Consider the mechanical system with three degrees of freedom. The positions of the particles are measured from their equilibrium positions. The system has a normal mode eigenvector

|13.98|
| b |
|13.98|.

If all particles start from their equilibrium positions and the leftmost and rightmost particles are given a velocity of 5.88m/s, the velocity of the middle particle is -40.85m/s, the system wil oscillate in a normal mode.

Determine the value of b.

To determine the value of b, we can use the information provided about the normal mode eigenvector and the given initial velocities.

Let's consider the equation of motion for the system's normal mode. The equation can be written as:

M * d^2X/dt^2 = K * X

where M is the mass matrix, K is the stiffness matrix, and X is the displacement vector.

Given that the system has three degrees of freedom, the mass matrix M will have dimensions of 3x3. The stiffness matrix K will also have dimensions of 3x3.

Let's assume that the displacement vector X has three components: x1, x2, and x3. Since the positions of the particles are measured from their equilibrium positions, the displacement vector X can be written as:

| x1 |
| x2 |
| x3 |

The normal mode eigenvectors represent the mode shapes of the system. The given normal mode eigenvector has components |13.98|, |b|, and |13.98| respectively. This means that the second particle has a displacement equal to b.

Now, let's consider the given initial velocities. It states that the leftmost and rightmost particles have velocities of 5.88 m/s each, and the middle particle has a velocity of -40.85 m/s.

The initial velocity vector V can be written as:

| v1 |
| v2 |
| v3 |

where v1 = 5.88 m/s, v2 = -40.85 m/s, and v3 = 5.88 m/s.

To determine the value of b, we can express the initial velocity vector V in terms of the displacement vector X. This can be done using the equation:

V = dX/dt

Integrating this equation with respect to time, we get:

X = ∫V dt

Integrating the components of V separately, we get:

| x1 |
| x2 |
| x3 | = ∫ | v1 | dt
| v2 |
| v3 |

Integrating the given velocities, we have:

| x1 |
| x2 |
| x3 | = ∫ | 5.88 | dt
| -40.85 |
| 5.88 |

Since the velocities are constant, integrating gives:

| x1 |
| x2 |
| x3 | = | 5.88 | * t
| -40.85 | * t
| 5.88 | * t

Where t represents time.

Now, comparing the components of the displacement vector X with the normal mode eigenvector, we can determine the value of b.

From the normal mode eigenvector, we know that x1/x3 = 13.98/13.98 = 1, and x2 = b.

Comparing these equations with the displacement vector components, we can conclude that the value of b is 1.