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Chapter 5 Design

Chapter 5 Design. Acceptable vibration levels (ISO) Vibration isolation Vibration absorbers Effects of damping in absorbers Optimization Viscoelastic damping treatments Critical Speeds Design for vibration suppression . 5.1 Acceptable levels of vibration.

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Chapter 5 Design

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  1. Chapter 5 Design • Acceptable vibration levels (ISO) • Vibration isolation • Vibration absorbers • Effects of damping in absorbers • Optimization • Viscoelastic damping treatments • Critical Speeds • Design for vibration suppression Mechanical Engineering at Virginia Tech

  2. 5.1 Acceptable levels of vibration • Each part or system in a dynamic setting is required to pass “vibration” muster • Military and ISO provide a regulation and standards • Individual companies provide their own standards • Usually stated in terms of amplitude, frequency and duration of test Mechanical Engineering at Virginia Tech

  3. Example 5.1.2 Dissimilar devices with the same frequency m=1 kg k=400 N/m c=8 Ns/m m=1000 kg k=400,000 N/m c=8000 Ns/m car CD drive Mechanical Engineering at Virginia Tech

  4. But: response magnitudes different • Magnitude plot will have the same shape • Time responses will have the same form for similar (but scaled) disturbancesBUT WITH DIFFERENT MAGNITUDES different Fig 5.3 Mechanical Engineering at Virginia Tech

  5. Section 5.2 Isolation • Important class of vibration analysis • Preventing excitations from passing from a vibrating base through its mount into a structure • Vibration isolation • Shocks on your car • Satellite launch and operation • Disk drives A major job of vibration engineers is to isolate systems from vibration disturbances or visa versa. Uses heavily material from Sections 2.4 on Base Excitation Mechanical Engineering at Virginia Tech

  6. m m Recall from Section 2.4 that the FBD of SDOF for base excitation is x(t) k c y(t) base Mechanical Engineering at Virginia Tech

  7. SDOF Base Excitation assumes the input motion at the base has the form The steady-state solution is just the superposition of the two individual particular solutions Mechanical Engineering at Virginia Tech

  8. Particular Solutions (sine input) With a sine for the forcing function, Mechanical Engineering at Virginia Tech

  9. Particular Solutions (cosine input) With a cosine for the forcing function, we showed Mechanical Engineering at Virginia Tech

  10. Magnitude X/Y Magnitude of the full particular solution Mechanical Engineering at Virginia Tech

  11. 40 z =0.01 z =0.1 30 z =0.3 z =0.7 20 X/Y (dB) 10 0 -10 -20 0 0.5 1 1.5 2 2.5 3 Frequency ratio r The magnitude plot of X/Y Mechanical Engineering at Virginia Tech

  12. Notes on Displacement Transmissibility • Potentially severe amplification at resonance • Attenuation only for r > sqrt(2) • If r< sqrt(2) transmissibility decreases with damping ratio • If r>>1 then transmissibility increases with damping ratio Xp=2Yz/r Mechanical Engineering at Virginia Tech

  13. m It is also important to look at the Force Transmissibility: x(t) FT k c y(t) base Mechanical Engineering at Virginia Tech

  14. 40 z =0.01 z =0.1 30 z =0.3 z =0.7 20 F/kY (dB) 10 0 -10 -20 0 0.5 1 1.5 2 2.5 3 Frequency ratio r Plot of Force Transmissibility Mechanical Engineering at Virginia Tech

  15. Isolation is a sdof concept; • Two types: moving base and fixed base • Three magnitude plots to consider TR= transmissibility ratio Moving base displacement Moving base force Fixed base force Mechanical Engineering at Virginia Tech

  16. For displacement transmissibility, isolation occurs as a function of stiffness • For stiffness such that the frequency ration is larger the root 2, isolation occurs, but increasing damping reduces the effect • For less then root 2, increased damping reduces the magnitude. Mechanical Engineering at Virginia Tech

  17. Example 5.2.1 Design an isolation mount Fig 5.6 • Design an isolator (chose k, c) to hold a 3 kg electronics module to less then 0.005 m deflection if the base is moving at y(t)=(0.01)sin(35t) • Calculate the force transmitted through the isolator Mechanical Engineering at Virginia Tech

  18. z r 1.73 1.74 1.76 1.84 2.35 4.41 0.01 0.05 0.1 0.2 0.5 1.2 T.R. Plot for moving base displacement 1.5 For T.R. =0.5 z 1 z =0.01 T.R. z =0.05 z =0.1 0.5 z =0.2 z =0.5 z =1.2 0 0 0.5 1 1.5 2 2.5 3 Frequency ratio r Mechanical Engineering at Virginia Tech

  19. From the plot, note that Mechanical Engineering at Virginia Tech

  20. Rattle Space Choice of k and c must also be reasonable As must force transmitted: Mechanical Engineering at Virginia Tech

  21. The transmitted force is Transmitted force, T.R., static deflection, damping and stiffness values must all be reasonable for the application. Mechanical Engineering at Virginia Tech

  22. Shock Isolation Shock pulse Pulse duration Increased isolation with increasing k Mechanical Engineering at Virginia Tech

  23. Figure 5.8 Shock Response Mechanical Engineering at Virginia Tech

  24. Shock versus Vibration Isolation In figure 5.8 for  = 0.5 requires for shock isolation to occur. • Thus shock isolation requires a bound on the stiffness • Also from Figure 5, high damping is desirable for shock attenuation. Mechanical Engineering at Virginia Tech

  25. Example 5.2.3 Design a system that is good for both shock and vibration isolation. • The design constraints are that we have the choice of 3 off the shelve isolation mounts: • 5 Hz, 6 Hz and 7 Hz each with 8% damping • The shock input is a 15 g half sine at 40 ms • The vibration source is a sine at 15 Hz • The response should be limited to 15 g’s and 76.2 mm, and 20 dB of vibration isolation Mechanical Engineering at Virginia Tech

  26. Simulation of the response to the shock input for all three mounts choices • From these numerical simulations, only the 7 Hz mount satisfies all of the shock isolation goals: • Less then 15 g’s • Less then 3 in deflections Fig 5.11 Fig 5.12 Mechanical Engineering at Virginia Tech

  27. Now consider the vibration isolation by plotting shock isolator design’s transmissibility: • For the 7 Hz shock isolator design, the reduction in Transmissibility is only 9.4 dB. • From this plot, and recalling Fig 5.7 less damping is required. • However, less damping is not possible Fig 5.13 Mechanical Engineering at Virginia Tech

  28. 5.3 Vibration Absorbers • Consider a harmonic disturbance to a single-degree-of freedom system • Suppose the disturbance causes large amplitude vibration of the mass in steady state • A vibration absorber is a second spring mass system added to this “primary” mass, designed to absorb the input disturbance by shifting the motion to the new added mass (called the absorber mass). Mechanical Engineering at Virginia Tech

  29. Absorber concept F(t) = F0sinwt x Primary mass (optical table) m ka xa k /2 absorber k/2 ma • Primary system experiences resonance • Add absorber system as indicated • Look at equations of motion (now 2 dof) Mechanical Engineering at Virginia Tech

  30. The equations of motion become: To solve assume a harmonic displacement: Mechanical Engineering at Virginia Tech

  31. The form of the response magnitude suggests a design condition allowing the motion of the primary mass to become zero: pick ma and ka to make zero All the system motion goes into the absorber motion Mechanical Engineering at Virginia Tech

  32. Choose the absorber mass and stiffness from: This causes the primary mass to be fixed and the absorber mass to oscillate at: As in the case of the isolator, static deflection, rattle space and force magnitudes need to be checked in each design Mechanical Engineering at Virginia Tech

  33. Other pitfalls in absorber design • Depends on knowing w exactly • Single frequency device • If w shifts it could end up exciting a system natural frequency (resonance) • Damping, which always exists to some degree, spoils the absorption let’s examine these: Mechanical Engineering at Virginia Tech

  34. Avoiding resonance (robustness) Mass ratio Original natural frequency of primary system before absorber is attached Natural frequency of absorber before it is attached to primary mass Stiffness ratio frequency ratio Mechanical Engineering at Virginia Tech

  35. Define a dimensionless amplitude of the primary mass Mechanical Engineering at Virginia Tech

  36. Normalized Magnitude of Primary Fig 5.15 Absorber zone w/w2 w/w1 Mechanical Engineering at Virginia Tech

  37. Robustness to driving frequency shifts • If w hits w1 or w2 resonance occurs • Using |Xk/F0|<1, defines useful operating range of absorber • In this range some absorption still occurs • The characteristic equation is: = Frequency dependence on mass and frequency ratio Mechanical Engineering at Virginia Tech

  38. Mass ratio versus frequency • Referring to fig. 5.16, as m increases, frequencies split farther apart for fixed b • thus if m is too small, system will not tolerate much fluctuation in driving frequency indicating a poor design • Rule of thumb 0.05< m <0.25 Fig 5.16 Mechanical Engineering at Virginia Tech

  39. 60 No vibration absorber With vibration absorber 40 20 Amplitude (Xk/F) 0 -20 -40 0 0.5 1 1.5 2 Frequency ratio (ra) Normalized Magnitude of the primary mass with and without the absorber m=0.1 and b=0.71 • Adding absorber increase the number of resonances (or modes) from one to two. • Smaller response of primary structure is at absorber natural frequency • Effective over limited bandwidth Mechanical Engineering at Virginia Tech

  40. 60 40 20 Amplitude (Xak/F) 0 -20 -40 0 0.5 1 1.5 2 Frequency ratio (ra) What happens to the mass of the vibration absorber? m=0.1 and b=0.71 • In the operational range of the vibration absorber the absorber mass has relatively large motion • Beware of deflection limits!! Mechanical Engineering at Virginia Tech

  41. Example 5.3.1: design an absorber given F0= 13 Nm=73.16 kg, k=2600 N/m, w = 180 cpm, xa< 0.02 m Mechanical Engineering at Virginia Tech

  42. Example 5.3.2 Compute the bandwidth of the absorber design in 5.3.1 3 roots Mechanical Engineering at Virginia Tech

  43. Comparing these 3 roots to the plot yields that: This is the range that the driving frequency can safely lie in and the absorber will still reduce the vibration of the primary mass. Mechanical Engineering at Virginia Tech

  44. 5.4 Damped absorber system Undamped primary Cannot be zero! Mechanical Engineering at Virginia Tech

  45. Magnitude of primary mass for 3 levels of damping As damping increases, the absorber fails, but the resonance goes away 1 Region of absorption Go to mathcad example 5.16 Mechanical Engineering at Virginia Tech

  46. 50 No VA 40 ca=0.01 30 ca=0.1 ca=1 20 10 Amplitude (Xk/F) 0 -10 -20 -30 0 0.5 1 1.5 2 Frequency ratio (ra) Effect of damping on performance • In the operational range of the vibration absorber decreases with damping • The bandwidth increases with damping • Resonances are decreased i.e. could be used to reduce resonance problems during run up • See Fig 5.19 m=0.25 and b=0.8 Mechanical Engineering at Virginia Tech

  47. Three parameters effect making the amplitude small • This curves show that just increasing the damping does not result in the smallest amplitude. • The mass ratio and also matter • This brings us to the question of optimization Mechanical Engineering at Virginia Tech

  48. With damping in the absorber: • Undamped absorber has poor bandwidth • Small damping extends bandwidth • But, ruins complete absorption of motion • Becomes a design problem to pick the most favorable m, b, z. Mechanical Engineering at Virginia Tech

  49. Viscous Vibration Absorber • Rotating machine applications • Rotational inertia, shaft stiffness and a fluid damper • Often called a Houdaille damper illustrated in the following: Primary system Viscous absorber x(t) xa k ca ma m Fig 5.22 Mechanical Engineering at Virginia Tech

  50. Houdaille Damper Fig 5.23 Equation of motion: Mechanical Engineering at Virginia Tech

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