. Find the solution of the vibrating membrane problem in the case where
7. Solve the vibrating membrane problem (symmetric case) 11(a,t) = 0 u(r,0) = f(r) u,(r, 0) = g(r) a) a = 1, c = 1, i(r) = Jo(air), g(r) = 0. b) a = 1, c = 1, f(r) = Jo(U3r), g(r) = 1-r'. 7. Solve the vibrating membrane problem (symmetric case) 11(a,t) = 0 u(r,0) = f(r) u,(r, 0) = g(r) a) a = 1, c = 1, i(r) = Jo(air), g(r) = 0. b) a = 1, c...
6.[10] Find the solution to the vibrating string problem governed by the given initial-boundary value problem: 9uxx = Utt 0<x< 1, t> 0 u(0,t) = 0) = u(tt,t), t> 0 u(x,0) = sin 4x + 7 sin 5x, 0<x< 1 uz (3,0) = { X, 0 < x < 1/2 r/2 < x <
1- Consider waves propagating in a vibrating quarter-circular membrane: at2 The displacement u(r, e t) is zero on the entire boundary at all times. a) Write down explicitly the three boundary conditions expressed above. b) Starting by the method of separation of variables, find the solution and show that it is given by ui (r, θ' t) = Σι Σ J1(A ct) sinde) [A, cos(JA Ct)+B, sin(vA ct)], where l is a positive even integer, and n is a positive...
9. Solve the vibrating membrane problem 11(r,θ,0) = f(r,0) b) F(r, θ) = 0. g(r, θ) = (1-rrsi20, a = c = 1. 9. Solve the vibrating membrane problem 11(r,θ,0) = f(r,0) b) F(r, θ) = 0. g(r, θ) = (1-rrsi20, a = c = 1.
5. (Vibrating Drum) The motion of a circular elastic membrane, such as the head of a drum, is governed by the two-dimensional wave equation in polar coordinates: tions satisfied by R(r), e(6), and T(t). Note: You do not need to solve these ODEs. 5. (Vibrating Drum) The motion of a circular elastic membrane, such as the head of a drum, is governed by the two-dimensional wave equation in polar coordinates: tions satisfied by R(r), e(6), and T(t). Note: You do...
Solve the circularly symmetric vibrating membrane PDE given as u_tt = ∇^2*u BC : u(1, θ, 0) = 0, 0 < t < ∞ ICs : u(r, θ, 0) = J_0*(2.4r) − 0.25*J_0*(14.93r), 0 ≤ r ≤ 1 u_t(r, θ, 0) = 0 Solve the circularly symmetric vibrating membrane PDE given as Utt = Dau BC : u(1,0,0) = 0, 0<t< oo ICs : u(r,0,0) = J.(2.4r) – 0.25J(14.93r), 0 <r <1 Ut(r,0,0) = 0
3. (20 pts). Find the solution to the vibrating-string problem: utt u(0,t) u(L,t) u2,0) 2,0) = 0 = 0 2 sin(27/L) + sin(31/L) sin(72/L) 0<<L, 0<t< 0<t< oo 0<t< 0<r<L 0<r<L
Problem 9. Find natural frequencies and principal coordinates of the two-degrees-of-freedom vibrating system: a) b) Problem 9. Find natural frequencies and principal coordinates of the two-degrees-of-freedom vibrating system: a) b)
please solve B? and C) The total energy for the vibrating string problem can be written as E = Kinetic Energy + Potential Energy = dx. Consider the case where u(r, t) satisfies the wave equation with the boundary con ditions ux(0,t) 4(L, t)-0. (a) Show that E is constant in time (b) Calculate the energy in 1 mode. (c) Show that the total energy is the sum of the energies contained in each mode
3) Consider the following vibrating system u" (1/4) 2u 2 cos (wt), u (0) 0, (0) 2 (a) Find transient and steady states of solution (b) Find the amplitude R of the steady state solution in terms of w and plot R versus w; (c) Find Rmax and wmax 3) Consider the following vibrating system u" (1/4) 2u 2 cos (wt), u (0) 0, (0) 2 (a) Find transient and steady states of solution (b) Find the amplitude R of...