<p dir="ltr">Vibration is ubiquitous in the modern world, making it a topic that cannot be avoided during design, manufacture, and maintenance. Systems, such as civil structures and suspension of cars, are normally designed to stay in the attenuation zone to avoid harsh vibrations. Designing and manufacturing systems with the desired natural frequency distribution is easy. However, it is much harder to maintain the frequency response since materials keep aging as time goes by. To counter the effect of aging and attenuate vibrations, this thesis designed a meta-material that is capable of reprogramming its natural frequency distribution by inserting various masses at different locations. This ability to specifically adjust the system's natural frequency distribution is what we define as "Digital Inertia Programming".</p><p dir="ltr">The model consists of 12 identical unit cells, with each unit cell comprising two types of springs. By determining whether to insert a mass into the unit cell at various locations, the model achieves its programmability to adjust its natural frequency distribution. A "Binary Representation" is used to label the patterns of mass inserted in the model. Each unit cell is represented by a binary bit and a total of 12 bits are used to indicate the presence of mass in each unit cell. In the thesis, we mainly discuss bilaterally symmetrical patterns to avoid unwanted twisting. For the 12 unit cells, we can obtain a total of 128 bilaterally symmetrical patterns, resulting in 896 independent natural frequencies for the model. The number of patterns and independent natural frequencies will increase exponentially with the increase of the number of unit cells in the model.</p><p dir="ltr">An ideal one-dimensional analytical metamaterial model is developed. Lagrange's method is used to determine the system's mass matrix and stiffness matrix directly from the kinetic energy and potential energy equations. The natural frequencies and mode shapes are then calculated from the eigenvalue equation. Based on free response analysis and sensitivity analysis, the model successfully showed great programmability on frequency distribution by varying the insert patterns, as well as changing the value of the variables in the model, such as the weight of the inserts, the weight of the top mass, the stiffness of the unit cell wall spring and the stiffness of the connecting spring. When continuously varying the parameter, the model's natural frequency distribution also changes continuously, giving a possibility to adjust the natural frequency distribution by carefully adjusting the weight of the mass inserted at each location. Lastly, a forced-response analysis is performed, and the amplitude of the model's frequency response is plotted. This provides a straightforward view of the changes in the band gaps and the overall stiffness of the model by altering the patterns with two inserts.</p><p dir="ltr">A two-dimensional model is developed based on the one-dimensional model. The model retains the same 12 unit cells setup as the one-dimensional model. Aiming to ensure stability, the rectangular-shaped unit cell is now configured as a combination of two triangles. Taylor expansion and small angle approximation are used to eliminate nonlinear terms and triangular function terms in the stiffness matrix respectively. The model again shows its programmability by adjusting the variables of the model. Since the results of asymmetrical patterns are bounded by the results of symmetrical patterns, including the asymmetrical patterns increases the model's precision. However, the symmetrical patterns already provide a good representation of the model. The rotational motion is added to the inserts in the model, which further increases the model's complexity. In the model, the mode shapes are characterized by the rotational motion of inserts and the horizontal motion of inserts, which correspond to a zero strain mode of the model. A linear regression model is trained based on 100 bilaterally symmetrical patterns to predict the second lowest natural frequencies of the two-dimensional model for both symmetrical and asymmetrical patterns. The success in the linear regression model indicates the potential for applying machine learning algorithms to the design of meta-materials in the future.</p>