URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=4DA00&collegejaar=2025&taal=enOmschrijving
In the Engineering Dynamics part of this course, concepts are introduced for the derivation and analysis of theoretical models, which describe the in-plane dynamics of simple mechanical systems.The dynamic behavior of these systems can be described by the equation of motion, the work-energy equation, or the linear/angular impulse-momentum equation. All three equations are based on Newton's second law. Depending on the problem, one of these equations may be preferred over the others.
The three types of equations are discussed for mechanichal systems that are modeled as particles, systems of particles, and rigid bodies.
Furthermore, attention is paid to the solutions of the equations of motion. In particular, the free vibrational behavior and forced vibrations of harmonically excited systems are analyzed.
In the Mathematics part of this course, three subjects are treated. Introduction linear algebra: matrix and vector computations, solving systems of linear equations, the determinant of a square matrix, and the inverse of a square matrix. Complex numbers: elementary computations with complex numbers, both in Cartesian and polar coordinates, and the geometric meaning of these computations in the complex plane, the complex exponential function, determination of all complex zeroes of a real polynomial. Differential equations: solving second order linear differential equations with constant coefficients, both in the homogeneous and inhomogeneous case.
Doelstellingen
General learning objectives- The student can derive dynamic and kinematic models of simple mechanical systems in the plane. Using a dynamic model, the student can obtain insight in the dynamic behavior of the mechanical system and solve simple dynamics problems.
- The student can apply mathematical concepts in dynamics problems. The student can solve sets of linear equations, use complex numbers, and solve linear homogeneous and inhomogeneous ordinary differential equations of second order, which describe the free and forced vibrational behavior of a dynamic mechanical system.
The simple mechanical systems in the plane, of which the dynamic behavior is modelled and analyzed, are (in increasing order of difficulty): a point mass (or particle), a system of point masses, and a rigid body. Three different concepts for describing dynamic models are used (all based on Newton’s second law): the equations of motion, the work-energy equation, and the impulse-momentum equations. Now, the specific learning objectives of the Engineering Dynamics part of the course can be formulated.
Point mass/particle:
- The student learns under which conditions he may model a mechanical system as a point mass.
- The student learns the relations between a position vector, a velocity vector, and an accelerations vector. If one of the vectors is known, the student can derive the two others.
- The student can describe the motion of a point mass in the plane using the following coordinate systems (kinematics): Cartesian (absolute and relative) coordinates, normal and tangential coordinates, and polar coordinates.
- The student learns to model several types of forces and can calculate these forces: gravity, normal force, Coulomb friction force, elastic spring force, and viscous damping force.
- The student can use his knowledge on forces to draw a Free-body diagram for a point mass. The student can combine Newton’s second law and the Free-body diagram to derive the equations of motion for a point mass.
- The student is introduced to the quantities work, kinetic energy, and potential energy (related to gravity and elastic forces) and can calculate these quantities. The student learns how these quantities can be combined with Newton’s second law to derive the work-energy equation, and can formulate this equation for a point mass.
- The student is introduced to the quantities linear momentum, linear impulse, angular momentum, and angular impulse, and can calculate these quantities. The student learns how these quantities can be combined with Newton’s second law to derive the (linear and angular) Impulse-momentum equations for a point mass, and can formulate this equation for a point mass.
- The student learns under which conditions he may model a mechanical system as a system of point masses. The student can calculate the center of mass of a system of point masses.
- By generalization of Newton’s second law and by using a Free-body diagram, the student can derive the equations of motion for the translational motion of a system of point masses. Using the moment-angular-momentum relation, the student can derive the equation of motion for the rotational motion.
- The student can derive the work-energy equation for a system of point masses.
- The student can derive the (linear and angular) impulse-momentum equations for a system of point masses.
- The student learns under which conditions he may model a mechanical system as a rigid body in the plane.
- Using kinematic relations, the student can describe the following motions of a rigid body in the plane: rectilinear and curvilinear translation, rotation, and general plane motion. He can apply the relative-velocity equation and the relative-acceleration equation and determine the instantaneous center of zero velocity.
- The student can calculate the mass moment of inertia of a rigid body about an axis normal to the plane and (if necessary) use the parallel-axis theorem for this. The student can calculate the center of mass of a rigid body in the plane.
- Using a Free-body diagram, the student can derive the equations of motion of a rigid body for all possible motions in the plane.
- The student can derive the work-energy equation for a rigid body in the plane after extending his knowledge on the earlier mentioned quantities work, kinetic energy, and potential energy.
- The student can calculate the (linear and angular) momentum of a rigid body in the plane. The student can derive the (linear and angular) impulse-momentum equations of a rigid body in the plane.
- The student can apply the equations of motion, the work-energy equation, or the impulse-momentum equations, to solve a simple dynamics problem.
- The student gains insight in which situations use of the equations of motion, use of the work-energy equation, or use of the impulse-momentum equation is preferred.
- The student can calculate the (un)damped eigenfrequency and the dimensionless damping coefficient of a single degree of freedom mechanical system. The student learns the concepts undercritical, critical, and overcritical damping. The student can analyze the free vibrational behavior of a single degree of freedom mechanical system.
- The student can analyze the forced vibrational behavior of a(n) (un)damped single degree of freedom mechanical system under harmonic excitation (both in the time domain and in the frequency domain).
Linear Algebra:
- Be able to carry out elementary matrix and vector computations, such as sum, product, difference, and transposition.
- Be able to solve a system of linear equations by writing it in matrix-vector form, and using the Gauss-Jordan elimination algorithm to obtain the row reduced echelon form. Be able to provide a parameter description of the general solution based on the row reduced echelon form.
- Know the definition of the inverse of a square matrix, and be able to use the inverse for solving systems of linear equations.
- Be able to compute the inverse of a square matrix, using the Gauss-Jordan algorithm.
- Be able to compute the determinant of a square matrix, and to interpret the meaning of the result for 2-by-2 and 3-by-3 matrices.
- Be able to check the invertibility of a square matrix, using its determinant or row reduced echelon form.
- Be able to carry out elementary computations with complex numbers, such as sum, difference, product, quotient, real and imaginary part, complex conjugate, modulus, and argument. Know the geometric meaning of these operations in the complex plane.
- Be able to carry out the conversion of complex numbers from Cartesian coordinates to polar coordinates and vice-versa.
- Know the definition of the complex exponential function, and be able to apply it for finding all complex n-roots of a given complex number.
- Be able to determine all complex zeros of a real polynomial of degree 2, and, in simple cases, also for polynomials of degree 3 and higher.
- Be able to find the solution of a second order linear differential equation with constant coefficients, both in the homogeneous case and for simple inhomogeneous cases.
Learning objectives for design
- The student can theoretically model the in-plane dynamics of a simple mechanical system using the equation of motion, the work-energy equation, or the linear/angular impulse and momentum equation. The student is able to characterize the dynamic properties of the system model based on its eigenvalues.
- This model can be used to accurately predict the dynamic behaviour of a design or prototype of a simple mechanical system. In this way, the student can assess the design specifications/requirements for the mechanical system related to its dynamic behavior