Final answer:
The student's question relates to solving a second-order linear constant coefficient differential equation that models underdamped oscillations in a damped mass-spring system, considering the condition where the natural frequency Ω₀ is greater than B/(2M). The general solution includes exponential decay combined with oscillatory motion, characterized by amplitude decay over time.
Step-by-step explanation:
The student is dealing with the study of oscillations in a model for the Millennium Bridge in London, which can be expressed by a second-order linear constant coefficient differential equation with forcing. This equation is a representation of the equation of motion for a damped mass-spring system. The general solution for the complementary homogeneous problem is sought in the case of underdamped oscillations, where the natural frequency of undamped oscillation, denoted as Ω₀ (Omega-zero), is greater than the damping ratio divided by twice the mass, expressed as B/(2M).
To solve the complementary homogeneous problem, we set the forcing term to zero. The resulting differential equation is MX′′ + BX′ + KX = 0. Using Ω₀ = √(K/M), and given that Ω₀ > B/(2M), the system is underdamped, which means the solution will involve oscillatory behavior with an exponential decay. The general solution in such cases is typically of the form X(t) = e^(-Bt/2M) (A cos(Ωt) + B sin(Ωt)), where Ω is the damped natural frequency given by Ω = √(Ω₀² - (B/(2M))²).
The natural frequency Ω₀, the damping coefficient B, and the spring constant K are all crucial to the behavior of the system. The general solution and the gain function are important concepts that help understand the system's response to external forces.