Mechanical Engineering at Virginia Tech 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 © D. J. Inman 1/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 2/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 3/51 Mechanical Engineering at Virginia Tech

But: response magnitudes different Magnitude plot will have the same shape Time responses will have the same form for similar (but scaled) disturbances BUT WITH DIFFERENT MAGNITUDES different Fig 5.3 © D. J. Inman 4/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Section 5.2 Isolation 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 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 © D. J. Inman 5/51 Mechanical Engineering at Virginia Tech

Recall from Section 2.4 that the FBD of SDOF for base excitation is x(t) m m k c y(t) base © D. J. Inman 6/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 7/51 Mechanical Engineering at Virginia Tech

Particular Solutions (sine input) With a sine for the forcing function, © D. J. Inman 8/51 Mechanical Engineering at Virginia Tech

Particular Solutions (cosine input) With a cosine for the forcing function, we showed © D. J. Inman 9/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Magnitude X/Y Magnitude of the full particular solution © D. J. Inman 10/51 Mechanical Engineering at Virginia Tech

The magnitude plot of X/Y 0.5 1 1.5 2 2.5 3 -20 -10 10 20 30 40 Frequency ratio r X/Y (dB) z =0.01 =0.1 =0.3 =0.7 © D. J. Inman 11/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 12/51 Mechanical Engineering at Virginia Tech

It is also important to look at the Force Transmissibility: x(t) m FT k c y(t) base © D. J. Inman 13/51 Mechanical Engineering at Virginia Tech

Plot of Force Transmissibility 0.5 1 1.5 2 2.5 3 -20 -10 10 20 30 40 Frequency ratio r F/kY (dB) z =0.01 =0.1 =0.3 =0.7 © D. J. Inman 14/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 15/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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. © D. J. Inman 16/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 17/51 Mechanical Engineering at Virginia Tech

T.R. Plot for moving base displacement 1.5 For T.R. =0.5 z 0.01 0.05 0.1 0.2 0.5 1.2 z r 1.73 1.74 1.76 1.84 2.35 4.41 1 T.R. z =0.01 z =0.05 z 0.5 =0.1 z =0.2 z =0.5 z =1.2 0.5 1 1.5 2 2.5 3 Frequency ratio r © D. J. Inman 18/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech From the plot, note that © D. J. Inman 19/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Rattle Space Choice of k and c must also be reasonable As must force transmitted: © D. J. Inman 20/51 Mechanical Engineering at Virginia Tech

The transmitted force is Transmitted force, T.R., static deflection, damping and stiffness values must all be reasonable for the application. © D. J. Inman 21/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Shock Isolation Shock pulse Pulse duration Increased isolation with increasing k © D. J. Inman 22/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Figure 5.8 Shock Response © D. J. Inman 23/51 Mechanical Engineering at Virginia Tech

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. © D. J. Inman 24/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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 © D. J. Inman 25/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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 © D. J. Inman 26/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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 © D. J. Inman 27/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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). © D. J. Inman 28/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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) © D. J. Inman 29/51 Mechanical Engineering at Virginia Tech

The equations of motion become: To solve assume a harmonic displacement: © D. J. Inman 30/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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 © D. J. Inman 31/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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 © D. J. Inman 32/51 Mechanical Engineering at Virginia Tech

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: © D. J. Inman 33/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 34/51 Mechanical Engineering at Virginia Tech

Define a dimensionless amplitude of the primary mass © D. J. Inman 35/51 Mechanical Engineering at Virginia Tech

Normalized Magnitude of Primary Fig 5.15 Absorber zone w/w1 w/w2 © D. J. Inman 36/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 37/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 38/51 Mechanical Engineering at Virginia Tech

Normalized Magnitude of the primary mass with and without the absorber m=0.1 and b=0.71 0.5 1 1.5 2 -40 -20 20 40 60 Frequency ratio (ra) Amplitude (Xk/F) No vibration absorber With vibration absorber 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 © D. J. Inman 39/51 Mechanical Engineering at Virginia Tech

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!! 0.5 1 1.5 2 -40 -20 20 40 60 Frequency ratio (ra) Amplitude (Xak/F) © D. J. Inman 40/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Example 5.3.1: design an absorber given F0= 13 N m=73.16 kg, k=2600 N/m, w = 180 cpm, xa< 0.02 m © D. J. Inman 41/51 Mechanical Engineering at Virginia Tech

Example 5.3.2 Compute the bandwidth of the absorber design in 5.3.1 3 roots © D. J. Inman 42/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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. © D. J. Inman 43/51 Mechanical Engineering at Virginia Tech

5.4 Damped absorber system Undamped primary Cannot be zero! © D. J. Inman 44/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech 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 © D. J. Inman 45/51 Mechanical Engineering at Virginia Tech

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 0.5 1 1.5 2 -30 -20 -10 10 20 30 40 50 Frequency ratio (ra) Amplitude (Xk/F) No VA ca=0.01 ca=0.1 ca=1 © D. J. Inman 46/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 47/51 Mechanical Engineering at Virginia Tech

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. © D. J. Inman 48/51 Mechanical Engineering at Virginia Tech

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 © D. J. Inman 49/51 Mechanical Engineering at Virginia Tech

Mechanical Engineering at Virginia Tech Houdaille Damper Fig 5.23 Equation of motion: © D. J. Inman 50/51 Mechanical Engineering at Virginia Tech

Frequency response of primary mass Figure 5.24 Design on m and z © D. J. Inman 51/51 Mechanical Engineering at Virginia Tech