## 3. Planar Modal Characteristic of Heavy Load Radial Tire
### System Implementation of the Experimental Modal Analysis  
Figure 6b shows the schematic diagram of the planar modal experiment of the heavy load radial tire and an experimental hammer test is implemented, the principal features of which are shown in Figure 6a. The system implementation of the experimental modal analysis consists of three main parts, including the excitation part, response part and the data analysis part. Vibration excitation is realized using a B&K hammer sensor (Naerum, Denmark), with the input force of the hammer sensor being measured by a force transducer. The PCB accelerometer sensors (PCB Piezotronics, Inc., Depew, NY, USA) are used to measure the resulting acceleration of the tire belt, the tire sidewall and the tire wheel illustrated in Figure 6c, in which the acceleration sensors are attached to the tire with a thin layer of mounting wax. Figure 6d presents the hardware system of the experimental modal analysis, in which by measuring the excitation force with the hammer sensor and the vibration response with the acceleration sensor, the input excitation force and output response acceleration are gathered using high-speed data acquisition equipment, namely a DEWE-43 (Austria). The experimental transfer function is calculated using spectrum analysis by combining the input excitation force and the output response acceleration. Two additional methods were implemented to further improve the data quality for the measured transfer function:  
1. Linear averaging was used to calculate the transfer function for every excitation point by averaging 10 repetitions of the transfer function measurement; and
2. The coherence function was checked to evaluate the quality of the measured transfer function to assess the measurement process for every excitation. The frequency transfer functions for a heavy-load radial tire with a large aspect ratio are shown in Figure 7 to illustrate the vibration characteristics, in which, Figure 7a illustrates the three-dimensional planar transfer function between the excitation at point A and the response. Figure 7b presents the coherence value calculated from the input force and output acceleration response as a means of checking the planar transfer function; furthermore the following conclusion can be made on the basis of Figure 7:  
1. The three-directional transfer function and coherence values for the excitation force and the response acceleration compared in Figure 7a show that the coherence value of the radial excitation force and the radial acceleration response have a difference of 1 for frequencies below 13 Hz. There is a similar difference of 1 between the coherence value for the radial excitation force and the tangential acceleration response for frequencies below 50 Hz. Beyond the above frequency  
Figure 6: Schematic diagram of the modal analysis system implementation; (**a**) experimental picture of the modal analysis; (**b**) schematic picture of the modal analysis; (**c**) experimental acceleration sensors; and (**d**) hardware implementation of the modal analysis. band, the coherence value of the radial excitation force and the lateral acceleration response fluctuate around the frequency of 75 Hz; and
2. The larger amplitude of the radial transfer function than the tangential and lateral transfer functions implies that planar radial deformation of the flexible belt is the main vibration source for heavy-load radial tires with a large aspect ratio. In addition to this, the sectional planar transfer function within 300 Hz can be divided into two parts: 0-160 Hz and 160-280 Hz. The sectional transfer function results for the coupling vibration between the flexible belt and the continuous sidewall, moreover, indicates that coupling vibration between the flexible belt and the continuous sidewall generally needs to be considered when undertaking vibration analysis of heavy-load radial tires with a large aspect ratio.