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=Report 1=

Problem R1


Derive the equation of the motion of a spring dashpot system in parallel with a mass and applied force $$ f(t)\,\!$$

First we examine the figure, and declare constants. $$ y \,\!$$ is the distance the coil expands, and k is the spring constant. Therefore, we are able to define a force $$ F_1\,\!$$ as

(1.0) $$ F_1= -ky\,\!$$.

The force is negative because in this model, the up position is negative and the down position is positive. This is a force opposing the motion of the mass, thus it is negative. Thereafter, we know Newton's second law as

$$ Force= mass* acceleration \,\!$$, or otherwise known as

$$ F=m\,\!$$$$ \frac{dV}{dt}\,\!$$

From calculus, it is also known that $$ \frac{dy}{dx}\,\!$$ can also be written as $$ y',\!$$. Furthermore, acceleration is known as

$$ \frac{dV}{dt}\,\!$$ $$ =\!$$ $$ \frac{d^2y}{dx^2}\,\!$$ $$ = \,\!$$ $$ y'',\!$$ (1.1)

If we sum the forces and equal them to zero we ca write Newtons second law for this system as shown below

$$ \sum{F}\!$$ $$ = \!$$$$ m\,\!$$$$ \frac{dV}{dt}\,\!$$ (1.2)

$$ my''+ky=0 \!$$ (Notice it is important to have all terms on the same side of the)

It is important to note, this force only includes the restoring force of the spring. It is now necessary to include the damper term from the dashpot. The damping constant will be defined as $$ C\!$$. It is also assumed the damping force is proportional to the velocity (Kreyszig, 2011). Therefore, we are able to write

$$ F_2= -c y',\!$$ (1.3)

and by summing forces one more time:

$$ \sum{F}\! = $$ $$ \frac{dV}{dt}\,\!$$ $$ \longrightarrow\!$$ $$F_1 + F_2=my \!$$ $$ \longrightarrow,\!$$$$ my+ cy' +ky =0 \!$$ (1.5)

To get the differential equation in standard form, the only thing required is to divide the equation by the mass.

Solution
$$ y''+ \frac{c}{m}y' +\frac{c}{m}y =0 \!$$

Since this is a parallel system, we are able to say the spring force relation and the damper force relation are equal to each other, which is the reason we don't differ from the $$ y's,\!$$ with subscripts.

Picture obtained from here Section 1 Lecture Notes