Overview

In this project you theoretically model a physical system and calculate its parameters using experimental data.

System

You will examine the motion of a ruler that oscillates back and forth after it is let go$.$ This exhibits damped oscillations $($also known as decaying oscillations$).$See the attached photo $1.$

Theoretical Modeling

Model Selection $(40$ pts$)$

We first have to pick a model for our beam to allow us to analyze the system. For this project, I will be selecting the Lumped parameter model for my system dynamic analysis.

Given the setup of the experiment shown in the image and video above, pick a model for the beam $($i$.$e$.,$ what are the boundary conditions and loads applied to the beam$).$ Justify your choice based on observations.

Mass $(20$ pts$)$

We saw in our work on lumped parameter modeling that we can find the equivalent mass of a beam based on the beam model and the parameters of the beam. In particular, the equivalent mass depends on the mass density of the material $(\backslash$rho $),$ the cross sectional area of the beam $($A$),$ and the length of the beam $($L$).$ Calculating these values requires us to know the exact geometry and material properties of the ruler, which are difficult to obtain $($especially since this ruler has two materials, a metal layer and a cork layer$).$ Instead, we can use a simple experiment to get these values.

Note that $\backslash$rho A is the linear density of the beam $($i$.$e$.,$ mass per unit length$).$ We can easily get this value by finding the total mass m$0$ and dividing by the total length L$0,$ i$.$e$.,\backslash$rho A$=$m$0/$L$0.$ As measured, our ruler has a total mass m$0=40\mathrm{.}47$ g and a total length L$0=18\mathrm{.}25$ in$.$ The length of the beam that moves is L$=12\mathrm{.}625$ in $($i$.$e$.,$ the portion of the beam that is not clamped is L$).$

$1.$Calculate the value of $\backslash$rho A using the measured data.

$2.$ Using the beam model from above, find the equivalent mass me of the portion of the beam that is moving.

Spring Coefficient $(40$ pts$)$

We saw in our work on lumped parameter modeling that we can also find the equivalent spring coefficient for a cantilever beam, which depends on the Young's modulus of the material $($E$),$ the area moment of inertia of the beam $($I$),$ and the other parameters from above. Again, calculating these values requires us to know the exact geometry and material properties of the ruler, which are difficult to obtain. To get the parameters EI we can instead perform a second simple experiment. We can see that the ruler is flexible enough that it bends under its own weight $($see below$).$ In other words, there is a distributed load w $($with units N$/$m$)$ acting along the length of the beam due to gravity. From solid mechanics, we can find an equation for the deflection at the free end of the beam $($Vmax$)$ depending on the boundary and loading conditions.

The grid paper behind the beam has steps of $5$ mm and the orange line is horizontal. In the attached Photo $2$ above, L$=12\mathrm{.}625$ in$.$Please see the attached photo $2$

Report the measured value of Vmax.

Calculate the values of the distributed load w$.$

Calculate the value of EI$.$

Calculate the equivalent spring coefficient ke$.$