Efekt rastezanja kod ugađanja klavira

undergraduate thesis

• Title (english): The stretching effect in piano tuning
• Author: Ružica Radoš
• Mentor: Neven Alujević (mentor)
• Granter: University of Zagreb; Faculty of Mechanical Engineering and Naval Architecture; Zagreb
• Defense date and country: 2023-03-01, Croatia
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Basic Technical Sciences; Technical Mechanics (Mechanics of Rigid and Deformable Bodies)

Abstract

U ovom radu objašnjeni su problemi pri ugađanju klavira koji nastaju zbog nezanemarive savojne krutosti klavirskih žica. Klavirski tehničar kontrolira tenziju žica tako da omjeri među frekvencijama budu zadovoljavajući. Kod konsonantnih intervala težit će se izostanku akustičkih udara, percipiranih kao periodičke promjene u glasnoći koje nastaju interferencijom dvaju tonova s malom razlikom u frekvencijama. Matematički model napete idealne žice (bez savojne krutosti) predviđa drugu prirodnu frekvenciju dvostruko višu od prve, treću trostruko višu od prve i tako dalje. Međutim, praksa pokazuje da klavirske žice imaju drugu vlastitu frekvenciju nešto višu od dvostruke prve. Zbog ovoga kod ugađanja klavira nastaje tzv. efekt rastezanja muzičke skale. Klavirski tehničar podešava napetost žice tako da se druga prirodna frekvencija osnovnog tona podudara s prvom prirodnom frekvencijom oktavu višeg tona nastojeći eliminirati akustičke udare. U ovom radu objašnjeni su uzroci efekta rastezanja koristeći matematički model za proračun prirodnih frekvencija grede opterećene konstantnom aksijalnom silom. Uspoređeni su rezultati dobiveni modelom grede opterećene konstantnom aksijalnom silom s modelom nategnute žice bez savojne krutosti, i ti rezultati uspoređeni s mjerenim fundamentalnim prirodnim frekvencijama tonova stvarnog klavira. Na primjeru slobodno oslonjene grede pokazano je da u takvom slučaju druga prirodna frekvencija nije dvostruka vrijednost prve, nego nešto viša. Pokazano je da s porastom natezne sile više prirodne frekvencije zaista teže prema vlastitim frekvencijama nategnute žice, međutim kod konačne natezne sile to nije slučaj te zbog toga nije moguće jednostavno izbjeći efekt rastezanja.

Abstract (english)

This thesis explains problems that occur in piano tuning process due to the non-negligible bending stiffness of piano strings. Normally, a piano technician adjusts tensions of the strings such that the ratios between their fundamental frequencies follow a certain harmonic pattern. In consonant intervals it is aspired towards the absence of acoustic beats, which are perceived as slow periodic variations in volume that occur due to the interference of two tones with slightly different frequencies. Typical mathematical models of an ideal string (without bending stiffness) under axial tension predict the second natural frequency to be twice the first, the third three times the first, and so on. However, piano wires under realistic conditions have the second natural frequency slightly higher than twice the first natural frequency. As a result, the musical scale during the piano tuning process becomes stretched. This is because a piano technician adjusts the tension of the string so that the second natural frequency of its octave lower counterpart matches the first natural frequency of the string currently tuned, trying to eliminate acoustic beats. In this thesis the causes of the stretching effect are explained using a mathematical model to calculate natural frequencies of a beam under constant axial tension. The results obtained using the model of a tensioned beam are compared to the results obtained using the model of an ideal string without bending stiffness, and those results are compared to measured fundamental natural frequencies of tones on a real piano. In the example of a pinned-pinned beam it is shown that the second natural frequency is not equal to twice the fundamental natural frequency, but slightly higher. It is shown that with the increase of the tensile force, the higher natural frequencies of a tensioned beam tend to the natural frequencies of a tensioned string. However, with a finite axial load that is not the case so that the stretching effect cannot be easily avoided.

• Keywords: Vibroakustika muzičkih instrumenata; Ugađanje klavira; Efekt rastezanja tonske ljestvice; Vibracije nategnute žice; Vibracije Euler-Bernoullijeve grede; Vibracije nategnute Euler-Bernoullijeve grede
• Keywords (english): Vibroacoustics of musical instruments; Piano tuning; Stretching effect; Vibrations of tensioned strings; Vibrations of Euler-Bernoulli beams; Vibrations of tensioned Euler-Bernoulli beams
• Language: croatian
• URN:NBN: urn:nbn:hr:235:103050
• Project: Number: IP-2019-04-5402; Title: Dinamika Aktivnih i Rotirajućih KONstrukcija; Title: Dynamics of Active and Rotating Structures; Acronym: DARS; Leader: Neven Alujević; Jurisdiction: Croatia; Funder: HRZZ; Funding stream: IP
• Study programme: Title: Mechanical Engineering; specializations in: Design, Process and Energy Engineering, Production Engineering, Engineering Modelling and Computer Simulation, Marine Engineering, Industrial Engineering and Management, Materials Engineering; Course: Design; Study programme type: university; Study level: undergraduate; Academic / professional title: sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka strojarstva (sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka strojarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2023-02-25 03:19:13