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2. Dick Chalmers 1931 - 1998. 3. Joel Leifer. 4. Henry Pusey. 5. Shock spectrum definitionsEquations: Conceptual and PracticalCalculate; Do loop, Filter, ResidualPlotting SS's with PV as ordinate on (4CP) log-log four coordinate paper. Proofs that stress is proportional to velocity for rods, beams, plates.Fundamental Maximum modal velocitiesShock data needs least range in velocitiesPV use: Civil/Structural, nuclear defense, Army Conventional Weapons Effect, Earthquake. Hall's chapter in 1143
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1. 1 SHOCK ANALYSIS USING THE PSEUDO VELOCITY SHOCK SPECTRUM PART 1
2. 2 Dick Chalmers 1931 - 1998
3. 3 Joel Leifer
4. 4 Henry Pusey
5. 5 Points to Learn
6. 6 More Important Points
7. 7 Shock Spectrum Definition
8. 8 ABSOLUTE ACCELERATION SHOCK SPECTRUM, SRS
9. 9 PVSS on 4CP
10. 10 FOUR COORDINATE PAPER
11. 11 4CP example. Lines slanting down to the right are constant g’s. Slanting down to left are constant displacement. Horizontal: pseudo velocity. Vertical: frequency.
12. 12 On 4CP, if you know any two, you can get the other two from
13. 13 SHOCK SPECTRUM EQUATION DWGS; THIS SYSTEM: A BOGY GETS THE SHOCK; RATTLES THE MASS
14. 14 Mental Model
15. 15 Free Body Diagram
16. 16 Shock Spectrum Equation
17. 17 Duhamel’s Integral Solution
18. 18 Aside: Undamped
19. 19 Calculating Details
20. 20 Residual Calculation
21. 21 PLOTTING PVSS ON 4CP DISPLACEMENT, VELOCITY, ACCELERATION
22. 22 ACCELERATION MEANING ON 4CP
23. 23 Response of a 20 Hz 15% damped SDOF to Explosion.. Notice max acceleration does not occur at same time as max displacement.
24. 24 Arguments for shock spectra with PV as ordinate on log-log 4CP
25. 25 Why Pseudo Velocity and not Absolute or Relative Velocities are Best For Shock Spectra
26. 26 Precedent for PVSS on 4CP
27. 27 MECHANICAL SHOCK
28. 28 Near Miss Explosion, Army Tank
29. 29 EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE WITH VELOCITY
30. 30 EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE WITH VELOCITY, II
31. 31 EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE WITH VELOCITY, III
32. 32 Gunfire Shock plotted as acceleration shock spectrum shows constant velocity character
33. 33 Railroad humping on acceleration shock spectrum shows constant velocity
34. 34 Existing Knowledge that stress proportional to modal velocity
35. 35 Stress is Proportional to Velocity
36. 36 Shock Spectrum Equation,Variables and Constants
37. 37 Strain in a rod
38. 38 F = ma
39. 39 Stress Velocity Rods
40. 40 More Stress Velocity Rods
41. 41 SUM UP STRESS VELOCITY IN RODS
42. 42 BEAMS STRESS VELOCITY EQUATIONS
43. 43 STRESS VELOCITY IN BEAMS
44. 44 MORE BEAM STRESS VELOCITY
45. 45 BEAM SHAPE FACTORS
46. 46 Ted Hunt’s Analysis (Frederick Vinton Hunt from Harvard)
47. 47 SEVERE VELOCITIES
48. 48 HIGH PVSS SHOWS SHOCK CAPACITY TO DELIVER ENERGY TO AN SDOF SYSTEM
49. 49 Before I Look at the Extremely Important Asymptotes Consider Half Sine and other Simple Shocks First
Then we’ll do:
3 Regions on the PVSS on 4CP
Important
Acceleration and Displacement Asymptotes
Severity
50. 50 Half Sine and Other Simple Shocks Include drop in analyzed signal. Have chuckle re Hdbk. SRS & drop.
You need to learn why: Drop needed for low frequency
No rebound
Other simple shocks same SS
Impact Velocity Change
Peak acceleration
Max displacement
51. 51 PVSS-4CP Example: 800 g, 1 ms half sine
52. 52 ZERO MEAN SIMPLE SHOCK THAT LAST SHOCK WAS A ZERO MEAN SIMPLE SHOCK.
ZERO MEAN ACCELERATION MEANS SHOCK BEGINS AND ENDS WITH ZERO VELOCITY
THE SHOCK INCLUDES THE DROP AND ANY REBOUND.
THE INTEGRAL OF THE ACCELERATION IS ZERO IF IT HAS A ZERO MEAN.
BY SIMPLE SHOCK I MEAN ONE OF THE COMMON SHOCKS:: HALF SINE, INITIAL PEAK SAW TOOTH, TERMINAL PEAK SAW TOOTH, TRAPEZOIDAL, HAVERSINE
53. 53 ZERO MEAN, SIMPLE SHOCK, PVSS-4CP HILL SHAPE WHEN A ZERO MEAN SIMPLE SHOCK PVSS IS PLOTTED ON 4CP IT HAS A HILL SHAPE:
THE LEFT UPWARD SLOPE IS A PEAK DISPLACEMENT ASYMPTOTE
THE RIGHT DOWNWARD SLOPE IS THE PEAK ACCELERATION ASYMPTOTE.
THE TOP IS A PLATEAU AT THE VELOCITY CHANGE DURING IMPACT.
54. 54 Collision and Kickoff Shocks Collision shock, (a car slamming into a wall)
Kickoff shock (environment on a ball when it is kicked)
Do not have a zero mean.
The collision starts with a high velocity and ends with zero velocity
The kickoff starts with zero velocity and ends with a high velocity.
These PVSS's on 4CP have damped max velocity change low frequency asymptote, vice down to the left maximum deflection asymptote line.
55. 55 Half sine equations
56. 56 HALF SINE SHOCK WITHOUT THE DROP I'm going to plot this half sine shock along with it's two integrals. Let's illustrate it with a moderately severe shock with a 100 ips velocity change and a peak acceleration of 200 g's. From the above formula, we find the duration to be: 2.035 ms. Assume a shock machine table could do this, and consider the integrals.
57. 57 BAD SHOCK SPECTRUM OF HALF SINE This what I want everyone to learn to expect. Because the velocity did not end at 0, this is an unrealistic SS. It's indicating that a 0.1 Hz SDOF would have a peak deflection of about 140 inches, 12 ft. No way. However notice that the velocity change of 100 ips shows up as it should and the curve at high frequency is assympotic to 200 g's, as it must.
58. 58 Correct concept. Half sine time history with drop, and integrals
59. 59 SS of 100 ips, 200 g half sine with drop
60. 60 COSINE RAMP TRAPEZOID SHOCK EQUATIONS
61. 61 HOW MUCH COSINE RAMP IS REASONABLE
62. 62 Trapezoid, 30% cosine ramp, 100 ips, 200 g with drop
63. 63 Terminal peak sawtooth, 200 g. 100 ips, with 10% cosine fall off.
64. 64 Initial peak shock time plot
65. 65 All the simple shocks are equally severe.
66. 66 And with 5% damping we get the same result
67. 67 HIGH AND LOW FREQUENCY ASYMPTOTES FOR PVSS ON 4CP TRADITIONAL RATIONALIZATIONS High frequency oscillators have very stiff springs and light masses. The mass follows the acceleration of the foundation; thus, these SDOFs record a peak absolute acceleration equal to the peak acceleration of the shock.
Low frequency oscillators are very flexible. If their foundations are given a very quick or short duration wiggle, the mass barely moves until the foundation motion is over. If the shock being analyzed is one that begins and ends with zero velocity, the peak relative deflection will be the peak shock displacement.
For intermediate values of the frequency, the peak pseudo velocity is often almost constant. In this region the pseudo velocity closely approximates the relative velocity. This region is the high PV region contains the frequencies where the shock has the greatest capacity to deliver energy to the SDOFs.
68. 68 Low Frequency Asymptote Carefully
In low frequency region, spring very soft, mass heavy. Mass just sits there during shock. Maximum spring stretch or zmax is just maximum y, ymax We plot ?zmax
Since ymax is constant, log of the pv here is straight line with positive slope.
69. 69 HIGH FREQUENCY ASYMPTOTE CAREFULLY In the PVSS on 4CP, we are plotting ?z; focus on that.
In high frequency region: mass very light and spring very stiff: Mass exactly follows input motion. Acceleration of mass equal to foundation. Max z, is spring force over stiffness, k.
On log log paper, since the maximum acceleration is a constant, pv is a straight line with a negative slope.
70. 70 PLATEAU, MID FREQUENCY The mid frequency, plateau region of the spectrum is the important region. It shows the velocity change of a simple pulse.
No one explains it well. And this is the really important part.
Pick up any civil structural textbook and go to the earthquake section and read the traditional explanations.
Vigness [1] declared without any explanation that four regions exist on the shock spectrum, and broke the center region into two parts; this is typical of our business.
This middle region is the most severe and thereby most important region of the of the shock's SS. Here the PV is at its maximum. Vigness's comments:
71. 71 VIGNESS ON MID FREQUENCY REGION Vigness's very important, yet somewhat weakly justified paper 1964 paper "Elementary Considerations of Shock Spectra" from the SVB n34, pp 211-222.
In paragraph 16 which is on page 213, "At some intermediate frequency, region B, peaks in the shock spectral curves indicate sustained frequencies of the shock motions.
At some lower frequency (generally), a section of the shock spectral curve, region V, remains at a constant velocity value. This corresponds to a frequency region over which the shock motion can be considered to be an impulsive (step) velocity change."
That's a very interesting few sentences, probably correct, but not justified. In his intro to the paper he warns that these are observations, and suggestions, not necessarily new. So maybe in 1964, many people had agreed on this.
72. 72 Roberts, W.H., “Explosive Shock,” Shock and Vibration, Bulletin 40, Part 2, Dec 1969, pp 1-10. This is a difficult paper for me to follow and has not received attention.
However he reasons, he gets many of the right answers. For example:
“It is reasonable to conclude therefore, by analogy to the discussion on the shock spectrum that vibration velocity measures internal stresses, not vibration acceleration.” This is exactly the result that our analyses lead to.
He uses Rinehart and Pearson, “Behavior of Metals Under Impulsive Loads”, 1954 Dover, which establishes 1200-2400 inches per second as theoretical wave velocity limits in structural materials. He cites his experience that structure tolerates 360 ips. He next reasons that mechanical components’ velocity limits are 60 – 120 ips.
His reasoning allows him to conclude that the 3 shock spectrum regions are logical.
Clarifying of how Robert’s thinks would help me.
73. 73 Robert’s Drawing
74. 74 UNDAMPED PLATEAU IMPORTANT CONVINCE YOURSELF Instant shock
Bogey and mass fall h
Shock over before spring compresses
Uundamped free vibration equation govern
75. 75 PLATEAU FINISH
76. 76 PLATEAU SUMMARY
77. 77 SUMMARIZE UNDAMPED ZERO MEAN SIMPLE SHOCK SS’s UNDAMPED SIMPLE DROP TABLE SHOCKS HAVE A FLAT CONSTANT PV PLATEAU AT THE VELOCITY CHANGE THAT TOOK PLACE DURING THE SHOCK.
THE HIGH FREQUENCY LIMIT OF THE PLATEAU IS SET BY THE MAXIMUM ACCELERATION OF THE SHOCK.
THE HIGH FREQUENCY ASYMPTOTE IS THE MAXIMUM ACCELERATION LINE.
THE LOW FREQUENCY LIMIT OF THE PLATEAU IS SET BY THE MAXIMUM DEFLECTION OF THE SHOCK.
THE LOW FREQUENCY ASYMPTOTE IS THE MAXIMUM DISPLACEMENT LINE
78. 78 ADJUST A SPECTRUM
79. 79 2g LINE CONCEPT UNDAMPED SIMPLE SHOCK NO REBOUND DROP HEIGHT IS WHERE THE PLATEAU INTERSECTS 2 g’s
80. 80 2 g Line for Drop Height of a Simple Pulse as a Function of Velocity Change
81. 81 Exception: Josh’s 200ips, 5 g Halfsine
82. 82 Velocity level when numerical value of acceleration in g’s equals numerical value of frequency in Hz
83. 83 Explosive shock time history and integrals
84. 84 Explosive shock PVSS on 4CP
85. 85 Bad Editing Example Pyroshock Example Himelblau Piersol 95-96 SVS Proceedings
86. 86 Pyroshock Shock Spectrum Very High Frequency Bad at 200 Hz
87. 87 This completes Part I. In Part II Experimental and computational proof, additional concepts. Pseudo velocity compared to relative velocity shock spectra indicating pitfalls for relative velocity.
Relative velocity low frequency problem.
Tests to determine which transient motion analyses method is the best indicator of damage potential.
The best damage potential analysis is the damped PVSS on 4CP.
