2. Feb 18, 2002 1/34 Mechanical Engineering at Virginia Tech What to bring and what to study One 8.5 X 11 formula sheet, one side only, no examples. Save the other side for test 2. Put your name on it and turn it in with the test. If you number the formulas. Suggest you use the same numbers as the text, you will be free to refer to them on the test. That is: “ from equation (1.37):

3. Feb 18, 2002 2/34 Mechanical Engineering at Virginia Tech Material (sections) Covered 1.1,1.2, 1.3, 1.4, 1.5, 1.7 2.1, 2.2 Log decrement from 1.6

4. Feb 18, 2002 3/34 Mechanical Engineering at Virginia Tech Also bring Paper, pencil, calculator No other resources allowed Your honor, but no anxiety Knowledge of all examples worked in class or presented in the text All assigned homework

5. Feb 18, 2002 4/34 Mechanical Engineering at Virginia Tech Expect 4 to 5 problems 1. An example covered in class 2. A homework problem 3. An example from the book, not covered in class 4. A problem involving combining parts of any of the above in “two steps” and/or 5. A derivation 25% (or 20%) each the last problem(4 and/or 5) intended to sort out the A’s and B’s

6. Feb 18, 2002 5/34 Mechanical Engineering at Virginia Tech Free-body diagram and equations of motion Newton’s Law:

7. Feb 18, 2002 6/34 Mechanical Engineering at Virginia Tech 2nd Order Ordinary Differential Equation with Constant Coefficients

8. Feb 18, 2002 7/34 Mechanical Engineering at Virginia Tech Periodic Motion x(0) Time usually sec Displacement amplitude Phase Maximum Velocity

9. Feb 18, 2002 8/34 Mechanical Engineering at Virginia Tech Frequency We often speak of frequency in Hertz, but we need rad/s in the arguments of the trigonometric functions.

10. Feb 18, 2002 9/34 Mechanical Engineering at Virginia Tech Amplitude & Phase from the ICs

11. Feb 18, 2002 10/34 Mechanical Engineering at Virginia Tech Other forms of the solution: See window 1.4, page 12 for relationships among these.

12. Feb 18, 2002 11/34 Mechanical Engineering at Virginia Tech Peak Values

13. Feb 18, 2002 12/34 Mechanical Engineering at Virginia Tech Spring-mass-damper systems From Newton’s law:

14. Feb 18, 2002 13/34 Mechanical Engineering at Virginia Tech Solution: Given m, c, k, x 0, v 0 find x ( t )

15. Feb 18, 2002 14/34 Mechanical Engineering at Virginia Tech

16. Feb 18, 2002 15/34 Mechanical Engineering at Virginia Tech Three possibilities:

17. Feb 18, 2002 16/34 Mechanical Engineering at Virginia Tech

18. Feb 18, 2002 17/34 Mechanical Engineering at Virginia Tech

19. Feb 18, 2002 18/34 Mechanical Engineering at Virginia Tech Underdamped 0 <  < 1 Reduces to undamped formulas for  = 0

20. Feb 18, 2002 19/34 Mechanical Engineering at Virginia Tech Potential and Kinetic Energy The potential energy of mechanical systems U is often stored in “springs” (remember that for a spring F=kx) The kinetic energy of mechanical systems T is due to the motion of the “mass” in the system M k x0x0 Mass Spring x=0 The potential energy of mechanical systems U is also gravitational:

21. Feb 18, 2002 20/34 Mechanical Engineering at Virginia Tech Conservation of Energy For a simply, conservative (i.e. no damper), mass spring system the energy must be conserved: At two different times t 1 and t 2 the increase in potential energy must be equal to a decrease in kinetic energy (or visa- versa). Equation of motion An expression for the natural frequency

22. Feb 18, 2002 21/34 Mechanical Engineering at Virginia Tech Deriving equation of motion M k x Mass Spring x=0

23. Feb 18, 2002 22/34 Mechanical Engineering at Virginia Tech Natural frequency If the solution is given by Asin(  t+  ) then the maximum potential and kinetic energies can be used to calculate the natural frequency of the system

24. Feb 18, 2002 23/34 Mechanical Engineering at Virginia Tech Static Deflection

25. Feb 18, 2002 24/34 Mechanical Engineering at Virginia Tech Combining Springs Equivalent Spring

26. Feb 18, 2002 25/34 Mechanical Engineering at Virginia Tech Harmonically Excited Systems Equations of motion ( c =0):

27. Feb 18, 2002 26/34 Mechanical Engineering at Virginia Tech Linear nonhomogenous ode: Solution is sum of homogenous and particular solution The particular solution assumes form of forcing function (physically the input wins) To be determinedDriving frequency

28. Feb 18, 2002 27/34 Mechanical Engineering at Virginia Tech Substitute into the equation of motion: Thus the particular solution has the form :

29. Feb 18, 2002 28/34 Mechanical Engineering at Virginia Tech Add particular and homogeneous solutions to get general solution:

30. Feb 18, 2002 29/34 Mechanical Engineering at Virginia Tech Apply the initial conditions to evaluate the constants Solving for the constants and substituting into x yields

31. Feb 18, 2002 30/34 Mechanical Engineering at Virginia Tech 2.2 Harmonic excitation of damped systems

32. Feb 18, 2002 31/34 Mechanical Engineering at Virginia Tech Substitute the values of A s and B s into x p : Add homogeneous and particular to get total solution: Note: that A and  will not have the same values as in Ch 1, as t gets large, transient dies out

33. Feb 18, 2002 32/34 Mechanical Engineering at Virginia Tech Magnitude: Non dimensional Form: Phase: Frequency ratio:

34. Feb 18, 2002 33/34 Mechanical Engineering at Virginia Tech Magnitude plot

35. Feb 18, 2002 34/34 Mechanical Engineering at Virginia Tech Phase plot  /2