Determine the general free vibration response of a two-mass three-spring threedamper system where \(m_{1}=m_{2}=m, k_{1}=k_{2}=k_{3}=k, c_{1}=c_{2}=c_{3}=c\) and \(c^{2} / \mathrm{km}=\) \(0.04\).

Understanding the dynamics of a two-mass three-spring system is fundamental for students diving into the realm of mechanical vibrations. This type of system involves two masses connected by springs in such a way that they can oscillate independently and also influence each other's vibrations.

The two-mass three-spring system is representative of many physical and engineering applications. To visualize, imagine a simple setup where two weights (masses) are tied to fixed points using three individual springs. The masses are free to move along a line, and the motion of each mass affects the other due to the connecting spring.

When studying the free vibration of such a system, we are interested in understanding what happens when the masses are disturbed from their equilibrium position and then allowed to vibrate without further external forces applied to them. This scenario is a cornerstone example for many complex vibration problems and provides insights into how systems respond dynamically to disturbances.

The equations of motion for a mechanical system are the foundation for analyzing its behavior over time. They are derived using Newton's second law of motion, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration \( F = ma \).

In our case, with a two-mass three-spring system, we must establish two separate equations that correspond to each mass and its interaction with the connected springs and dampers. These equations enable us to model the movement of the masses and predict how they will react following a disturbance. A key point here is to impose equilibrium conditions at the beginning, simplifying the equations by eliminating terms that relate to a stationary state.

Once we have created these equations, they can reveal the system's dynamics under free vibrations and allow us to explore the consequence of altered system parameters such as mass, spring stiffness, or damping.

The characteristic equation is a pivotal concept in the analysis of differential equations, especially in the study of mechanical vibrations. In essence, the characteristic equation represents the roots that are associated with the solution of the differential equation governing the system's vibration.

When we have our system of equations expressing the accelerations and velocities of the masses in terms of displacements and the damping coefficient, we are in a position to form the characteristic equation. Solving this equation will yield the natural frequencies and damping ratios of the system, which are crucial for understanding the vibrational modes.

The roots of the characteristic equation explain how the system's response will decay or grow over time and are indispensable for determining the system’s free vibration response. Finding these roots can often be the most mathematically challenging part of analyzing a system.

Differential equations are the language of change and dynamics in physics and engineering. They are indispensable in mechanical vibrations for describing how systems evolve in time. When we apply differential equations to a two-mass three-spring system, we create a relationship between the current state of the system and its rate of change.

The motion of each mass in our system can be expressed through second-order differential equations, which account for the relationship between displacement, velocity, acceleration, damping, and stiffness. These equations encapsulate the complexities of the system’s motion and form the basis for predicting how the masses will respond after being disturbed from their rest position.

Mastering differential equations allows students to move from a simplistic understanding of systems at rest to a dynamic analysis of systems in motion, making it a cornerstone of vibration analysis.

Damping and stiffness are two fundamental properties that influence how a system responds to vibrations. Damping refers to the force that opposes the motion, dissipating energy and causing the vibration amplitude to decrease over time. Meanwhile, stiffness is a measure of a system's resistance to deformation; the stiffer the system, the higher the force required to displace it.

In the scenario of a two-mass three-spring system, both properties play crucial roles. Here, the springs provide the stiffness, and the dampers provide the damping. The balance between these properties determines whether the system will oscillate indefinitely (undamped), gradually come to rest (over-damped or critically damped), or oscillate with reducing amplitude (under-damped).

Understanding the interplay between damping and stiffness is essential for predicting the long-term behavior of the system. This knowledge also serves as a stepping stone to more complex analyses such as forced vibrations and resonance phenomena.