1.4 Developing and Solving the Eq. of Motion

Figure 1.4.1 Mass-spring System and Free Body Diagram

The Eq. of motion can be developed using the shown free body diagram. The term mü represent the inertial force developed opposite to the motion of the mass. From the dynamic equilibrium,

Eq. 1.7

The solution of this differential Eq. can be written in the form,

Eq. 1.8

Which can be re-written in the form,

Eq. 1.9

where C is the amplitude of motion ω is the angular frequency of motion, and β is the phase shift.,

Eq. 1.10

where u_o and v_o are the initial displacement and the initial velocity respectively. The angular frequency ω can be defines as,

Eq. 1.11

where f is the frequency in Hz. The period T can be defined as the reciprocal of the frequency,

Eq. 1.12

and the phase shift can be defined as,

Eq. 1.13

the velocity can be obtain by differentiating the displacment, equation 1.9,

Eq. 1.14

similarly, the acceleration can be obtained by differentiating the velocity, equation 1.14,

Eq. 1.15

The displacement, velocity and acceleration can be plotted in the below chart.

Try to change the parameters to see how the response changes!

Show Tips and Notes

Figure 1.4.2 Reationships between displacement, velocity and acceleration.

The term β/ω represents the time shift between the displacement, velocity and acceleration.

Velocity is in advance of the displacement by 𝝅/2.

Acceleration is in advance of the displacement by 𝝅.

By the time the displacment is maximum, the velocity is zero and the acceleration is maximum in the reverse direction.

By the time the velocity is maximum, both displacement and acceleration are zero.