12 Transmissibility Curves and Vibration Isolation Version 3

TRANSMISSIBILITY TRANSMISSIBILITY CURVES AND VIBRATION ISOLATION The following is the graph of the transmissibility for a one-dimension spring isolation mounting.

AMPLIFICATION  _____________  ISOLATION

70% 80%

90%

The vertical axis shows the ratio of the transmitted transmitted force to the driving force. Transmissibilities greater  than 1.0 indicate amplification: the transmitted force is greater than than the driving force. Transmissibilities less than 1.0 indicate effective isolation: the transmitted force is less than the driving force. The horizontal axis shows the ratio of the driving, or forcing, frequency to the natural frequency of the spring-mass system. A frequency ratio greater than 1.0 indicates that that the forcing frequency is higher than the natural frequency; less than 1.0 indicates that the forcing frequency is lower than the natural frequency. Where the frequency ratio is 1.0 is sometimes sometimes referred to as resonance. resonance.

Transmissibility curves and vibration isolation

page 1 of 4.

Damping

Damping is the process of removing energy from an oscillating system. Car suspension systems have coil springs, which act as the vibration isolators, and they also have dampers (mis-named shock absorbers –  usually a sealed tube containing oil and a piston) which prevent any up and down oscillations of the car   body continuing unchecked. The graph on the previous page shows the effect of adding damping to a spring isolation system. The transmissibility is considerably reduced at resonance but there is also a raising of the transmissibility (i.e. worse isolation) at frequency ratios greater than 1.414.

Practical isolation

Practical isolation designs use zero damping and a natural frequency (arrived at by the choice of spring stiffness) which is a third or quarter of the driving frequency (giving a frequency ratio of 3 or 4 respectively.

Forcing frequency

The forcing frequency is usually due to something rotating (e.g. fan or rotating drum which is not in  perfect balance) or reciprocating (e.g. pistons in a compressor). Rotation rates in rev/min can be expressed in hertz. E.g. 600 rev/minute = 10 rev/second = 10 Hz

Natural frequency

The natural frequency of oscillation of a spring of stiffness k when loaded with a mass m is given by:   f  0

=

1

k 

2π  

m

If the static deflection is  xs , millimetre, when the spring is loaded, then   f  0

15 .8

=

1  x s

Transmissibility

The full equation for the transmissibility of a mass spring system is given by: 2

   f     1 + 4ξ     f           T  =     f         f     1 −     + 4ξ     f          f             2

0

2

2

2

2

0

0

where ξ is the damping ratio (see the numbers ranging from 0 to 1 on the curves on page 1) For an undamped system ξ = 0 and the above expression then becomes: T  =

Transmissibility curves and vibration isolation

1

    f    2  1 −        f  0     

2

page 2 of 4.

Isolation

In practical terms, interest centres on the isolation provided by the spring mass system. If the transmissibility is 0.2, the isolation is 1 – 0.2 = 0.8. Another way is to quote the 0.2 transmissibility as 20%, therefore the isolation is 80%. Example An isolation of 90% is required. Therefore the transmissibility is 10% or  T = 0.1 T 

=

0.1

=

1

    f    2  1 −         f  0     

2

square root the right side 0.1

±1     f    2  1 −         f  0     

=

invert  both sides

10

    f    2  1 −         f      0    =  ±1

so 2

±10

   f     = 1 −         f  0  

rearrange 2

   f          =    f  0  

1 10

= −9

or 

+11

the minus value is ignored

   f          =   f     0  

11

=

3.32

So a frequency ratio of 3.32 will give 90% isolation If the driving force is due to a rotating machine running at 900 r.p.m. = 15 Hz

   f     15    3 . 32 = =     f  0    f  0  

giving

  f  0

= 4.52

Hz

The static deflection of the mounted system can be derived from: 1   f  0 15 .8  x s =

4.52

= 15 .8

1  x s

Transmissibility curves and vibration isolation

giving

 x s

=

12 .2 mm

page 3 of 4.

If the mass of the motor etc is 80 kg, the weight is 80

9.81 = 784.8 N



For this to give a static deflection of 12.2 mm, t he spring constant k must be:  F  784 .8 k  64 .3  N/mm  x 12 .2 =

=

=

If this stiffness is due to four springs supporting the platform carrying the motor, then the stiffness of each spring is: 64.3 4 = 16.1 N/mm = 16.1 kN/m 

Exercises

1.

The total stiffness of the springs of a vibration isolation platform is 200 kN/m. Calculate the natural frequency if the mass of the platform and load is: (a) 20 kg (b) 50 kg (c) 100 kg

2.

Calculate the static deflexion in each case in question 1 (assume  g = 9.81 m/s2 ). Verify your results against the formula   f  n

= 15 .8

1  x s

where  x s is in mm.

3.

A vibration isolation platform has a transmissibility of 0.07. (a) What is the percentage isolation. (b) What ratio of forcing frequency to undamped natural frequency would give this isolation.

4.

A vibration isolation platform has a percentage isolation of 87%. (a) What is its transmissibility. (b) What ratio of forcing frequency to undamped natural frequency would give this isolation.

5.

A compressor and electric motor are mounted on an undamped vibration isolation platform and  produce a static deflexion of 10 mm. Calculate: (a) the natural frequency of the assembly; (b) (i) the transmissibilty; and (ii) the percentage isolation if the working compressor/motor produces a forcing frequency of 30 Hz. (c) the lowest forcing frequency for which the isolation is 95%

6.

An electric motor and rotary fluid pump are mounted on an undamped vibration isolation platform. When running, the principle source of vibration is the pump which is turning at 900 r.p.m. Calculate: (a) The designed natural frequency of the platform in order to have 90% isolation. (b) The designed static deflexion of the platform in order to have 90% isolation. (c) The total stiffness of the platform springs, if the combined mass of the platform, motor and   pump is 200 kg (assume g = 9.81 m/s2 ). (d) The stiffness of each spring if there are 4 springs supporting the platform.

Answers

1.

(a) 15.9 Hz (b) 10.1 Hz

3.

(a) I = 93% (b) ratio = 3.91

5.

(a) 5.0 Hz

6.

(a) T = 0.10 , ratio = 3.32 ,  f  n = 4.52 Hz (b) 12.2 mm

(b) (i) 0.029

(c) 7.1 Hz

(ii) 97.1%

2.

(a) 0.98 mm

(b) 2.5 mm

4.

(a) T = 0.13

(b) ratio = 2.95

(c) 161 kN/m

(d) 40.3 kN/m

(c) 4.9 mm

(c) 22.9 Hz

Transmissibility curves and vibration isolation

page 4 of 4.