Adisorn Owatsiriwong, D.Eng.

**Introduction**

In many engineering applications, mechanical systems are subjected to dynamic loads that can induce vibrations. These vibrations, if not properly controlled, can lead to reduced performance, increased wear, and even structural failure. One effective way to mitigate vibrations is by using a **Tuned Mass Damper (TMD)**, a device that is designed to absorb and dissipate vibrational energy from the primary structure.

In this blog post, we explore the concept of a 2-degree-of-freedom (2DOF) mass-spring system, where the primary mass experiences unwanted vibrations due to an external harmonic force. To suppress these vibrations, we introduce a secondary mass connected to the primary mass via a spring, forming a coupled system. The goal is to optimize the properties of the secondary mass and spring (specifically their mass and stiffness) to minimize the vibration amplitude of the primary mass.

**The System Setup**

Consider a simple mechanical system consisting of two masses connected by springs, as illustrated in the figure below.

**Figure 1**: **Schematic of a 2DOF Mass-Spring System with a Tuned Mass Damper**

****Understanding the Problem****

The primary mass m1 is subjected to an external harmonic force F(t), which causes it to oscillate. Without any damping or additional control, this oscillation can reach significant amplitudes, potentially leading to undesirable consequences.

To counteract this, a secondary mass m2 is introduced, connected to m1 via a spring with stiffness k2. The secondary system (comprising m2 and k2 is tuned to a specific frequency that allows it to effectively absorb vibrational energy from the primary mass. The goal is to adjust the mass ratio m_2/m_1 and the stiffness ratio k_2/k_1 to achieve the best vibration suppression for m1.

****Key Considerations****

- **Natural Frequencies**: The effectiveness of the TMD depends on how well it is tuned to the natural frequency of the primary system or the frequency of the external force.

- **Mass and Stiffness Ratios**: The ratios m2/m1 and k2/k1 are critical parameters that determine the vibration suppression capability of the TMD.

- **Damping**: While not the focus of this study, damping elements can further enhance the performance of the TMD by dissipating energy more effectively.

*****Equilibrium Equations*****

The above equation can be solved by ode45 function (based on Runge-Gutta integration) in MATLAB or direct integration methods.

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