The Essence of System Vibrations
The dynamically loaded structural vibration system assumes its own unique natural frequency based on stiffness and mass. The design engineer's duty and responsibility is to analyze the proposed system to determine the resulting natural frequencies and the quality of the system. The engineer (designer) can then modify the design to achieve the required performance prior to construction, before the fact (on paper), not after the fact when the maintenance issues can become overwhelming.
The Natural Frequency
The true essence of vibration is the natural frequency of the total system:
The natural frequency fn of a single degree of freedom system (one spring, one mass) is expanded to any number of degrees of freedom (spring, mass points). Every mass point constitutes a degree of freedom; all of the expansion from single to multiple degrees of freedom is done by finite element analysis (FEA).
Equation (1) shows the stiffness (k) is directly proportional and the mass (m) is inversely proportional to the natural frequency and as the square root of the ratio - not linearly, as myth has it. The FEA must be properly designed and include the steel structure and foundation geometry and member properties; the major problem lies in the foundation.
Myths of "more mass" and/or "foundation can be ignored" are erroneous. All of the system's natural frequencies are erroneous if the foundation is ignored in the system design. The foundation is a spring-mass system and must be included in the designed and holistically analyzed system.
The Frequency Ratio
The FEA, loaded with the driving forces, will develop the FFT frequency domain output plot (see Figure 1); the spikes are problematic high maintenance resonance locations.
The system is then investigated to resolve resonance locations (this is all done on paper, before the fact). Arrange all natural frequencies fn in ascending order from the first (lowest) frequency. Determine the frequency ratios, R (in Equation 2) of the first (fundamental) frequency and the critical spikes of Figure 1 of the total holistic system where fd is the driving frequency and fn is the natural frequency:
R = (2)
If the ratio, R, of the first mode is:
•a) Between zero (a static structure) and about 0.7, expect minimum maintenance.
•b) Between 0.7 and 1.3, the system is doomed by resonance and high maintenance.
•c) Greater than 1.0, the system will suffer start and stop resonance, high maintenance problems and questionable performance.
The following may not be readily obvious: Using ratio R of the lowest frequency at less than 0.7, all other frequencies being higher will operate better because higher natural frequencies will develop lower Rs, which in turn move toward lesser dynamic effects and make good pump systems better.
Amplification Factors (AF)
Let's look at an example (see Figure 2).
If R = 0.8 (bottom), an AF of 7.5 (off the chart) requires all loads be amplified by 7.5. That includes nearby motors, bearings, shafts, bolted joints, piping, and welds, etc.; if you sized a bearing to support a 2-ton load and it is operating with a 15-ton load at a rated speed, then it will fail and result in high maintenance.
Foundation by Others is a Myth
The structural design, done by one party (perhaps an OEM) with a note on the drawing "Foundation by others," completely violates unstated principles of the holistic design and operation of the system. This may have been done in the interest of shirking responsibility and/or saving money, perhaps not understanding the workings of a dynamic system. The prototype naturally operates holistically, regardless of how it is designed.
A Sudden Change
Your system operated fine, but suddenly something went wrong. Did a foundation crack, settle and change internal member forces? What are you dealing with?
•a) Thermal loads, expansion/contraction?
•b) Mass changes due to processes and/or moving of equipment?
•c) Modified operating speeds?
•d) Change of machinery?
Such changes, subtle as they may appear, will change natural frequencies fns, frequency ratios Rs and amplification factors AFs. Knowing how the system will respond to the change is important-with more or less amplification.
Everything is a Spring - Soil, Concrete, Steel
1. The substructure is a spring-mass system of concrete stiffness (k), mass (m) and mass-less soil of stiffness (k). It must be loaded symmetrically, totally resistant to any uplift and/or overturning due to machine processes, wind and/or seismic forces (these are different from the dynamic loads being discussed). Soil cannot develop a tension force, so care must be exercised to ensure positive (or compressive) contact pressures over the entire contact surface between concrete and soil.
The foundation is the most important and costly part of the system to repair or modify; it is out of sight, but should not be out of mind. The foundation must be designed to resist all loads and the imposed soil pressure with minimum deflection. A substructure designed "by others" is not included in a holistic FEA analysis; therefore, it is "out of mind," creating a very problematic decision with high maintenance consequences. The substructure and the superstructure must be designed and analyzed as an integrated whole (a holistic system).
The amplified loads, service loads times amplification factor, are then extended through the foundation to the soil. Flexibility in the foundation slab allows varying soil pressures, soil spring constants and settlements, which will modify the natural frequencies fn of the design. The vital design details required of the foundation system are omitted here with the understanding that a qualified professional engineer of record is consulted.
2. The soil is a spring; it must be included in the design of the system. The spring constant to be used in the design is dependent upon and developed from the modulus of subgrade reaction of the soil. The modulus of subgrade reaction is a measure of soil capacity lbs/sq ft/inch of elastic settlement for the required dynamic conditions.
Soil spring constants vary depending upon the type of soil and its condition. The geotechnical engineer, with guidance and input from the professional engineer of record, must determine the value of the modulus of subgrade reaction to be used. Similar to steel and concrete, soil characteristics must remain elastic.
3. The concrete foundation slab is a spring and mass system; it must be a continuous monolithic (single pour piece), rigid, non-flexible slab. High strength concrete is beneficial to reinforced concrete for rigidity. Adding water to the mix above the design quantity at any time will diminish your anticipated strength and the dynamic characteristics of the concrete.
Size the concrete footing large enough to carry the amplified loads at a soil pressure less than required by the modulus of subgrade reaction. As the foundation slab is to be rigid, its thickness could approach 3-ft (or more) in least dimension, at which time "mass concrete design procedure" is required to reduce heat generated, control expansion and contraction when cooling.
Eliminate varying thicknesses, pipe chases, stress risers, etc. within the slab. Do not exceed the allowable dynamic soil spring constant (soil pressure) developed above. Design the foundation slab, with all contributing loads, symmetrically placed to develop uniform soil pressures (soil spring constants). Reinforce and cast the motor and other pedestals, appendages, etc. on top of the main slab.
4. The steel superstructure is also a spring-mass system; it must be designed as a dynamic system. Static design procedures are not acceptable and become high maintenance issues.
Summary
This information is assembled for guidance of how to create and what to look for in a dynamic foundation and structural system to avoid resonance and how to design a vibration system. Of particular interest, Figure 3 (horizontal axis) lists machine speeds, cycles per minute (CPM); the vertical axis lists permissible amplitudes (INCHES) for various operating conditions. The amplitudes listed are total for the system. Subjective guidelines are described within the figure.
Figure 3 is also a guideline to use when beginning a new design or investigating an existing system for a particular response. The particular required amplitude is the total amplitude of the system, structure and foundation. It should be obvious that stiffness and rigidity are a vital requirement to making good pump systems better.
Pumps & Systems, December 2007