25.3: A "Larger" Spring-Mass System- n Degrees of Freedom

We now consider the equilibrium of the system of \(n\) springs and masses shown in Figure \(25.13\). (This string of springs and masses in fact is a model (or discretization) of a continuum truss; each spring-mass is a small segment of the truss.) Note for \(n=2\) we recover the small system studied in the preceding sections. This larger system will serve as a more "serious" model problem both as regards existence and uniqueness but even more importantly as regard computational procedures. We then consider force balance on mass 1 , \[\begin{aligned} &\sum \text { forces on mass } 1=0 \\ &\Rightarrow f_{1}-k_{1} u_{1}+k_{2}\left(u_{2}-u_{1}\right)=0, \end{aligned}\] and then on mass 2, \[\begin{aligned} &\sum \text { forces on mass } 2=0 \\ &\Rightarrow f_{2}-k_{2}\left(u_{2}-u_{1}\right)+k_{3}\left(u_{3}-u_{2}\right)=0, \end{aligned}\] and then on a typical interior mass \(i\) (hence \(2 \leq i \leq n-1\) ) \[\begin{aligned} &\sum \text { forces on mass } i=0(i \neq 1, i \neq n) \\ &\Rightarrow f_{i}-k_{i}\left(u_{i}-u_{i-1}\right)+k_{i+1}\left(u_{i+1}-u_{i}\right)=0, \end{aligned}\] and finally on mass \(n\), \[\begin{aligned} &\sum \text { forces on mass } n=0 \\ &\Rightarrow f_{n}-k_{n}\left(u_{n}-u_{n-1}\right)=0 \end{aligned}\]

Figure 25.13: System of \(n\) springs and masses.

We can write these equations as

\[\begin{aligned} & \begin{array}{llll}\left(k_{1}+k_{2}\right) u_{1} & -k_{2} u_{2} & 0 \ldots & =f_{1}\end{array} \\ & -k_{2} u_{1}+\left(k_{2}+k_{3}\right) u_{2} \quad-k_{3} u_{3} \quad 0 \ldots \quad=f_{2} \\ & \begin{array}{llll}0 & -k_{3} u_{2} & +\left(k_{3}+k_{4}\right) u_{3} & -k_{4} u_{4}\end{array}=f_{3} \\ & \ldots 0 \quad-k_{n} u_{n-1} \quad+k_{n} u_{n}=f_{n} \\ & \left(\begin{array}{ccccccc}k_{1}+k_{2} & -k_{2} & & & & & \\-k_{2} & k_{2}+k_{3} & -k_{3} & & & 0 & \\& -k_{3} & k_{3}+k_{4} & -k_{4} & & & \\& & & & & & \\& & \ddots & \ddots & \ddots & & \\0 & & & & & \\u_{n-1} \\u_{n}\end{array}\right) \quad\left(\begin{array}{c}u_{1} \\u_{2} \\u_{3} \\\vdots \\f_{n-1}\end{array}\right)\left(\begin{array}{c}f_{1} \\f_{2} \\f_{3} \\\vdots \\f \\f_{n-1}\end{array}\right) \\ & \begin{array}{ccc}K & u & f \\n \times n & n \times 1 & n \times 1\end{array} \end{aligned}\]

which is simply \(A u=f(A \equiv K)\) but now for \(n\) equations in \(n\) unknowns.

In fact, the matrix \(K\) has a number of special properties. In particular, \(K\) is sparse \(-K\) is mostly zero entries since only "nearest neighbor" connections affect the spring displacement and hence the force in the spring \({ }^{5}\); tri-diagonal - the nonzero entries are all on the main diagonal and diagonal just below and just above the main diagonal; symmetric \(-K^{\mathrm{T}}=K\); and positive definite (as proven earlier for the case \(n=2)-\frac{1}{2}\left(v^{\mathrm{T}} K v\right)\) is the potential/elastic energy of the system. Some of these properties are important to establish existence and uniqueness, as discussed in the next section; some of the properties are important in the efficient computational solution of \(K u=f\), as discussed in the next chapters of this unit.

\({ }^{5}\) This sparsity property, ubiquitous in MechE systems, will be the topic of its own chapter subsequently. \[\begin{aligned} & \begin{array}