Design for Vibration Suppression

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1 Design for Vibration Suppression Outlines: 1. Vibration Design Process. Design of Vibration Isolation Moving base Fixed base 3. Design of Vibration Absorbers

2 Vibration Design Process Actual system Modeling Analysis Solve EOM to predict dynamic characteristics and vibration response Design Criteria Nature of input Acceptable amplitude level, displacement, velocity, or acceleration Make design decision Choose physical parameters Change system Isolations Absorbers Test

3 Nature of input / Design objectives Nature of Input 1. Vibration: has some oscillatory features. Shock: sharp, aperiadic, and relatively short time Design objectives 1. To protect the device from motion of its point of attachment (moving base). To protect the point of attachment (ground) from vibration of the mass (isolate vibration from the source) 3. To reduce vibration of the mass Isolation Absorber

4 Acceptable level of vibration (1) Level of vibration: displacement, velocity or acceleration Standards are tested in terms of rms values x rms 1/ 1 lim T x ( t) dt T 0 T x( t) Asin t v( t) x ( t) n A cos t 1 T A x lim ( A sin nt) dt t T 0 n x v A A n a( t) x ( t) A sin t n a A n

5 Acceptable level of vibration ()

6 Vibration isolation (moving base) Vibration isolator Objective: To isolate a device from the source of vibration (moving base). (To reduce vibration of machine transmitted through moving base.) Applications: Automobile suspension Table isolator

7 Base excitation (review) EOM m x cx kx cy ky x x x y y n n n n Harmonic motion of the base FBD y( t) Y cos t Re[ Ye The response in complex form is z p ( t) Xe jt jt ] Substituting z p (t) into EOM yields Z n ( n) j ( n n ) j Y 1 (r) j 1 r (r) Y j ; r n

8 Displacement transmissibility (1) Y T Y j r r j r Z ) ( ) ( 1 ) ( 1 From Displacement transmissibility (T.R.) ) ( 1 ) ( 1 ) ( r j r r j Y Z Input Output T ) ( ) (1 ) ( 1 ) ( r r r Y Z T Tells how motion is transmitted from the base to the mass at various driving frequencies

9 Displacement transmissibility () T( ) Z Y 1 (r) (1 r ) (r) r = 0, T.R. = 1 Isolation occurs when r > Isolation region To increase r (decrease n ), m or k are changed In the isolation region, T.R. decreases with decreased In addition to T.R., another design criteria such as transmitted forces and installation space need to be considered.

10 Force transmissibility (1) EOM m x cx kx cy ky Force transmitted to the mass F( t) k( x y) c( x y ) mx FBD F( t) m X cos( t ) m Y T( ) cos( t ) F T cos( t ) Force transmissibility F T ky r 1 (r) (1 r ) (r) Tells how much force is transmitted from the base to the mass at various driving frequencies.

11 Force transmissibility () The force transmitted does not necessarily fall off for r > The force transmitted increase dramatically for r >, as the damping increases.

12 Example 5..1 (1) To protect the electronic control module from fatigue and breakage, it is desirable to isolate the module from the vibration induced in the car body by road and engine vibration. mass of the module 3kg. y(t) = 0. 01sin(35t) m. Desire to keep the displacement of module less than m at all times. Design values for isolator, calculate the magnitude of transmitted force.

13 Example 5..1 ()

14 Conclusion (Design problem) Given: m,, X, Y 1. Calculate T.R.. Select - calculate n, - calculate k, c 3. Select k, c from catalogue 4. Calculate FT 5. Calculate static deflection (used space)

15 Vibration isolation (Fixed base) Vibration isolator Objective: To isolate the source of vibration (machine) from the other system. To reduce vibration (force) transmitted from the machine to base. Applications: Washing machine Generator or the other machines

16

17 Force transmissibility (1) F(t) EOM m x cx kx F 0 cos t F0 x nx n x cos t m The steady-state response of EOM is FBD x( t) X cos( t ) F0 k Where X ( 1 r ) (r) ; r n kx (t) c x(t) Force transmitted to base is F T ( t) kx( t) cx ( t) F T ( t) kx cos( t ) cx sin( t )

18 Force transmissibility () Im kx F T ( t) kx cos( t ) cx sin( t ) Magnitude of F T (t) is cx t- Re F T ( t) ( kx) ( cx ) X k c 0 F T ( t) k c (1 r F ) k (r) Force transmissibility (T.R.) F T ( t) 1 (r) F0 (1 r ) (r) Tells how much force is transmitted from the machine to the base at various driving frequencies.

19 Force transmissibility (3) F T ( t) 1 (r) F0 (1 r ) (r) Isolation region Disp. T.R. (moving base) and Force T.R. (fixed base) have the same form Design consideration for both cases are the same Isolation occurs when r > To increase r (decrease n ), m and k are changed In the isolation region, T.R. decreases with decreased Some damping is also desirable to reduce vibration at resonance

20 Force transmissibility (4) Isolation region For a large freq. ratio (r > 3) and small damping ( < 0.) T.R. is not effected by damping Damping term can be neglected in common design 1 T. R. r 1

21 Construction of design curves (1) 1 T. R. (r > 3, < 0.) r 1 Reduction in T.R.: R = 1 T.R. r k m 1 R R n 60 g( R) (1 R) R (1 R) n = speed in rpm k mg is static deflection log n 1 log log R R

22 Construction of design curves () log n 1 log log R R

23 Example 5.. The disk drive motor is mounted to the computer chassis through an isolation pad (spring). The motor has a mass of 3 kg and operates at 5000 rpm. Calculate the value of stiffness of the isolator needed to provide a 95% reduction in force transmitted to the chassis (considered as ground). How much clearance is needed between the motor and the chassis. Circuit boards

24 Example: washing machine A model of a washing machine is illustrated in the figure. A bundle of wet clothes form a mass of 10 kg (m 0 ) and causes a rotating unbalance. The rotating mass id 0 kg (including m 0 ) and the diameter of the washer basket (e) is 50 cm. Assume that the spin cycle rotates at 300 rpm. Let k be 1000 N/m and =0.01. (a) Calculate the force transmitted to the sides of the washing machine. (b) The quantities m, m 0 e and are all fixed by the previous design. Design the isolation system so that the force transmitted to the side of the washing machine is less than 100 N.

25 Shock Isolator 1.0 Reduction of the acc. through isolation occurs only if the T.R. falls below 1. Shock isolation enforces a boundary on the stiffness. Increasing the damping greatly reduces the max. acc.

1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!

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