This paper aims the automatic ball balancers (ABBs) used in passive balancing devices for suppressing vibration of the eccentric rotor. The system model describes which equipped with a skew-mounted ABB with angular deviation. The dynamic equilibrium equations of the system are deduced from the perspective of three-dimensional (3D) dynamics. The results obtained are consistent with those derived from the Euler-Lagrange equations. It is exciting that the spatial dynamics method reveals the spatial geometric characteristic of dynamic balance positioning of the balls when the system is balanced with vibration suppression. The spatial property emerges the perpendicular line from each ball to the rotating spindle of the system must pass through the central axis of the orbit perpendicular to the ABB plane. This characteristic is a general rule that can be used to explain the phenomenon of the stable equilibrium positions of the balls in all previously studied cases.

We report a facile route for the green synthesis of trimethylchitosan nitrate-capped silver nanoparticles (TMCN-AgNPs) with positive surface charge. In this synthesis, silver nitrate, glucose, and trimethyl chitosan nitrate (TMCN) were used as silver precursor, reducing agent, and stabilizer, respectively. The reaction was carried out in a stirred basic aqueous medium at room temperature without the use of energy-consuming or expensive equipment. We investigated the effects of the concentrations of NaOH, glucose, and TMCN on the particle size, zeta potential, and formation yield. The AgNPs were characterized by UV-visible spectroscopy, photon correlation spectroscopy, laser Doppler anemometry, transmission electron microscopy, X-ray diffraction, and X-ray photoelectron spectroscopy. The catalytic activity of the TMCN-AgNPs was studied by the reduction of 4-nitrophenol using NaBH₄ as a reducing agent. We evaluated the antibacterial effects of the TMCN-AgNPs on Acinetobacter baumannii, Escherichia coli, Pseudomonas aeruginosa, and Staphylococcus aureus using the broth microdilution method. The results showed that both gram-positive and gram-negative bacteria were killed by the TMCN-AgNPs at very low concentration (< 6.13 μg/mL). Moreover, the TMCN-AgNPs also showed high antibacterial activity against clinically isolated multidrug-resistant A. baumannii strains, and the minimum inhibitory concentration (MIC) was ≤ 12.25 μg/mL.