Vibration Transmissibility Analyses of Asymmetric Cantilevered Structures with Screw Fastening by Mode Equivalent Factors
Authors:   Tokyo Roki Co., Ltd., JAPAN, Oh Gyoko 
Published: April 2022

The majority of fatigue failures in automotive components are due to vibrations, and it is important to analyse vibration response properties. Using the measured data of a symmetric sample and four kinds of asymmetric screw fastening cantilevered structures consisting of a steel block, copper weights, steel pipes, and joints modelling the components, vibration parameters of the first translational resonance, the second translational resonance, and the rotating direction resonance peaks were identified through analytical models of viscous damping. The influence degrees of the second and the rotating direction resonances were taken into account using mode equivalent factors of the amplitudes, and a simplified analytical method with high accuracy was proposed and verified. The samples which were supplied for measurements had a main mass m and a main length l, as well as added masses mad and added lengths lad. The measured vibration response data were summarized, and the transmissibility curves were plotted in figures according to the resonance frequency in a lateral excitation. To treat the sample as one mass point and as the Voigt model, a physical model was built and the equation of motion was set up. Equations of vibration transmissibilities of the system were obtained by solving the equations of motion. The equations were directly used for the translational mode 1. For the translational mode 2, the maximum vibration amplitudes were not always at the tip of the sample, and a mode equivalent factor η2 was used. The equation for the composed transmissibility was written as T=T1+η2 T2, where η2 = 0 at 𝑓1. For the rotating mode 𝜗¸ and 𝜑, and the third translational mode 3, mode equivalent factors η𝜗, η𝜑, and η3 could also be used, and the equation for the total transmissibility was written as 𝑇 ＝ 𝑇1+𝜂𝜗𝑇𝜗+𝜂2𝑇2+𝜂𝜑𝑇𝜑+𝜂3𝑇3 ＝ 𝑇1+Σ𝜂𝑛𝑇𝑛, where ηn = 0 at 𝑓≤𝑓n-1, and subscript n indicates the mode. The vibration parameters were identified using the measured peak data, and the theoretically calculated transmissibility curves using the parameters were compared with the measured curves. The calculated curves accorded closely with the measured curves over frequency. The frequency ratios f2/f1 of the second translational response peak to the first one were 5.6-11.2, and their transmissibility ratios T2/T1 were 0.02-0.18. The lowest transmissibilities of the valley between the two resonance peaks were 0.3-0.49, and excessive damping did not occur. In the samples with rotating vibration around the vertical axis, the f1 and the f2 declined when the added mass increased, but the spring coefficient did not greatly change. In the samples with rotating vibration around the node in the vertical direction, the f2 declined greatly when the added mass increased, and the spring coefficient also declined greatly. By changing the parameters of the model, vibration characteristics of similar systems can be estimated and predicted theoretically with the mode equivalent factors, indicating general versatility of this method.