# Problem Definition

A crane is experiencing a load P at the end of its cantilever beam (with length L, width B and height H).  A cable is connected from the vertical pole to the beam at an angle $\Theta$ (Theta).

We would like to determine the torque T, which acts along the cable given an uncertain load P and an uncertain density $\rho$ of the steel in the beam. The mass of the cantilever beam itself, which acts at L2, needs to be considered as well.

The mass of the steel beam will be calculated using equation 1, taking density and gravity into account. The torque T can be obtained using equation 2, which is derived from the summation of all forces given a clockwise positive momentum.

1 $M=V\times \varrho =B\times H\times L\times \varrho \times g$

2 $T={P\times g\times {1 \over 2}\times (B\times H\times L\times \varrho \times g) \over \sin(\Theta )}$

# Input Data

Length L = 5 m

Width B = 3 m

Height H = 3 m

Angle Theta = 30 °

Gravity g = 9.81

Load P: uniform distributed with lower bound= 10 kg and upper bound = 1000 kg

Density $\rho$  : uniform distributed with lower bound= 7 $g/cm^{3}$ and upper bound = 8 $g/cm^{3}$

# Creating constant parameters

To create a constant parameter, go to the Input folder, right-click on Parameters and click on add parameter.

In the new window, you are able to name your constant and add a definition.

Once you are done, click on finish.

You can now insert the value of your parameter in the Values field .

# Creating random variables

To create a random variable right-click on Random Variable and click on add random variable.

In the new panel you can name your random variable and add a description.

In the next step Cossan-X provides you with a list of distributions to choose from (in this case Uniform)

Next, specify your input values (in this case we use upper and lower bounds).

Once you are finished save your random variable.

# Functions

We will be using a function within our Matlab Script so we need to create one beforehand.

Right-click on Functions and click on add function. You can now name and describe your function.

In the function definition field you can input your equation.

By using Ctrl-Space Cossan-X provides you with a list of your input parameters which you can then select.

Note: Operators have to be expressed with a dot in front of them e.g  .*

Note: It is not necessary to write the left side of the equation.

Once you are finished, save your function.

# Matlab File

To create a Matlab script go to the Evaluator folder in the overview panel, right-click on Matlab Files and click on add Matlab File. You can now name and describe your Matlab File.

The Input panel shows you all created parameters, random variables and functions.

In order to create a Matlab Script, we need define an output (in this case T). On the right side of the Output panel click on Plus (+) and name your output.

In a second step, we need to write a Matlab code to define our equation. Navigate to Script/Function at the bottom of the panel and click on it.

The new panel shows you a code editor in which you can write your Matlab script.

Cossan-X already created a for loop so that the equation can be run multiple times with different values for the given random variables.

Equation 2 has to be adapted to a Matlab-readable code and needs to include n (number of iterations), by using Toutput(n) and Tinput(n) to specify out - and inputs, respectively.

By using Ctrl-Space Cossan-X provides you with a list of your input parameters which you can then select.

Once you have finished save your Matlab script.

# Physical Model

Before you start your analysis you need to create a physical model. In the overview panel navigate to Models, right-click on Physical Models and select add physical model.

You can now name and describe your physical model.

The panel gives you an overview over your inputs, output and Matlab files. Check if all your inputs are defined as you specified them above.

Save the physical model and press the green play button in the top-right corner of the panel.

A new window opens which gives you three different options for running an analysis.

1.     Design of Experiment

2.     Sensitivity Analysis

3.     Uncertainty Quantification

You can now also name your analysis (it won’t start if the name already exists).

# Analyses

## Design of Experiment Analysis

The Design of Experiments analysis gives you two options to choose –deterministic and design of experiment.

In this case you will run a deterministic analysis to get a single value for T based on single values for our random variables.

The next window gives you the possibility to run the analysis on a cluster, which we will not do.

Click on finish to start the analysis.

## Sensitivity Analysis

The sensitivity analysis will provide you with three different options to choose from – global sensitivity, local sensitivity and a gradient.

In this case, we will run a global sensitivity analysis.

Our output of interest is set to T (The variable we specified as our output before) by default, so click next.

In the next window we will leave all settings on default.

Click on finish to start the analysis.

## Uncertainty Analysis

The Uncertainty Analysis provides you with five different options to choose from.

In this case we will run a Monte Carlo simulation.

Increase the number of samples per batch to 1000 and leave the other termination criteria set on default.

This will run our model one time for 1000 samples.

Click next and click on finish to start the analysis.

# Statistical Results

## Deterministic Analysis

Deterministic analysis is often to better understand the safe limits of operation within a system or mechanism. In the cargo crane scenario we are identifying the upper bounds of torque generated before a fault occurs. Anything exceeding this value will cause of the system.

Once the analysis has been performed we see that the upper boundaries for our system before failure is 4.2388e+4 $N \over m^{2}$

## Sensitivity Analysis

Sensitivity analysis allows for the identification of crucial components within a model, by analysing how uncertainty in the output of a model relates to the uncertainty of its inputs. Global sensitivity analysis is a particular approach that uses a representative set of samples to explore the design space.

The output demonstrated highlights the contribution of each random variable to output T. This can be visualised in the form of a bar chart:

From this we can determine that the primary contributor to our model is P (load). With the upper boundary for our system identified via the deterministic analysis, and understanding the role of P in our model, these details will help guide our understanding of the uncertainty analysis.

## Uncertainty Analysis

An uncertainty analysis allows us to identify the role uncertainty plays in our model. This is achieved by running a Monte-Carlo simulation, randomly simulating multiple data points within our random variables. The output is provided in a matrix, however, there are useful ways of visualising the data – the first is a scatterplot, and the second is parallel co-ordinates view.

The first scatterplot we view is the interaction between our output T and our random variable P. Since our sensitivity analysis confirmed the importance of P in our model, this is primary variable to focus on. We see a positive correlation between T and P, and from the plot we can estimate that a load 400Kg would sufficient to induce a failure as T would exceed the determined limit of 4.2388e+4 $N \over m^{2}$.

Despite there being a positive trend, it’s hard to determine the overall effect of $\rho$ on T. This is confirmed by our sensitivity analysis highlighting the low contribution $\rho$ has in our model.

This is a parallel co-ordinates view of our results, using only 10 Monte-Carlo simulations to simplify our discussion. This approach allows us to identify the interaction between variables P and $\rho$, and our output T. By tracking the lines from T that are beneath our limit of 4.2388e+4 $N \over m^{2}$, we can see the potential combinations of P and $\rho$ that are viable. Any lines above this value present combinations of P and $\rho$ that exceed the determined capacity.