MESB 374 System Modeling and Analysis Translational Mechanical System ME 375 – Spring 2003 MESB 374 System Modeling and Analysis Translational Mechanical System

Translational Mechanical Systems ME 375 – Spring 2003 Translational Mechanical Systems Basic (Idealized) Modeling Elements Interconnection Relationships -Physical Laws Derive Equation of Motion (EOM) - SDOF Energy Transfer Series and Parallel Connections Derive Equation of Motion (EOM) - MDOF In this and next class, we will talk about the modeling and analysis of translational mechanical systems. The basic topics are listed here. By the way, section 3.1 of the textbook gives us a good review of basic concepts of mass, weight, force and systems of units. I wish that you may read it. At first, we would like to introduce some basic, idealized modeling elements. It should be kept in mind that we may use bunches of these elements to model a single object in realty. For example, a series of mass, spring and damper may be used to model a single flexible beam. Then we will discuss the physical laws used to describe the interconnection relationship . By using these physic laws, for example Newton’s second law, we may obtain the equation of motion of system To learn it step by step, we firstly discuss systems with single degree of freedom. System with multiple degree freedom will be discussed later. Also, we will discuss the topic of energy transfer and how to simply to model in case of series or parallel connections.

Key Concepts to Remember ME 375 – Spring 2003 Key Concepts to Remember Three primary elements of interest Mass ( inertia ) M Stiffness ( spring ) K Dissipation ( damper ) B Usually we deal with “equivalent” M, K, B Distributed mass  lumped mass Lumped parameters Mass maintains motion Stiffness restores motion Damping eliminates motion Throughout this course, we will always three basic elements, inertia element, or mass, spring element which is used to model the stiffness of a object, element dissipating energy or damper. Usually we use M to denote mass, K to denote stiffness, B to denote damping. For system with single degree of freedom, M, K, B are scalars. For system with multiple degree of freedom, they are matrices. As we discussed in the first lecture, all the system in realty are distributed system. If we want to model them in lumped models, we need to find the equivalent mass, stiffness and damping. Now let us look at the physical meaning of these three elements. At first, what is mass? How will mass affect the motion? How can we measure the mass? Mass maintains the motion. It means that if there is no external force acting on an object, then it will maintain its current situation. If it moves, then it keep on moving at the same speed. If it original stops, then it will never move until a force acts on it. Stiffness can restore the motion. It is related to potential energy. Damping can eliminate the motion, which means to resist the motion of a object. Does it mean that damping eliminate energy? No. we all know that energy can only be changed from one form to another and never be eliminated. In our case, damping usually change the energy to heat (Kinetic Energy) (Potential Energy) (Eliminate Energy ? ) (Absorb Energy )

Variables x : displacement [m] v : velocity [m/sec] ME 375 – Spring 2003 Variables x : displacement [m] v : velocity [m/sec] a : acceleration [m/sec2] f : force [N] p : power [Nm/sec] w : work ( energy ) [Nm] 1 [Nm] = 1 [J] (Joule) Now let’s look at some basic variables used to describing motion. Displacement is usually defined with respect to a reference frame. Velocity is used to describe how fast the displacement changes with respect to time. Mathematically, it is time derivative of displacement. Acceleration is used to describe how fast velocity changes. Hence, it is Vectors, scalars

Basic (Idealized) Modeling Elements ME 375 – Spring 2003 Basic (Idealized) Modeling Elements Spring Stiffness Element Reality 1/3 of the spring mass may be considered into the lumped model. In large displacement operation springs are nonlinear. K Linear spring  nonlinear spring  broken spring !! Idealization Massless No Damping Linear Stores Energy Hard Spring Potential Energy Soft Spring

Basic (Idealized) Modeling Elements ME 375 – Spring 2003 Basic (Idealized) Modeling Elements Damper Friction Element Mass Inertia Element x2 x1 fD B x f2 M f1 f3 Dissipate Energy fD Stores Kinetic Energy

Interconnection Laws Newton’s Second Law Newton’s Third Law ME 375 – Spring 2003 Interconnection Laws Newton’s Second Law Lumped Model of a Flexible Beam K,M x Newton’s Third Law Action & Reaction Forces x K M K M Massless spring E.O.M.

ME 375 – Spring 2003 Modeling Steps Understand System Function, Define Problem, and Identify Input/Output Variables Draw Simplified Schematics Using Basic Elements Develop Mathematical Model (Diff. Eq.) Identify reference point and positive direction. Draw Free-Body-Diagram (FBD) for each basic element. Write Elemental Equations as well as Interconnecting Equations by applying physical laws. (Check: # eq = # unk) Combine Equations by eliminating intermediate variables. Validate Model by Comparing Simulation Results with Physical Measurements

Vertical Single Degree of Freedom (SDOF) System ME 375 – Spring 2003 Vertical Single Degree of Freedom (SDOF) System B,K,M x xs f Define Problem The motion of the object g Input Output Develop Mathematical Model (Diff. Eq.) Identify reference point and positive direction. M K M B g f x M Draw Free-Body-Diagram (FBD) Write Elemental Equations From the undeformed position From the deformed (static equilibrium) position Validate Model by Comparing Simulation Results with Physical Measurement

Energy dissipated by damper ME 375 – Spring 2003 Energy Distribution EOM of a simple Mass-Spring-Damper System We want to look at the energy distribution of the system. How should we start ? Multiply the above equation by the velocity term v : Ü What have we done ? Integrate the second equation w.r.t. time: Ü What are we doing now ? x K M B f Change of kinetic energy Energy dissipated by damper Change of potential energy

Example -- SDOF Suspension (Example) ME 375 – Spring 2003 Example -- SDOF Suspension (Example) Simplified Schematic (neglecting tire model) Suspension System Minimize the effect of the surface roughness of the road on the drivers comfort. From the “absolute zero” From the path From nominal position g M x K B x x p

Series Connection Û Springs in Series K1 K2 x1 x2 fS KEQ K1 K2 x1 x2 ME 375 – Spring 2003 Series Connection Springs in Series K1 K2 x1 x2 fS KEQ Û K1 K2 x1 x2 fS xj fS

Series Connection Û Dampers in Series x1 x2 x1 fD x2 fD fD fD B1 B2 ME 375 – Spring 2003 Series Connection Dampers in Series x1 x2 x1 fD x2 fD Û fD fD B1 B2 BEQ

Parallel Connection Û Springs in Parallel x1 x2 fS K1 K2 x1 x2 fS fS ME 375 – Spring 2003 Parallel Connection Springs in Parallel x1 x2 fS K1 K2 x1 x2 fS Û fS KEQ

Parallel Connection Û Dampers in Parallel x1 x2 x2 fD x1 BEQ B1 B2 fD ME 375 – Spring 2003 Parallel Connection Dampers in Parallel x1 x2 x2 fD x1 BEQ B1 B2 Û fD fD

Horizontal Two Degree of Freedom (TDOF) System ME 375 – Spring 2003 Horizontal Two Degree of Freedom (TDOF) System DOF = 2 K Absolute coordinates FBD K K Newton’s law

Horizontal Two Degree of Freedom (TDOF) System ME 375 – Spring 2003 Horizontal Two Degree of Freedom (TDOF) System K Static coupling Absolute coordinates Relative coordinates Dynamic coupling

Two DOF System – Matrix Form of EOM ME 375 – Spring 2003 Two DOF System – Matrix Form of EOM K K Input vector Absolute coordinates Output vector Mass matrix Damping matrix Relative coordinates Stiffness matrix SYMMETRIC NON-SYMMETRIC

MDOF Suspension Suspension System ME 375 – Spring 2003 MDOF Suspension Simplified Schematic (with tire model) Suspension System TRY THIS x p

MDOF Suspension Suspension System ME 375 – Spring 2003 MDOF Suspension Simplified Schematic (with tire model) Suspension System Assume ref. is when springs are Deflected by weights Car body Suspension Wheel Tire Road Reference

Example -- MDOF Suspension ME 375 – Spring 2003 Example -- MDOF Suspension Draw FBD Apply Interconnection Laws x2 M2 FS2 FD2 FS2 FD2 FS2 FD2 FS2 FD2 x1 M1 FS1 FD1 FS2 FD2 FS1 FD1

Example -- MDOF Suspension ME 375 – Spring 2003 Example -- MDOF Suspension Matrix Form Mass matrix Damping matrix Stiffness matrix Input Vector