MECH $4510$ DYNAMIC SYSTEMS $$ PROJECT$1$ DUE $10/24/2024$ MECH $4510$ Project $1$ Develop Generic MCK Model $1$ Rev $101024$ Prof.Murat Inalpolat Develop Generic MATLAB and SIMULINK M$,$C$,$K Models Purpose The intent of this project is to develop generic models in MATLAB and SIMULINK $($and verified by closed$-$form solution $$ supplied on Page $7$ of this document$)$ to address the response of a simple single degree of freedom mechanical mass, spring, dashpot system due to external forces and$/$or initial conditions of displacement and velocity. In addition to the generic model, a specific m$,$ c$,$ k system will be evaluated with parameters associated with randomly generated set of numbers. Model for Evaluation The model used for evaluation is the single degree of freedom lumped mass model defined by second order differential equation with constant coefficients. This model is shown in Figure $1.$ m k c x$($t$)$ f$($t$)$ Figure $1$ Single Degree of Freedom Model The equation of motion describing this system can easily be shown to be $)$t$($fxk dt dx c dt xd m $22=++$ or $=++)$t$($fxkxcxm $(1)$ where m is the mass, c is the damping and k is the stiffness with the displacement, velocity and acceleration and the forcing function as shown. Parameters such as the natural frequency, damped natural frequency, damping factor, critical damping, and techniques for estimating damping are important terms that have been defined as part of the development of class notes. A $7-$digit randomly generated system identification number $($XY$-$Z$-$ULTW$)$ is used to define the mass, damping and stiffness and other random numbers will be used as the initial displacement $()$ and initial velocity $(),$ respectively according to Table $1.$ All units are SI$.$ Please generate a random number using one of Matlab$$s random generator commands $($for example, X $=$ randsample$($n$,$k$)$ or use one of the commands $$rand$$ or $$randn$$ or $$randperm$,$ etc.$)$ between $0$ and $9$ for each of the digits X$,$Y$,$ Z$,$ U$,$ L$,$ T$,$ and W of the system identification number. Table $1$ Parameters for Single Degree of Freedom Model. System Characteristics Mass Damping Stiffness Randomly generated Number$($s$)$ XY$+10$ Z$+50$ ULTW Initial displacement $($m$)$ D$=$ Random # between $1$ and $20=1000$ m Initial velocity $($m$/$s$)$ V$=$ Random # between $1$ and $40=1000$ m$/$s MECH $4510$ DYNAMIC SYSTEMS $$ PROJECT$1$ DUE $10/24/2024$ MECH $4510$ Project $1$ Develop Generic MCK Model $2$ Rev $101024$ Prof.Murat Inalpolat For example, System ID that consist of randomly generated $7$ digits $12-3-6789$ with randomly generated numbers D$=7$ and V$=30$ for initial conditions would result in a model of M$=12+10=22$ kg$,$ C$=3+50=53$ N$.$s$/$m$,$ K$=6789$ N$/$m $22536789()$ ; $(0)0\mathrm{.}007(0)0\mathrm{.}03/$ x x x f t x m and x m s $++===($use all values as given $$ no need to make unit conversions$)$ Note: You can use Matlab$$s random number generating function $($e$.$g$.,$ randn$)$ to generate D and V along with the other digits for the system identification number. Perform the following tasks in the order specified as follows: $1.$ Clearly identify the parameters for your model $$ M$,$C$,$K and initial conditions $(5$ pts$)2.$ Compute n$,$ and identify system characteristics $($overdamped$,$ underdamped, etc.$)(5$ pts$)3.$ Solve $($Hand solution, not Matlab$)$ the second order differential equation, under the influence of initial conditions but no external forcing, using the classical ODE $($assumed solution$)$ approach $(20$ pts$)4.$ Solve $($Hand solution, not Matlab$)$ the second order differential equation, under the influence of initial conditions but no external forcing, using the Laplace approach $(20$ pts$)5.$ Use MATLAB$$s DSOLVE command to solve the ODE and PLOT the response; supply the details and the DSOLVE script used. $(10$ pts$)6.$ Prepare a Matlab Simulink model $($see the later pages of this sheet as well as the tutorials under BB for some help$)$ and plot the response due to initial conditions; supply Simulink diagram$$s screenshot; make sure all details included with all variables and constants have readable values. Axes should be properly labeled with units included $(20$ pts$).7.$ Plot each one of the four different solutions obtained from the previous parts of the question in parts $3,4,5,$ and $6($separate plots for each of the solutions obtained in parts $3,4,5,$ and $6)$ separately. Add a separate fifth plot and provide an overlay of all $4$ solutions together $($solution obtained in parts $3,4,5,$ and $6).$ You are expected to have $5$ plots overall. Comment on similarities and discrepancies and the corresponding reasons for any discrepancies in the overlay plot. Use proper labels, a legend, and units for the plot and make sure results can be clearly seen. $((4$pts x $4$ plots$)+4$ pts for clear$/$reasonable explanations of the results $=20$ pts$).$ Report$($INDIVIDUAL Report$)$ Mark your report with a simple header with your name and student ID $$ this is a formal report with clear, concise presentation.