ME 440 Intermediate Vibrations June 16, 2018 ME 440 Intermediate Vibrations Th, January 22, 2009 M1051 ECB 2, 2hr EES seminars: 2-4 pm on Jan 20 4-6 pm on Jan 20 © Dan Negrut, 2009 ME440, UW-Madison

Before we get started… Last Time: Today: ME440 Logistics June 16, 2018 Before we get started… Last Time: ME440 Logistics Syllabus Grading scheme Quick overview of topics covered in ME440 Started Chapter 1, “Fundamentals of Vibrations” Today: HW Assigned: 1.9 + problem stated in today’s presentation + problem stated in handout (to be emailed to you) HW due in one week Covering material out of 1.8, 1.9, 1.10 Equivalent Mass, Damping Elements, Periodic Functions 2

Example, Equivalent Spring June 16, 2018 Example, Equivalent Spring Find the equivalent spring 3

Example, Equivalent Spring June 16, 2018 Example, Equivalent Spring Find the equivalent spring in the direction of  4

Nonlinear Springs What if you are dealing with nonlinear springs? June 16, 2018 Nonlinear Springs What if you are dealing with nonlinear springs? Go back to the linear spring case by carrying out a local linearization Works only when the deformation of the spring is small (because you linearize about a point, see next slides…) This idea of linearization goes way beyond linearizing springs, apply it to any function Do you always *have* to linearize? No, but then might not be able to find analytical solution to your problem Fall back on numerical method for finding approximation of solution In 440 though, we try to keep things linear 5

June 16, 2018 Linearity, We Like You… A linear process is a process in which there is a linear dependency between the input and output Most of the processes in nature are nonlinear Linear systems are easy to deal with, and over the years the engineering community got a good understanding of this type of systems 6

Linearization How one goes about performing a “Linearization” June 16, 2018 Linearization How one goes about performing a “Linearization” At the system level: assumptions are made to allow application of laws that lead to a linear set of equations At the equation level: the complicated equations that govern the physics associated with the model are linearized (simplified) Linearization of the equations requires linearization of functions that show up in the equations Where should you linearize? At system or equation level? Linearize as late as possible in the solution process V = i R. 7

Function Linearization: how to get it June 16, 2018 Function Linearization: how to get it We like linear functions. How do we get one? Intuitively: approximate the dependency through a straight line 8

Function Linearization: how to get it June 16, 2018 Function Linearization: how to get it Mathematically speaking… Two concepts are important: 1) Place of linearization r (where you carry out linearization) 2) The Taylor series expansion of the function f The linearization of f at r is simply: 9

June 16, 2018 Example Linearize sin() at r=/4 10

New Topic: Inertia (Mass) Elements June 16, 2018 New Topic: Inertia (Mass) Elements Discrete masses: Point mass Has translation only, therefore kinetic energy is Rigid body Has both translation and rotation, therefore kinetic energy is 11

June 16, 2018 Equivalent Mass For systems with 1DOF, equivalent mass is something that conceptually is very similar to the equivalent spring idea: What would be the mass, associated with a generalized coordinate that characterizes the DOF of the system, that would lead to a kinetic energy identical to that of the actual system? Why do you keep asking questions like the one above? You’ll get a straight answer in Chapter 6, when we’ll talk about Lagrange’s equations 12

Deriving the Equations of Motion (EOM) for One-DOF Systems June 16, 2018 Deriving the Equations of Motion (EOM) for One-DOF Systems EOM determined following four steps: STEP 1: Identify the displacement variable of interest STEP 2: Write down the defining kinematic constraints STEP 3: Get equivalent mass/moment of inertia Kinetic energy of actual system and that of the simplified 1-DOF system (expressed in terms of the time derivative of displacement variable of interest) should be the same Step 4: Get equivalent force/torque Equate virtual power between actual system and the simplified 1-DOF system in terms of the displacement variable of interest Discussion serves as justification for need of equivalent mass. Examples covered later in the course. 13

Example (from textbook): June 16, 2018 Example (from textbook): Start with a system with four masses (see (a)). Find the equivalent mass as in (b) Note the generalized coordinate that is used in conjunction with the equivalent mass 14

Another Example (from textbook): June 16, 2018 Another Example (from textbook): For the 1DOF system below, find it’s equivalent mass If the generalized coordinate that captures this degree of freedom is the angle  If the generalized coordinate that captures this degree of freedom is the displacement x 15

New Topic: Damping Elements June 16, 2018 New Topic: Damping Elements In real life, systems don’t vibrate forever, or if they do, there should be something pumping energy into the system Energy initially associated with an oscillatory motion is gradually converted to heat and/or sound This mechanism is known as damping Most common damping mechanism: Viscous Damping Coulomb friction Material or Solid or Hysteretic Damping 16

June 16, 2018 Viscous Damping (VD) Experienced by systems vibrating in a fluid medium such as air, water, oil Resistance offered by the fluid to the moving body causes energy to be dissipated Amount of energy dissipated depends on: Fluid viscosity Vibration frequency Relative velocity of the vibrating body wrt that of the fluid Typically damping force is proportional to relative velocity Shape (geometry) characteristics 17

VD Between Parallel Plates June 16, 2018 VD Between Parallel Plates The most common damping force expression: Linear form, c is a constant coefficient, v is relative velocity Why this expression? Justify its use for damping force acting between two plates with relative motion, viscous fluid in between 18

VD Between Parallel Plates (Contd) June 16, 2018 VD Between Parallel Plates (Contd) Symbols used:  – fluid viscosity  – shear stress dev. in the fluid layer at a distance y of the fixed plate v – plate relative horizontal velocity; no velocity in the vertical direction u – velocity of intermediate fluid layers; assumed to change linearly 19

Homework Problem (due on Jan. 29) June 16, 2018 Homework Problem (due on Jan. 29) Read Example 1.39 in the textbook. It shows that based on some assumptions, the damping coefficient can be computed as (see figure below): For the assignment: 1) List all the assumptions that you believe were made in order to arrive to expression of c reported above 2) Which one of these assumptions seems to be the weakest of them all (which is the weakest link in the argument that leads to this value of the damp. coeff. c)? 20

Damping through Friction June 16, 2018 Damping through Friction More precisely, through Coulomb friction Several other friction models are in use beside Coulomb friction (see, for instance, LuGre model1) We’ll stick to the Coulomb model Damping force is constant in magnitude and opposite to relative velocity between bodies in contact Proportional to the normal contact force between bodies Caused by rubbing surfaces that are dry or without sufficient lubrication 1: Nice paper on LuGre model implementation for numerical simulation available at: http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCNDDM000002000004000281000001&idtype=cvips&gifs=yes 21

Damping through Friction (Cntd) June 16, 2018 Damping through Friction (Cntd) Equations of Motion for Friction here… FBD: 22

Damping through Friction (Cntd) June 16, 2018 Damping through Friction (Cntd) Finding the solution is not difficult, but tricky We’ll see in Chapter 2 that it should assume an expression of the form: The solution looks like that for half of period, then A and B change, since the direction of the force changes… Revisited in chapter 2. 23

Material of Solid or Hysteretic Damping June 16, 2018 Material of Solid or Hysteretic Damping Materials are deformed, energy is absorbed and dissipated by the material Friction between internal planes, which slip and slide as the deformations take place Stress-strain diagram shows hysteresis loop, i.e., Area of this loop denotes energy lost per cycle due to damping Rubber-like materials do this without permanent deformation 24