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1) Answer the following questions for harmonic oscillator with the given parameters and initial c...
Problem 4. The Fast Decay of Critically Damped Simple Harmonic Oscillator. A simple harmonic oscillator (a...
Problem 4. The Fast Decay of Critically Damped Simple Harmonic Oscillator. A simple harmonic oscillator (a box with mass m attached to a Hook's spring of coefficient k with linear air friction of coefficient n) is described by mx"(t) + n2'(t) + ku(t) = 0 where m, n, k > 0. (a) Write down the solutions for three cases and their long term limits 1. Overdamped: when friction is strong 1 > 4mk 2. Underdamped: when friction is weak 72...
5. A 2 kg mass is attached to a spring whose constant is 30 N/m, and the entire system is submerg...
5. A 2 kg mass is attached to a spring whose constant is 30 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 12 times the instaataneous velocity (a) Write the second-order linear differential equation to umodel the motion (b) Convert the second-order linear differential equation from part (a) to a first-order linear system (c) Classify the critical (equilibrium) point (0.0) (d) Sketch the phase portrait (e) Indicate the initial condition x(0)-(...
For the harmonic oscillator with mel, C=8, K:16 ; Xo = 5, Vo=4, do the following...
For the harmonic oscillator with mel, C=8, K:16 ; Xo = 5, Vo=4, do the following a. Find the position function x(t) and determine whether the motion is overdamped, critically damped or underdamped. b. Find the undamped (when c-0) position function ult) = Co Cos(wot-do)
Answer last four questions 1. A spring-mass-damper system has mass of 150 kg, stiffness of 1500...
Answer last four questions
1. A spring-mass-damper system has mass of 150 kg, stiffness of 1500 N/m and damping coefficient of 200 kg/s. i) Calculate the undamped natural frequency ii) Calculate the damping ratio iii) Calculate the damped natural frequency iv) Is the system overdamped, underdamped or critically damped? v) Does the solution oscillate? The system above is given an initial velocity of 10 mm/s and an initial displacement of -5 mm. vi) Calculate the form of the response and...
A damped osillator has a mass (m = 2.00kg), a spring (k = 10.0N/m), and a...
A damped osillator has a mass (m = 2.00kg), a spring (k = 10.0N/m), and a damping coefficient b = 0.102kg/s. undamped angular frequency of the system is 2.24rad/s. If the initial amplitude is 0.250m, How many periods of motion are necessary for the amplitude to be reduced to 3/4 it initial value? is this system underdamped, critically damped, or overdamped
1. The change of position of the center of mass of a rigid body in a mechanical system is being monitored. At time t 0, when the initial conditions of the system were x = 0.1 m and x -0m/s, a ste...
1. The change of position of the center of mass of a rigid body in a mechanical system is being monitored. At time t 0, when the initial conditions of the system were x = 0.1 m and x -0m/s, a step input of size 10 N began to apply to the system. The response of the system was represented by this differential equation: 2r + 110x + 500 x = 10 a) Write the order of the system, its...
6. A mass of 2 kilogram is attached to a spring whose constant is 4 N/m, and the entire system is...
6. A mass of 2 kilogram is attached to a spring whose constant is 4 N/m, and the entire system is then submerged in a liquid that inparts a damping force equal to 4 tines the instantansous velocity. At t = 0 the mass is released from the equilibrium position with no initial velocity. An external force t)4t-3) is applied. (a) Write (t), the external force, as a piecewise function and sketch its graph b) Write the initial-value problem (c)Solve...
[1] 25 pts. A damped single degree of freedom system without applied forces is oscillating due to...
solve for #2
[1] 25 pts. A damped single degree of freedom system without applied forces is oscillating due to a certain unknown initial conditions. Derive a response equation x(t) for the following four cases. a. 5 pts. 0 (no damping) b. 10 pts. 0<1 (underdamped) c. 5 pts. >1 (overdamped) d. 5 pts. ๕-1 (critically damped) Here the is the damping ratio of the oscillating system. [2] 5 pts. For the same system of underdamped case with initial conditions...
For the following ODEs, find roots λ of their characteristics equation and determine the type of damping
a) For the following ODEs, find roots λ of their characteristics equation and determine the type of damping (overdamped, undamped, underdamped, critically damped). For undamped and underdamped cases, find their angular frequency and period (14 points)i) y''+6 y'+10 y=0ii) 4 y''+8 y'+3 y=0iii) y''+8 y'+16 y=0b) Find the general solution for the following ODE 16 y''+8 y'+y=0(6 points)c) Find the particular solution of b) if y(0)=2 and y'(0)=2.5 .(5 points)
(d) A 4-kg mass is suspended from a spring with a constant k 25, and a dashpot with various level...
(d) A 4-kg mass is suspended from a spring with a constant k 25, and a dashpot with various levels of damping viscosity is present. The mass is displaced 0.5 m from its equilibrium and released. Determine the displacement y(t) of the mass if (i) c-15 i) c20, (iii) c-25, and (iv) c 30 In each case, state whether the system is overdamped, critically damped, or underdamped, and sketch the solution curve.
(d) A 4-kg mass is suspended from a...
Problem 4. The Fast Decay of Critically Damped Simple Harmonic Oscillator. A simple harmonic oscillator (a box with mass m attached to a Hook's spring of coefficient k with linear air friction of coefficient n) is described by mx"(t) + n2'(t) + ku(t) = 0 where m, n, k > 0. (a) Write down the solutions for three cases and their long term limits 1. Overdamped: when friction is strong 1 > 4mk 2. Underdamped: when friction is weak 72...
5. A 2 kg mass is attached to a spring whose constant is 30 N/m, and the entire system is submerged in a liquid that imparts a damping force equal to 12 times the instaataneous velocity (a) Write the second-order linear differential equation to umodel the motion (b) Convert the second-order linear differential equation from part (a) to a first-order linear system (c) Classify the critical (equilibrium) point (0.0) (d) Sketch the phase portrait (e) Indicate the initial condition x(0)-(...
For the harmonic oscillator with mel, C=8, K:16 ; Xo = 5, Vo=4, do the following a. Find the position function x(t) and determine whether the motion is overdamped, critically damped or underdamped. b. Find the undamped (when c-0) position function ult) = Co Cos(wot-do)
Answer last four questions
1. A spring-mass-damper system has mass of 150 kg, stiffness of 1500 N/m and damping coefficient of 200 kg/s. i) Calculate the undamped natural frequency ii) Calculate the damping ratio iii) Calculate the damped natural frequency iv) Is the system overdamped, underdamped or critically damped? v) Does the solution oscillate? The system above is given an initial velocity of 10 mm/s and an initial displacement of -5 mm. vi) Calculate the form of the response and...
1. The change of position of the center of mass of a rigid body in a mechanical system is being monitored. At time t 0, when the initial conditions of the system were x = 0.1 m and x -0m/s, a step input of size 10 N began to apply to the system. The response of the system was represented by this differential equation: 2r + 110x + 500 x = 10 a) Write the order of the system, its...
6. A mass of 2 kilogram is attached to a spring whose constant is 4 N/m, and the entire system is then submerged in a liquid that inparts a damping force equal to 4 tines the instantansous velocity. At t = 0 the mass is released from the equilibrium position with no initial velocity. An external force t)4t-3) is applied. (a) Write (t), the external force, as a piecewise function and sketch its graph b) Write the initial-value problem (c)Solve...
solve for #2
[1] 25 pts. A damped single degree of freedom system without applied forces is oscillating due to a certain unknown initial conditions. Derive a response equation x(t) for the following four cases. a. 5 pts. 0 (no damping) b. 10 pts. 0<1 (underdamped) c. 5 pts. >1 (overdamped) d. 5 pts. ๕-1 (critically damped) Here the is the damping ratio of the oscillating system. [2] 5 pts. For the same system of underdamped case with initial conditions...
a) For the following ODEs, find roots λ of their characteristics equation and determine the type of damping (overdamped, undamped, underdamped, critically damped). For undamped and underdamped cases, find their angular frequency and period (14 points)i) y''+6 y'+10 y=0ii) 4 y''+8 y'+3 y=0iii) y''+8 y'+16 y=0b) Find the general solution for the following ODE 16 y''+8 y'+y=0(6 points)c) Find the particular solution of b) if y(0)=2 and y'(0)=2.5 .(5 points)
(d) A 4-kg mass is suspended from a spring with a constant k 25, and a dashpot with various levels of damping viscosity is present. The mass is displaced 0.5 m from its equilibrium and released. Determine the displacement y(t) of the mass if (i) c-15 i) c20, (iii) c-25, and (iv) c 30 In each case, state whether the system is overdamped, critically damped, or underdamped, and sketch the solution curve.
(d) A 4-kg mass is suspended from a...
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