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Multidegree-of-Freedom (MDOF) Systems and Longitudinal Vibration in Bars
Published in Dhanesh N. Manik, Vibro-Acoustics, 2017
First, an undamped MDOF system without force excitation is studied to introduce the basic concept of multiple natural frequencies and their corresponding mode shapes. A detailed physical understanding of the concept of mode shapes is presented by using complex exponential and mode shapes. These mode shapes that are expressed in the form of a modal matrix can be used to decouple the equations corresponding to the MDOF with force excitation, under some specific conditions known as proportional damping. In proportional damping the damping matrix is proportional to the mass and stiffness matrices. This holds good for the vast majority of the cases and, therefore, the response of these MDOFs can be obtained for any arbitrary excitation. Although we have limited the discussion here to natural frequencies and mode shapes of MDOF discrete systems, the same discussion can be easily carried forward to determine the force response. The discussion presented here can also form the basis for experimental modal analysis.
Dynamic investigations of EDZs from Bátaapáti radwaste repository based on passive seismoacoustic measurements
Published in Vladimir Litvinenko, EUROCK2018: Geomechanics and Geodynamics of Rock Masses, 2018
Damping was solved by using Rayleigh damping. With this type of damping, the damping matrix that relates the damping force and velocity of the system is expressed in terms of the stiffness and mass matrix of the system. The damping becomes proportional to the mass and stiffness of the system. [C]=α[M]+β[k]
Dynamic investigations of EDZs from Bátaapáti radwaste repository based on passive seismoacoustic measurements
Published in Vladimir Litvinenko, Geomechanics and Geodynamics of Rock Masses: Selected Papers from the 2018 European Rock Mechanics Symposium, 2018
Damping was solved by using Rayleigh damping. With this type of damping, the damping matrix that relates the damping force and velocity of the system is expressed in terms of the stiffness and mass matrix of the system. The damping becomes proportional to the mass and stiffness of the system. [C]=α[M]+β[K]
Research on torsional vibration characteristics of reciprocating compressor shafting and dynamics modification
Published in Mechanics of Advanced Materials and Structures, 2020
Jian Liu, Xiaodong Sun, Xiao Zhang, Xiaobing Hou
In general, proportional damping and mode damping are used to describe the damping of small damping systems for the convenience of decoupling the equations. Here, proportional damping means that the damping matrix C is proportional to the inertia matrix I or the stiffness matrix K, or it is proportional to the linear combination of the both of them. However, proportional damping is just appropriate for the condition that C is linear combinations of I and K. Therefore, it is common that mode damping directly is used to describe the damping of the system, such as equation (12). Where, ηii = ξiiωni, and ξii is the damping ratio of the ith order canonical coordinate mode. Similarly, it is also assumed that the mode damping ratio is the same, which can make the damping matrix in regular coordinate transform to diagonal matrix, so that it will be easy to decouple. This method can get a good approximate solution when the system damping is small enough. The torsional vibration system of compressor shafting is multi-degree of freedom and small damping vibration system generally. Therefore, mode damping is adopted, and damping ratio is regard as 0.01.
Multi-hazard loss analysis of tall buildings under wind and seismic loads
Published in Structure and Infrastructure Engineering, 2018
Ilaria Venanzi, Oren Lavan, Laura Ierimonti, Stefano Fabrizi
The P-delta effect is taken into account by adding the structural geometric stiffness matrix to the structural stiffness matrix, as in Rutenberg (1981). A damping ratio of for each mode is considered. The damping matrix is computed from mass and stiffness matrix, by adopting the Rayleigh assumption.