MATLAB $-$ Assignment #$2$

Problem #$1$

Your engineering firm has tasked you with conducting a simple vibration analysis of new shock

absorbers that are being rolled out. The shocks can be modeled as a mass$-$spring$-$damper

system, as shown in Figure $1.$

Figure $1$: The Mass$-$Spring Damper System

Mathematically, the resulting damped motion of the vehicle, after it hits a bump in the road, can

be described as:

x$($t$)=$xe$^(-\backslash$zeta $\backslash$omega $\_($n$)$t$)$cos$(\backslash$sqrt$(1-\backslash$zeta $^(2))\backslash$omega $\_($n$)$t$-\backslash$phi $)$

Where x$($t$)$ is the time$-$variant displacement of the vehicle from some equilibrium position, x is

the initial magnitude of the displacement, $\backslash$zeta is the damping ratio, $\backslash$omega $\_($n$)$ is the angular frequency

and $\backslash$omega is the phase shift. The angular frequency can be further expanded as:

$\backslash$omega $\_($n$)=2\backslash$pi f$\_($n$)$

Where f$\_($n$)$ is the natural frequency. You have three different shock absorbers to model, which vary only in the magnitude of their damping ratios. Those values are: $\backslash (\backslash$zeta$=0\mathrm{.}1$ ; $0\mathrm{.}3\backslash )$ and $0\mathrm{.}5.$ The remaining parameters are shown in Table $1.$

Table $1$: Shock Absorber Parameters

Substituting these values into the displacement equation, we can find $\backslash ($ X$($t$)\backslash )$ for any time, $\backslash ($ t $\backslash ).$ However, we are not interested in a discrete solution to the problem. To analyze the transient behavior, we will have to obtain the results over a finite period of time i$.$e$.$ multiple values of $\backslash ($ t $\backslash ).$ Determine the variation in displacement of the system over the period $\backslash (0\backslash$leq t $\backslash$leq $3\backslash$mathrm$\{$~s$\}\backslash ).$

Tasks

Using MATLAB, create a vector of time values, where each value will correspond to a solution of the displacement equation. Once you have obtained a vector of solutions $($numerical displacement values$),$ these can be plotted against time, allowing you to analyze the performance of each of the three shock absorbers$.$

Using MATLAB's plot and subplot commands, create separate graphs for the transient response of the three shocks, within the same figure. Create a fourth graph in the same figure that contains all three plots, superimposed on top of one another. All of the plots in the figure should be arranged as a $\backslash (2\backslash$times $2\backslash )$ grid. Also, include: a legend; axis titles and graph titles. Tasks

Using MATLAB, create a vector of time values, where each value will correspond to a solution of the displacement equation. Once you have obtained a vector of solutions $($numerical displacement values$),$ these can be plotted against time, allowing you to analyze the performance of each of the three shock absorbers$.$

Using MATLAB's plot and subplot commands, create separate graphs for the transient response of the three shocks, within the same figure. Create a fourth graph in the same figure that contains all three plots, superimposed on top of one another. All of the plots in the figure should be arranged as a $\backslash (2\backslash$times $2\backslash )$ grid. Also, include: a legend; axis titles and graph titles.

Things to Think About

The time vector should be of the form: $\backslash (\backslash$boldsymbol$\{$t$\}=\backslash$mathbf$\{$t $0\}\backslash )$ : dt: $\backslash ($ t $\backslash$boldsymbol$\{$f$\}\backslash ),$ where to represents the initial time $\backslash (($t$=0),\backslash$mathrm$\{$dt$\}\backslash )$ is the time$-$step and $\backslash ($ t f $\backslash )$ is the final time $\backslash (($t$=3\backslash$mathrm$\{$~s$\})\backslash ).$ The size of the time$-$step can be varied. What size of time$-$step would you use? What happens to the simulation as you decrease the value of dt$?$

You may encounter an error when inserting the time vector directly into the displacement equation. What does the error message mean? How can you resolve it$?$ Hint: Research element$-$wise multiplication of vectors in MATLAB.