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- 1. For the system below: a. find the differential equation in terms of 0 (for small values of 0) b. The damped natural frequency (od), and the damping ratio () (leave the answer in terms of M, C and I). c. For M=.026 lb-sec2 / in, 1 = 10 in, C = .0323 lb-sec / in, 0(0) = 10° and (0)=0, find the equation for time response 0(t) and sketch a plot of this response. M www ის CQ1/ An over damped system consists of two first-order processes operating in 16, T2 = 4). Find the equivalent values of T and for the second order system and prove that the T1 = 16, T2 = 4 ). - series, if the (T1 =2. A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t)=C₁e Pt cos (@₁t-a₁). Also, find the undamped position function u(t)= Cocos (@ot-ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). 1 m = 2, c = 4, k = 6, x₁ = 4, V₁ = 0 x(t) = which means the system is (1). (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than or…
- A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e -p' cos (@,t-a,). Also, find the undamped position function u(t) = Co cos (@ot – ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c=0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). c= 4, k = 6, xo =3, vo = 0 m =One of the tail rotor blades of a helicopter was damaged, and due to the repair, there is an extra 2 kg mass 0.40 m from the axis of rotation. The equivalent system before the repair had a mass of 102 kg and stiffness of 93.8 kN/m. Assume the rotor was well-balanced before the damage. The repaired helicopter tail has a damping ratio of 0.15. Find the steady-state response of the tail when the blades rotate at 1500 rpm.Find the response of the mass if it is subject to the force in the plot. The damping ratio for the system is 0.1. You do not have to expand the series nor carry out any multiplication in the response equation. Hint: ao = 0. You may use this value and do not have to calculate ao- 2a 50 Mass m m = 10 kg Massless Rigid Rod a =0.3 m Massless Circular Cord -50 Length { = 0.625n m Diameter d = 0.01 m 0 1 Time (s) -2 -1 2 3 Modulus E = 0.001 GPa Force (N)
- A mass-spring system is driven by the external force g(t) = 2 sin3t + 10 cos 3t. The mass equals 1, the spring constant equals 5, and the damping coefficient equals 2. If the mass is initially located at y (0) =-1, with initial velocity y' (0) = 5, find its equation of motion to find y (1). Round up the answer to the second decimal place point.A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position x and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, runderdamped. If it is underdamped, write the position function in the form x(t) =C₁ e Pl cos (@t-α₁). Also, find the undamped position function u(t) = Cocos (ot) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). m = 16, c=40, k = 169, x = 5, Vo = 16 x(t)=, which means the system is (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures radians. Use angle measures greater than or equal to 0 and less than or equal to 2x.)A mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e - pt cos (0,t-a,). Also, find the undamped position function u(t) = Co cos (@ot - ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c= 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). C = 1 m = - 4 c= 3, k= 8, x, = 7, vo = 0 x(t) = 14 e -4t - 7 e -8t , which means the system is overdamped. (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater…
- Figure below shows a single degree of freedom system. The system has a moment of inertia I =37 kg.m² about point O, the stiffness and damping values are k =548 N/m and C =113 N.s/m, respectively. Considering a =1.6 m and b =0.7 m, find the damped natural frequency of the system in rad/s. a b C wwwA mass m is attached to both a spring (with given spring constant k) and a dashpot (with given damping constant c ). The mass is set in motion with initial position xo and initial velocity vo. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form x(t) =C, e - pt cos (0,t-a,). Also, find the undamped position function u(t) = Co cos (@ot - a0) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c= 0). Finally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). 1 m = - 4' c= 3, k= 8, xo =7, vo = 0 x(t) = which means the system is (Use integers or decimals for any numbers in the expression. Round to four decimal places as needed. Type any angle measures in radians. Use angle measures greater than or equal to and ess than or…Q1: For the system shown in figure (1), find the values of a, b, and K, to obtain the required response of damping ratio (5 = 0.5) and natural frequency (O,=4) R(s) + C(s) s+b K. s+a 1 s(s+3)(s+8) Figure (1)