Time period of transverse vibration of skew plate with parabolic temperature variation

Abstract

Time period of natural transverse vibration of a nonhomogeneous skew (parallelogram) plate with variable thickness and temperature field has been investigated on clamped CCCC and combination of clamped and simply supported CSCS edge conditions. The thickness variation on the plate is assumed to be linear in two dimensions, and the temperature variation on the plate is considered to be parabolic in two dimensions. For nonhomogeneity, authors considered circular variation in density. The Rayleigh–Ritz technique is applied to solve the differential equation of motion. A comparative analysis of frequency modes of the present study with the available published result is also given to support the present findings. The convergence study of obtained results is also presented with the help of figures.

Time Period of Thermal-Induced Vibration of Skew Plate with Two-Dimensional Circular Thickness

“…In order to validate the findings of the present study as well as to present the objective of the study, a comparison of frequency modes obtained in the present study was done with the following: (i) Frequency modes of parallelogram plate obtained in [11] on CCCC and CSCS edge condition corresponding to tapering parameters ϖ 1 , ϖ 2 and aspect ratio (a/b) (ii) Frequency modes of rectangle plate obtained in [17,18] on CCCC, SSSS, and CSCS edge condition corresponding to tapering parameters ϖ 1 , ϖ 2 and on CCCC, SSSS, CSSS, CSCS, and CCCS edge conditions corresponding to a thermal gradient κ e results are reported in tabular form (refer Tables 6 and 7 for parallelogram plate and refer Tables 8 and 9 for rectangle plate). For comparison of frequency modes of rectangle plate and obtained in [17,18], the authors exclude the nonhomogeneity φ in the present study as it was not considered in [17,18].…”

“…A comparison of frequency modes of the present study (parallelogram plate) and the frequency modes obtained in [11] is presented in Table 7 on CCCC and CSCS edge Table 3: Time period of parallelogram plate on various edge conditions vs nonhomogeneity φ for a fixed value of aspect ratio (a/b) � 1.5. [11] corresponding to tapering parameters ϖ 1 , ϖ 2 for a fixed value of thermal gradient κ � 0.0, nonhomogeneity φ � 0.0, skew angle θ � 30 °, and aspect ratio (a/b) � 1.5. Bold font values are obtained in [11].…”

“…Table 6 presents the comparison of frequency modes of the present study (parallelogram plate) and the frequency obtained in [11] on CCCC and CSCS edge conditions corresponding to tapering parameters ϖ 1 , ϖ 2 , for a fixed value of thermal gradient (HTML translation failed) and nonhomogeneity φ � 0.0, skew angle θ � 30 °, and aspect ratio (a/b) � 1.5. Table 6 clearly shows that the frequency obtained in the present study (parallelogram plate) is less Table 2: Time period of parallelogram plate on various edge conditions vs thermal gradient κ for a fixed value of aspect ratio (a/b) � 1.5. when compared with the frequency modes obtained in [11] with the increasing value of tapering parameters ϖ 1 , ϖ 2 on both CCCC and CSCS edge conditions.…”

“…[11] corresponding to tapering parameters ϖ 1 , ϖ 2 for a fixed value of thermal gradient κ � 0.0, nonhomogeneity φ � 0.0, skew angle θ � 30 °, and aspect ratio (a/b) � 1.5. Bold font values are obtained in [11].…”

“…Ritz method has been applied in [10] to investigate vibrational behavior (first three modes) of nonhomogeneous, nonuniform circular plates on clamped, simply supported, and free edges and presented the various plate parameters impact on frequencies. Effect of circular variation in density on time period of parallelogram plate having two-dimensional thickness and temperature on CCCC and CSCS edge condition has been computed in [11] using Rayleigh-Ritz method, where C and S stand for clamped and simply supported edge conditions, respectively. Modeling on time period of vibrational frequencies of nonuniform skew plate made up of nonhomogeneous material has been presented in [12] using Rayleigh-Ritz method, and effect of various plate parameters on time period has been computed.…”