Answer to Question #140576 in Differential Equations for Nikhil Singh

Question

The differential equation of damped vibrating system under the action of an external periodic force is d^2x/dt^2+ 2r(dx/dt)+ n^2x= acospt

Answer 1

"\\displaystyle\n\\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} + 2r\\left(\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)+ n^2x= a\\cos(pt) \\\\\n\n\\textsf{Let}\\, x = f(t) \\\\\n\nf"(t) + 2rf'(t) + n^2f(t) = a\\cos(pt)\\\\\n\n\\textsf{The complementary function is derived}\\\\\n\\textsf{when}\\, \nf"(t) + 2rf'(t) + n^2f(t) = 0\\\\\n\n\\text{The auxiliary equation is}\\\\\n\n\nm^2 + 2rm + n^2 = 0\\\\\n\n\nm = \\frac{-2r \\pm \\sqrt{4r^2 - 4n^2}}{2} = -r \\pm \\sqrt{r^2 - n^2}\\\\\n\n\\textsf{Since}\\, n > r\\\\\n\nm = -r \\pm j\\sqrt{n^2 - r^2}\\\\\n\n\\textsf{Recall that if the auxiliary equation is}\\\\\nm = \\alpha \\pm j\\beta, \\, \\textsf{the general solution can be}\\\\\n\\textsf{written as}\\\\\n\ny = e^{\\alpha t}(A\\cos(\\beta t) + B\\sin(\\beta t))\\\\\n\n\\therefore \\textsf{The complementary function of the given DE is}\\\\\n\nf(t) = e^{-rt}(C\\cos(\\sqrt{n^2 - r^2}t + D\\sin(\\sqrt{n^2 - r^2}))\\\\\n\n\n\\textsf{This is a system with a frequency of}\\\\ \\frac{2\\pi}{\\sqrt{n^2 - r^2}}\\, \\textsf{per units of time and a}\\\\\n\\textsf{period of}\\, \\sqrt{n^2 - r^2} \\, \\textsf{units of time and an amplitude}\\\\\n\\textsf{that is decreasing with time.}\\\\\n\n\\textsf{The effect of the}\\, f'(t)\\, \\textsf{in the differential}\\\\\n\\textsf{equation is to introduce damping}\\\\\\textsf{into the oscillatory motion so}\\\\\\textsf{causing it to decay.}\\\\\n\n\n\\textsf{The coefficient}\\, e^{-rt} \\, \\textsf{introduced damping}\\\\\n\\textsf{into the system, and since}\\\\\n\\textsf{as}\\, t \\rightarrow \\infty, e^{-rt} \\rightarrow 0 \\implies f(t)\\rightarrow 0,\\,\\textsf{the vibrations}\\\\\n\\textsf{are ultimately damped out.}\\\\\n\n\n\\textsf{The particular integral is of the form}\\\\\n\nf(t) = A\\cos(pt) + B\\sin(pt)\\\\\n\nf'(t) = -Ap\\sin(pt) + Bp\\cos(pt)\\\\\n\n\nf''(t) = -Ap^2\\cos(pt) - Bp^2\\sin(pt)\\\\\n\n\n -Ap^2\\cos(pt) - Bp^2\\sin(pt) + 2r(-Ap\\sin(pt) + Bp\\cos(pt)) + n^2(A\\cos(pt) + B\\sin(pt)) = a\\cos(pt)\\\\\n\n\n(-Ap^2 + 2rBp + An^2)\\cos(pt) + (-Bp^2 - 2rAp + Bn^2)\\sin(pt) = a\\cos(pt)\\\\\n\n\\textsf{Comparing coefficients}\\\\\n\n-Ap^2 + 2rBp + An^2 = a\\\\\n\n-Bp^2 - 2rAp + Bn^2 = 0\\\\\n\nA(n^2 - p^2) + 2rBp = a\\hspace{0.5cm}(1)\\\\\n\nB(n^2 - p^2) - 2rAp = 0\\hspace{0.5cm}(2)\\\\\n\n(1) \\times B \\, \\textsf{yields}\\\\\nAB(n^2 - p^2) + 2rB^2p = aB\\hspace{0.5cm}(3)\\\\\n\n(2) \\times A \\, \\textsf{yields}\\\\\nAB(n^2 - p^2) - 2rA^2p = 0\\hspace{0.5cm}(4)\\\\\n\n\\textsf{Subtract}\\, (4) \\, \\textsf{from}\\, (3)\\\\\n\n2rA^2 + 2rB^2 = aB\\\\\n\n\nA^2(n^2 - p^2) + 2rABp = aA\\hspace{0.5cm}(5)\\\\\n\nB^2(n^2 - p^2) - 2rABp = 0\\hspace{0.5cm}(6)\\\\\n\n\\textsf{Add}\\, (5)\\, \\textsf{and}\\, (6)\\\\\n\n(A^2 + B^2)(n^2 - p^2) = aA\\hspace{0.5cm}(7)\\\\\n\n2r(A^2 + B^2) = aB\\hspace{0.5cm}(8)\\\\\n\n\\implies A^2 + B^2 = \\frac{aB}{2r}\\\\\n\n\\textsf{Inserting in}\\, (7)\\\\\n\n\\frac{aB}{2r}\\cdot(n^2 - p^2) = aA\\\\\n\nB(n^2 - p^2) = 2rA\\\\\n\n\\implies B = \\frac{2rA}{n^2 - p^2}\\\\\n\\textsf{By}\\, (5) \\,\\textsf{and} \\, (6)\\\\\n\nA + \\frac{2rpB}{n^2 - p^2} = \\frac{a}{n^2 - p^2}\\\\\n\n-\\frac{2rpA}{n^2 - p^2} + B = 0\\\\\n\n-\\frac{4rp^2 A}{(n^2 - p^2)^2} + \\frac{2rpB}{n^2 - p^2} = 0\\\\\n\n\nA\\left(1 + \\frac{(2rp)^2}{(n^2 - p^2)}\\right) = \\frac{a}{n^2 - p^2}\\\\\n\nA\\left(\\frac{(n^2 - p^2)^2 + 4r^2p^2}{(n^2 - p^2)^2}\\right) = a\\\\\n\n\nA = \\frac{a(n^2 - p^2)}{(n^2 - p^2)^2 + 4r^2p^2}\\\\\n\nB =\\frac{2r}{n^2 - p^2}\\\\ A = \\frac{2arp}{(n^2 - p^2)^2 + 4r^2p^2}\\\\\n\n\\therefore f(t) = \\frac{a(n^2 - p^2)}{(n^2 - p^2)^2 + 4r^2p^2}\\cos(pt) + \\frac{2arp}{(n^2 - p^2)^2 + 4r^2p^2}\\sin(pt)\\\\\n\\begin{aligned}\nf(t) &= \\frac{a}{(n^2 - p^2)^2 + 4r^2p^2}\\sqrt{(2rp)^2 + (n^2 - p^2)}\\cos\\left(pt - \\arctan\\left(\\frac{2r}{n^2 - p^2}\\right)\\right)\\\\\n&=\\frac{a}{\\sqrt{(n^2 - p^2)^2 + 4r^2p^2}}\\cos\\left(pt - \\arctan\\left(\\frac{2r}{n^2 - p^2}\\right)\\right)\n\\end{aligned}\\\\\n\\textsf{Thus, the Particular integral is of the form}\\\\ b\\cos(pt - \\alpha),\\, \\textsf{where}\\, b^2 = \\frac{a^2}{(n^2 - p^2)^2 + 4r^2p^2}\\\\\n\n\n\\textsf{And the solution to the differential equation is}\\\\\n\nx = e^{-rt}(C\\cos(\\sqrt{n^2 - r^2}t + D\\sin(\\sqrt{n^2 - r^2})) + \\frac{a}{\\sqrt{(n^2 - p^2)^2 + 4r^2p^2}}\\cos\\left(pt - \\arctan\\left(\\frac{2r}{n^2 - p^2}\\right)\\right)\\\\"