EXAMPLE $3-3$ Consider the spring$-$mass$-$dashpot system mounted on a massless cart as shown in Figure $3-3.$ Let us obtain mathematical models of this system by assuming that the cart is standing still for $t<mn>0</mn>$ and the spring$-$mass$-$dashpot system on the cart is also standing still for In this system, $u(t)$ is the displacement of the cart and is the input to the system. At $t=0,$ the cart is moved at a constant speed, or $u^{}=$ constant. The displacement $y(t)$ of the mass is the output. $($The displacement is relative to the ground.$)$ In this system, $m$ denotes the mass, $b$ denotes the viscous$-$friction coefficient, and $k$ denotes the spring constant. We assume that the friction force of the dashpot is proportional to $y^{}-u^{}$ and that the spring is a linear spring; that is$,$ the spring force is proportional to $y-u.$

For translational systems, Newton's second law states that

$ma={\displaystyle \underset{?}{\overset{?}{}}}F$

What is the mathematical model of this system

Find the transfer function of this system