What to bring and what to study

1. 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): Mechanical Engineering at Virginia Tech

2. Material (sections) Covered • 1.1,1.2, 1.3, 1.4, 1.5, 1.7 • 2.1, 2.2 • Log decrement from 1.6 Mechanical Engineering at Virginia Tech

3. 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 Mechanical Engineering at Virginia Tech

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

5. Free-body diagram and equations of motion • Newton’s Law: Mechanical Engineering at Virginia Tech

6. 2nd Order Ordinary Differential Equation with Constant Coefficients Mechanical Engineering at Virginia Tech

7. Periodic Motion amplitude x(0) Displacement Maximum Velocity Phase Time usually sec Mechanical Engineering at Virginia Tech

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

9. Amplitude & Phase from the ICs Mechanical Engineering at Virginia Tech

10. Other forms of the solution: See window 1.4, page 12 for relationships among these. Mechanical Engineering at Virginia Tech

11. Peak Values Mechanical Engineering at Virginia Tech

12. Spring-mass-damper systems • From Newton’s law: Mechanical Engineering at Virginia Tech

13. Solution: Given m, c, k, x0, v0 find x(t) Mechanical Engineering at Virginia Tech

14. Mechanical Engineering at Virginia Tech

15. Three possibilities: Mechanical Engineering at Virginia Tech

16. Mechanical Engineering at Virginia Tech

17. Mechanical Engineering at Virginia Tech

18. Underdamped 0 < z < 1 Reduces to undamped formulas for z = 0 Mechanical Engineering at Virginia Tech

19. Potential and Kinetic Energy The potential energy of mechanical systems U is often stored in “springs” (remember that for a spring F=kx) x=0 x0 k M Spring Mass The potential energy of mechanical systems U is also gravitational: The kinetic energy of mechanical systems T is due to the motion of the “mass” in the system Mechanical Engineering at Virginia Tech

20. Conservation of Energy For a simply, conservative (i.e. no damper), mass spring system the energy must be conserved: Equation of motion At two different times t1 and t2 the increase in potential energy must be equal to a decrease in kinetic energy (or visa-versa). An expression for the natural frequency Mechanical Engineering at Virginia Tech

21. Deriving equation of motion x=0 x k M Spring Mass Mechanical Engineering at Virginia Tech

22. Natural frequency If the solution is given by Asin(wt+f) then the maximum potential and kinetic energies can be used to calculate the natural frequency of the system Mechanical Engineering at Virginia Tech

23. Static Deflection Mechanical Engineering at Virginia Tech

24. Combining Springs • Equivalent Spring Mechanical Engineering at Virginia Tech

25. Harmonically Excited SystemsEquations of motion (c =0): Mechanical Engineering at Virginia Tech

26. 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 determined Driving frequency Mechanical Engineering at Virginia Tech

27. Substitute into the equation of motion: Thus the particular solution has the form: Mechanical Engineering at Virginia Tech

28. Add particular and homogeneous solutions to get general solution: Mechanical Engineering at Virginia Tech

29. Apply the initial conditions to evaluate the constants Solving for the constants and substituting into x yields Mechanical Engineering at Virginia Tech

30. 2.2 Harmonic excitation of damped systems Mechanical Engineering at Virginia Tech

31. Substitute the values of As and Bs into xp: Add homogeneous and particular to get total solution: Note: that A and f will not have the same values as in Ch 1, as t gets large, transient dies out Mechanical Engineering at Virginia Tech

32. Magnitude: Non dimensional Form: Phase: Frequency ratio: Mechanical Engineering at Virginia Tech

33. Magnitude plot Mechanical Engineering at Virginia Tech

34. Phase plot p/2 Mechanical Engineering at Virginia Tech