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Table 8 Non-dimensional natural frequencies of free-free rods with A(x) = Ao sin³ (ax+b) for various values of a (L₁=0, L₂=1, b=1) a=0.25m-0.5b 3.171275 6.298229 Mode 1 2 3 4 5 6 Mode 1 2 3 4 5 6 9.434833 12.573918 15.714004 18.854591 b=0.9 3.606218 6.568999 Table 9 Non-dimensional natural frequencies of free-free rods with A(x) = Ao sin³ (ax+b) for various values of b (L₁ = 0, L₂=1, a = - 2b) 9.624896 12.719210 Mode 1 2 15.831289 18.952812 3 4 5 6 a=0.5m-b 3.237508 6.333403 9.458556 12.591784 15.728324 18.866537 solutions by matlap program b=1.0 3.470891 6.475015 9.557056 12.666805 15.788776 18.917112 n=1 3.250523 6.342610 9.465160 12.596870 15.732444 18.869993 a=0.75-1.5b 3.331630 6.387328 9.495522 12.619786 15.750829 18.885339 I need the graph and the code for solving equations using Matlab methods and the method of solving according to the program find the derivation of A(x) = A. Sin (ax +b) and draw the graph bel, b = 1.1 3.361601 6.405441 9.508098 12.629357 15.758539 18.891788 lu and L. Aydin / Exact longitudinal vibration characteristics of rods with variable cross-sec a=-26 3.470891 6.475015 Table 10 in-dimensional natural frequencies of free-free rods with A(x) = Ao sin" (ar + b) for rious values of n (L₁ = 0, L2=1, an-2b, b= 1) n=2 3.360328 6.406605 9.509269 12.630355 15.759385 18.892514 9.557056 12.666805 15.788776 18.917112 b=1.2 3.276077 6.355043 9.473321 12.602950 15.737291 18.874025 n=3 3.470891 6.475015 9.557056 12.666805 15.788776 18.917112 n=4 3.582093 6.547675 9.608466 12.706197 15.820605 18.943781