Pumps & Systems , August 2007
With the wide use of variable frequency drives in the pump industry and increasing unit size, it is becoming more difficult to design mechanical systems free from natural frequencies within operating speed range. If such an occurrence is allowed in the field, a resulting resonance condition threatens to significantly impact performance and longevity of the equipment.
Traditional treatment methods that involve structural modifications are often time consuming and expensive. Blocking the problem frequencies in the variable frequency drive limits the use of the system by the user. One possible solution is an installation of a dynamic vibration absorber (DVA).
The DVA has certain advantages over other methods of vibration suppression. It is external to the machine structure, so no re-installation of equipment is necessary. Unlike with structural modifications, when the final effect is unknown until mass-elastic properties of the machine components have been modified, a DVA can be designed and tested before installation. It can be adjusted in the lab environment with predictable field results.
In many scenarios, this offers an economical vibration reduction solution. Figure 1 schematically shows one possible methods of installation of a DVA on a pump.
History and Principle of Operation
The dynamic vibration absorber was invented in 1909 by Hermann Frahm (US Patent #989958, issued in 1911), and since then it has been successfully used to suppress wind-induced vibration and seismic response in buildings. Characteristics of DVA were studied in depth by Den Hartog (1985).
In the industry, it has been primarily used to suppress vibration caused by a resonance condition in machinery. A DVA, sometimes referred to as a tuned mass damper, consists of a spring-mass system installed on a vibrating machine. In its classic form, its natural frequency is tuned to match the natural frequency of the machine it is installed on. Because of this tuning a DVA exerts a force on the main system that is equal and opposite to the excitation force, canceling vibration at the resonant frequency.
In modern applications, the goal is to assure the performance within specifications over a wide frequency range while minimizing the size of the device. A DVA is viewed by many engineers as a single frequency device. Analysis and an application example (presented below) demonstrate how vibration suppression in a wide frequency range can be achieved.
Dynamic Model
For simplicity, we will consider a dynamic model for a machine as a single degree of freedom system consisting of a single mass and a single spring. We will use a similar model for the dynamic vibration absorber. When the DVA is installed on the main system, the result is a two degree of freedom system whose dynamic model is shown in Figure 2.
In this system, the coordinate x1 corresponds to the displacement of the main mass M, and the coordinate x2 corresponds to the displacement of the absorber mass m. The main system's stiffness is represented by the equivalent spring K, while the absorber system has the spring k. The absorber system has a viscous damping element c while the main system is considered undamped. The main system is excited by a periodic force F that in rotating machines is usually represented by residual imbalance force, but could be any periodic excitation originating in the machine, such as vane passing excitation in centrifugal pumps.
The system above is described mathematically by a system of two ordinary differential equations. Employing some standard math (Den Hartog, pp. 93-96), the formula for the system response can be derived. We will plot the results of this formula with varying absorber parameters.
First, a few variables and dimensionless ratios must be introduced, since the results will be easier to handle in this form:
Now we are ready to plot the results. First, we will evaluate the effect of an undamped dynamic absorber with the absorber tuned to the main system natural frequency, so that the tuning ratio f = 1 (damping ratio ζ = 0). These results are shown in Figure 3.
It is notable how the dynamic absorber cancels vibration at the resonance frequency. Instead, it creates two new natural frequencies, one below and one above the original natural frequency. This happens because with the absorber the system has two degrees of freedom and hence two corresponding natural frequencies. The width between the two new natural frequencies depends on the mass ratio μ. Figure 3 shows the response with two different mass ratios.
With a larger absorber mass the natural frequencies sit wider apart, so a wider safe operating range around the original resonant frequency can be achieved. However, the large absorber mass very quickly becomes impractical, especially for large machinery. Figure 4 shows the two new natural frequencies in relation to the mass ratio of the absorber.
By changing the tuning ratio of the absorber, the position of the two new natural frequencies and a usable operating speed range between them can be further adjusted. Figure 5 shows the effect of tuning on the natural frequencies of the combined system with an undamped absorber (damping ratio ζ = 0).
Two curves represent two absorber systems: one with the standard tuning ratio f = 1 (blue lines), and the other one with the tuning ratio f = 1.4, representing an over tuned absorber system (magenta lines). The over tuned absorber creates a slightly higher low natural frequency, but significantly extends the range into the area of high frequencies.
Figure 6 shows the two natural frequencies of the combined system in relation to the tuning ratio. By varying tuning and mass ratios, a necessary operating speed range free of natural frequencies can be achieved with an undamped DVA.
This is important because an undamped absorber is simple to design and manufacture and its adjustment is less complicated than in a damped absorber that is described below. The tradeoff is that for a wide frequency range a required undamped absorber may become quite large.
