1. ME 440 Intermediate Vibrations Tu, Feb. 3, 2009 Sections 2.1-2.2, 2.6-2.8 © Dan Negrut, 2009 ME440, UW-Madison

2. Before we get started… Last Time: Frequency Spectrum Complex form of the Fourier Series Expansion Went through three examples Last one was the propeller blade example Today: Covering material out 2.1-2.2, potentially 2.6-2.8 HW Assigned: 2.35, 2.45, 2.69 out of the book 2

3. Chapter 2: What Are We Up To? Title of Chapter captures the essence: Free Vibration of Single Degree of Freedom Systems “Free” means that there is no forcing term What makes the system vibrate? Motion is due to a set of nonzero initial conditions: 3 “Single Degree of Freedom” As simple as it gets… Dealing with a system like one in figure (a) System in (b) already has two degrees of freedom…

4. What are we up to? (Cntd) Understand how to derive equations of motion (EOM) for 1DOF systems Once we’ll have the EOMs, we’ll solve them First then, how do we derive the EOM? Newton’ Second Law Always works, our workhorse. Abbreviated N2L. Work and Energy Methods Very handy for conservative systems Lagrange’s Equation and Hamilton’s Principle Will get limited mileage in this course, used a lot in Physics 4

5. Kinetic and Potential Energy: Conservation of energy Key assumption for conservation of energy: We are dealing with a conservative system Force acting on the system derived from a potential U(x) 5

6. Newton’s 2 nd Law - Translation  is the sum of external forces acting on the rigid body Newton’s Second Law is the acceleration of the center of mass (CM) of the rigid body relative to a fixed point in space m is the mass of rigid body x y z 6

7. Newton’s 2nd Law – Rotation About a Fixed Axis Assumes that axis of rotation is fixed in space M 0 is the sum of external moments acting about rotation axis I 0 is the mass moment of inertia about the rotation axis (units kg-m 2 ) Notation: Newton’s 2 nd Law Applied to Rotation: Kinetic Energy: 7

8. Mass Moment of Inertia Parallel-Axis Theorem Sphere Mass rotating about point 0 Hollow cylinder Rectangular Prism 8

9. General Planar Motion A body moving in a plan is described by a set of 3 DOFs Motion in the x direction Motion in the y direction Rotation of angle  Equations of motion (EOM): G – is center of mass (important!!!) M G – total torque about point G A consequence of Newton’s 2 nd Law 9

10. General Planar Motion (continued) The previous quantities are defined as: The vector M G is the total torque: 10

11. Energy Approach to Derivation of EOM For a conservative system, conservation of energy leads to 11 For a nonconservative system,

12. Conservative systems A system in which only conservative forces are active. Conservative forces are derived from a potential energy function U(x): Here the x represents the position where the force is to be calculated One consequence of Newton’s second Law for conservative systems: Sum of kinetic and potential energy is constant (conservation of energy) 12

13. Example m x f 1) What are the names of the following force functions? 2) Which ones represent conservative forces? a) f = -bv b) f = -kx c) f = -µmg (v>0) = µmg (v<0) d) f = -kx 2 e) f = C 13

14. Lagrange’s Equations Assume system has ndof degrees of freedom; array of generalized coordinates is 14 The second order differential equation that captures the time evolution of each degree of freedom q j is obtained as follows:

15. Hamilton’s Principle Gets some mileage particularly in elastodynamics 15

16. Example (N2L) Determine the EOM for solid cylinder. Assume rolling without slip. 16

17. EOM determined following four steps: STEP 1: Identify the displacement variable of interest STEP 2: Write down kinematic constraints (if present) STEP 3: Get equivalent mass/moment of inertia Equate kinetic energy between actual system and the simplified 1-DOF system in terms of the displacement variable of interest 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 Deriving the Equations of Motion (EOM) for One-DOF Systems (340 Vintage) 17

18. Example (revisited) Determine the EOM for solid cylinder using the equivalent mass approach 18

19. Short Excursion: A Word on the Solution of Ordinary Differential Equations Classical analytic techniques Laplace transforms Numerical solution Usually found using MATLAB, or some other software package (Maple, EES, Sundials, etc.) MATLAB demonstrated later in this lecture (or next…) 19