$3\mathrm{.}3$ Determine the equation governing the system studied in Example $3\mathrm{.}13$ by carrying

out a force balance.

EXAMPLE $3\mathrm{.}13$ Governing equation for a translating system with a pretensioned or

precompressed spring

We revisit the pretensioned spring system shown in Figure $2\mathrm{.}15$ and add to the spring

combination a mass $m$ as shown in Figure $3\mathrm{.}12.$ We shall use Lagrange's equations to

derive the governing equation of motion for vertical translations $x$ of the mass about the

static$-$equilibrium position of the system and examine how the initially horizontal spring

with linear stiffness $k_1$ affects the natural frequency of this system. The equation of

motion will be derived for "small" amplitude vertical oscillations, that is$,|\frac{x}{L}|1.$

From Eq$.(2\mathrm{.}41)$ it is found that when $|x|L|1||,$ the total stiffness $k_v$ in the vertical

direction, in the notation of Figure $3\mathrm{.}12,$ is given by

$k_v=k_2+{}_o\frac{k_1}{L}$

where ${}_o=\frac{T_1}{k_1}$ when the spring is initially in tension. Then the expression for the

potential energy is

$V(x)=\frac{1}{2}{k}_v{x}^2=\frac{1}{2}(k_2+{}_o\frac{k_1}{L})x^2=\frac{1}{2}(k_2+\frac{T_1}{L})x^2$

The expression for the kinetic energy is

$T(x)=\frac{1}{2}m{x}^{}{?}^2$

Noting that the dissipation function $D=0$ and that the generalized force $Q_1=0,$ we

substitute Eqs. $($b$)$ and $($g$)$ into Eq$.(3\mathrm{.}44)$ to obtain the following governing equation of

motion

$m{x}^{}+(k_2+\frac{T_1}{L})x=0$

Figure $3\mathrm{.}12.$ Single degree$-$of$-$freedom system with the horizontal

spring under an initial tension $T_1.$

From Eq$.($d$),$ we recognize the natural frequency to be

${}_n=\sqrt[\phantom{2}]{\frac{k_2+\frac{T_1}{L}}{m}}$

It is seen that the effect of a spring under tension, which is initially normal to the direc$-$

tion of motion, is to increase the natural frequency of the system.

If the spring of constant $k_1$ is initially compressed instead of being in tension, then we

can replace $T_1$ by $-T_1$ and Eq$.($e$)$ becomes

${}_n=\sqrt[\phantom{2}]{\frac{k_2-\frac{T_1}{L}}{m}}$

From Eq$.(f),$ it is seen that the natural frequency can be made very low by adjusting the

compression of the spring with stiffness $k_1.$ At the same time, the spring with stiffness $k_2$

can be made stiff enough so that the static displacement of the system is not excessive.

This type of system is the basis of at least one commercial product. ${?}^{14}$