### Coil Springs Calcs

thanks for the techical question. Many of our p eng type contributors can give you a more mathematical technical answer but, generally.

Most screeners use vertical coil springs which will elimate plus minus 90% of the inertia or motion of the moving body down thru the vertical line of the coil spring into the structure.

When the screen is operating that is one thing, but when it shuts down it must handle the ERRATIC FORCES of the 2 bearing heavy flywheel wgts which are trying in the last 30 seconds of ramp down on shutdown to HOP JUMP AND SOMEHOW come to a stop eventually. This is the nature of the beast on a 2 brg circle throw design unit. So the COIL SPRINGS must be able to handle that shutdown dynamic without breaking and we typically would have a 4" keeper cone on the spring TOP MOUNTING PLATE to help keep the springs with the machine vs having the screen box jump off the COIL SPRINGS.

Do not be afraid to consider the use of FIRESTONE MOUNTS, vertical rubber mounts which work very nicely and much quieter.

Also, plan C .......consider ROSTA TYPE SCISSOR MOUNTS.....EXCELLENT....which dissipate the vibration on shutdown in a horizontal plane vs down thru the structure and is superb if you are worried about structural vibration. ■

### Selection Of Coil Spring For Vibratory Screen

Dear Chrome,

Firstly let me comment that the 0.188 inch screen stroke at 900 rpm (giving 2.2 “g” acceleration) seems very low, and would not in normal applications provide efficient screening. Refer VSMA for suggested stroke v speed combinations.

However ignoring this for the moment:

As guru George points out, when selecting support springs for vibrating equipment a primary objective is to achieve good isolation efficiency i.e.: => 90%, and preferably => 95%, with about 98% considered commercially perfect.

In simple terms the greater the spring compression under load (the softer the spring), the more efficient will be its isolation efficiency, and the lower the operating frequency of the screen, the greater will be the compression required to achieve a nominated efficiency.

In your case a frequency of 900 rpm would require a spring compression under load of 0.49 inches to give a 90% isolation efficiency, 0.93 inches to give 95%, and 2.26 inches would give 98%. The spring dimensions you describe (with 1.47 inches static deflection) would achieve 97% isolation, which would be satisfactory.

Furthermore as you point out we must ensure that the spring will not bottom out during operation, and particularly during the increased motion at shut down as the screen coasts through the natural frequency of the spring. Normally a minimum allowance of 1.25 to 1.5 inches from the solid (fully compressed) spring height to the compressed height under load will adequately cater for this. Once again the spring dimensions you nominate seem satisfactory with approx 3.8 inches from solid to compressed height.

It also should be noted the higher the isolation efficiency of the spring – the lower will be its natural frequency – and the less will be the bounce at rundown. ■

### Re: Selection Of Coil Spring For Vibratory Screen

Thanks for looking at the situation guys.

John, I am curious as to what method you are using to specify the deflection-dampening relationship. I have found several formulas that relate to transmissibility but not to an actual deflection of a spring system.

I looked into some vibration control catalogs and used the formulas presented to arrive at the following:

The spring alone has a natural freq of around 52Hz. For the spring-mass system I used the formula for natural frequency of 3.13*[(K/W)^.5] where K is the spring-rate and W is the system weight.

So Fn = 3.13*(195/(2265/8)^.5 = 2.59Hz for a natural frequency of the system.

I used the equation for Transmissibility of 1/[((Fd/Fn)^2)-1] where Fd is the disturbing frequency of 900rpm ~ 15Hz and Fn is the the 2.59Hz solved for above. The Transmissibility I got was around 0.03 which is far into the region of isolation.

According to the commentary in the catalogs when the Fd/Fn ratio is around 1 is when you get the amplification of output to input we see in springs where the screens bounce 4-5 times what they do normally as they pass through the natural frequency of the system.

Could you provide a bit of explanation of how you arrive at the figures you quote? I found one of your earlier posts where you list rpms from 750 to 1800 and give a deflection in mm required to achieve 95% isolation but there was no method shown. ■

### Re: Selection Of Coil Spring For Vibratory Screen

G’day Chrome

My rough workings follow

THE NATURAL FREQUENCY OF A SPRING

The natural frequency of a spring (fn) determines its efficiency as an isolator.

Effective isolators have a low natural frequency.

Natural Frequency (fn) = 188 / SQRT effective deflection (ins)

or on our case

= 188 / SQRT(1.45) = 156.13 rpm

INSULATION RATIO

The insulation ratio (z) is the ratio of the imposed frequency (fd) to natural frequency (fn)

Insulation Ratio (z) = fd / fn

or in our case

= 900 / 156.13 = 5.76

TRANSMISSIBILITY

Transmissibility (T) is the amount of vibration energy that is transmitted to the support structure.

Transmission (T) = 1 / (z^2 - 1)

or in our case

= 1 / (5.76 ^2 – 1) = 0.03

ISOLATION EFFICIENCY

Isolation efficiency is the amount of vibration energy prevented from being transmitted through the spring.

Isolation Efficiency = 100 – (T x 100)

or in our case

= 100 – (0.03 x 100) = 97% isolation efficiency

------------------------------------------------------

p.s. to determine the required loaded spring deflection to achieve a nominated efficiency, simply transpose the formula:

for example, what loaded spring deflection is required to achieve 97% isolation efficiency at a 900 rpm imposed frequency

Required transmission (T) = (100 – isolation %) / 100 = (100 – 97) / 100 = 0.03

Required Natural Frequency of Spring (fn) = Fd / SQRT((1 / T) + 1)

= 900 / SQRT((1 / 0.03) + 1) ≈ 156 rpm

Required Spring Deflection = (188 / fn)^2 = (188 / 156)^2 = 1.45” ■

### Re: Selection Of Coil Spring For Vibratory Screen

Ok, I see it now. I didn't realize the 0.03 I got for a value of Transmissibility was where the 97% Isolation came from, duh

We were using the same formulas just slightly rearranged since you left frequency in terms of RPM while I was using Hz and you had the SQRT of static deflection in the denominator while I has its reciprocal components in the numerator of the Fn formula.

For anyone else following along in this thread in my formula:

3.13*[(K/W)^.5]

notice how I had to get the term "W" by dividing the total sprung mass by the number of springs supporting (i.e. 2265/8). The inverse of the K/W term is by definition the static deflection. ■

### Significance Of Vibration Acceleration.

Firstly let me comment that the

__0.188 inch screen stroke at 900 rpm (giving 2.2 “g” acceleration)__seems very low, and would not in normal applications provide efficient screening. Refer VSMA for suggested stroke v speed combinations.

However ignoring this for the moment:

As guru George points out, when selecting support springs for vibrating equipment a primary objective is to achieve good isolation efficiency i.e.: => 90%, and preferably => 95%, with about 98% considered commercially perfect.

In simple terms the greater the spring compression under load (the softer the spring), the more efficient will be its isolation efficiency, and the lower the operating frequency of the screen, the greater will be the compression required to achieve a nominated efficiency.

In your case a frequency of 900 rpm would require a spring compression under load of 0.49 inches to give a 90% isolation efficiency, 0.93 inches to give 95%, and 2.26 inches would give 98%. The spring dimensions you describe (with 1.47 inches static deflection) would achieve 97% isolation, which would be satisfactory.

Furthermore as you point out we must ensure that the spring will not bottom out during operation, and particularly during the increased motion at shut down as the screen coasts through the natural frequency of the spring. Normally a minimum allowance of 1.25 to 1.5 inches from the solid (fully compressed) spring height to the compressed height under load will adequately cater for this. Once again the spring dimensions you nominate seem satisfactory with approx 3.8 inches from solid to compressed height.

It also should be noted the higher the isolation efficiency of the spring – the lower will be its natural frequency – and the less will be the bounce at rundown.

Dear Mr. John

First I must thank you for a very informative post.

Please be kind enough to clarify the following:-

a. I would like to know how you arrived at 2.2 g acceleration from RPM and Stroke.

b. How can the Vibration Acceleration data be used to determine Stroke-Speed Combinations

c. How does the size and nature of the material to be screened affect the design requirement of Vibration acceleration

Thanks a Lot,

Regards,

Keshav ■

### Re: Selection Of Coil Spring For Vibratory Screen

Hi Keshav,

To try and answer your questions:

(Qa). I would like to know how you arrived at 2.2 g acceleration from RPM and Stroke.

(A).

**g = (stroke (mm) x rpm^2) / 1,800,000**

(this formula has been simplified and is not exact - but is close enough for all practical purposes here)

Therefore in our example of 0.188 inch (4.8mm) screen stroke at 900 rpm

g = (4.8 x 900^2) / 1,800,000 = 2.2

(Qb).* How can the Vibration Acceleration data be used to determine Stroke-Speed Combinations&*

(Qc).

*How does the size and nature of the material to be screened affect the design requirement of Vibration acceleration*

(A) These two questions are best answered together.

As a starting point:

- The combined effect of correct stroke and screening frequency are amoungst the most important considerations in achieving good screening efficiency
- It is these two factors (stroke and frequency) that determine screen acceleration ('g' factor), and this factor should be held within certain limits.
- However it is quite possible to also achieve the same 'g' factor with poor stroke and/or frequency selections, and thus have inefficient screening.

In other words the 'g' factor needs to be within certain limits - but being within those limits does not in itself imply good screening conditions. It is the the actual stroke and frequency selection that is important.__Stroke__

If the screen stroke is too small capacity will be reduced and material may clog or wedge in the openings. The stroke however should not be so great that it will interfere with stratification and tend to throw near size particles out of the apertures before they have had a chance to adjust themselves and pass through. Too much stroke will also decrease the life of the screening medium, bearings, and screen frame components.

A practical rule of thumb is that if material is jumping violently instead of flowing, chances are the stroke is too great.

The magnitude of stroke is usually selected in relation to the largest split size required. i.e.: the top deck aperture size. The thinking here is that the top deck with the largest particles and aperture sizes will require a greater stroke than the lower decks - and if the top deck does not screen efficiently - then the lower decks will not have the opertunity to do so.__Frequency__

The frequency of vibration will effect screening by increasing or decreasing the number of impacts between the screening medium and the material. The speed selected must be fast enough to maintain the necessary travel rate and liveliness in the oversize bed.

The selected speed however is not always correct for maximum capacity or highest efficiency, but is directly related to the required stroke. This is done to obtain acceptable bearing life, and to keep stresses in such components as deck frames and side plates etc. to within established limits for good life.

The sensitivity of bearing life expectancy to speed cannot be over emphasised. This effect can be appreciated by considering that for a given stroke, a 10% increase in speed will shorten bearing life expectancy by 50%, and an increase of 20% will reduce life expectancy to 25% of the initial value.

In brief the stroke and frequency of vibration are closely related, and for every size and kind of material there will be an ideal combination. If decreased stroke is selected the frequency should be increased, and where increased stroke is used the frequency must be reduced for the reasons outlined above.

Every manufacturer will have his own thoughts and standards for stroke and speed combinations, but a good starting point is the table listing in the VSMA manual which will prove satisfactory for most applications. However these may sometimes need to be modified since the products handled differ physically with regard to shape, irregularity of surface, moisture content, and the percentage of clays and micron size fines present.__The ‘g’ factor__

A limit must be placed on the “g” factor due to mechanical design of the screen – but circular motion screens generally run at around 3.5 – 4.0g while linear motion screens from say 4.5 to around 5.5g. The ideal is to aim for the lowest 'g' that will satisfactorily do the required job.

In summary the steps I would use to determine stroke speed combinations are:

- Select the screen stroke in relation to top deck aperture sizi (refer VSMA manual)
- Select the accompanying screen frequency (rpm). This can be done from the VSMA table, but if differing check that it does not exceed allowable 'g' factor

Hope this answers some of your questions, but always happy to discuss further if required

Kind Regards ■

### Thanks John

To try and answer your questions:

(Qa). I would like to know how you arrived at 2.2 g acceleration from RPM and Stroke.

(A).

**g = (stroke (mm) x rpm^2) / 1,800,000**

(this formula has been simplified and is not exact - but is close enough for all practical purposes here)

Therefore in our example of 0.188 inch (4.8mm) screen stroke at 900 rpm

g = (4.8 x 900^2) / 1,800,000 = 2.2

(Qb).* How can the Vibration Acceleration data be used to determine Stroke-Speed Combinations&*

(Qc).

*How does the size and nature of the material to be screened affect the design requirement of Vibration acceleration*

(A) These two questions are best answered together.

As a starting point:

- The combined effect of correct stroke and screening frequency are amoungst the most important considerations in achieving good screening efficiency
- It is these two factors (stroke and frequency) that determine screen acceleration ('g' factor), and this factor should be held within certain limits.
- However it is quite possible to also achieve the same 'g' factor with poor stroke and/or frequency selections, and thus have inefficient screening.

In other words the 'g' factor needs to be within certain limits - but being within those limits does not in itself imply good screening conditions. It is the the actual stroke and frequency selection that is important.__Stroke__

If the screen stroke is too small capacity will be reduced and material may clog or wedge in the openings. The stroke however should not be so great that it will interfere with stratification and tend to throw near size particles out of the apertures before they have had a chance to adjust themselves and pass through. Too much stroke will also decrease the life of the screening medium, bearings, and screen frame components.

A practical rule of thumb is that if material is jumping violently instead of flowing, chances are the stroke is too great.

The magnitude of stroke is usually selected in relation to the largest split size required. i.e.: the top deck aperture size. The thinking here is that the top deck with the largest particles and aperture sizes will require a greater stroke than the lower decks - and if the top deck does not screen efficiently - then the lower decks will not have the opertunity to do so.__Frequency__

The frequency of vibration will effect screening by increasing or decreasing the number of impacts between the screening medium and the material. The speed selected must be fast enough to maintain the necessary travel rate and liveliness in the oversize bed.

The selected speed however is not always correct for maximum capacity or highest efficiency, but is directly related to the required stroke. This is done to obtain acceptable bearing life, and to keep stresses in such components as deck frames and side plates etc. to within established limits for good life.

The sensitivity of bearing life expectancy to speed cannot be over emphasised. This effect can be appreciated by considering that for a given stroke, a 10% increase in speed will shorten bearing life expectancy by 50%, and an increase of 20% will reduce life expectancy to 25% of the initial value.

In brief the stroke and frequency of vibration are closely related, and for every size and kind of material there will be an ideal combination. If decreased stroke is selected the frequency should be increased, and where increased stroke is used the frequency must be reduced for the reasons outlined above.

Every manufacturer will have his own thoughts and standards for stroke and speed combinations, but a good starting point is the table listing in the VSMA manual which will prove satisfactory for most applications. However these may sometimes need to be modified since the products handled differ physically with regard to shape, irregularity of surface, moisture content, and the percentage of clays and micron size fines present.__The ‘g’ factor__

A limit must be placed on the “g” factor due to mechanical design of the screen – but circular motion screens generally run at around 3.5 – 4.0g while linear motion screens from say 4.5 to around 5.5g. The ideal is to aim for the lowest 'g' that will satisfactorily do the required job.

In summary the steps I would use to determine stroke speed combinations are:

- Select the screen stroke in relation to top deck aperture sizi (refer VSMA manual)
- Select the accompanying screen frequency (rpm). This can be done from the VSMA table, but if differing check that it does not exceed allowable 'g' factor

Hope this answers some of your questions, but always happy to discuss further if required

Kind Regards

Thank You for devoting your time and effort for that post John. I think I am getting a hang of 'g' now. Thanks a lot

Keshav ■

## Selection of Coil Spring for Vibratory Screen

I am evaluating a old set of drawings for a vibratory screen with the following specifications:

Vibrating Frame Weight (LF): 2265 pounds

Inclined 2 bearing "unbalanced" style

Counterweight (CW): 85 pounds total

Eccentricity of Counterweight from Shaft (R): 2.5"

Shaft Speed: 900rpm

Using the VSMA and general engineering formulas I calculated:

Static Moment = 212.5 inch-pounds

Dynamic Moment = 425 inch-pounds

Stroke = 0.188 inches

gFactor = 2.2g

The drawing indicates using coil springs with the following specs:

Wire Diameter: 0.5 inch Oil-Tempered Steel Wire ASTM A 229 MB Grade

Coil Mean Diameter: 3.415 inches

Coil OD: 3.915 inches

Free Length: 12 inches

Spring Rate: 195 pounds per inch

Total Coils: 13.5

Active Coils: 11.5

There are 8 springs used, 2 in each "corner". From this I calculated the static deflection to be about 1.45 inches. Typically it seems like springs are chosen that deflect about 30% of their maximum to handle the static load but that leaves unanswered how to properly size the spring to handle the oscillating load.

I

thoughtabout using the 0.188 stroke but during start/stop the screens of this type exhibit MUCH more travel than the operating stroke at full speed. I alsothoughtabout using the 2.2g acceleration and ensuring the spring could deflect enough to handle the 622 pound transient force that such an acceleration produces on this much mass per the (Force = mass * acceleration) formula but the acceleration is for such a short amount of time the spring doesn't "seem" to have enough time to "react" and deflect the 2.9 inches the spring rate would calculate for the resultant force. Likely because the spring rate formula assumes it is a "static" load for ease of calculation.What is the rule of thumb or procedure to choose the proper spring to handle the dynamic loading in this situation or with vibratory screens in general? Spring manufacturers ask what the L1 and L2 loads are but I hesitate to give them an L2 load by using F=ma or just giving them the deflection @ L2 from the estimated stroke.

Thanks for wading through all this explanation. ■