Dynamics and Vibrations





Planar kinematics for particle motion: Student should be able to use Cartesian, polar and path-coordinate kinematics to define the velocity and acceleration components of a material point in motion.  Student will learn to use coordinate transformations to shift back and forth between the three coordinate systems (Cartesian, polar and path). Student should be able to mathematically differentiate functions of time and space coordinates to determine desired functional forms.


Physical modeling of particle dynamics (1 DOF):   You should be able to identify the fundamental components of mechanical systems into generalized lumped mass (inertia) M, stiffness K, damping C elements.  Determine the degrees of freedom and/or the constraints present on the system.  Establish the equivalence of Kinetic and Potential (Strain) Energies in Conservative systems. You should be able to derive the fundamental equations governing the motion of lumped-parameter (1 DOF and 2 DOF) mechanical systems in general plane motion.  Fundamental knowledge of the kinematics and kinetics of planar rigid body motion: rectilinear motion and rotational motion about a rigid axis. Concepts of relative velocity and acceleration should be mastered.


Mathematical modeling of 1 DOF mechanical systems:  Student should be able to determine analytically the dynamic response (Solutions) of 1DOF systems described by the linear ODE and given initial conditions. Be able to explain the concept of natural frequency wn. Determine the free (transient) response to initial conditions and the dynamic response to Impulse and Step loads. Be able to discuss the concepts of transient and steady state responses, and the effect of viscous damping ratio (and logarithmic decrement) on the amplitude and decay speed of system response.  Derive the dynamic response to periodic (harmonic) external forcing functions and discuss about the regimes of operation: below, close to, or above its natural frequency. Be able to obtain the Frequency Response Function (FRF) for sustained periodic excitations and explain the effects of system parameters and frequency on the Amplitude of motion and Phase lag.  Use FRF for appropriate design considerations and reliable operation of vibrating systems.


Mathematical modeling of 2 DOF mechanical systems:  Student should be able to derive the EOMS for 2- or M-DOF lumped parameter systems. You should be able to linearize the EOMs about an equilibrium or operating point and determine the linear system of ODEs: . For undamped 2-DOF systems Student should be able to determine analytically the eigenvalues and eigenvectors of . Be able to explain the concept of modal (natural) coordinates and mode shapes. Student should be able to use the transformation  to uncouple the EOMS in physical coordinates and determine (analytically) the free and forced response of 2-DOF systems to arbitrary initial conditions, step and periodic loads.


Numerical modeling of mechanical systems: Student should be able to use computational software to solve linear and nonlinear algebraic and differential equations describing the motion of 1- or M-DOF systems. You should be able to apply knowledge gained in MEEN 357 to select appropriate numerical techniques with due consideration for time steps and procedures (algorithms) ensuring accurate, numerically stable, and cost efficient system response. Student should be able to interpret numerical calculations (predictions) to explain system behavior (motion), identify possible failure mechanisms due to excessive amplitudes of motion or reaction forces, etc.


Luis San Andrés

Course Instructor