Study On the Bistable Vibration Behaviour of a Rod Fastened Rotor-Bearing System

Based on the Lagrange equation, the motion equation of a rod fastened rotor-bearing system considering the damping of the contact interface is established. The numerical method is employed for numerical analysis. The bifurcation diagram, time series, frequency waveform, phase spectrum and Poincare map are used to illustrate the nonlinear dynamic behaviour. The transient responses during acceleration and deceleration are calculated to reveal the dynamic behaviour of the system. The numerical results hold that since the oil film is nonlinear, the system presents obvious bistable behaviour and a jumping phenomenon. In addition, a test bench of the rod fastened rotor-bearing system is built. The bistable behaviour and jumping phenomenon are experimentally proven, and the effect of the eccentric distance of the rotor on the bistable behaviour is experimentally explored. The results of this paper can be used for the basic design and fault diagnosis of rod fastened rotors.

The approximate calculation of the natural frequencies of a Stockbridge type vibration damper and analysis of natural frequencies' sensitivity to the structural parameters

Vibration damper is widely used in overhead transmission lines to alleviate aeolian vibration. Its natural frequencies are important parameters for a vibration damper. In this paper, the approximate calculation formulas of natural frequencies of the one-side subsystem of a Stockbridge type vibration damper were derived and the design sensitivity analysis of the natural frequencies was studied using partial differential equations with respect to each concerned parameter including the length of the steel strand, the mass of the counterweight, the eccentric distance, and the radius of gyration of the counterweight. Through a case study that considered a variation of up to ±30 % in the values of the design parameters, the exact calculation and approximate calculation results of the natural frequencies were analysed, and the sensitivity of the vibration damper's natural frequencies to the design parameters was studied. The results show that, within the range of the parameters used in this study, the approximately calculated first-order frequency is lower than the exact values, whereas the approximately calculated second-order frequency is larger than the exact values. The sensitivity analysis indicates that the first-order frequency is highly sensitive to the steel strand's length, whereas it is moderately sensitive to the counterweight's mass and slightly sensitive to the eccentric distance and the gyration radius of the counterweight; the second-order frequency is highly sensitive to the steel strand's length and the gyration radius of the counterweight, moderately sensitive to the counterweight's mass, and slightly sensitive to the eccentric distance. It will provide theoretical guidance and approximate analysis method in engineering for the design of the vibration damper.

Effect of Eccentric Distance on Successive Dual-droplet Impacting a Super-hydrophobic Tube

Droplet impact is a common but significant phenomenon in industry. The CLSVOF (coupled level set and volume-of-fluid) method is used to numerically study the successive dual-droplet impacting a super-hydrophobic tube. For the impact velocity of 1.0 m/s, the effect of the eccentric distance on dynamic characteristics is analysed, the corresponding eccentric distances are 0.5, 1.0 and 2.0, respectively. In addition, the break-up during rebound is analysed with velocity field and pressure nephogram. Results show that, the eccentric distance hinders the spread during the initial period of spreading. With the increase in eccentric distance, more liquid gathers at the eccentric side and the liquid film might rebound easily without break-up under the same impact velocity. The break-up during rebound mainly depends on the local airflow and pressure difference. The high-pressure zone near the solid-liquid interface moves towards the eccentric side with the increase in eccentric distance.

General eccentric distance sum of graphs

For [Formula: see text], we define the general eccentric distance sum of a connected graph [Formula: see text] as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of a vertex [Formula: see text] in [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This index generalizes several other indices of graphs. We present some bounds on the general eccentric distance sum for general graphs, bipartite graphs and trees of given order, graphs of given order and vertex connectivity and graphs of given order and number of pendant vertices. The extremal graphs are presented as well.

Further results on Zagreb eccentricity coindices

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.