PSI - Issue 71

Faisal Hussain et al. / Procedia Structural Integrity 71 (2025) 248–255

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spectral, were employed to record the inherent beam frequencies. 4. Results and Discussion

Table 2 displays the non-dimensional frequency values used to describe the natural frequencies of a cantilever beam. These values are utilized to better understand the dynamic reaction of structures when tested by external excitations. A computer program was created using MATLAB to create the global stiffness matrix [ K* ] and mass matrix [ M* ] for a system with 2 mobilities at each node in the system. The non-dimensional natural frequencies were calculated for the presumed values of stiffness parameters by considering the response amplitudes (θ) in radians. It was observed that all four modes of non-dimensional natural frequencies were within the range of ideal values i.e. 1.849, 4.537, 7.249, and 9.566. A method based on statistics was used to determine the complex joint variables that represent the relationships among many subsystems or variables. Table 2 displays the non dimensional natural frequencies that have been identified. Table 2. Natural Frequencies (Non-Dimensional)

Assumed values of stiffness parameters

Response Amplitude in radian Ideal Value

Non - dimensional natural frequency

 

 

 

 

 

1.849 1.6277 1.6277 1.6278 1.6279 1.6280 1.6281 1.6283 1.6286

4.537 4.3253 4.3254 4.3261 4.3264 4.3266 4.3271 4.3273 4.3284

7.249 7.0190 7.0221 7.0273 7.0534 7.0685 7.0895 7.1230 7.1831

9.5660 9.4211 9.4213 9.4216 9.4216 9.4219 9.4232 9.4237 9.4249

12.1313 12.0819 12.0823 12.0827 12.0830 12.0834 12.0859 12.0911

K 1 = 0.97 x 10 4 N/m K 2 = 1.26 x 10 4 N-m/rad K 4 = 1.33 x 10 5 N-m/rad 3

0.0042 0.0063 0.0074 0.0087 0.0095 0.0105 0.0111 0.0122

12.0954 Fig.6 depicts the acceleration and frequency of the modeled system in frequency and time domain analysis considering the time limitation. It can be observed that there are variations in acceleration, facilitating the assessment of exerted forces and torques. A spectrogram, also known as a frequency-time graph, as shown in Fig.7 offers a succinct depiction of the sequential progress of the frequency characteristics of a signal or system. The graph depicts time (x-axis) and frequency (y-axis). The amplitude level for every component of frequency is commonly depicted using color or shading. The graph is crucial for studying signals in contexts that change over time, providing valuable insights into the contributions of different frequencies to the total signal at various points in time. It is necessary to comprehend the changing frequency components to diagnose problems, recognize trends, and enhance system performance.

Fig. 6. Time Domain Response

Fig.7.Frequency Domain Response

Fig.8. Hammer Impact Analysis

Fig.9 Frequency Variation Amplitude