If we have a spring attached to a wall with an object on the other side, the differential equations describing the system are:
$$x'=v$$
$$v'=-\frac{k}{m}x-\frac{b}{m}v$$
Where: x is position of the object, m is the mass of it, k is the spring stiffnes and b is constant of friction.
My questions:
- Could you please draw the equations of the Runge-Kutta method in this case?
- How much does the situation changes if we change to wall to another object?
- How much does it change if we more than one springs attached to the objects?
I am sorry for the many subquestions but can hardly understand this whole.
Best Answer
For subquestion (1), this site already has How-tos.
For subquestions (2) and (3), here is a scheme.
Suppose you have $n$ mass points that can move along the $x$ axis. Let $m_1,\ldots,m_n$ denote their (constant) masses. Let $x_1,\ldots,x_n$ denote the displacements of the mass points from their rest position.
No springs, no dampers yet.
At this stage, your system of equations looks like this: $$\underbrace{\begin{pmatrix}m_1\\&m_2\\&&\ddots\\&&&m_n\end{pmatrix}}_{\mathbf{M}} \underbrace{\begin{pmatrix}\ddot{x}_1\\\ddot{x}_2\\\vdots\\\ddot{x}_n\end{pmatrix}}_{\mathbf{\ddot{x}}} +\underbrace{\begin{pmatrix}\cdot&\cdot&\cdots&\cdot\\\cdot&\cdot&\cdots&\cdot\\\vdots&\vdots&\ddots&\vdots\\\cdot&\cdot&\cdots&\cdot\end{pmatrix}}_{\mathbf{B}} \underbrace{\begin{pmatrix}\dot{x}_1\\\dot{x}_2\\\vdots\\\dot{x}_n\end{pmatrix}}_{\mathbf{\dot{x}}} +\underbrace{\begin{pmatrix}\cdot&\cdot&\cdots&\cdot\\\cdot&\cdot&\cdots&\cdot\\\vdots&\vdots&\ddots&\vdots\\\cdot&\cdot&\cdots&\cdot\end{pmatrix}}_{\mathbf{K}} \underbrace{\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}}_{\mathbf{x}} = \underbrace{\begin{pmatrix}f_1\\f_2\\\vdots\\f_n\end{pmatrix}}_{\mathbf{f}} $$ Here, empty matrix entries are to be interpreted as zeros.
$\mathbf{M}$ is the mass matrix (and it's diagonal here), $\mathbf{B}$ is the damping matrix (empty yet), $\mathbf{K}$ is the stiffness matrix (empty yet), $\mathbf{x}$ is the time-dependent vector of the displacements that you want to describe, and $\mathbf{f}$ is a time-dependent vector of forces: $f_1$ acts directly on the first mass point (mass $m_1$), $f_2$ acts directly on the second, and so on. You may assume $\mathbf{f}$ to be zero; but we will sometimes think of adding some forces, that's why I have given names to them.
For simplicity, let us suppose that we have only three mass points, $n=3$. This means that I do not need to write those $\ddots$ dots again.
Now let us add (linear) parts. Say, you have a spring with stiffness $k_1$, and that spring shall be mounted between mass point $m_3$ and the wall (or the ground, whatever, something fixed in any case). To represent this, we could add to $f_3$ the term $-k_1 x_3$, but everything that has to do with some $x_i$ shall be moved over to the left-hand side, so we write: $$\underbrace{\begin{pmatrix}m_1\\&m_2\\&&m_3\end{pmatrix}}_{\mathbf{M}} \underbrace{\begin{pmatrix}\ddot{x}_1\\\ddot{x}_2\\\ddot{x}_3\end{pmatrix}}_{\mathbf{\ddot{x}}} +\underbrace{\begin{pmatrix}\cdot&\cdot&\cdot\\\cdot&\cdot&\cdot\\\cdot&\cdot&\cdot\end{pmatrix}}_{\mathbf{B}} \underbrace{\begin{pmatrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\end{pmatrix}}_{\mathbf{\dot{x}}} +\underbrace{\begin{pmatrix}\cdot&\cdot&\cdot\\\cdot&\cdot&\cdot\\\cdot&\cdot&k_1\end{pmatrix}}_{\mathbf{K}} \underbrace{\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}}_{\mathbf{x}} = \underbrace{\begin{pmatrix}f_1\\f_2\\f_3\end{pmatrix}}_{\mathbf{f}} $$ Note that $k_1$ lands on the main diagonal of $\mathbf{K}$ in the row (and column) corresponding to $m_3$.
Let us add another part. A linear damper with damping constant $b_1$ shall be mounted between mass points $m_1$ and $m_2$. Again, we could think of this as adding to $f_1$ the expression $-b_1(\dot{x}_1-\dot{x}_2)$ and adding to $f_2$ the expression $-b_1(\dot{x}_2-\dot{x}_1)$, but that stuff is to be moved over to the left-hand side, therefore we get: $$\underbrace{\begin{pmatrix}m_1\\&m_2\\&&m_3\end{pmatrix}}_{\mathbf{M}} \underbrace{\begin{pmatrix}\ddot{x}_1\\\ddot{x}_2\\\ddot{x}_3\end{pmatrix}}_{\mathbf{\ddot{x}}} +\underbrace{\begin{pmatrix}b_1&-b_1&\cdot\\-b_1&b_1&\cdot\\\cdot&\cdot&\cdot\end{pmatrix}}_{\mathbf{B}} \underbrace{\begin{pmatrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\end{pmatrix}}_{\mathbf{\dot{x}}} +\underbrace{\begin{pmatrix}\cdot&\cdot&\cdot\\\cdot&\cdot&\cdot\\\cdot&\cdot&k_1\end{pmatrix}}_{\mathbf{K}} \underbrace{\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}}_{\mathbf{x}} = \underbrace{\begin{pmatrix}f_1\\f_2\\f_3\end{pmatrix}}_{\mathbf{f}} $$ Notice the pattern $\begin{pmatrix}b&-b\\-b&b\end{pmatrix}$ added to the rows and columns of $\mathbf{B}$ that are associated with the mass points that we connect with the damper. You will see this again.
Let us add more: A linear damper with damping constant $b_2$ between mass point $m_1$ and ground, and a spring with stiffness $k_2$ between mass points $m_2$ and $m_3$: $$\underbrace{\begin{pmatrix}m_1\\&m_2\\&&m_3\end{pmatrix}}_{\mathbf{M}} \underbrace{\begin{pmatrix}\ddot{x}_1\\\ddot{x}_2\\\ddot{x}_3\end{pmatrix}}_{\mathbf{\ddot{x}}} +\underbrace{\begin{pmatrix}b_1+b_2&-b_1&\cdot\\-b_1&b_1&\cdot\\\cdot&\cdot&\cdot\end{pmatrix}}_{\mathbf{B}} \underbrace{\begin{pmatrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\end{pmatrix}}_{\mathbf{\dot{x}}} +\underbrace{\begin{pmatrix}\cdot&\cdot&\cdot\\\cdot&k_2&-k_2\\\cdot&-k_2&k_1+k_2\end{pmatrix}}_{\mathbf{K}} \underbrace{\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}}_{\mathbf{x}} = \underbrace{\begin{pmatrix}f_1\\f_2\\f_3\end{pmatrix}}_{\mathbf{f}} $$ Note that the matrix entries are added up when the patterns overlap.
Let us be a bit sophisticated now and add a spring with stiffness $k_3$ between mass points $m_1$ and $m_3$, bypassing $m_2$ somehow. Oh, and $m_2$ should get a damper $b_3$ with the other side fixed. No problem: $$\underbrace{\begin{pmatrix}m_1\\&m_2\\&&m_3\end{pmatrix}}_{\mathbf{M}} \underbrace{\begin{pmatrix}\ddot{x}_1\\\ddot{x}_2\\\ddot{x}_3\end{pmatrix}}_{\mathbf{\ddot{x}}} +\underbrace{\begin{pmatrix}b_1+b_2&-b_1&\cdot\\-b_1&b_1+b_3&\cdot\\\cdot&\cdot&\cdot\end{pmatrix}}_{\mathbf{B}} \underbrace{\begin{pmatrix}\dot{x}_1\\\dot{x}_2\\\dot{x}_3\end{pmatrix}}_{\mathbf{\dot{x}}} +\underbrace{\begin{pmatrix}k_3&\cdot&-k_3\\\cdot&k_2&-k_2\\-k_3&-k_2&k_1+k_2+k_3\end{pmatrix}}_{\mathbf{K}} \underbrace{\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}}_{\mathbf{x}} = \underbrace{\begin{pmatrix}f_1\\f_2\\f_3\end{pmatrix}}_{\mathbf{f}} $$ So this is the scheme. Things become more complicated when objects can move in more than one dimension and when parts can be rotated, but in the end, system matrices will be assembled from element matrices.