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# Dynamic Systems with Vibrations MECH 330

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This 3 page Class Notes was uploaded by Antonia Quitzon on Tuesday October 13, 2015. The Class Notes belongs to MECH 330 at Kettering University taught by Pinhas Barak in Fall. Since its upload, it has received 134 views. For similar materials see /class/222303/mech-330-kettering-university in Mechanical Engineering at Kettering University.

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Date Created: 10/13/15

2005 Catalog Data Textbook References Coordinator MECH 330 Dynamic Systems I Credit 404 FourLectureHours Prerequisites MATH204 Differential Equations amp Laplace Transforms MECH310 Mechanics III MECH23l Signal Analysis for Mechanical Systems OR EE210 Circuits I OR EE212 Applied Electrical Circuits MECH IYLD ign i 39 39 r r I MECH322 Fluid Mechanics MATH305 Numerical Methods amp Matrices OR MATH307 Matrix Algebra Corequisites A study of mathematical modeling of mechanical electrical hydraulic and multidiscipline engineering system using bondgraphtechnique yielding state space equations Derivation of the Equations of Motion EOM of single Degree of Freedom SDOF and 2DOF using Lagrange Equation andor Newton Second Law NSL Determine transfer functions and frequency transfer function response for first and second order systems A study of linear mechanical vibrations for SDOF and 2DOF systems and of their vibration isolation Determine characteristic equation stability eigenvalues of systems Develop computer code in order to simulate analyze real engineering systems in the time and frequency domain using Matlab Barak P Mathematical Modeling ofMechanical and Multidiscipline Systems John Wiley amp Sons Inc Kamop Dean C Margoles Donald L and Rosenberg Ronald C System Dynamics iModeling and Simulation of M echatronic Systems 3rd Edition John Wiley amp Sons Thomson William T and Dillou Dableh Marie Theory of Vibration with Applications 5Lh Edition Prentice Hall Strum and Ward Laplace Transform Solution of Di erential Equations PrenticeHall Kreyszig Advanced Engineering Mathematics John Wiley amp Sons Inc Pinhas Barak Room 2217 CS Mott Engineering amp Science Center 810 7627840 pbarakketteringedu l Course Learning Objectives Upon completion of this course Dynamics Systems I the student will be able to N E 4 V39 0 gt1 9 0 identify system components their symbols terminology attributes constitutive equations and interactions based on a uni ed approach ME PO s A C P model mechanical electrical hydraulic and multidiscipline systems using bondgraph technique ME PO s A B C E P S derive the Equations of Motion EOM in state space form from bondgraph models of mechanical electrical hydraulic and multidiscipline systems with MultiInputMulti Output MIMO variablesME PO s A B C L S derive the equations of motion of Single Degree of Freedom SDOF and a 2DOF mechanical system using Lagrange equation andor Newton Second Law NSLME PO A determine transfer functions of first and second order systems using Laplace transformation pairs from the tdomain to the sdomainME PO A derive the characteristic equation of a first and second order system solve for the eigenvalues natural frequency if any and evaluate the stability of the systemME PO s A B C L S estimate the value of a function ft at t gt0 and t gtoo using the Initial Value Theorem IVT and the Final Value Theorem FVT ME PO A evaluate the time response to deterministic inputs for first and second order systems ME PO s A B C E J investigate and analyze the vibration isolation of SDOF and 2DOF mechanical systems in the time and frequency domain ME PO s A B C E S develop a computer code to simulate and analyze and design real engineering systems using Matlab softwareME PO s A B C E L P S take the second course in systems engineering entitled MECH 430 7 Dynamic Systems 11 dealing with more advanced multidomain systems and closedloop control systems Prerequisites by Topic 1 Differential equations 2 Laplace Transform and it s application to the solution of differential equations 3 Electrical circuit charge force potential current capacitors resistors and inductors 4 Fundamentals of mechanical and mechatronics design sensors and actuators 5 Principle of superposition for linear system and application of Cramer s Rule to the solution of a set of linear algebraic equations 6 Application and basics of Newtonian mechanics and physical laws 7 Kinematics and kinetics of a particle and a rigid body in 2D 8 Concept of kinetic and potential energy Corequisites by Topic Matrices and determinants and application to the solution of linear systems 2 Fluid motion uid properties ow regimes pressure variation Topics Covered Week 1 2 Topic I J quot A Uni ed Approach 7 Basic concepts and Governing metinn CIR components the tetrahedron of state examples Bond Graph Technique 7 Basic Components Model Nine nodes constitutive equations generic mechanical electrical hydraulic signals active bonds causality assignment examples Making a BondGraph Model 39 39 39 T 39 quot 39 and quot 39 Examples Making a BondGraph Model Electrical Hydraulic and 39 quot39 quot 39 quot Examples State Space Formulation Introduction standard form for linear system equations causality assignment to systems augmented bond graph state energy variables input power variables state vector input vector derivative causality cases examples State Space Formulation continued Derivation of state space equations in physical terms for mechanical electrical hydraulic and multidiscipline systems Derivation of the Linear Time Invariant LTI input and output equations Identi cation of the ABC and D matrices from the LTI equations examples Transfer FunctionTransfer Matrix and Time Response to Step Input of Low Order Systems Introduction Laplace transform pairs splane characteristic equation via the Amatrix transfer function for Single Input Single Output SISO and MIMO systems Characteristic equation via transfer function for first and second order system Attributes of First Order System Stability eigenvalue time constant time response to a unit step input examples Attributes of a Second Order System Stability natural frequency damping ratio eigenvalues stability via eigenvalues underdamped UD critically damped CD overdamped OD and undamped UND Location of the Eigenvalues in SPlane for the Four Cases UD CD OD and UND Effect of the type of eigenvalues on the time response to a unit step input for secondorder system Initial Value Theorem IVT and Final Value Theorem FVT Identify transient and steady state response for first and second order systems examples Steadv State Frequencv Transfer Function Time response to a sinusoidal input with forcing frequency Transmissibility ratio amplitude ratio second order mechanical system vibration analysis transmissibility ratio first order system electrical system examples Computer Simulation using Matlab Laboratory sessions application examples F 39 39 in Linear 39 39 39 Vibrations SDOF SDOF 7 Mechanical System deriving the EOM using NSL and Lagrange equation Classification of a SDOF vibration free damped vibration forced undamped vibration forced damped vibration logarithmic decrement transfer function characteristic 3

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