# Critical Speed

Critical speed is the natural frequency of vibration of a pump. All centrifugal and vertical turbine pumps have rotors and structures that can vibrate in response to excitation forces.

When the frequency of the excitation forces is close to the natural frequencies of the structures, resonance can occur and excessive and damaging vibration levels can be reached. These natural frequencies of vibration usually occur in one or more of the following modes:

- Rotor lateral vibration
- Rotor torsional vibration
- Structure lateral vibration

Computational methods using application-specific programs, or finite element analysis (FEA) programs may be required to produce accurate results. The actual distribution of the structure mass and stiffness can be difficult to determine, affecting the accuracy of the calculation.

- Pump manufacturers can calculate or determine by test the natural frequency of the pump assembly. However, in a field installation, the vibrating structure comprises, in addition to the pump, the foundation, the mounting, the piping and its supports, and may include the driver and flexible shafting.

- The natural frequency of the vibrating structure is determined by the stiffness of the total structure and by its equivalent mass. It may, therefore, differ significantly from the natural frequency of the pump alone.

- Accurate measurement of critical speed can be made by externally exciting the pump with an instrumented hammer and measuring the resulting vibration.

- This can be done with the pump stationary or operating.
- If done when stationary, neither the dynamic effects of liquid motion and resulting excitation forces are included. nor the support provided by the impeller wearing rings.
- If done during operation, special computer software is necessary to filter out the vibration frequencies that are caused by normal operation.

There is a radial deflection when the pump operates off of is best efficiency point (BEP)

We calculate the magnitude of the deflection from the basic formula:

If the centrifugal pump is of a double ended design with sag occurring between two bearings, the bending formula will change slightly. The new formula looks like this:

- Y = The deflection in inches or millimeters
- W = Force on the impeller, in pounds or Newtons (includes the weight of the shaft)
- L = The length of the shaft from the center of the inboard bearing to the center of the impeller (in inches or millimeter. For double ended pumps it is the length of the shaft between the bearing supports.)
- E = The modulus of elasticity of the shaft material (lbs./ square inch or Newtons / square millimeter)
- I = The moment of inertia for solid shafts is (π d
^{4}/ 64). For tubular shafts we would use (π (d0^{4}– di^{4}) / 64) - Substituting (π d
^{4}/ 64) for “I” in the first formula, we get:

We use this formula to make comparisons between competitor pumps specified for the same application. We do this by eliminating the non variables from the formula. The non-variables are:

- W = The force on the shaft will be the same in the designs we are comparing.
- 3 or 384 = This is a factor that describes the shaft support method and load distribution. We eliminated it because the pumps we are comparing are similar in construction.
- E = The modulus of elasticity is similar for all common shaft materials.
- π = 3.1416 (does not change with application).
- 64 = is a constant

This leaves us with Y = L^{3}/D^{4}

I reviewed this formula with you because we are going to use the same formula to learn the first critical speed of a centrifugal pump.

At this point it is important to note that any object made from an elastic material (and metal is an elastic material) has a natural period of vibration. This happens because the pump rotating assembly is not absolutely uniform around the center-line of the shaft. We get variations in the density of the materials as well as manufacturing tolerances and casting irregularities contributing to the problem.

This eccentricity produces deflection when the rotating assembly rotates at the speed the centrifugal force exceeds the elastic restoring forces. At this speed the assembly will vibrate as if it were unbalanced, and could fail the seal, bearings or fatigue the shaft itself. The lowest speed at which this happens is called the first critical speed.

The first critical speed is linked to the pump’s static deflection. We can calculate this deflection by going back to the original formula and substituting the weight of the rotating assembly for the “W” in the formula. You can use either pounds or Newtons.

It should also be noted that this critical speed can be very destructive in mixer and agitator applications because of their very high L^{3}/D^{4} numbers.

Now that you have calculated the static deflection (sag) of the shaft as measured at the impeller, we will use this number to calculate the first critical speed of the pump. For all practical purposes you can calculate the first critical speed by using one of the following formulas:

- Nc = Critical speed
- Y = The deflection that we calculated.

To maintain internal clearances of the wear rings in a closed impeller pump and to prevent the impeller from hitting the volute or back plate in an open impeller pump, most pump companies would like to limit shaft deflection to between 0.005 and 0.006 inches (0,125 and 0,150 mm.). Putting these desirable numbers into the formula we get:

As you can see, these numbers are well in excess of the 1750 or 1450 rpm. that we normally use for centrifugal pump speed. They are, however, lower than the higher speed pumps that run at 3500 rpm. or 3000 rpm. This means that higher speed pumps and variable speed pumps will experience shaft deflection as they pass through, or run at these critical speeds.

Since operation off of the pump’s best efficiency point (**BEP**) is common for centrifugal pumps, you will be experiencing shaft loads well in excess of those noted in the above examples; meaning that your critical speed will actually be experienced at a much lower rpm. than noted.

The numbers we calculated reference a shaft running in air. In actual practice the impeller and a major portion of the rotating assembly is immersed in liquid that provides a hydrodynamic support to help stabilize the assembly. Pump people call this hydrodynamic stabilizing the “Lomakin Effect.”

Shaft packing provided an additional stabilization affect, but it was lost when the modern pumps were converted to mechanical face seals. Closed impeller pumps continue to retain some of the effect in their wear rings. This is in fact the major cause of wear ring wear.

In addition to the radial force created by passing through a critical speed the rotating assembly is subjected to additional radial loads that include:

- Misalignment between the pump and its driver.
- Bent or warped shafts.
- An unbalanced rotating assembly.
- Operating off of the best efficiency point (BEP).
- Pressure surges and water hammer.
- Corrosion and erosion of the rotating parts, especially the impeller.
- Thermal growth.
- Some centrifugal pumps are belt driven.
- Piping misalignment.
- Cavitation.

All of these radial forces will have a major affect on the life of the seal and bearings as well as the shaft itself. Since it is almost impossible to calculate all of these changing forces in advance, it is important for you to stabilize the shaft as best you can to hold the deflection to an absolute minimum. Your options include:

- Eliminate shaft sleeves and use only solid, corrosion resistant shafts. This will make a major difference in any piece of rotating equipment.
- You can increase the shaft diameter by up-grading the centrifugal pump power end to a more robust model. Many pump and after market suppliers have adapters and up-grade kits readily available.
- Stabilize the shaft with a sleeve or journal bearing in the packing chamber and move the mechanical seal closer to the precision bearings. You can use any suitable material for the sleeve bearing with carbon, Ryertex, and Teflon® being the most popular. Most people prefer to use split mechanical seals with these stabilization bushings.

Changing the shaft material will not help. All the common shaft materials have just about the same modulus of elasticity:

- In USCS units the modulus is 28 to 30 x 10
^{6}psi. - In SI units the modulus is 1,96 to 2,10 x 10
^{6}Kg/ cm^{2}

If you are purchasing a new pump try to purchase solid, larger diameter or shorter shafts when ever possible. An L^{3}/D^{4} number of less than 60 (2 in the metric system) is as good a guide as any thing else you can use.

Converting packed pumps to a mechanical seal presents a major shaft stabilization problem to the pump manufacturer. Some day the ANSI (American National Standards Institute) and ISO (International Standards Organization) standards will be modified to compensate for this change.

Between now and then you will have to provide your own stabilization if you want to achieve satisfactory seal and bearing life.