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物理代写|振动力学代写Vibration Mechanics代考|EGR350 Closely Distributed Natural Modes of a Symmetric Structure

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物理代写|振动力学代写Vibration Mechanics代考|Closely Distributed Natural Modes of a Symmetric Structure

There are numerous symmetric structures in engineering. For example, airplanes and ground vehicles must be symmetric about a plane parallel to their moving direction. That is, they have mirror-symmetry. The other example of symmetry covers the bladed disc of an aero-engine in Fig. 1.8a and the radio-telescopic antenna of satellite Spektru-R in Fig. 1.8b, where both structures are exactly identical after they are rotated through an angle of $2 \pi / n$ about a central axis and exhibit the cyclic symmetry or a compound term cyclosymmetry. The designs of those symmetric structures not only show the beauty in appearance, but also meet the requirements of their functions.
It is well known that the system symmetry enables one to simplify the dynamic computation of the system. For a structure with mirror symmetry, it is easy to divide an arbitrary dynamic load to the structure into two parts, namely, a symmetric part and an anti-symmetric part. They excite the symmetric vibration and anti-symmetric vibration, respectively. Hence, one can take a half of the structure and compute respectively the symmetric and anti-symmetric vibrations, and then sum up the two vibrations to get the vibration due to an arbitrary dynamic load. For a structure with cyclosymmetry, the past decades since the 1970 s have seen several approaches to simplify the computation of natural vibrations of the structure. The approaches enable one to compute the natural vibrations of the structure from those of one of $n$ sectors. The bladed disc of an aero-engine or a turbo-engine usually has so many blades that $n>>10$ holds true in practice. These approaches, therefore, can save a great amount of computational cost.

In recent years, engineers have not worried so much about the computational cost with an integration of those approaches based on structure symmetry and fast development of computational techniques. Yet, they have encountered some new problems from structure symmetry as follows.

物理代写|振动力学代写Vibration Mechanics代考|Problems of Concern

(1) Problem 5A: Poor repeatability of modal tests of a symmetric structure
The engineering consultations that the author attended dealt with several kinds of symmetric structures. The problems often proposed by engineers were about the natural modes closely distributed in frequency domain and the poor repeatability in modal tests. For instance, the modal test of a bladed disc of a turbo-engine usually led to many sets of natural vibrations with very close frequencies but different mode shapes. One mode shape looked like the rotation of the other mode shape about the central axis by an angle. However, these mode shapes exhibited very poor repeatability in further modal test. As well known, the poor repeatability of industrial products in tests before use is a tough issue. This is particularly a big issue for any space structures in ground vibration tests before launch. Then, what is the reason of the above problem?
(2) Problem 5B: Natural vibrations of a cyclosymmetric structure with the dynamic deformation at the central axis

In the previous researches on the natural vibrations of cyclosymmetric structures, most publications dealt with annular structures shown in Fig. 1.8a, where the central axis of the structure is either clamped or hollowed, and no dynamic deformation needs to be studied. In practice, however, a cyclosymmetric structure may undergo the dynamic deformation at the central axis. A typical example is the radio-telescopic antenna of satellite Spektr-R in Fig. 1.8b. Then, it is reasonable to check the difference between the two kinds of cyclosymmetric systems, and whether the research on the natural vibrations of cyclosymmetric structures of the first kind can be extended to the second kind.

Among the early studies on the natural vibrations of cyclosymmetric structures of the first kind, D. L. Thomas in the UK analyzed the standing waves in the structures and pointed out the existence of standing waves along the circumferential direction. Many people thought that the standing waves in a cyclosymmetric structure of the second kind would be no longer those along the circumferential direction since the central DoFs of the cyclosymmetric structure of the second kind were not fixed. Therefore, it is difficult to extend the vibration analysis of cyclosymmetric structures of the first kind to those of the second kind.

物理代写|振动力学代写振动力学代考|关注的问题

(1)问题5A:对称结构模态试验的可重复性差

(2)问题5B:具有中心轴动变形的环对称结构的自然振动

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