Assoc. Prof. František Palèák, PhD., ÚAMM 02010
2-5520 Theory of Mechanisms
for bachelors study in 2rd year-classis, winter semester
Course description
Newton and Euler dynamics of a particle and rigid body. Structural properties of multibody systems. Velocity acceleration, angular velocity and acceleration relative to a fixed and moving frame. Principles of linear and angular momentum balance and work-energy. Examples include particle ballistics , pendula, wheels, pulleys, gears, mass-spring-dashpot, mechanisms, dynamic balance and vehicle dynamics.
Course objectives
For students to learn how to use particle and rigid body kinematics and dynamics to make predictions about the forces and motions of objects and multibody systems in machines.
- Importance of dynamics for engineering practice. Historical overview about development of dynamics. Intergration of statics and dynamics (kinematics and kinetics) presented by quantities expressed as couples of line vectors (resultant force , resultant moment, or torque ), (instataneous angular velocity of a body, instataneous velocity of a bodys’s point ), (linear momentum of a body, angular momentum of a body as moment of momentum ). Parameters of multibody systems structures. Local and global position coordinates of links, local mobility of body in geometrical constraint. Mobility of correct and incorrect mechanisms.
lecture_tm_II_1.pdf
- Kinematics and dynamics of a particle. Unit vectors of a body’s local position frame with origin moving along prescribed curve. Instantaneous velocity, tangential and normal accelerations of a body’s point. Euler’s equation for velocity of a point of rotating body. Kinematic quantities for translational and rotational motion of a body. Dynamics of a particle. Newton’s laws. Principles of work-energy, linear and angular impulse-momentum for a particle. Relative and constrained motions of a particle. Free-body diagrams.
lecture_tm_II_2.pdf
- Principle of linear impulse and momentum for a system of particles. Vibrations of single-dof linear mechanical systems. Undamped free and forced vibration. Viscous damped free vibration.
- Forced vibrations of single-dof linear mechanical system with viscous damping, with harmonic excitation, with uniform, or non uniform amplitude of excitation force, with kinematic harmonic excitation. Transmission of vibrations into ground.
- Part 1 of planar rigid-body kinematics. Instantaneous slew centre (OSO) of a body. Replacement of general motion of a body by rolling of movable centrode against fixed centrode. Application of principle of mutually rolling centrodes in mechanisms. Dynamics of translational and rotational motion of a rigid-body.
lecture_tm_II_5.pdf
- Moments and deviation moments of inertia. Ellipsoid of inertia. Dynamic reactions in bearings (e.g. geometrics constraints, joints) of a shaft. Counterbalance of shafts, critical rpm of a shaft.
- Part 2 of planar rigid-body kinematics. Poissont‘s decomposition of general planar motion of the body to the fictive translation represented by reference point and to the fictive rotation about reference point. Development of general formula for time derivation of the vector in different space like is expressed. Dynamics of general planar motion of a rigid-body.
- Part 3 of planar rigid-body kinematics. Decomposition of the body general planar motion to the simultaneous fictive carying motion and fictive local relative motion. (body motion with respect to rotating axes). Resulting angular velocity of the body in the multibody system during simultaneous motions. Resal’s and resulting angular acceleration of the body in the multibody system during simultaneous motions. Resulting velocity of the body in the multibody system during simultaneous motions. Coriolis’es and resulting acceleration of the body in the multibody system during simultaneous motions.
- Turning and centering acceleration of the point during spherical motion of the body. Euler’s angles for precession, nutation and local rotation. Euler’s kinematics equations, applications of the spherical motion in mechanisms. Dynamics of spherical motion of a rigid-body. Gyroscopic moment. Force-free and heavy gyroscope. Utilization of gyroscopic effect in engineering.
- Poissont‘s decomposition of general spatial motion of a body to the fictive translation of a body represented by reference point and to the fictive spherical motion of a body about reference point. Description of general spatial motion of a body by instantaneous tangential screw motion of a body wrt axis of viration. Applications of general spatial motion of a body in mechanisms. Dynamics of general spatial motion of a rigid-body.
- Dynamics of multibody systems. Lagrange’s equations. Principle of virtual work.
- Collision mechanics, direct, skew and eccentric impact.
- Dynamics of bodies with varying mass. Application on motion of racket.
Lab topics
1 DOF vibrations, normal modes, slider-crank kinematics, and dynamics, counterbalance and gyroscope phenomenology.
Class/laboratory schedule
Each week two 50-minute lectures and one 50-minute recitation (14 weeks). Two 50-minute excercises (13 weeks). One 2 hours lab during the semester.
Contribution of course to meeting the professional component
Basic engineering sciences with experimental experience.
Course outcomes
Upon completion of the course, students should be able to:
- Draw free-body diagrams, distinguishing forces from inertial effects.
- Describe particle motion in 1-D, 2-D and 3-D employing Cartesian and path coordinates, and moving reference frames.
- Characterize 2-D and 3-D kinematics of rigid bodies, including vector angular velocities and accelerations.
- Apply Newton/Euler laws to the motion of particles and rigid bodies
- Use the principles of linear/angular impulse-momentum and work-energy to solve dynamics problems.
- Recognize simple harmonic motion for 1-degree-of-freedom mechanical systems.
Outcome Assessment
The instructor will assess the six outcomes of the course by considering student results in specific questions on homeworks, exams and laboratory reports.
Literature
- Ruina, A., Pratap, R.: Introduction to Static and Dynamic, Cornell University, Oxford University press, 2002
- Meriam, J.L., Kraige L.G.: Engineering Mechanics: Dynamics, John Wiley & Sons, Fifth Edition, 2002
- Hibbeler, R.C.: Engineering Mechanics: Statics and Dynamics Prentice-Hall, Englewood Cliffs, Seventh Edition, 1995
