ME 440 Intermediate Vibrations

1. ME 440Intermediate Vibrations Th, Feb. 12, 2009 Sections 2.3, 2.6.5 © Dan Negrut, 2009ME440, UW-Madison

2. Before we get started… • Last Time: • Logarithmic Decrement • Coulomb Friction • Example, 1DOF motion • Today: • HW Assigned: 2.71, 2.80 (due on Feb. 19) • For 2.71, assume that at point O there is a revolute joint between the bar and ground. Also, assume that motion is in the horizontal plane • Examples • Motion of pendulum • Torsional vibration • Rayleigh’s method 2

3. New Topic:Inverted Pendulum • Consider uniform rigid bar pivoted at one end • Center of gravity indicated in figure • Using N2L: • Can’t digest this (nonlinear), assume small oscillations to linearize… 3

4. Inverted Pendulum (Cntd) • Three cases identified based on the sign of coefficient of  • Case 1: • Characteristic Equation (CE) has two imaginary solutions (conjugate) • Motion assumes form of stable oscillations, A1 and A2 determined from ICs: • Natural frequency: 4

5. Remarks: • 1) This form of solution suggests that as t !1,  keep growing • 2) Note that if , then pendulum doesn’t move from initial configuration: Inverted Pendulum – Case 2: • EOM reduces to • Solution assumes form • Use ICs to get C1 and C2 : 5

6. Consequently, solution must look like Inverted Pendulum – Case 3: • Define • Characteristic Equation (CE) becomes: • Solve for B1 and B2 based on ICs to get • Solution suggests that (t) increases exponentially with time leading to unstable motion • Essentially, the restoring moment of the springs is less than the nonrestoring moment due to gravity… 6

7. J0 – mass moment of inertia about z axis • For displacement  the restoring torque (due to weight) is Moving on to:Compound Pendulum: EOM • Compound Pendulum: Any rigid body pivoted at a point O other than its center of mass G • Body oscillates about pivot point under its own gravitation force • EOM: • Small Oscillation Assumption leads to: • Resulting natural frequency: 7

8. Comparison:Compound Pendulum vs. Simple Pendulum • EOM: • Small Oscillation Assumption leads to: • Question of Interest: • What should be the length L of the simple pendulum so that its natural frequency is identical to that of the compound pendulum? 8

9. New Topic:Free Vibration of an Undamped Torsional Spring • Framework, general considerations: • If a rigid body oscillates about a specific reference axis, the resulting motion is called torsional vibration. • The displacement is measured in terms of an angular coordinate. • The restoring moment may be due to the torsion of an elastic member or to the unbalanced moment of a force or couple. • Examples: • Torsion pendulum (restoring moment: torsion due to elastic member) • Compound pendulum (restoring moment: due to unbalanced couple) 9

10. Torsion Pendulum • From strength of materials: Equivalent torsional spring constant… • There exists a restoring moment: 10

11. Torsion Pendulum: EOM • Assume • Shaft centerline coincides with the mass center of the disk at the point of attachment (otherwise the shaft would also be in bending) • Disk of diameter D is rigid • EOM: • Equivalently, • Natural frequency: 11

12. [AO1]Example, Deriving EOM + Application 12

13. [AO2]Example, Deriving EOM + Application 13

14. [AO3]Example, Deriving EOM + Application 14

15. Example: Deriving EOM + Application [AO4] 15

16. [AO5]Example: Ballistic Pendulum 16

17. Example: Gun Recoil - 1 [AO6] 17

18. Example: Gun Recoil - 2 [AO7] 18

19. Example: Railroad Car [AO8] 19

20. Example: Motorcycle Shock Absorber [AO9] 20

21. Example: Door Tuning [AO10] • What is the damping coefficient such that the system is critically damped? • A person entering the door (when it is originally closed) kicks the door to cause it to open. What angular velocity must his kick impart to cause the door to open to 75deg? (Assume it is critically damped.) 21