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3. Flexible body analysis using vibration mode
When a deformed object returns to its original shape, it vibrates. The shaking of the bridge in the bridge and truck example above is also a vibration. No matter what, a vibration means an object flexing. As such, a simulation can use the finite element method, which can deal with flexible bodies, to perform an analysis of vibrations.
366Hz and 710Hz in <Figure 6> introduced in the part 1 are simple vibrations at a single frequency. These frequencies are called natural frequencies or the mode. And all natural frequencies have their own vibration form, and these vibration forms are called mode shapes.
<Figure 8> shows the mode shape of a tuning fork. Theoretically, all physical objects have an infinite number of natural frequencies and vibration shapes from low frequencies to high frequencies. But factoring in such an infinite number of natural frequencies and vibration shapes in a simulation would require an immense amount of computational power. As such, the property of, “a vibration can express any complex deformation as the sum of mode shapes” is utilized. In other words, these mode shapes can be used when simulating the deformation of a flexible body over time. In fact, many dynamic simulations offer such a method.
Results of FFT on the vibrations of a tuning fork
A layman’s explanation of the above can be provided by a quote from French physicist Fourier, who stated, “A single complex wave is actually the sum of multiple simple waves.” This statement means that no matter how complex the wave, it can always be divided up into multiple simple waves, and when these simple waves are combined together, they become identical to the complex wave.
<Figure 7> shows how a complex wave (C) is separated into the two simple waves (A) and (B).
Figure 8 Natural frequency mode shapes for a tuning fork as calculated by a simulation
Most of the methods offered involve component mode synthesis, which is called CMS. As the objects analyzed by traditional finite element method grew larger and involved more complicated shapes, scaling issues with the models arose, so the CMS method was developed as an alternative reduced model.
In addition, since there are many cases of flexible bodies being connected with other objects through joints and the like in dynamic simulations, the mode shape at the location of the joint is very important. Since the CMS method provides both a constraint mode and a static correction mode , thereby including the mode shapes at the location of joints, they are also effective for dynamic simulations.
Static correction mode is also called fixed interface normal mode, and depending on the situation, may also be called the “constraint mode” on job sites.
3.1 Strengths of using vibration modes for flexible body analysis
3.1.1 Reduced computational complexity
CMS was developed for the reduction of finite element models. Here, reduction means a reduction in computational complexity. For example, a detailed model, simplified model, and reduced model can all be created for the same flexible body in accordance with the degree of computational complexity.
If the detailed model is a model that uses many small elements in a manner similar to the model used for strength analysis, then the simplified model is a model that uses fewer elements, as it uses elements that are larger than the elements used in detailed model. And the reduced model is a model that has been reduced through the use of methods such as CMS as explained above. In order to create a reduced model, there must first be a detailed model or simplified model. Once a detailed model or simplified model is made, CMS is used to create a reduced model.
Figure 9 Creation of a reduced model using CMS
In terms of computational complexity, the biggest difference is that while modes are used when making calculations with a reduced model as shown above, the calculations for an unreduced detailed model or simplified model require the use of all of the elements. While computational complexity depends on the number of modes used in a reduced model, unreduced models must use all of their elements, so computational complexity for unreduced models increases in proportion to the number of elements. For example, if there is a theoretical computational load of 600,000 for an unreduced model having 100,000 elements, then the theoretical computational load for the relevant reduced model could be 100.
Figure 10 Comparison of computational complexity based on model
As such, the use of flexible bodies using mode (reduced model) greatly reduces computational complexity, so it allows for shorter computation times when arriving at solutions.
(RFlexGen can create a reduced model in RecurDyn)
3.1.2 Reduction in effort needed to create flexible bodies
The meshes used in reduced models allow for sufficiently accurate results even if they are not as dense as the meshes used for strength analyses (detailed model). When reducing a model, it is sufficient to use the simplified model as shown above.
If the subject is high frequency noise or acoustics, then a detailed model may be needed, but in general, the frequencies at issue for machinery are often lower than such a threshold. In such cases, creating and using a mesh that properly expresses rigidity and mass alone is not a problem. In particular, bending and torsion is extremely accurate for the low frequencies in the 0 ~ 100Hz range that are most often used in everyday life, so flexible bodies created by the reduction of simplified models through CMS can be used to solve machinery vibration issues.
When such flexible bodies using modes (reduced models) are utilized, users can greatly reduce the amount of effort required to create a flexible body . In addition, when efficient auto mesh functions are used, the amount of effort required is reduced even further.
(The auto mesh function in Mesher can create meshes in RecurDyn)
3.2 Limitations of using vibration modes for flexible body analysis
The reason why it is possible to represent a given vibration shape as the combination of multiple mode shapes as shown above is because of the principle of superposition. The principle of superposition only applies to linear systems. As such, flexible body analysis using vibration modes are good to use within the scope of a linear system.
Cases that may exceed the scope of a linear system include cases where the materials are not linear materials such as metals but hyper elastic materials such as rubber, cases such as plastic deformations where major deformations with an uneven rate of deformation occur, and cases where contact occurs in areas that are difficult to predict. In such cases, a simplified model or a detailed model must be used instead of a reduced model.
4. Utilization of flexible body analysis using vibration modes
Despite the limitations set forth above, flexible body analysis using vibration modes have an extremely wide scope of application. This is because flexible body analysis using vibration modes (analysis using reduced models) are positioned between rigid body analyses that do not consider flex at all and flexible body analyses that have not been reduced. As such, the strengths of both can be utilized while the disadvantages avoided.
In terms of usability, as explained in 3.1.2 above, the degree of detail in the flexible body that is used to create the reduced model does not have a large impact on the accuracy of the results, so flexible bodies can be created rapidly when creating a reduced model.
In terms of vibration analysis, if a frequency range of only 0~20Hz is applicable when only rigid bodies are used, then the use of a flexible body using vibration modes would allow for the range to be expanded to around 100~200Hz. As such, this allows for real world use applications such as analyses of vehicle vibrations or passenger comfort.
In terms of flexible body analysis, reducing a large-scale model not only allows for faster calculations, but also allows for a higher degree of accuracy for results for real world use applications. Although analyses using vibration modes are unsuitable in theory for cases exceeding the scope of a linear system due to a large amount of deformation, there is generally not much deformation involved when designing machines, so most designs can utilize these analyses. As such, they can be used for automobiles, frames for operated machines such as construction equipment, frames for automated factory equipment, and other parts for machinery.
In addition, as shown above, reduced models can be used in dynamic simulations as well, allowing for quick and accurate analyses of models that include both rigid bodies and flexible bodies. Although detailed models are widely used — especially in models where the non-linear characteristics are important — in many cases reduced models also offer meaningful results given the amount of effort and time required. It is my hope that this work assists in the greater utilization of simulations going forward by explaining what reduced models are, in addition to their strengths and weaknesses.
Figure 11 Expansion of a multibody dynamic simulation using a reduced model
Written by Taero Cha (Director of China Business Division)
