1. ME 440 Intermediate Vibrations Tu, March 3, 2009 Single DOF Harmonic Excitation © Dan Negrut, 2009 ME440, UW-Madison

2. Before we get started… Last Time: Discussed Design Problem Covered Response of Damped System Under Rotating Unbalance Today: HW Assigned (due March 10): 3.35, 3.36 For 3.36,  is provided in the picture Material Covered: Examples Beating phenomena Support excitation 2

3. Example [AO1] 3

4. Example [AO1] (Cntd) 4 Lateral stiffness, one leg From Mechanics of Materials:  : Maximum deflection Bending Moment caused by  Bending Stress caused by 

5. New Topic: Beating Phenomenon For undamped system, if forcing frequency is close to, but not exactly equal to, the natural frequency of the system, a phenomenon known as beating may occur. Vibration amplitude builds up and then diminishes in a regular pattern. Go back to the undamped forced vibration: 5 Solution expressed in one of the following two equivalent forms:

6. Beating Phenomenon (Cntd) Assume zero initial conditions: 6 Response becomes: Use next the following trigonometric identity (“sum to product” rule): The response of the system can then be written as

7. Beating Phenomenon (Cntd) Assume forcing frequency slightly smaller than natural frequency: 7 Assume  is small positive quantity, multiplied by 2 for convenience only The following then hold Then the time evolution of system can be equivalently expressed as

8. Beating Phenomenon (Cntd) No force excitation case: 8 Time evolution of mass m: Force excitation at frequency  (close to  n ): Time evolution of mass m:

9. Beating Phenomenon Comments Since  is small, the function sin(  t) varies slowly Period  equal to 2  / , which is large… 9 The way to interpret the quantity in square parentheses above: A very small varying amplitude for a vibration that otherwise is characterized by a frequency  (the frequency of the excitation force) Nomenclature: Period of Beating: Frequency of Beating:

10. New Topic: Support Excitation Framework of the problem at hand: You have a machine positioned somewhere on the floor The floor vibrates (assumed up and down motion only) How is the machine going to oscillate (vibrate) in response to this excitation of the support (base, floor)? Why’s relevant? Maybe you can select the values of k and c and isolate the vibration of the floor, get the machine to stay still in spite of the floor vibrating… 10 The perspectives from which you can tackle the problem: Investigating the absolute motion of the machine Motion described relative to a fixed (therefore inertial) reference frame Investigating the relative motion of the machine Motion described relative to the motion of the support

11. The “Absolute Motion” Alternative Notation used: x(t) captures the motion of the machine y(t) captures the motion of the base 11 EOM: Apply N2L for body of mass m: Leads to Equivalently,

12. The “Absolute Motion” Alternative (Cntd) Motion of the floor considered known You can always measure the vibration of the floor… Assumed to be of the form 12 If floor motion not in this form but some other general periodic function use Fourier Series Expansion and then fall back on the principle of superposition Based on the assumed expression of the floor motion, EOM becomes Note that this looks as though the mass m is acted upon a force whose expression is the RHS of the EOM above:

13. The “Absolute Motion” Alternative (Cntd) 13 Steady-state response of the mass is then given by NOTE: Phase angle  1 will be the same for both terms above It depends on the values m, c, k, . It does not depend on the amplitude of the excitation Expression of x p (t) can be further “massaged” to assume a more compact form (see next slide)

14. The “Absolute Motion” Alternative (Cntd) 14 Equivalent form of response The angles are defined as

15. The “Absolute Motion” Alternative (Cntd) 15 Amplification factor finally evaluated as Recall the important question here How can you have a small X even when Y is large? In practice, given m (mass of machine) and frequency of oscillation , you choose those values of k and c that minimize the value of the amplification ratio above…

16. Example: Vehicle Vibration 16