# Calculation of Frequencies of Truncated Pyramids

Calculation of Frequencies of Truncated Pyramids STRUCTURES Calculation of Frequencies of. For the square pyramid, we have Truncated Pyramids Development of the Theories for the Longitudinal, Flexural and and, as before, for the cone Torsional Vibrations of a Truncated Cone 3. Torsional Vibrations By H. D. Conway, M.A., B.Sc., Ph.D. To calculate the frequency of torsional vibra­ tions, the differential equation to be solved is For a complete pyramid, a=0 and p =nπ N a previous paper by the author ('The Calcu­ where n is any positive integer. The fundamental lation of Frequencies of Vibration of a Trun­ frequency of longitudinal vibrations is, as before, cated Cone', AIRCRAFT ENGINEERING, Vol. where ф = angle of twist. given by the equation XVIII, July 1946, pp. 235-236) expressions for C = torsional stiffness. the fundamental frequencies of longitudinal, J = polar second moment of area at flexual and torsional vibrations were obtained. a distance x from the free end. The methods used have now been extended to Since the normal cross-sections of the pyramids This is twice the value obtained for a uniform include the vibrations of truncated pyramids are regular figures, we write cylinder of cross-section equal to that of the encastré at one end and free at the other, and the pyramid at the encastré end. C = yN(a+mx)4 results obtained are given in this paper. Special J = δ(a+mx)4p (16) attention is given to the pyramids having equi­ 2. Flexural Vibrations lateral triangle and square cross-sections. where y and δ are constants dependent only upon The differential equation to be solved for the the particular cross-section and N is the shearing The material of which the pyramids are made case of flexural vibrations is modulus of the material. For the equilateral tri­ is assumed to be homogeneous and isotropic. The taper is also assumed to be small. FIG. 1 shows angle, square and circle, y is and π/2 the equilateral triangle and square pyramids, the respectively and § is and π/2 respec­ vibrations considered being tively. where I is the second moment of area of the cross- (1) Longitudinal, in the direction ox; Writing ф=ф sin cot and inserting ξ=x/1, we section at distance x from the free end. (2) Flexural, normal to the plane xoy; write equation (15) in the form Since, as before, every normal cross-section of (3) Torsional, about the axis ox. a pyramid is a regular figure, we may write 1. Longitudinal Vibrations A=a(a +mξ)2,I=B(a +mξ)4 (8) 1 1 The differential equation to be solved in con­ with the same notation as before and with a and ft where nexion with the calculation of the frequency of as constants. For harmonic motion, z=z sin cot This is the same equation as derived for the longitudinal vibrations is and by substitution in equation (7), we have truncated cone. The frequency of torsional vibra­ tions can therefore be written as before where p — density of material. A = cross-sectional area at distance x where from the free end. u = extension at distance x from the This is the equation previously obtained for free end. For the complete pyramid a=0 and we have the truncated cone and the solution follows as E = Young's modulus. tan p =p (19) 3 3 before. The frequency equation for the complete g = acceleration due to gravity. The first few roots of this equation are For harmonic motion we may write u—u sin wt pyramid obtained is written in the form of the p =4•493, 7•725, 10•904 (20) where u is a function of x only and sin cot a func­ summation tion of t only. The fundamental frequency of torsional vibra­ Introducing the non-dimensional abscissa tions of a triangular pyramid is therefore we may write the cross-sectional area The first few roots of this equation are given by A—f(ξ). Substituting in equation (1) we obtain For the square pyramid, we have where For the equilateral triangle, square and circle, and, for the cone a=√3 , 4 and Π respectively and Now, since every normal cross-section of the 4/3 and π/4 respectively. Using these values, we pyramid is a regular figure—e.g., square or equi­ have for the frequency of flexural vibrations of a lateral triangle—we may write complete triangular pyramid A=f(ξ)=a(a +mξ) 2 (3) where and a is a constant de­ pendent only upon the shape of the cross-section. Substituting equation (3) in equation (2), we may write This is precisely the same equation as obtained for the special case of the truncated cone. It therefore follows, as in the case of the longi­ tudinal vibrations of a uniform cylinder, that the frequency is independent of the shape of the cross-section. Proceeding in exactly the same manner as for the truncated cone, the frequency of longitudinal vibrations is given by the equation 148 Aircraft Engineering http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

# Calculation of Frequencies of Truncated Pyramids

, Volume 20 (5): 1 – May 1, 1948
1 page      /lp/emerald-publishing/calculation-of-frequencies-of-truncated-pyramids-pm8vOJZeWE
Publisher
Emerald Publishing
ISSN
0002-2667
DOI
10.1108/eb031635
Publisher site
See Article on Publisher Site

### Abstract

STRUCTURES Calculation of Frequencies of. For the square pyramid, we have Truncated Pyramids Development of the Theories for the Longitudinal, Flexural and and, as before, for the cone Torsional Vibrations of a Truncated Cone 3. Torsional Vibrations By H. D. Conway, M.A., B.Sc., Ph.D. To calculate the frequency of torsional vibra­ tions, the differential equation to be solved is For a complete pyramid, a=0 and p =nπ N a previous paper by the author ('The Calcu­ where n is any positive integer. The fundamental lation of Frequencies of Vibration of a Trun­ frequency of longitudinal vibrations is, as before, cated Cone', AIRCRAFT ENGINEERING, Vol. where ф = angle of twist. given by the equation XVIII, July 1946, pp. 235-236) expressions for C = torsional stiffness. the fundamental frequencies of longitudinal, J = polar second moment of area at flexual and torsional vibrations were obtained. a distance x from the free end. The methods used have now been extended to Since the normal cross-sections of the pyramids This is twice the value obtained for a uniform include the vibrations of truncated pyramids are regular figures, we write cylinder of cross-section equal to that of the encastré at one end and free at the other, and the pyramid at the encastré end. C = yN(a+mx)4 results obtained are given in this paper. Special J = δ(a+mx)4p (16) attention is given to the pyramids having equi­ 2. Flexural Vibrations lateral triangle and square cross-sections. where y and δ are constants dependent only upon The differential equation to be solved for the the particular cross-section and N is the shearing The material of which the pyramids are made case of flexural vibrations is modulus of the material. For the equilateral tri­ is assumed to be homogeneous and isotropic. The taper is also assumed to be small. FIG. 1 shows angle, square and circle, y is and π/2 the equilateral triangle and square pyramids, the respectively and § is and π/2 respec­ vibrations considered being tively. where I is the second moment of area of the cross- (1) Longitudinal, in the direction ox; Writing ф=ф sin cot and inserting ξ=x/1, we section at distance x from the free end. (2) Flexural, normal to the plane xoy; write equation (15) in the form Since, as before, every normal cross-section of (3) Torsional, about the axis ox. a pyramid is a regular figure, we may write 1. Longitudinal Vibrations A=a(a +mξ)2,I=B(a +mξ)4 (8) 1 1 The differential equation to be solved in con­ with the same notation as before and with a and ft where nexion with the calculation of the frequency of as constants. For harmonic motion, z=z sin cot This is the same equation as derived for the longitudinal vibrations is and by substitution in equation (7), we have truncated cone. The frequency of torsional vibra­ tions can therefore be written as before where p — density of material. A = cross-sectional area at distance x where from the free end. u = extension at distance x from the This is the equation previously obtained for free end. For the complete pyramid a=0 and we have the truncated cone and the solution follows as E = Young's modulus. tan p =p (19) 3 3 before. The frequency equation for the complete g = acceleration due to gravity. The first few roots of this equation are For harmonic motion we may write u—u sin wt pyramid obtained is written in the form of the p =4•493, 7•725, 10•904 (20) where u is a function of x only and sin cot a func­ summation tion of t only. The fundamental frequency of torsional vibra­ Introducing the non-dimensional abscissa tions of a triangular pyramid is therefore we may write the cross-sectional area The first few roots of this equation are given by A—f(ξ). Substituting in equation (1) we obtain For the square pyramid, we have where For the equilateral triangle, square and circle, and, for the cone a=√3 , 4 and Π respectively and Now, since every normal cross-section of the 4/3 and π/4 respectively. Using these values, we pyramid is a regular figure—e.g., square or equi­ have for the frequency of flexural vibrations of a lateral triangle—we may write complete triangular pyramid A=f(ξ)=a(a +mξ) 2 (3) where and a is a constant de­ pendent only upon the shape of the cross-section. Substituting equation (3) in equation (2), we may write This is precisely the same equation as obtained for the special case of the truncated cone. It therefore follows, as in the case of the longi­ tudinal vibrations of a uniform cylinder, that the frequency is independent of the shape of the cross-section. Proceeding in exactly the same manner as for the truncated cone, the frequency of longitudinal vibrations is given by the equation 148 Aircraft Engineering

### Journal

Aircraft Engineering and Aerospace TechnologyEmerald Publishing

Published: May 1, 1948