Question:

(c) Suppose there is a small amount of friction included in the model by introducing the “drag” force F = -βx′₃ on the third mass at position x₃(t). In this case the equation becomes:

m₃x″₃ = -k₃(x₃ − x₂) − k₄x₃ − βx′₃ for β = 0.04.

State the six different real (not complex) terms the solution could contain.
Estimate the time it will take for the longest lasting (most persistent) terms in your solutions to decay to around 5% of their initial values.

Answer:

Step 1: Understanding the System

The system consists of three coupled masses with linear springs. The inclusion of a damping term introduces exponential decay into the vibrational modes of the system.

The governing equation becomes a set of second-order ordinary differential equations (ODEs) with damping. Since there are 3 second-order ODEs, and each has 2 linearly independent solutions, the system has:
⇒ 6 real solution terms.

Step 2: Nature of the Solutions

Because the damping is small (β = 0.04), the system is underdamped. Therefore, each mode has the form:
e−γtcos(ωt) and e−γtsin(ωt)
These are oscillatory terms that decay exponentially.

The decay rate γ is given by:
γ = β / (2m₃)

Assuming m₃ = 1 unit mass:
γ = 0.04 / 2 = 0.02

Step 3: Estimating Decay to 5%

We use the condition:
e−γt = 0.05 (i.e., 5% of the original amplitude)

Taking natural logarithm:
t = ln(0.05) / (−γ) = ln(0.05) / (−0.02)
t ≈ −2.996 / −0.02 = 149.8 ≈ 150 time units

Conclusion:

• Six real terms: 3 damped vibrational modes × 2 solutions per mode
• Time to decay to 5% amplitude: about 150 time units