1. Consider a spring-mass-damper system with equation of motion given by: 2! x!+8x! + 26x =...
1. Consider a spring-mass-damper system with equation of motion given by: 2! x!+8x! + 26x = 0 .
i) Compute the solution if the system is given initial conditions x0 = 0 and v0 = -3 m/s
j) Compute the solution if the system is given initial conditions x0 =1 m and v0 = −2 m/s
k) Compute the solution if the system is given initial conditions x0 = −1 m and v0 = 2 m/s
2. Compute the solution to ! x!+8x! +16x = 0 for x0 =1 mm and v0 = −1 mm/s.
2. Compute the solution to ! x!+8x! +16x = 0 for x0 =1 mm and v0 = −1 mm/s.
Homework Answers
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1. Consider a spring-mass-damper system with equation of motion
given by: 2! x!+8x! + 26x =...
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed...
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed by the equation of motion where m is the mass, c is the damping coefficient of the damper, k is the stiffness of the spring, F is the net force applied on the mass, and x is the displacement of the mass from its equilibrium point. In this problem, we focus on a mass-damper-spring system with m = 1 kg, c-4 kg/s, k-3 N/m,...
Consider the following differential equation which describes a spring-mass-damper system më + ci + kx =...
Consider the following differential equation which describes a spring-mass-damper system më + ci + kx = cos(2nt) where c 1.9 and k = 3.1. The initial conditions are given as x(0) = 0 and 2(0) = 0 and the time step is 0.1 s. 1. Assuming that m - 0, use the Runge-Kutta 4th order method to find (a) x(0.1) and (b) *(0.1). 2. Assuming that m 1, use Euler's method to find (a) 2(0.2) and (b) X(0.3).
Consider a mass-spring-damper system whose motion is described by the following system of differe...
Consider a mass-spring-damper system whose motion is described by the following system of differentiat equations [c1(f-k)+k,(f-х)-c2(x-9), f=f(t), y:' y(t) with x=x( t), where the function fit) is the input displacement function (known), while xit) and yt) are the two generalized coordinates (both unknown) of the mass-spring-damper systenm. 1. Identify the type of equations (e.g. H/NH, ODE/PDE, L/NL, order, type of coefficients, etc.J. 2. Express this system of differential equations in matrix form, assume f 0 and then determine its general...
Spring mass damper system with forced response, the forced system given by the equation For damping...
Spring mass damper system with forced response, the forced system given by the equation For damping factor:E-0.1 ; mass; m-| kg: stiffness of spring; k-100 Nm; f-| 00 N; ω Zun; initial condition: x (0)-2 cms; r(0) = 0. fsincot Task Marks 1. Write down the reduced equation into 2first orderns Hand written equations differential equations 2. Rearrange equation (1) with the following generalized equation 250, x+osinor calculations 3. Calculate the value of c calculations Hand calculations 4. Using the...
2. (30 pts) Consider the system of Figure 1 (m-2 kg, k 50 N/m, 0-30°). a) Obtain the equation of motion. b) Compute the initial conditions such that the system oscillates at only one frequency wh...
2. (30 pts) Consider the system of Figure 1 (m-2 kg, k 50 N/m, 0-30°). a) Obtain the equation of motion. b) Compute the initial conditions such that the system oscillates at only one frequency when Fa)-2sin10 c) Calculate the response of the system for F)-2sin10/, xo-0,-10 m/s. d) Calculate the response of the system for F)-108t), xo-0, -10 m/s. c) Calculate the response of the system for F(i)-2sin10+108(-2), x0-0, ao-10 m/s. nt Ft) Figure 1. Mass-spring system
2. (30...
(4) Consider the 2nd order equation for a mass-spring-damper system, mx'' + bx' + kx =...
(4) Consider the 2nd order equation for a mass-spring-damper system, mx'' + bx' + kx = f(t) a) Assuming f(t) is a step function, find the Laplacian transform, X(s) (include terms for the initial conditions xo, vo). b) Assume m = 1, b = 5, and k = 6, and x(0) = 3, x’(0) = 0. Find the time-domain solution (take the inverse transform). (5)Find the Laplace transform of y(t) from the differential equation, assuming u(t) is a step function....
3. The motion of a 1DOF mass-spring-damper system (see Figure 1) is modeled by the following seco...
3. The motion of a 1DOF mass-spring-damper system (see Figure 1) is modeled by the following second order linear ODE: dx,2 dt n dt2 (0) C dt where is the damping ratio an wn is the natural frequency, both related to k, b, and m (the spring constant, damping coefficient, and mass, respectively) (a) Use the forward difference approximations of (b) Using Δt andd to obtain a finite difference formula for x(t+ 2Δ) (like we did in class for the...
Consider a mass-spring-dashpot system in which the mass is m = 4 lb-sec^2/ft, the damping constant...
Consider a mass-spring-dashpot system in which the mass is m = 4 lb-sec^2/ft, the damping constant is c =24 lb-sec/ft, and the spring constant is k=52lb/ft. The motion is free damped motion and the mass is set in motion with initial position x0=5ft and the initial velocity v0= -7ft/sec. Find the position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped.
Consider the mass-spring-damper system depicted in the figure below, where the input of the system is...
Consider the mass-spring-damper system depicted in the figure below, where the input of the system is the applied force F(t) and the output of the system is xít) that is the displacement of the mass according to the coordinate system defined in that figure. Assume that force F(t) is applied for t> 0 and the system is in static equilibrium before t=0 and z(t) is measured from the static equilibrium. b m F Also, the mass of the block, the...
Problem Set A Problem 6. (20%) A ordinary differential equation for a mass-damper-spring system is following....
Problem Set A Problem 6. (20%) A ordinary differential equation for a mass-damper-spring system is following. The mass m 1, damping coetfic e initial position y(o) O, and the initial velocity i constant k 3 and force 10, all are in appropriate units. Th 1, spring zero, within the time range of O to 20 unit of time, use Matlab find the solution of function y(t)? Hint: you need to convert the 2nd order ODE into two 1st order ODEs....
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed by the equation of motion where m is the mass, c is the damping coefficient of the damper, k is the stiffness of the spring, F is the net force applied on the mass, and x is the displacement of the mass from its equilibrium point. In this problem, we focus on a mass-damper-spring system with m = 1 kg, c-4 kg/s, k-3 N/m,...
Consider the following differential equation which describes a spring-mass-damper system më + ci + kx = cos(2nt) where c 1.9 and k = 3.1. The initial conditions are given as x(0) = 0 and 2(0) = 0 and the time step is 0.1 s. 1. Assuming that m - 0, use the Runge-Kutta 4th order method to find (a) x(0.1) and (b) *(0.1). 2. Assuming that m 1, use Euler's method to find (a) 2(0.2) and (b) X(0.3).
Consider a mass-spring-damper system whose motion is described by the following system of differentiat equations [c1(f-k)+k,(f-х)-c2(x-9), f=f(t), y:' y(t) with x=x( t), where the function fit) is the input displacement function (known), while xit) and yt) are the two generalized coordinates (both unknown) of the mass-spring-damper systenm. 1. Identify the type of equations (e.g. H/NH, ODE/PDE, L/NL, order, type of coefficients, etc.J. 2. Express this system of differential equations in matrix form, assume f 0 and then determine its general...
Spring mass damper system with forced response, the forced system given by the equation For damping factor:E-0.1 ; mass; m-| kg: stiffness of spring; k-100 Nm; f-| 00 N; ω Zun; initial condition: x (0)-2 cms; r(0) = 0. fsincot Task Marks 1. Write down the reduced equation into 2first orderns Hand written equations differential equations 2. Rearrange equation (1) with the following generalized equation 250, x+osinor calculations 3. Calculate the value of c calculations Hand calculations 4. Using the...
2. (30 pts) Consider the system of Figure 1 (m-2 kg, k 50 N/m, 0-30°). a) Obtain the equation of motion. b) Compute the initial conditions such that the system oscillates at only one frequency when Fa)-2sin10 c) Calculate the response of the system for F)-2sin10/, xo-0,-10 m/s. d) Calculate the response of the system for F)-108t), xo-0, -10 m/s. c) Calculate the response of the system for F(i)-2sin10+108(-2), x0-0, ao-10 m/s. nt Ft) Figure 1. Mass-spring system
2. (30...
3. The motion of a 1DOF mass-spring-damper system (see Figure 1) is modeled by the following second order linear ODE: dx,2 dt n dt2 (0) C dt where is the damping ratio an wn is the natural frequency, both related to k, b, and m (the spring constant, damping coefficient, and mass, respectively) (a) Use the forward difference approximations of (b) Using Δt andd to obtain a finite difference formula for x(t+ 2Δ) (like we did in class for the...
Consider the mass-spring-damper system depicted in the figure below, where the input of the system is the applied force F(t) and the output of the system is xít) that is the displacement of the mass according to the coordinate system defined in that figure. Assume that force F(t) is applied for t> 0 and the system is in static equilibrium before t=0 and z(t) is measured from the static equilibrium. b m F Also, the mass of the block, the...
Problem Set A Problem 6. (20%) A ordinary differential equation for a mass-damper-spring system is following. The mass m 1, damping coetfic e initial position y(o) O, and the initial velocity i constant k 3 and force 10, all are in appropriate units. Th 1, spring zero, within the time range of O to 20 unit of time, use Matlab find the solution of function y(t)? Hint: you need to convert the 2nd order ODE into two 1st order ODEs....
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