Work done by a cutting tool, Function modeling and 'Mass-Spring-Damper' analysis using SIMULINK

AIM: To find work done by a cutting tool, create a model to implement a given equation and analyze the example 'Mass-Spring-Damper' using Simulink. 

OBJECTIVE: 

1. To determine the work done as the tool moves through a distance of 100 mm using the given data.

2. To implement the equation `z = 3x^2 - 5y +18` and store result values using To File block.

Given array data sets are;

x = [22 3 55.6 9 12 10] and y = [5 31 95 56.3 77 61]

3. To run the Simulink example 'Suspension System Comparison' and determine:

- Derive the governing equation for the model.

- 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.

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

SOLUTION:

Part #1: A Simulink model for the given problem is shown below.

Input #1: A plot for the Force vs. Distance is displayed below for d = 100 mm.

Output #1: A plot for the work done at each time instance is displayed below for d = 100 mm.

Part #2: A Simulink model for the given function is shown below.

Code #2: The MATLAB program to solve the given equation is shown below.

% Program to solve the given equation.

% Input variables:

x = [22 3 55.6 9 12 10];
y = [5 31 95 56.3 77 61];

% Output:

z = (3*x.^2) - (5*y) + 18

Output #2: Similar values are obtained in workspace output as that of the Simulink model display.

A plot for the output values is displayed below according to given input arrays.

Part #3:

THEORY: A mass-spring-damper (MSD) system is a discretized model of any dynamic system.

The mass of the dynamic system is lumped into a single point mass in the MSD system. The inertial effect of the dynamic system is related through this lumped mass. The stored energy/internal energy of the dynamic system is modelled as a one-dimensional spring in the MSD system. The spring is able to store energy inside when it is stretched or compressed from its original length. The energy dissipated out of the dynamic system is modelled through a one-dimensional damper in the MSD system. The viscous damper, for instance, is able to dissipate energy as heat outside the dynamic system.

#3.1: Deriving the governing equation for the model.

From the figure; considering the free body diagram (FBD):

`ma = Sigma F = F - kx - bv`

`ma = F - kx - bv`

`ma + bv + kx = F`

For equilibrium condition,  `F = 0`

`ma + bv + kv = 0`   i.e.   `m frac[d^2x][dt^2] + bfrac[dx][dt] +kx = 0`

This is the required expression.

where;

`a = ddotx = frac{d^2x}{dt^2}` = Acceleration (m/s²)

`v = dotx = frac{dx}{dt}` = Velocity (m/s)

`x` = Displacement (m)

`m` = Mass (kg)

`b` = Damping coefficient [N/(m/s)]

OR

At Summation block,

`[- kx - bv] frac[1][m] = a`

`-kx - bv = ma`

`ma + bv + kx = 0`

#3.2:

- Change the value of mass, damper and spring for the Simscape model and plot the result.

For b = 20 N/(m/s), k = 250 N/m, m = 10 kg                                                                                    

For b = 40 N/(m/s), k = 100 N/m, m = 25 kg

- Change the value of Mass x0=2 in the Simulink model and plot the results.

#3.3: Make the response from Simscape and Simulink as two separate displays.

- Simscape output (b = 10 N/(m/s), k = 400 N/n, m = 3.6 kg)

- Simulink output (b = 10 N/(m/s), k = 400 N/m, m = 3.6 kg)

CONCLUSION: From the above results, we can infer that:

1. Net Work done is the area under the curve (Force vs Distance) and found to be 5780 kN.mm i.e. 5780 N.m

2. All values are displayed in the workspace and stored in the file z.mat

3.

   3.1 Governing equation was derived in two ways

   3.2 Increasing the damping coefficient value decreases the amplitude of the output, Incrementing the initial condition increases the oscillations 

   3.3 Simulink and Simscape outputs are same irrespective of their environments

REFERENCES:

1. Image Source

2. Mass-Spring-Damper in Simulink and Simscape