefficients between disk and frame are f v and f s, respectively. 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 damped and undamped natural frequencies; 3 evaluate the dimensionless damping factor; 4 evaluate the static equilibrium position of the system; 5 write the expression of point O displacements in the free-motion and in the steady state condition (show also a qualitative but realistic plot of the time-dependent displacements); 6 to write the maximum amplitude of the steady state response of the bar as a function of the mechanical parameters of the system; 7 show the procedure to verify that that no slippage occurs between disk and supporting frame (in the steady state motion); 8 explain how it is possible to mitigate the vibration amplitude of point O by adjusting the parameter J O.