FUNDAMENTAL FREQUENCY OF VIBRATION IN MEMBRANES AND PLATES, ANALYTICAL AND NUMERICAL APPROACHES

• Main Author: Cahyono, Agung
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/50756
• Institution: Institut Teknologi Bandung

Musical instruments can produce sound because there are elements that vibrate. The vibrating part of the instrument can vary, depending on the type of musical instrument. The resulting wave vibrations are generally in the form of harmonic waves, in particular a standing wave. The pitch of the resulting tone depends on the frequency, or rather the fundamental frequency of the object. In this thesis we examine the fundamental frequencies of tones produced by the vibrations of strings and rods (1-dimensional problems), as well as membrane and plate vibrations (2-dimensional problems). The analytical fundamental frequency can be obtained through the implementation of variable separation technique, i.e. by solving the corresponding eigenvalue problem. In this case, the fundamental frequency is obtained as a product of the smallest eigenvalue and the wave celerity. Thus, the fundamental frequency depends on the geometric shape of the vibrating object, as well as the parameters which depend on the physical properties of the object material. If the spatial domain is not simple, the analytical solution becomes difficult, but a numerical approach is still possible. In this thesis, the discussion of fundamental frequencies is determined in two ways; analytically and numerically (the finite difference method). The study was carried out for 1-dimensional problems, namely the vibration of string with fixed ends, and the vibration of rod with free ends. As well as for 2-dimensional problems, namely membrane vibration and plate vibration with fixed-end boundaries. The analytical and numerical results show good agreement for all four cases considered. This shows that the finite difference numerical schemes can be used as an alternative way of determining the eigenvalues, which also means the fundamental frequency of the related vibration. Thus, the numerical approach can be an alternative for determining the fundamental frequency vibration of a musical instrument with non-simple geometric shape.