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PROBLEM 1 Consider the mass-spring system with the given forcing function 2 +5x 4 cos(0.8t) Compute...
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' +...
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' + 8y = 100 cos 3t + 4et assuming that the mass is released from rest at the equilibrium position. (This forcing function is not very realistic.) (b) Assume the equation from part (a) describes a mass-spring-dashpot system with a dashpot containing honey. Imagine that the honey is changed...
Use DUHAMEL INTEGRAL / CONVOLUTION INTEGRAL to solve. DO NOT USE FOURIER SERIES. Problem 4- Consider...
Use
DUHAMEL INTEGRAL / CONVOLUTION INTEGRAL to solve. DO NOT USE
FOURIER SERIES.
Problem 4- Consider a simple damped mass-spring system under a general forcing function p(t) such that: Find the solution x(t) for the periodic forcing function described below: p(t) = Fo [1-cos (? t/2to)1 for 0-t-to (0)-0 for to
a-d please 6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy"...
a-d please
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' + 8y = 100 cos 3t + 4et assuming that the mass is released from rest at the equilibrium position. (This forcing function is not very realistic.) (b) Assume the equation from part (a) describes a mass-spring-dashpot system with a dashpot containing honey. Imagine that the honey...
Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m...
Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m as a function of time is as follows: x = Xocoswt) + cos(Wnt) ωη where xo is the initial displacement equals to 0.1 m, čo is the initial velocity equals to 1 m/s, and Wr is the natural frequency of the system equals to 4 rad/s. Calculate the acceleration (second time derivative of displacement) of the mass after 1 s with a time step...
Problem 2.31: An underdamped mass-spring-dashpot system is subject to a periodic force F(t) of a ...
Problem 2.31: Please complete all of the following
Problem 2.31: An underdamped mass-spring-dashpot system is subject to a periodic force F(t) of a period T and a saw-tooth form, as shown in Fig. P2.31. Assume ζ 0.1. AF(t) T" 2T 3T Figure P2.31 Periodic loading of saw-tooth shape (a) Obtain the Fourier series expansion for the force. (b) Find the Fourier series expansion of the system's steady-state response. (c) For T/T, = 0.5, where T, is the natural period of...
Problem 1. Consider the following mass, spring, and damper system. Let the force F be the...
Problem 1. Consider the following mass, spring, and damper system. Let the force F be the input and the position x be the output. M-1 kg b- 10 N s/m k 20 N/nm F = 1 N when t>=0 PART UNIT FEEDBACK CONTROL SYSTEM 5) Construct a unit feedback control for the mass-spring-damper system 6) Draw the block diagram of the unit feedback control system 7) Find the Transfer Function of the closed-loop (CL) system 8) Find and plot the...
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...
Consider a single degree of freedom (SDOF) with mass-spring-damper system
Consider a single degree of freedom (SDOF) with mass-spring-damper system subjected to harmonic excitation having the following characteristics: Mass, m = 850 kg; stiffness, k = 80 kN/m; damping constant, c = 2000 N.s/m, forcing function amplitude, f0 = 5 N; forcing frequency, ωt = 30 rad/s. (a) Calculate the steady-state response of the system and state whether the system is underdamped, critically damped, or overdamped. (b) What happen to the steady-state response when the damping is increased to 18000 N.s/m? (Hint: Determine...
1. A 1 kg mass is attached to a spring of spring constant k = 4kg/82, The spring-mass system is a...
Differntial Equations Forced Spring Motion
1. A 1 kg mass is attached to a spring of spring constant k = 4kg/82, The spring-mass system is attached to a machine that supplies an external driving force of f(t) = 4 cos(wt). The systern is started from equilibrium i.e. 2(0) = 0 and z'(0) = 0. There is no damping. (a) Find the position x(t) of the mass as a function of time (b) write your answer in the form r(t)-1 sin(6t)...
Now, let us consider that the forcing function is not zero anymore, i.e. let us consider the causal system described by 0)2 (0)1 4. Derive a difference equation model of (2) with sampling period T...
Now, let us consider that the forcing function is not zero anymore, i.e. let us consider the causal system described by 0)2 (0)1 4. Derive a difference equation model of (2) with sampling period T
Now, let us consider that the forcing function is not zero anymore, i.e. let us consider the causal system described by 0)2 (0)1 4. Derive a difference equation model of (2) with sampling period T
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' + 8y = 100 cos 3t + 4et assuming that the mass is released from rest at the equilibrium position. (This forcing function is not very realistic.) (b) Assume the equation from part (a) describes a mass-spring-dashpot system with a dashpot containing honey. Imagine that the honey is changed...
Use
DUHAMEL INTEGRAL / CONVOLUTION INTEGRAL to solve. DO NOT USE
FOURIER SERIES.
Problem 4- Consider a simple damped mass-spring system under a general forcing function p(t) such that: Find the solution x(t) for the periodic forcing function described below: p(t) = Fo [1-cos (? t/2to)1 for 0-t-to (0)-0 for to
a-d please
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' + 8y = 100 cos 3t + 4et assuming that the mass is released from rest at the equilibrium position. (This forcing function is not very realistic.) (b) Assume the equation from part (a) describes a mass-spring-dashpot system with a dashpot containing honey. Imagine that the honey...
Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m as a function of time is as follows: x = Xocoswt) + cos(Wnt) ωη where xo is the initial displacement equals to 0.1 m, čo is the initial velocity equals to 1 m/s, and Wr is the natural frequency of the system equals to 4 rad/s. Calculate the acceleration (second time derivative of displacement) of the mass after 1 s with a time step...
Problem 2.31: Please complete all of the following
Problem 2.31: An underdamped mass-spring-dashpot system is subject to a periodic force F(t) of a period T and a saw-tooth form, as shown in Fig. P2.31. Assume ζ 0.1. AF(t) T" 2T 3T Figure P2.31 Periodic loading of saw-tooth shape (a) Obtain the Fourier series expansion for the force. (b) Find the Fourier series expansion of the system's steady-state response. (c) For T/T, = 0.5, where T, is the natural period of...
Problem 1. Consider the following mass, spring, and damper system. Let the force F be the input and the position x be the output. M-1 kg b- 10 N s/m k 20 N/nm F = 1 N when t>=0 PART UNIT FEEDBACK CONTROL SYSTEM 5) Construct a unit feedback control for the mass-spring-damper system 6) Draw the block diagram of the unit feedback control system 7) Find the Transfer Function of the closed-loop (CL) system 8) Find and plot the...
Differntial Equations Forced Spring Motion
1. A 1 kg mass is attached to a spring of spring constant k = 4kg/82, The spring-mass system is attached to a machine that supplies an external driving force of f(t) = 4 cos(wt). The systern is started from equilibrium i.e. 2(0) = 0 and z'(0) = 0. There is no damping. (a) Find the position x(t) of the mass as a function of time (b) write your answer in the form r(t)-1 sin(6t)...
Now, let us consider that the forcing function is not zero anymore, i.e. let us consider the causal system described by 0)2 (0)1 4. Derive a difference equation model of (2) with sampling period T
Now, let us consider that the forcing function is not zero anymore, i.e. let us consider the causal system described by 0)2 (0)1 4. Derive a difference equation model of (2) with sampling period T
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