Chapter 12, Problem 37P

Mechanical/Aerospace Engineering

Consider the three mass-four spring system in Fig. P12.37. Determining the equations of motion from ${\displaystyle \sum F_x}=ma$, for each mass using its free-body diagram results in the following differential equations:

${\ddot{x}}_1+\left(\frac{k_1+k_2}{m_1}\right)x_1-\left(\frac{k_2}{m_1}\right)x_2=0$

${\ddot{x}}_2-\left(\frac{k_2}{m_2}\right)x_1+\left(\frac{k_2+k_3}{m_2}\right)x_2-\left(\frac{k_3}{m_2}\right)x_3=0$

${\ddot{x}}_3-\left(\frac{k_3}{m_3}\right)x_2+\left(\frac{k_3+k_4}{m_3}\right)x_3=0$

FIGURE P12.37

Where $k_1=k_4=10N/m,k_2=k_3=30N/m,\text{and}{\text{m}}_1=m_2=m_3=2kg$. Write the three equations in matrix form:

$0=\left[\text{Acceleration vector}\right]+\left[k/m\text{matric}\right]\left[\text{displacement vector }x\right]$

At a specific time when x 1 5 0.05 m, x 2 5 0.04 m, and x 3 5 0.03 m, this forms a tridiagonal matrix. Solve for the acceleration of each mass.