spring pulled const velo

A working note for system of two masses and one spring pulled with constant velocity

A mass $m_i$ connected with mass $m_j$ via a spring with constant $k_{ij}$ and normal length $l_{ij}$. Position of $m_i$ relative to $m_j$ is

$$\tag{1} x_{ij} = x_i - x_j $$

and the distance between mass $m_i$ and $m_j$ is

$$\tag{2} r_{ij} = \sqrt{(x_i - x_j)^2}. $$

Notice that $r_{ij}$ is always positive, while $x_{ij}$ can be positive or negative.

Spring force on $m_i$ due to $m_j$ is

$$\tag{3} S_{ij} = -k_{ij}(r_{ij} - l_{ij}) u_{ij}, $$

where the unit vector is simple

$$\tag{4} u_{ij} = \frac{x_i - x_j}{r_{ij}}. $$

Eqns (2) and (4) can be extended from this 1-d system to 2-d and 3-d systems.

Newton’s second law of motion

$$\tag{5} \sum F = m\ddot{x} $$

and Eqn (3) will produce

$$\tag{6} m_i \ddot{x}_i = -k_{ij}(r_{ij} - l_{ij}) u_{ij} $$

with $u_{ij}$ and $r_{ij}$ are given in Eqns (2) and (4).

Suppose that $m_i$ is fixed and $m_j$ is pulled with constant velocity $v_j$ then position of $m_j$ is simply

$$\tag{7} x_j(t) = x_{j,0} + v_j t, $$

while position of $m_i$ can be obtained after solving Eqn (6), which can be explcitly in the form of

$$\tag{8} \ddot{x}_i = - \frac{k_{ij}}{m_i} \left( \sqrt{(x_i - x_j)^2} - l_{ij} \right) \left( \frac{x_i - x_j}{\sqrt{(x_i - x_j)^2}} \right). $$

With $n$ is index of time $t$, advancement of $t$ can be formulated as

$$\tag{9} t^{n + 1} = t^n + \Delta t $$

with $n = 0, 1, 2, ..$. Eqn (7) is rewriten as

$$\tag{10} x_j^{n+1} = x_j^n + v_j \Delta t $$

and also Eqn (8) as

$$\tag{11} \ddot{x}_i^n = - \frac{k_{ij}}{m_i} \left( \sqrt{(x_i^n - x_j^n)^2} - l_{ij} \right) \left( \frac{x_i^n - x_j^n}{\sqrt{(x_i^n - x_j^n)^2}} \right). $$

Using Euler’s method ¹ new velocity of mass $m_i$ can be calculated

$$\tag{12} \dot{x}_i^{n+1} = \dot{x}_i^n + \ddot{x}_i^n \Delta t $$

and also its new position

$$\tag{13} x_i^{n+1} = x_i^n + \dot{x}_i^n \Delta t. $$

And

$$\tag{14} x_j^0, \ \ x_i^0, \ \ t^0 $$

are the initial conditions. Simulation can be performed using Eqns (9) – (14). Additional initial condition can also be applied, e.g. if the spring is in its normal length, then

$$\tag{15} x_i^0 = x_j^ + l_{ij} $$

should hold.