Mechanical Engineering Assignment Help

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Problems Problem 1 :

Dynamics of a mass-spring-damper system

x

k b

m

x0 & v0 F t   F0 sin t 

A mass-spring-damper system is subject to an external sinusoidal force F t 

with amplitude

F0 , and angular frequency  . The position of particle, x , is zero when the spring is neither compressed or stretched. The mass of the particle is m , the damping coefficient is b and the spring constant is k . Initial condition is expressed as x(0)  x0 and v(0)  v0 . Derive the equation of motion for this mass-spring-damper system.

Problem 2 : Solver for a mass-spring-damper system using Euler method You are familiar with Euler method used to obtain the trajectory of a ball dropping and bouncing in Homework #3. In this homework, you will create a function that use Euler method to derive the particle trajectory (two column matrix for time and position each) in mass-spring-damper system from t=0 to 50 sec. The input parameters of the function are coefficients ( m , b , k , F0 and  ) and initial conditions ( x0 and v0 ). Use time increment t  0.1sec . Function name (and m-file name) should be ‘MSDSE_your_kerberos_name’ and upload it to 2.003 MIT Server site. You also submit the hardcopy of your code with appropriate comments.

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Problem 3 : Solver for a mass-spring-damper system with using Runge-Kutta method Using Runge-Kutta algorithm to solve the differential equation of this system, generate a m-file function with input parameters ( m , b , k , F0 ,  , x0 and v0 ) to calculate particle trajectory (two column matrix for time and position each) from t=0 to 50 sec. You may use either Runge-Kutta 23 or 45 algorithm (See ode23 or ode45 in the MATLAB help, and use one of them in your m-code.). Use t  0.1sec for both the time step size for the solver (See also odeset in the MATLAB help) and the time step for the output time from the solver (See input argument for tspan in ode23 or ode45). Function name (and m-file name) should be ‘MSDSRK_your_kerberos_name’ and upload it to 2.003 MIT Server site. You also submit the hardcopy of your code with appropriate comments.

Problem 4 : Trajectory of a mass-spring-damper system with different parameters and initial conditions For the following cases, plot the time response of the particle. Plot the trajectories with Euler and Runge-Kutta solvers on a single graph with appropriate legends. Compare the results for these two approaches. Submit a hardcopy of your trajectory plots. m  1 kg ,

i)

b  0.5 N  sec/m ,

k  1 N/m , F0  1N ,   3 rad/sec ,

x(0)  0m

and v(0)  0m / sec ii) m  1 kg ,

b  0.5 N  sec/m ,

k  1 N/m ,

F0  0 N ,   0 rad/sec ,

x(0)  1m

and

b  0 N  sec/m ,

k  1 N/m ,

F0  1N ,  1 rad/sec ,

x(0)  0m

and

v(0)  0m / sec iii) m  1 kg , v(0)  0m / sec matlabassignmentexperts.com

Solutions Problem 1 : Dynamics of mass-spring-damper system

From the above diagram,

Problem 2 : Solver for mass-spring-damper system with Euler method First, 2nd order differential equation is split into two 1st order differential equations as did in Homework #3.

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With Eq.(3), we also obtain the discrete version of above differential equations. x n,1  x1 nt 

&

x n, 2   x2 nt 

x  n  1,1  x n,1  x n, 2   t x  n  1, 2   x n, 2   a  t

where

 m The solver of mass-spring-damper system with Euler method is implemented as below. Explanation of each command line is a

1

F

sin   nt   0b  x n, 2   k  x n,1

included in the following codes. function O=MSDSE(m,b,k,F0,w,x0,v0) % % Solver for Mass-Sprring-Damper System with Euler Method % ----- Input argument ----% m: mass for particle % b: damping coefficient % k: spring constant % F0: amplitude of external force % w: angular freuency of external force % x0: initial condition for the position x(0) % v0: initial condition for the velocity v(0) % ----- Output argument ----% t: time series with given time step from ti to tf. % x: state variable matrix for corresponding time t matrix

% Define time step in Euler method

dt=0.1;

% Make time series with given time step

t=[0:dt:50]';

% Initialize output matrix % the 1st column: the position of the particle % the 2nd column: the veleocity of the particle % the 1st row has initial conditions

x(1,:)=[x0 v0];

% Start the simulation with Euler method matlabassignmentexperts.com

x=zeros(length(t),2);

for i=1:length(t)-1

% Apply Euler method % x(n+1)=x(n)*v(n)*dt

x(i+1,1,1)=x(i,1)+x(i,2)*dt;

% v(n+1)=v(n)*a(n)*dt %where a(n)=1/m*(F0sin(wt)-bv(n)-kx(n))*dt

x(i+1,2)=x(i,2)+(1/m)*(F0*sin(w*t(i))-b*x(i,2)-

k*x(i,1))*dt; end % Extract only particle position trajectory

O=[t,x(:,1)];

end

Problem 3 :

Solver for mass-spring-damper system with Runge-Kutta method

Unlike Euler method, you don’t need to solve differential equation itself in MATLAB. (You can also make your own code for Runge-Kutta algorithm for yourself.) However, the subfunction should be needed to implement Runge-Kutta algorithm. (Single m-file can have several functions. The first one is the primary function, and others are subfunction.) Subfunction includes several 1st order differential equations by splitting higher order differential equation. In our case, two 1st order equations are used as described in problem 6.2. With this subfunction, the solver (either ‘ode23’ or ‘ode45’) solves differential equations actually. The handler of function which has dynamic equation description, time span which the solver calculates between, and initial condition at the simulation starting time are at least specified as the input arguments when the solver runs. More detailed syntax description of either ‘ode23’ or ‘ode45’ is shown in MATLAB help. The solver of mass-springdamper system with Runge-Kutta method is implemented as below. Explanation of each command line is included in the following codes. function O=MSDSRK(m,b,k,F0,w,x0,v0) % % Solver for Mass-Sprring-Damper System with Runge-Kutta Method % ----- Input argument ----% m: mass for particle % b: damping coefficient % k: spring constant % F0: amplitude of external force % w: angular freuency of external force matlabassignmentexperts.com

% x0: initial condition for the position x(0) % v0: initial condition for the velocity v(0) % ----- Output argument ----% t: time series with given time step from ti to tf. % x: state variable matrix for corresponding time t matrix

% define time steps for solver and display

dt=0.1;

% set both initial time step size and maximum step size % for Runge-Kutta solver

options=odeset('InitialStep',dt,'MaxStep',dt);

% set time span to be generated from Runge-Kutta solver % from 0 sec to 50 sec with 0.1 sec time step

td=[0:dt:50];

% Solve differential equation with Runge-Kutta solver

[t,x]=ode45(@(t,X)MSD(t,X,m,b,k,F0,w),td,

[x0;v0],options); % Extract only particle position trajectory

O=[t x(:,1)];

end

function dX=MSD(t,X,m,b,k,F0,w) % % With two 1st order diffeential equations, % obtain the derivative of each variables % (position and velocity of the particle) % % t: current time % X: matrix for state variables %

The first column : the particle position

%

The second column : the particle velocity

% m,b,k,F0,w: parameters for the system %

% Apply two 1st order differential equations % dx(n+1)/dt<-v(n)

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dX(1,1)=X(2,1); % dv(n+1)/dt<-1/m*(F0sin(wt)-bv(n)-kx(n)) dX(2,1)=(1/m)*(F0*sin(w*t)-b*X(2,1)-k*X(1,1));

Problem 4 : i)

end

Trajectory of mss-spring-damper system with different parameters and initial conditions

Following code is used to generate below plot.

>> Oe=MSDSE(1,0.5,1,1,3,0,0); >> Or=MSDSRK(1,0.5,1,1,3,0,0); >>

plot(Oe(:,1),Oe(:,2),'r',Or(:,1),Or(:,2),'b--','LineWidth',2);

axis

tight; grid on;

>> xlabel('\bfTime (sec)'); ylabel('\bfParticle Position (m)'); >>

title({'\bf

Mass-Spring-Damper

System

Simulation';

'm=1kg,

b=0.5Nsec/m, k=1N/m, F_0=1m,

\omega=3rad/sec, x_0=0m, & v_0=0m/sec'}) >> legend('\bfEuler','\bfRunge-Kutta','Location','NorthEast')

It’s the response of the sinusoidal excitation for the particle when it is stationary, and no spring is compressed or stretched. Comparing two results from Euler method and Runge- Kutta method, they produce similar simulation results. The plot is shown as below:

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ii) Following code is use to generate below plot. >> Oe=MSDSE(1,0.5,1,0,0,1,0); >> Or=MSDSRK(1,0.5,1,0,0,1,0); >>

plot(Oe(:,1),Oe(:,2),'r',Or(:,1),Or(:,2),'b--','LineWidth',2);

axis

tight; grid on;

>> xlabel('\bfTime (sec)'); ylabel('\bfParticle Position (m)'); >>

title({'\bf Mass-Spring-DamperSystemSimulation';'m=1kg,

b=0.5Nsec/m, k=1N/m, F_0=0m, \omega=0rad/sec,

x_0=1m, & v_0=0m/sec'}) >> legend('\bfEuler','\bfRunge-Kutta','Location','NorthEast')

It’s the response with no external force and no damping when the spring is initially stretched. Comparing two results from Euler method and Runge-Kutta method, Runge- Kutta method is more accurate than Euler method, based on the analytical solution of this system. The plot is shown as below:1

iii) Following code is use to generate below plot. >> Oe=MSDSE(1,0,1,1,1,0,0); >> Or=MSDSRK(1,0,1,1,1,0,0); >> plot(Oe(:,1),Oe(:,2),'r',Or(:,1),Or(:,2),'b--','LineWidth',2); axis

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tight; grid on;

>> xlabel('\bfTime (sec)'); ylabel('\bfParticle Position (m)'); >>

title({'\bf

Mass-Spring-Damper

System

Simulation';

'm=1kg,

b=0Nsec/m, k=1N/m, F_0=1m,

\omega=1rad/sec, x_0=0m, & v_0=0m/sec'}) >> legend('\bfEuler','\bfRunge-Kutta','Location','NorthWest')

It’s the response of external force with the system resonance frequency when the spring is initially stretched. Resonance happens very quickly, which means the particle position oscillates from  to  . However, numerical simulation has limitation to express the infinity. Therefore, as time goes on, oscillations for both methods become larger, but Euler method one has smaller oscillation, compared with Runge-Kutta one.

So, Euler method is not powerful near the singularity. The plot is shown as below: Mass-Spring-Damper System Simulation m=1kg, b=0Nsec/m, k=1N/m, F 0=1m,  =1rad/sec, x 0 =0m, & v0=0m/sec

Euler Runge-Kutta

80

Particle Position (m)

60 40 20 0

-20 -40 -60 -80 0

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5

10

15

20

25 30 Time (sec)

35

40

45

50

function O=MSDSE(m,b,k,F0,w,x0,v0) % % Solver for Mass-Sprring-Damper System with Euler Method % ----- Input argument ----% m: mass for particle % b: damping coefficient % k: spring constant % F0: amplitude of external force % w: angular freuency of external force % x0: initial condition for the position x(0) % v0: initial condition for the velocity v(0) % ----- Output argument ----% t: time series with given time step from ti to tf. % x: state variable matrix for corresponding time t matrix % Define time step in Euler method dt=0.1; % Make time series with given time step t=[0:dt:50]'; % Initialize output matrix % the 1st column: the position of the particle % the 2nd column: the veleocity of the particle x=zeros(length(t),2); % the 1st row has initial conditions x(1,:)=[x0 v0]; % Start the simulation with Euler method matlabassignmentexperts.com

for i=1:length(t)-1 % Apply Euler method % x(n+1)=x(n)*v(n)*dt x(i+1,1,1)=x(i,1)+x(i,2)*dt; % v(n+1)=v(n)*a(n)*dt %where a(n)=1/m*(F0sin(wt)-bv(n)-kx(n))*dt x(i+1,2)=x(i,2)+(1/m)*(F0*sin(w*t(i))-b*x(i,2)-k*x(i,1))*dt; end % Extract only particle position trajectory O=[t,x(:,1)]; end function O=MSDSRK(m,b,k,F0,w,x0,v0) % % Solver for Mass-Sprring-Damper System with Runge-Kutta Method % ----- Input argument ----% m: mass for particle % b: damping coefficient % k: spring constant % F0: amplitude of external force % w: angular freuency of external force % x0: initial condition for the position x(0) % v0: initial condition for the velocity v(0) % ----- Output argument ----% t: time series with given time step from ti to tf. % x: state variable matrix for corresponding time t matrix matlabassignmentexperts.com

% define time steps for solver and display dt=0.1; % set both initial time step size and maximum step size % for Runge-Kutta solver options=odeset('InitialStep',dt,'MaxStep',dt); % set time span to be generated from Runge-Kutta solver % from 0 sec to 50 sec with 0.1 sec time step td=[0:dt:50]; % Solve differential equation with Runge-Kutta solver [t,x]=ode45(@(t,X)MSD(t,X,m,b,k,F0,w),td,[x0;v0],options); % Extract only particle position trajectory O=[t x(:,1)]; end function dX=MSD(t,X,m,b,k,F0,w) % % With two 1st order diffeential equations, % obtain the derivative of each variables % (position and velocity of the particle) % % t: current time % X: matrix for state variables % The first column : the particle position % The second column : the particle velocity % m,b,k,F0,w: parameters for the system % matlabassignmentexperts.com

% Apply two 1st order differential equations % dx(n+1)/dt<-v(n) dX(1,1)=X(2,1); % dv(n+1)/dt<-1/m*(F0sin(wt)-bv(n)-kx(n)) dX(2,1)=(1/m)*(F0*sin(w*t)-b*X(2,1)-k*X(1,1)); end

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