governing differential equation and function

1. The force on a cutting tool of a shaping machine varies over the length of the cut as follows:

Distance (mm) 0 20 40 60 80 100

Force (kN) 60 72 65 53 44 50

Determine the work done as the tool moves through a distance of 100 mm.

AIM:-

calculate the work done when tool moves 100mm.

MAIN BLOCKS USED:-

integrator:-it is used to integrate the constant or signal.

FORMULA USED:-

Work done = force * distance.

but in our condition force is varying with respect to distance, therefore integrated.

SIMULINK MODEL:-

RESULTS:-

the work-done when tool moves 100mm is 5780 KN-mm.

2. Create two array data sets x = [22 3 55.6 9 12 10] and y = [5 31 95 56.3 77 61] from the command window. Create a Simulink model to implement equation z=3x²-5y+18 Store result values using To File block.

AIM:-

Create two array data sets x = [22 3 55.6 9 12 10] and y = [5 31 95 56.3 77 61] from the command window. Create a Simulink model to implement equation z=3x²-5y+18 Store result values using To File block.

MAIN BLOCKS USED:-

TO FILE:-The To File block writes input signal data into a MAT-file. The block writes to the output file incrementally, with minimal memory overhead during the simulation. If the output file exists when the simulation starts, the block overwrites the file. The file automatically closes when you pause the simulation or the simulation completes. If simulation terminates abnormally, the ToFile block saves the data it has logged up until the point of the abnormal termination.

INPUTS;-

input arrays are created.

SIMULINK BLOCK DIAGRAM:-

RESULTS:-

3. Go to Matlab Help. Type the keyword ‘Mass-Spring-Damper in Simulink and Simscape’. Open the model and solve the following problems:

1. Derive the governing equation for the model

Freebody diagram:-

This diagram is used to illustrate the forces acting on a body and to calculate the resultant reaction of a body. The below figure defines how forces acting on a body.

From the above figure

Xo = initial displacement in body (mm)

M = mass of the body (Kg).

K = stiffness of spring (N/mm).

C = damping co-efficient (N-s/mm).

Arrow indications show forces applying on the body

The second law of newton

In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object.

F = ma

Applied force = -K*xo-C*xo.

Mass * acceleration = M*xo:

M*xo: =-K*xo-C*xo …...………eq (1)

GOVERNING DIFFERENTIAL EQUATION:-

The governing differential equation(GDE) is a mathematical equation describes how the values of unknown variables change when only one or more variables are known.

From eq(1)

GDE of mass,spring and damper is

M*xo: + C*xo. + K*xo = 0 ...…………eq (2)

M(dxo2d2t)+C(dxdt)+K=0">M(dxo2d2t)+C(dxdt)+K=0M(dxo2d2t)+C(dxdt)+K=0

Assume

xo = e^ᵧt

1. = ᵧ *e^ᵧt

xo: = ᵧ ^2 *e^ᵧt

by submitting above conditions in equations(2)

M* ᵧ ^2 + C* ᵧ+ K = 0

2. Change the value of mass, damper, and spring for the Simscape model and plot the result. Change the value of Mass x0=2 in the Simulink model and plot the results.

original values of the mass, spring, and damper are shown below

results of the original values are shown below

changed the mass, damper and spring properties as below

results are shown in below

in Simulink model the displacement is changed from 1 to 2

and the results are shown below

3. Make the response from Simscape and Simulink as two separate displays.

example model has been plotted in two different displays.

1.SIMSCAPE

2.SIMULINK