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5. (a) Consider n masses and n springs under the action of gravity, with a fixe...

Oct 6, 2024

Solution by Steps

step 1

To find the incidence matrix

\mathbf{A}

for the system of masses and springs, we consider a

3 \times 3

case where each mass is connected to its adjacent springs. The incidence matrix can be represented as:

\mathbf{A} = \begin{bmatrix} 1 &amp; 0 &amp; 0 \\ -1 &amp; 1 &amp; 0 \\ 0 &amp; -1 &amp; 1 \end{bmatrix}

step 2

The transpose of the incidence matrix

\mathbf{A}^{T}

is obtained by switching rows and columns:

\mathbf{A}^{T} = \begin{bmatrix} 1 &amp; -1 &amp; 0 \\ 0 &amp; 1 &amp; -1 \\ 0 &amp; 0 &amp; 1 \end{bmatrix}

step 3

For the elastic rod in the stretched configuration, the equilibrium ODE for displacement

u(x)

is given by:

-\frac{d}{dx}\left(C^{*}(x) \frac{du(x)}{dx}\right) = f^{*}(x), \quad u(0) = 0, \quad \left.\frac{du}{dx}\right|_{x=1} = 0

step 4

The continuous analog of the discrete relation

\mathbf{e} = \mathbf{A x}

can be expressed as:

e(x) = \frac{du(x)}{dx}, \quad \text{with the boundary condition } u(0) = 0

step 5

The continuous version of the constitutive law

\mathbf{y} = \mathbf{C e}

is:

y(x) = C^{*}(x) e(x)

Here,

C^{*}(x)

is a continuous function representing stiffness, replacing the discrete matrix

\mathbf{C}

with spring constants

k

step 6

The continuous version of the equilibrium condition

\mathbf{A}^{T} \mathbf{y} = \mathbf{f}

becomes:

\int_0^1 C^{*}(x) \frac{du}{dx} dx = f^{*}(x)

The boundary condition at

x=1

indicates that there is no displacement at the free end, implying that the internal force must balance the external force

Answer

The incidence matrix

\mathbf{A}

and its transpose

\mathbf{A}^{T}

are defined, and the equilibrium ODE for the elastic rod is established with boundary conditions.

Key Concept

Understanding the relationship between discrete and continuous systems in mechanics.

Explanation

The transition from discrete matrices to continuous functions allows for modeling real-world systems more accurately, especially in the context of elasticity and forces.