# 33928 - Machine Mechanics M

• Campus: Bologna
• Corso: Second cycle degree programme (LM) in Mechanical Engineering (cod. 5724)

## Learning outcomes

The student learns topics concerning the modelling and dynamical analysis of mechanical systems, with emphasis on multibody systems and mechanical vibrations.

## Course contents

CONTENTS

The course will provide skills and tools that will enable the student to model, analyze and simulate the dynamic behavior of mechanical systems. In particular, the student will gain fundamental knowledge related, on the one hand, to rigid-body kinematics and dynamics, with particular reference to spatial multi-body systems, on the other to flexible-body dynamics, with particular reference to the vibration of linear systems with many degrees of freedom. The student will also learn how to model and solve applied dynamic problems by using a widely-used engineering software (Matlab).

SYLLABUS

Part I. Rigid-body kinematics.

1. Introductory concepts. Freedoms and constraints: systems, body models, freedoms, constraints, forms of constraints. Vectors, matrices and calculus: vector representation and operations, differentiation, eigenproblems, symmetric matrices. Linearization: linear functions, numerical solution of nonlinear equations, linear ordinary differential equations (ODE), linear ODE with constant coefficients, numerical integration of differential equations.
2. Kinematic fundamentals: finite motions. Coordinate transformations. Rotation matrices: definition, properties of the rotation matrix, elementary rotation matrices, composition of rotation matrices. Description of the orientation: three successive rotations about fixed axes, three successive rotations about body-fixed axes, non-commutativity of finite rotations. Homogeneous transformation matrices: homogeneous transformations, displacement analysis of a 3-dof serial robot, MATLAB implementation.
3. Kinematic fundamentals: infinitesimal motions. Particle kinematics: Cartesian coordinates, intrinsic coordinates. Rigid-body kinematics: angular velocity, vector derivatives with respect to different reference frames, angular acceleration, vector second derivatives with respect to different reference frames, instantaneous kinematics of rigid bodies, MATLAB implementation.
4. Kinematics applications. Rolling constraints: wheel on an axle, integrability of velocity constraint equations. Kinematics of a vehicle in planar motion: tricycle model, bicycle model, steering and instant centre of rotation, MATLAB model. Kinematics of the four-bar linkage: mechanism model, position analysis, differential kinematic analysis, algebraic kinematic analysis, MATLAB model.

Part II. Rigid-body dynamics.

1. Rigid-body dynamics fundamentals. Forces and moments: gravity forces, elastic forces, contact friction forces, aerodynamic and hydrodynamic forces, impulsive forces, resultant force and moment. Momentum and inertia forces: linear and angular momentum, inertia matrix, inertia forces. Laws of motion: dynamic equilibrium of a rigid body, dynamic equilibrium of a multi-body system by free-body diagrams. Power, work and energy: power of forces and moments, conservative forces and potential energy, kinetic energy of a rigid body, work of inertia forces, principle of conservation of energy. Principle of virtual work: virtual work of external forces, conservative and non-conservative forces, principle of virtual work.
Lagrange equations: Lagrange equations for holonomic systems, dynamics of 1-d.o.f. systems.
2. Rigid-body dynamics applications. Dynamics of the four-bar linkage: model, kinetic and potential energy, D’Alembert equations, Lagrange equation, MATLAB model. Dynamics of a ground vehicle in planar motion: single-track model, vehicle kinematics, D’Alembert equations, energy equation, motor torque, MATLAB model. Dynamics of a 3-d.o.f. serial robot: model, kinematic analysis, external and internal forces, inverse dynamics, energetic balance, MATLAB model. Dynamics of a quadcopter: parametrization of the pose, rotor aerodynamics, actuation forces and torques, inverse and forward dynamics, MATLAB model.

Part III. Mechanical vibrations.

1. Mechanical vibrations: 1-dof systems. First-order 1-dof systems: model, free response, response to a constant excitation. Second-order 1-dof systems: model, free response of undamped systems, free response of underdamped systems, damping estimation by logarithmic decrement, response to an impulsive force, step response, response to a general excitation – convolution integral, response to harmonic excitation, response to periodic excitation, harmonic excitation due to imbalances, whirling of rotating shafts – Jeffcott rotor, transmitted force, base excitation, numerical examples.
2. Mechanical vibrations: multi-dof systems. Modeling of multi-d.o.f. systems: elastic and kinetic energy, dissipation function, equations of motion, coupling. Undamped multi-d.o.f. systems: free motion, natural frequencies, modal vectors, general form of the free response, rigid-body modes, orthogonality of the modal vectors, modal equation of motion and modal response. Damped multi-d.o.f. systems: exact solution, state-form formulation, approximate solution. Response to harmonic excitation: multi-d.o.f. systems, 2-d.o.f. systems, modal analysis. Vibration reducing devices: principle of operation, model, dynamic vibration absorber.
3. Mechanical vibrations application: bounce, pitch and roll dynamics of a vehicle (optional). Suspension models: ride properties of a vehicle, classification of suspension models. Quarter-car model: single-dof quarter-car model, wheel hop, two-dof quarter-car model. Half-car model for pitch and bounce: two-dof half-car model for pitch and bounce, natural frequencies and modal vectors, Olley criteria. Half-car model for roll and bounce: two-dof half-car model for roll and bounce, anti-roll bar.

• H. Baruh, APPLIED DYNAMICS, 2015, CRC Press (required main text, in English).
• U. Meneghetti, A. Maggiore, E. Funaioli, LEZIONI DI MECCANICA APPLICATA ALLE MACHINE - TERZA PARTE: DINAMICA E VIBRAZIONI DELLE MACCHINE, 2011, Pàtron Editore (supplementary reading in Italian, mainly for Part III of the Course, "Mechanical vibrations").

## Teaching methods

Classes are based on theoretical lectures with application examples. The application examples deal with problems related to the dynamics of machines, to be solved by means of the Matlab numerical-computing software.

Classes are taught in Italian, though slides are in English.

## Assessment methods

The examination consists of a test in a computer lab. During the lab test, the student will be asked to answer to theoretical questions and to solve numerical problems related to the dynamics of machines by means of the Matlab numerical-computing software.

According to the University Regulations, each examination determines a positive or negative evaluation. A positive assessment involves the assignment of a numeric grade equal to or greater than 18/30. A negative assessment does not result in a numeric grade, but in a descriptor (i.e. withdrawn or rejected) reported on the examination record. The negative evaluation is not included in the student academic record and it does not affect the average score that determines her or his final degree grade. The examination, if passed with a positive outcome, cannot be repeated.

If a positive grade does not meet the expectations of a student, the latter may ask its cancellation and the repetition of the examination. The instructor complies, in general, with the following rules:

• a student may request the cancellation of a positive grade only within the date specified in the public communication of the examination outcome;
• a cancelled positive grade can in no way be recovered;
• a student may request the cancellation of a positive grade at most once.

## Teaching tools

On the website Virtual Learning Environment Platform , students may find:

- slides of the course lectures (in English);
- exercises and application examples;
- exam problems.