Energy flow and performance evaluation of inerter-based vibration isolators mounted on finite and infinite flexible foundation structures

This study investigates the vibration power flow behavior and performance of inerter-based vibration isolators mounted on finite and infinite flexible beam structures. Two configurations of vibration isolators with spring, damper, and inerter as well as different rigidities of finite and infinite foundation structures are considered. Both the time-averaged power flow transmission and the force transmissibility are studied and used as indices to evaluate the isolation performance. Comparisons are made between the two proposed configurations of inerter-based isolators and the conventional spring-damper isolators to show potential performance benefits of including inerter for effective vibration isolation. It is shown that by configuring the inerter, spring, and damper in parallel in the isolator, anti-peaks are introduced in the time-averaged transmitted power and force transmissibility at specific frequencies such that the vibration transmission to the foundation can be greatly suppressed. When the inerter is connected in series with a spring-damper unit and then in-parallel with a spring, considerable improvement in vibration isolation can be achieved near the original peak frequency while maintaining good high-frequency isolation performance. The study provides better understanding of the effects of adding inerters to vibration isolators mounted on a flexible foundation, and benefits enhanced designs of inerter-based vibration suppression systems.

An Accurate Estimation of the Eigenfrequency of an Offshore Wind Turbine Considered as the Stepped Euler-Bernoulli Beam in Three-Spring Flexible Foundation Using the Power Series Method

The present study employs the power series method (PSM) to accurately predict the natural frequencies of eleven offshore wind turbines (OWT). This prediction is very important as it helps in the quick verification of experimental or finite element results. This study idealizes the OWT as a stepped Euler-Bernoulli beam carrying a top mass and connected at its bottom to a flexible foundation. The first part of the beam represents a monopile and the transition piece while its second part is a tower. The foundation is modeled using three springs (lateral, rotational, and cross-coupling springs). This work’s aim is at improving therefore the previous researches, in which the whole wind turbine was taken as a single beam, with a tower being tapered and its wall thickness being negligible compared to its diameter. In order to be closer to real-life OWT, three profiles of the tapered tower are explored: case 1 considers a tower with constant thickness along its height. Case 2 assumes a tower’s thickness being negligible compared to its mean diameter, while case 3 describes the tower as a tapered beam with varying thickness along its height. Next, the calculated natural frequencies are compared to those obtained from measurements. Results reveal that case 2, used by previous researches, was only accurate for OWT with tower wall thickness lower than 15 mm. Frequencies produced in case 3 are the most accurate as the relative error is up to 0.01%, especially for the OWT with thicknesses higher or equal to 15 mm. This case appears to be more realistic as, practically, wall thickness of a wind tower varies with its height. The tower-to-pile thickness ratio is an important design parameter as it highly has impact on the natural frequency of OWT, and must therefore be taken into account during the design as well as lateral and rotational coupling springs.

Effects of Simulation Length and Flexible Foundation on Long-Term Response Extrapolation of a Bottom-Fixed Offshore Wind Turbine

This paper deals with statistical and modeling uncertainty on the estimation of long-term extrapolated extreme responses in a monopile offshore wind turbine. The statistical uncertainty is addressed by studying the effect of simulation length. Modeling uncertainty is explored by evaluating the effects of considering a rigid and flexible foundation. The soil's flexibility is taking into account by considering the improved apparent fixity method. To identify the most relevant environmental conditions, the modified environmental contour method is used. The analysis focuses on the fore-aft shear force (FASF) and the fore-aft bending moment (FABM) at the mudline. The results show that using a simulation length of 10-min, does not provide sufficient accuracy. It was found that for the FASF, simulation lengths of at least 30-min are required to achieve an accuracy of about +/-5%. For the FABM, it was found that both the extrapolations made with 20-min and 30-min simulations achieved similar levels of accuracy of about 20%. Meanwhile, the results obtained from 10-min simulations reached deviations of about 40%. Finally, from the comparison made between a rigid and flexible foundation, it was found that the extrapolated responses exhibit maximum deviations up to around 5% and 10% for the FASF and the FABM, respectively. Also, for the FABM, it was observed that the consideration of a flexible foundation causes the critical wind speed to shift from 16.5 m/s (rigid) to 18 m/s (flexible).

Dynamic Interaction of Two Independent SDOF Oscillators Supported by a Flexible Foundation Embedded in a Half-Space: Closed-Form Analytical Solution for Incident Plane SH-Waves

This paper presents a closed-form analytical solution for the dynamic response of two independent SDOF oscillators standing on one flexible foundation embedded in an elastic half-space and excited by plane SH waves. The solution is obtained by the wave function expansion method and is verified by comparison with the results of the special cases of a rigid foundation and the published research result of a flexible foundation. The model is utilized to investigate how the foundation stiffness influences the system response. The results show that there will be a significant interaction between the two independent structures on one flexible foundation and the intensity of the interaction is mainly dependent on foundation stiffness and structural stiffness. For a system with more flexible foundation, strong interaction will exist between the two structures; larger structural stiffness will also lead to a strong interaction between the two structures. When the structural mass and the structural stiffness are all larger, the flexible foundation cannot be treated as a rigid foundation even if the foundation stiffness is many times larger than that of soil. This model may be useful to get insight into the effects of foundation flexibility on the interaction of two independent structures standing on one flexible foundation.

Efficiency considerations for Sommerfeld effect attenuation

In rotating machinery with a flexible foundation used for vibration isolation, the drive (motor) is designed to provide maximum power at the critical speed so that the rotor can be coasted up to a supercritical operating speed. Over the time, rotors develop some degree of unbalance and it may be impossible to balance the rotors in-situ, such as in a wind turbine. Therefore, oversized motors are used in anticipation of some unbalance. To promote passage through resonance, without altering the motor, various kinds of additional hardware and/or control mechanisms have been proposed in the past. On the other hand, simply increasing the foundation (or structural) damping reduces the amplitude of unbalance induced vibrations and the motor power requirement at the critical speed. This appears to be the simplest possible solution and has been proposed in relevant mathematical research. However, it is shown in this article that increasing foundation damping significantly reduces the overall system efficiency at the supercritical operating speed and it should not be at all considered as a meaningful engineering solution. In fact, the loss of efficiency due to the increased foundation damping is so large that it may not be compensated by a redesigned motor.