2 点吊り物理振子の振動解析: 弾性紐の効果 木ノ内 智貴*1 舟田 敏雄*1 桜井 賢人*1 大庭 勝久*1 青木 悠祐*1 宮内 太積*2 望月 孔二*3 An Analysis of Mode Coupling in Three Modes of Bifilar Suspension Phys

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点吊り物理振子の振動解析: 弾性紐の効果 木ノ内 智貴* 舟田 敏雄* 桜井 賢人* 大庭 勝久* 青木 悠祐* 宮内 太積* 望月 孔二*3 An Analysis of Mode Coupling in Three Modes of Bifilar Suspension Physical-Pendulum: Effects of Elastic Strings Toshiki KINOUCHI* Toshio FUNADA* Kento SAKURAI* Katsuhisa OHBA* Yusuke AOKI* Tatsumi MIYAUCHI* and Kouji MOCHIZUKI*3 Abstract: A bifilar suspension pendulum with a uniform density bar may swing in two vertical planes as mode and or make torsional oscillation (mode 3 about a vertical axis. The period and pattern of oscillation depend upon the ratios of the string length and of the distance of the supporting points to the distance between two end points of the strings attached to an upper wall. When a cuboid (wooden chip is used in place of the bar, another effect due to elastic strings is found in experiments, so a new model spring-pendulum is proposed here to clarify mechanical effects in numerical simulation. Keywords: Bifilar Suspension Pendulum, Elasticity of Strings, Nonlinear Coupling of Oscillation Modes mode (ブランコ mode があり 鉛直軸回りの円木の 捩れ振動は mode 3 (捩れ振動 mode に分類される はじめに 3.0 対称 点吊り振子 (並進振子 を免震床に応用するための.5 検討[] に刺激されて 点吊り振子の静止平衡状態周りの.0 振子運動について 運動方程式を導出して初期値問題を.5 数値解析し 実験した (Fig., Table, Fig.[] [7] その.0 振子運動には 紐と円木の成す鉛直面内で揺れる mode (遊動円木 mode とそれとは垂直な鉛直面内で揺れる 0.5 0.0 0.0 0. 0. 0.3 0.4 0.5 0.6 Fig. Period T versus c for mode (solid line, (dashed, 3 (dotted. L = 0.443 (below, L = 0.54 (middle, and L = 0.648 (above. Marks are (c, Te for experimental Te. Cross points show coupling/resonance between the modes. 棒を木片 (直方体 に換えた実験装置 (Fig.3 で測定さ れた周期は Fig.4 の点で示され 別報[8], [9] で求められた 理論曲線 (Fig.4 の曲線 とよく一致している Fig. Experimental apparatus for the bifilar suspension pendulum with a bar of m = 0.6659 kg and length b = 0.899 m in a > c. When a = c, the bar swings horizontally as a part of parallelogram. Table Specification of the apparatus (Fig., a bar of length b and m = 0.6659 kg suspended by two ropes of length L at two points which are distant by a from the center. a [m] 0.6 b [m] 0.4495 c [m] 0.0875, 0.65, 0.375, 0.35, 0.3875, 0.465, 0.5375 L [m] 0.443, 0.54, 0.648 Fig.3 Experimental apparatus for the bifilar suspension pendulum with a cuboid of size La, ba and da. The attachment of the cuboid to the strings may change its * 電子制御工学科: Department of Digital Engineering. * 機械工学科: Department of Mechanical Engineering. *3 電気電子工学科: Department of Electrical & Electronics Engineering. moment of inertia, and then the elasticity of strings gives rise to various oscillations, as a new spring-pendulum.

.0.5.0 0.5 0.0 0.0 0. 0.4 0.6 0.8.0. Fig.4 Experimental period versus c (half distance of the attached points at the upper wall. Mode is red curve, node is green one and mode 3 is blue one, for L =, a = 0.044, b a = 0.088, L a = 0.09 and d a = 0.03. moment mode (Fig.4 (Fig. mode c / mode,, 3 (c = a (a c [0] model [] [5] model model [3] ( model 3 mode x y z (x, y, z (I C C (x 0, y 0, z 0 c O (x 0 + c, y 0, z 0, C (x 0 + c, y 0, z 0 ( G (x g, y g, z g O a A (x, y, z A (x, y, z ( A A = a, A G = a, GA = a, A C = L, A C = L, OG = r g O (r, θ y, θ z z-x θ yg y-z θ xg G (x g, y g, z g : x g = r g sin(θ yg + x 0 + c, y g = r g cos(θ yg sin(θ xg + y 0, (. z g = r g cos(θ yg cos(θ xg + z 0 G z-x θ y x-y θ z A (x, y, z A (x, y, z (.4 : x = x g a cos(θ y cos(θ z, y = y g a cos(θ y sin(θ z, (. z = z g + a sin(θ y, x = x g + a cos(θ y cos(θ z, y = y g + a cos(θ y sin(θ z, (.3 z = z g a sin(θ y, f 00 = (x x + (y y + (z z 4a = 0, f 0 = (x x 0 + (y y 0 + (z z 0 L = 0, f 0 = (x x 0 c + (y y 0 + (z z 0 L = 0, L = ( a + c a c cos(θ y cos(θ z + rg +r g (a cos(θ xg cos(θ yg sin(θ y +(c a cos(θ y cos(θ z sin(θ yg a cos(θ y cos(θ yg sin(θ xg sin(θ z, L x = ( a + c a c cos(θ y cos(θ z + rg (.4 r g (a cos(θ xg cos(θ yg sin(θ y +(c a cos(θ y cos(θ z sin(θ yg a cos(θ y cos(θ yg sin(θ xg sin(θ z, f 03 = z z = sec(θ z tan(θ y, x x f 04 = y y = tan(θ z, x x f 05 = z z = csc(θ z tan(θ y y y f 0 = 0 L = L f 0 = 0 L x = L { L L (r g, θ yg, θ xg, θ y, θ z, L x L x (r g, θ yg, θ xg, θ y, θ z (.5 L (i,j,k,l,m = i+j+k+l+m L / r i g θ j yg θ k xg θ l y θ m z ( energy potential energy energy energy Lagrange L : L = m (ẋ g + ẏ g + ż g + J y θ y + J z θ z cos (θ y + m gz g k (L L 0x k (L x L 0x

= m ṙ g + m ( r g θ yg + cos (θ yg θ xg + J y θ y + J z θ z cos (θ m y cos(θ yg rg sin(θ yg θ xg + m ṙ g [ẋ 0 sin(θ yg +cos(θ yg (ẏ 0 sin(θ xg +ż 0 cos(θ xg ] + m r g ṙ g θyg + m cos(θ yg r g ẍ 0 [ m r g sin(θ xg sin(θ yg ÿ 0 m cos(θ xg r g sin(θ yg z 0 + m r g θxg cos(θ yg (ẏ 0 cos(θ xg ż 0 sin(θ xg θ ] + m r θ g yg + gm cos(θ xg r g sin(θ yg yg sin(θ yg (ẏ 0 sin(θ xg + ż 0 cos(θ xg + m + k (L L 0x L (0,,0,0,0 (ẋ 0 + ẏ0 + ż0 + m g(r g cos(θ xg cos(θ yg + z 0 + k (L x L 0x L (0,,0,0,0 x = 0, (.3 k (L L 0x k (L x L 0x (.6 m cos(θ yg r θ g xg + m r g ṙ g θxg cos(θ yg 5 : r g m r θ g xg θyg cos(θ yg sin(θ yg + m r g ÿ 0 cos(θ xg cos(θ yg m r g z 0 cos(θ yg sin(θ xg + m g cos(θ yg r g sin(θ xg m ( r g cos(θ yg r g θ xg r g θ yg + sin(θ yg ẍ 0 + cos(θ yg sin(θ xg ÿ 0 + cos(θ xg cos(θ yg z 0 m g cos(θ xg cos(θ yg + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (.7 θ yg m (cos(θ yg r g sin(θ yg θ xg + r g ṙ g θyg + cos(θ yg r g ẍ 0 r g sin(θ xg sin(θ yg ÿ 0 cos(θ xg r g sin(θ yg z 0 + r g θ yg + m g cos(θ xg r g sin(θ yg + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (0,,0,0,0 x = 0, (.8 θ xg m ( r g cos(θ yg θxg + r g ṙ g θxg cos(θ yg r g θ xg θyg cos(θ yg sin(θ yg +r g ÿ 0 cos(θ xg cos(θ yg r g z 0 cos(θ yg sin(θ xg + m r g g cos(θ yg sin(θ xg + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0, (.9 θ z θ y J z θz + k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0, (.0 J y θy + k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (. x 0 = 0, y 0 = 0, z 0 = 0 r g, θ yg, θ xg, θ z, θ y : m r g m cos(θ yg r g θ xg m r g θ yg + +m sin(θ yg ẍ 0 + m cos(θ yg sin(θ xg ÿ 0 + m cos(θ xg cos(θ yg z 0 m g cos(θ xg cos(θ yg + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (. + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0, (.4 J z θz + k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0, (.5 J y θy + k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (.6 θ z Lagrange (.6 θ z = 0 4 : m r g m r g θ xg cos(θ yg m r g θ yg + m sin(θ yg ẍ 0 + m cos(θ yg sin(θ xg ÿ 0 + m cos(θ xg cos(θ yg z 0 gm cos(θ xg cos(θ yg + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (.7 m r g θ yg + m r g θ xg cos(θ yg sin(θ yg + m r g ṙ g θyg + m r g ẍ 0 cos(θ yg m r g ÿ 0 sin(θ xg sin(θ yg m r g z 0 cos(θ xg sin(θ yg + m r g g cos(θ xg sin(θ yg + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (0,,0,0,0 x = 0, (.8 m r g θ xg cos(θ yg + m r g ṙ g θxg cos(θ yg m r g θ xg θyg cos(θ yg sin(θ yg + m r g ÿ 0 cos(θ xg cos(θ yg m r g z 0 cos(θ yg sin(θ xg + m g cos(θ yg r g sin(θ xg + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0, (.9 J y θy + k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (.0 θ yg 4 : m r g m r g θ xg + m ÿ 0 sin(θ xg + m z 0 cos(θ xg m r g g cos(θ xg + k (L L 0x L (,0,0,0,0

+ k (L x L 0x L (,0,0,0,0 x = 0, (. m r g θ xg + m r g ṙ g θxg + m r g ÿ 0 cos(θ xg m r g z 0 sin(θ xg + m r g g sin(θ xg + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0, (. J z θz + k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0, (.3 J y θy + k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (.4 θ xg 4 : gm cos(θ yg m r g θ yg + m r g + m sin(θ yg ẍ 0 + m cos(θ yg z 0 + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (.5 m r g θ yg + m r g ṙ g θyg + m cos(θ yg r g ẍ 0 m r g sin(θ yg z 0 + gm r g sin(θ yg + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (0,,0,0,0 x = 0, (.6 J z θz + k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0, (.7 J y θy + k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (.8 θ y 4 : m r g + m r g θ yg m r g θ xg cos(θ yg + m sin(θ yg ẍ 0 + m ÿ 0 cos(θ yg sin(θ xg + m z 0 cos(θ xg cos(θ yg m g cos(θ xg cos(θ yg + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (.9 m r g θ yg + m r g θ xg cos(θ yg sin(θ yg + m r g ṙ g θyg + m r g ẍ 0 cos(θ yg m r g ÿ 0 sin(θ xg sin(θ yg m r g z 0 cos(θ xg sin(θ yg + m r g g cos(θ xg sin(θ yg + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (0,,0,0,0 x = 0, (.30 m r g θ xg cos(θ yg + m r g ṙ g θxg cos(θ yg m r g θ xg θyg cos(θ yg sin(θ yg + m r g ÿ 0 m cos(θ xg cos(θ yg cos(θ yg r g z 0 sin(θ xg + m r g g cos(θ yg sin(θ xg + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0, (.3 J z θz + k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0 (.3 Lagrange 3 mode Lagrange (.6 mode. 3. Mode ( mode mode θ xg = 0, θ z = 0, L [r g, θ yg, 0, θ y, 0] (.6-(. Lagrange 3 (r g, θ yg, θ y : L = m ( ṙg + rg θ yg + ṙ g ẋ 0 sin(θ yg +ṙ g ż 0 cos(θ yg + ẋ 0 + ẏ0 + ż0 +r g θyg (ẋ 0 cos(θ yg ż 0 sin(θ yg + J y θ y + m g(cos(θ yg r g + z 0 k (L L 0x k (L x L 0x, (3. ( m r g r g θ yg + sin(θ yg ẍ 0 + cos(θ yg z 0 gm cos(θ yg + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (3. m (r g θ yg + r g ṙ g θyg + r g ẍ 0 cos(θ yg r g z 0 sin(θ yg + m gr g sin(θ yg + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (0,,0,0,0 x = 0, (3.3 J y θy + k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (3.4 L, L x : L = ( a + c a c cos(θ y + r g +r g (a cos(θ yg sin(θ y +(c a cos(θ y sin(θ yg, (3.5 L x = ( a + c a c cos(θ y + r g r g (a cos(θ yg sin(θ y +(c a cos(θ y sin(θ yg (3.6 θ yg = 0, θ y = 0, r g = r g0 : gm + m z 0 + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (3.7 m r g0 ẍ 0 + k (L L 0x L (0,,0,0,0 + k (L x L 0x L (0,,0,0,0 x = 0, (3.8 k (L L 0x L (0,0,0,,0 + k (L x L 0x L (0,0,0,,0 x = 0 (3.9 : L 0 = L x0 = a a c + c + r g0 L 00 (3.0

(3., (3.3 : gm + m z 0 = 0, m r g0 ẍ 0 = 0 (3. (3.4 3. Mode ( mode mode θ z = 0, θ yg = 0, θ y = 0 (.6-(. Lagrange (r g, θ xg : L = m ( ṙg + rg θ xg + ṙ g ẏ 0 sin(θ xg + ṙ g ż 0 cos(θ xg + ẋ 0 + ẏ0 + ż0 +r g θxg (ẏ 0 cos(θ xg ż 0 sin(θ xg + m g(cos(θ xg r g + z 0 k (L L 0x k (L x L 0x, (3. ( m r g θ xg + r g + sin(θ xg ÿ 0 + cos(θ xg z 0 gm cos(θ xg + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (3.3 m (r g θ xg + r g ṙ g θxg + r g ÿ 0 cos(θ xg r g z 0 sin(θ xg + m r g g sin(θ xg + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0 (3.4 L, L x : L = L x = a a c + c + r g (3.5 θ xg = 0, r g = r g0 : gm + m z 0 + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (3.6 m r g0 ÿ 0 + k (L L 0x L (0,0,,0,0 + k (L x L 0x L (0,0,,0,0 x = 0 (3.7 : L 0 = L x0 = a a c + c + r g0 (3.8 (3.3, (3.4 : gm + m z 0 = 0, m r g0 ÿ 0 = 0 (3.9 3.3 Mode 3 ( mode mode 3 θ xg = 0, θ yg = 0, θ y = 0 (.6-(. Lagrange (r g, θ z : L 3 = m (ṙ g + ṙ g ż 0 + ẋ 0 + ẏ 0 + ż 0 + J z θ z + m g(r g + z 0 k (L L 0x k (L x L 0x, (3.0 m ( r g + z 0 m g + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (3. J z θz + k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0 (3. L, L x : L = L x = a + c a c cos(θ z + rg (3.3 θ z = 0, r g = r g0 : m z 0 m g + k (L L 0x L (,0,0,0,0 + k (L x L 0x L (,0,0,0,0 x = 0, (3.4 k (L L 0x L (0,0,0,0, + k (L x L 0x L (0,0,0,0, x = 0 (3.5 : L 0 = L x0 = a a c + c + r g0 (3.6 (3., (3. : gm + m z 0 = 0, (3.7 4 [] 3 mode moment [] [7] 0 [8], [9] model 3 mode mode, [] [5] (.7- (. 3 mode (Fig. mode mode mode, 3 mode 3,

mode,, 3 Lagrange [6] mode [] : 38 :. B-, II,, 000 (000, pp.635-636(. [] : 44 (00, pp.55-60. [3] : 44 (00, pp.83-88. [4] : 3 mode 45 (0, pp.63-68. [5] : 4 0 3 3 ( 9 0:50 :4 909 ( 3 3 CD/ROM pp.68-69. [6] : 60 ( 60 8 (A07 3 4 ( OS5 OS5-0:45 :00 8 60 No. 3- ( 0, pp.403-404. [7] : mode 60 3 9 ( 4 9:30-9:45 OS-4 OS-0 OS-0.pdf [8] : ( 46 (0, in press. [9] : ( 46 (0, in press. [0] D. Zhou, T. Ji: Dynamic characteristics of a generalised suspension system International Journal of Mechanical Sciences 50 (008, pp.30-4. [] : 0 0 9 ( 5 ( [OS] G00 9 3 ( 9:00-0:00 W4 G00033 0 DVDROM G00033.pdf. [] : 3 mode ( 46 (0, in press. [3] : 3 mode ( 46 (0, in press. [4] : 3 mode (3 46 (0, in press. [5] : 46 (0, in press. [6] : (3: 46 (0, in press.

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