The following set of differential equations can be used to represent that behavior of a simple...
The following set of differential equations can be used to represent that behavior of a simple spring-mass system, with x1(t) the mass’s position and x2(t) its velocity:
For the initial condition of x1(0) = 1.0, x2(0) = 0, and a step size 0.1 seconds, determine the values x1(0.3) and x2(0.3) using (a) Euler’s method, (b) the modified Euler’s method.
Homework Answers
Consider the differential equations that represent the behaviour of a simple spring-mass system.
Here,
is the mass’s position
is the velocity
(a)
Write the equations for calculating the new values from the old values using Euler’s method.
Substitute 0.1 for 
and 
for 
.
…… (1)
Substitute 0.1 for 
and 
for 
.
…… (2)
Determine 
using the initial condition by substituting 0 for t in equation (1) and using the initial conditions.
Determine 
using the initial condition by substituting 0 for t in equation (2) and using the initial conditions.
Determine 
using the initial condition by substituting 0.1 for t in equation (1) and using the value of 
.
Determine 
using the initial condition by substituting 0.1 for t in equation (2) and using the value of 
.
Determine 
using the initial condition by substituting 0.2 for t in equation (1) and using the value of 
.
Determine 
using the initial condition by substituting 0.2 for t in equation (2) and using the value of 
.
Thus, the value of 
is 
.
The value of 
is 
.
(b)
Write the equations for calculating the new values from the old values using modified Euler’s method.
Substitute 0.1 for 
and 
for 
.
…… (3)
Substitute 0.1 for 
and 
for 
.
…… (4)
Determine 
.
Determine 
.
Determine 
using the initial condition by substituting 0 for t in equation (3) and using the initial conditions.
Determine 
using the initial condition by substituting 0 for t in equation (4) and using the initial conditions.
Determine 
.
Determine 
.
Determine 
by substituting 0.1 for t in equation (3).
Determine 
by substituting 0.1 for t in equation (4).
Determine 
.
Determine
. 
Determine 
by substituting 0.2 for t in equation (3).
Determine 
by substituting 0.2 for t in equation (4).
Thus, the value of 
is 
.
The value of 
is 
.
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The following set of differential equations can be used to represent that behavior of a simple...
The following set of differential equations can be used to represent the behavior of a simple...
The following set of differential equations can be used to represent the behavior of a simple spring-mass system, with ?1(?) the mass’s position and ?2(?) the mass’s velocity: ??1 = ?2 ?? ??2 = −?1 ?? For the initial condition of ?1(0) = 1.0, ?2(0) = 0, and a step size of 0.1 seconds, determine the values ?1(0.3) and ?2(0.3) using Euler’s method.
A Mechanical system is shown in Fig. 1.1. a) Obtain a set of simultaneous integro-differential equations,...
A Mechanical system is shown in Fig. 1.1. a) Obtain a set of simultaneous integro-differential equations, in terms of velocity, to represent the system, where k is spring Constant, m is the mass and b is damper. (Hint: draw two Free Body Diagrams and obtain two equations in terms of x1 and x2 only). X1 - Ft mi m2 M TTTTTTTTYYTTTTTTTTTYYTTTTTTT Fig 1.1 b) What components and energy source do you need to use in order to build an electric...
2. Now let's investigate how the various methods work when applied to an especially simple differential equation, x...
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Solve for a, b and c. Please write clearly.
Thanks
9. (20%) System Differential Equations X = [X1 ; X2] Initial condition X1(0)=1, X2(0) = 1 find the solutions X by (a) Laplace transform method (6%) (b) Diagonalization transform method (7%) (c) Elimination method (7%)
a-represent system in state space form? b-find output response y(t? c-design a state feedback gain controller? 3- A dyn...
a-represent system in state space form?
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Differential Equations Question Method of Elimination - Initial Value Problems. Find the solution of the following...
Differential Equations Question
Method of Elimination - Initial Value Problems.
Find the solution of the following IVPs.
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Find the value of x(0.5) for the initial value problem at = thx(0)=1 using Euler's method with step size h 0.05 Find the value of x(0.4) for the coupled first order differential equations toge...
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differential equations step by step work A 1 kg mass is suspended from a spring with...
differential equations step by step work
A 1 kg mass is suspended from a spring with constant k = 6 in a medium with damping coefficient y = 5. The mass is pulled 1 meter from equilibrium and pushed with an initial velocity of - 1 m/s. Determine a formula for the displacement of the mass after t seconds. Validate your solution using Matlab.
A Mechanical system is shown in Fig. 1.1. a) Obtain a set of simultaneous integro-differential equations, in terms of velocity, to represent the system, where k is spring Constant, m is the mass and b is damper. (Hint: draw two Free Body Diagrams and obtain two equations in terms of x1 and x2 only). X1 - Ft mi m2 M TTTTTTTTYYTTTTTTTTTYYTTTTTTT Fig 1.1 b) What components and energy source do you need to use in order to build an electric...
2. Now let's investigate how the various methods work when applied to an especially simple differential equation, x' x (a) First find the explicit solution x(t) of this equation satisfying the initial condition x(0) = 1 (now there's a free gift from the math department... (b) Now use Euler's method to approximate the value of x(1)e using the step size At = 0.1. That is, recursively determine tk and xk for k 1,.., 10 using At = 0.1 and starting...
Question 22 1 pts Problem 22: Numerical solution of Ordinary differential equations Consider the following initial value problem GE:+15y = 1.C:y(0) -0.5 Carry out two-steps of the modified Euler (trapezoidal) method solution from the initial condition with a time step of At = 0.1. and the predicted solutions is y(0.2)-0.20 None of the above. y(0.2) - -0.75 y(0.2)-1.27 y(0.2)=0.25
Solve for a, b and c. Please write clearly.
Thanks
9. (20%) System Differential Equations X = [X1 ; X2] Initial condition X1(0)=1, X2(0) = 1 find the solutions X by (a) Laplace transform method (6%) (b) Diagonalization transform method (7%) (c) Elimination method (7%)
a-represent system in state space form?
b-find output response y(t?
c-design a state feedback gain controller?
3- A dynamic system is described by the following set of coupled linear ordinary differential equations: x1 + 2x1-4x2-5u x1-x2 + 4x1 + x2 = 5u EDQMS 2/3 Page 1 of 2 a. Represent the system in state-space form. b. For u(t) =1 and initial condition state vector x(0) = LII find the outp (10 marks) response y(t). c. Design a state feedback gain...
Differential Equations Question
Method of Elimination - Initial Value Problems.
Find the solution of the following IVPs.
(13) x1 = 4x1 - x2 + 3e2t x1(0) = - x2 = 2x1 + x2 + 2t X2(0) = 42
Find the value of x(0.5) for the initial value problem at = thx(0)=1 using Euler's method with step size h 0.05 Find the value of x(0.4) for the coupled first order differential equations together with initial conditions with step size 0.1: 2. dt t+x 3. dx dt = y, dy dt x(0) = 1.2 and --ty +xt2 + y(o) 0.8
Find the value of x(0.5) for the initial value problem at = thx(0)=1 using Euler's method with step size h...
2. Coupled Differential Equations (40 points) The well-known van der Pol oscillator is the second-order nonlinear differential equation shown below: + au dt 0. di The solution of this equation exhibits stable oscillatory behavior. Van der Pol realized the parallel between the oscillations generated by this equation and certain biological rhythms, such as the heartbeat, and proposed this as a model of an oscillatory cardiac pacemaker. Solve the van der Pol equation using Second-order Runge Kutta Heun's method with the...
differential equations step by step work
A 1 kg mass is suspended from a spring with constant k = 6 in a medium with damping coefficient y = 5. The mass is pulled 1 meter from equilibrium and pushed with an initial velocity of - 1 m/s. Determine a formula for the displacement of the mass after t seconds. Validate your solution using Matlab.
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