第四章 蝸桿蝸輪之接觸分析
4.4 範例
表 4.1 所示為嚙合之蝸桿蝸輪組之主要設計參數，根據先前所推 導的數學模式與兩嚙合方程式，配合本論文所發展之齒面接觸分析程 式，即可分析蝸桿蝸輪組之接觸情形。本節主要分為兩個部分，第一 部份是理想裝配狀況下的接觸情形，第二部分則是考慮具有裝配誤差 下的接觸情形。茲舉兩個例子，以說明蝸桿蝸輪組在不同齒形參數與 各種裝配條件下，運動誤差、瞬時接觸齒數、平均接觸比和接觸比的 變化情形。
表 4.1 蝸桿蝸輪組之主要設計參數
蝸桿 1 蝸桿 2 蝸輪 1 蝸輪 2 蝸輪 3 法向模數 m(mm/齒) 4.0 4.0 4.0 4.0 4.0
齒數 3 3 29 29 29
法向壓力角 20° 22.5° 20° 20° 20°
與蝸桿之交錯角 γ 30° 45° 60°
範例一：
以 ZK 型蝸桿和 ZN 型蝸輪進行齒輪組之嚙合測試，蝸桿之法向 壓力角分別為 20°與 22.5°，蝸輪之法向壓力角為 20°，假設在理想裝 配條件下，探討蝸桿蝸輪組在 30°、45°和 60°之交錯角的運動誤差、
瞬時接觸齒數、平均接觸比和接觸比。
依據本章所推導並發展之電腦輔助齒輪接觸分析模擬，各蝸桿蝸 輪之搭配組合的齒輪組在理想裝配條件下之接觸情形如表(4.2)至(4.7) 所示。在沒有裝配誤差下，蝸桿蝸輪組之運動誤差(K.E.)皆為零，其 運動誤差及接觸齒數如圖(4.2)至(4.7)所示。本研究係探討非 90°交錯 角之蝸桿蝸輪組嚙合，由分析之結果顯示，蝸桿蝸輪組在 60°交錯角 以內，，其平均接觸比與接觸比隨著交錯角之增大而變大。
範例二：
以 ZK 型蝸桿和 ZN 型蝸輪進行齒輪組之嚙合測試，其中蝸桿之 法向壓力角為 22.5°，而蝸輪之法向壓力角則為 20°。假設蝸桿蝸輪 組具有中心距之裝配誤差及水平和垂直軸向之裝配誤差，來探討蝸桿 蝸輪組在 45°交錯角之下的運動誤差、瞬時接觸齒數、平均接觸比、
與接觸比。
依據本章所發展之電腦輔助齒輪接觸分析模擬，蝸桿蝸輪組之接 觸情形如表(4.8)至表(4.19)所示，由分析之結果顯示，齒輪組在有裝 配誤差狀況下，會有運動誤差產生。圖(4.8)至圖(4.19)則表示蝸桿蝸 輪組在具有裝配誤差下進行嚙合傳動時，其接觸齒數和運動誤差的情 形。其運動誤差值會隨著不同的裝配誤差組合而改變。表 4.20 為蝸 桿蝸輪組之裝配條件與運動誤差值之關係表，由結果可知，運動誤差 值隨著交錯角之增加而越來越大。並且，當蝸桿蝸輪組之交錯角越來
表 4.2 理想裝配條件下蝸桿 1 與蝸輪 1 之接觸情形
φ′1(deg.) φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -224.8 -23.2604 22.4239 179.8742 -238.9561 8.0646 -239.1751 -224.8504 -23.2604 0.0000
-200.0 -20.6897 22.4239 179.8742 -212.4282 8.0646 -212.6472 -200.0195 -20.6917 0.0000 -160.0 -16.5517 22.4239 179.8742 -169.7651 8.0646 -169.9842 -160.0196 -16.5537 0.0000 -120.0 -12.4138 22.4239 179.8742 -127.2014 8.0646 -127.4204 -120.0195 -12.4158 0.0000 -80.0 -8.2759 22.4239 179.8742 -84.7494 8.0646 -84.9684 -80.0195 -8.2779 0.0000 -40.0 -4.1379 22.4239 179.8742 -42.4184 8.0646 -42.6374 -40.0195 -4.1400 0.0000
0.0 0.0000 22.4239 179.8742 -0.2148 8.0646 -0.4338 -0.0195 -0.0020 0.0000 40.0 4.1379 22.4239 179.8742 41.8577 8.0646 41.6387 39.9805 4.1359 0.0000 80.0 8.2759 22.4239 179.8742 83.7986 8.0646 83.5795 79.9805 8.2738 0.0000 120.0 12.4138 22.4239 179.8742 125.6093 8.0646 125.3903 119.9805 12.4118 0.0000 160.0 16.5517 22.4239 179.8742 167.2936 8.0646 167.0745 159.9805 16.5497 0.0000 208.4 21.5548 22.4239 179.8742 217.5520 8.0646 217.3330 208.3631 21.5548 0.0000
3.61
圖 4.2 理想裝配條件下蝸桿 1 與蝸輪 1 之接觸齒數與運動誤差
表 4.3 理想裝配條件下蝸桿 2 與蝸輪 1 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -209.9 -21.7911 23.9915 177.3102 -227.2390 7.0487 -232.7296 -210.6472 -21.7911 0.0000
-200.0 -20.6897 23.9915 177.3102 -216.7127 7.0487 -222.2034 -200.7727 -20.7696 0.0000 -160.0 -16.5517 23.9915 177.3102 -174.1105 7.0487 -179.6012 -160.7727 -16.6317 0.0000 -120.0 -12.4138 23.9915 177.3102 -131.5782 7.0487 -137.0688 -120.7726 -12.4937 0.0000 -80.0 -8.2759 23.9915 177.3102 -89.1275 7.0487 -94.6181 -80.7726 -8.3558 0.0000 -40.0 -4.1379 23.9915 177.3102 -46.7682 7.0487 -52.2588 -40.7726 -4.2179 0.0000 0.0 0.0000 23.9915 177.3102 -4.5085 7.0487 -9.9991 -0.7726 -0.0799 0.0000 40.0 4.1379 23.9915 177.3102 37.6460 7.0487 32.1554 39.2274 4.0580 0.0000 80.0 8.2759 23.9915 177.3102 79.6914 7.0487 74.2008 79.2274 8.1959 0.0000 120.0 12.4138 23.9915 177.3102 121.6259 7.0487 116.1353 119.2274 12.3339 0.0000 160.0 16.5517 23.9915 177.3102 163.4498 7.0487 157.9592 159.2274 16.4718 0.0000 200.0 20.6897 23.9915 177.3102 205.1646 7.0487 199.6739 199.2273 20.6097 0.0000 225.3 23.2257 23.9915 177.3102 231.4817 7.0487 225.9911 224.5153 23.2257 0.0000
3.62
表 4.4 理想裝配條件下蝸桿 1 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -222.2 -22.9932 22.4239 179.8742 -254.1506 8.0646 -254.3696 -222.2677 -22.9932 0.0000
-200.0 -20.6897 22.4239 179.8742 -228.3273 8.0646 -228.5463 -200.0195 -20.6917 0.0000 -160.0 -16.5517 22.4239 179.8742 -181.9484 8.0646 -182.1674 -160.0195 -16.5537 0.0000 -120.0 -12.4138 22.4239 179.8742 -135.8000 8.0646 -136.0191 -120.0195 -12.4158 0.0000 -80.0 -8.2759 22.4239 179.8742 -90.0631 8.0646 -90.2822 -80.0196 -8.2779 0.0000 -40.0 -4.1379 22.4239 179.8742 -44.8696 8.0646 -45.0887 -40.0196 -4.1400 0.0000 0.0 0.0000 22.4239 179.8742 -0.2844 8.0646 -0.5034 -0.0195 -0.0020 0.0000 40.0 4.1379 22.4239 179.8742 43.6903 8.0646 43.4713 39.9805 4.1359 0.0000 80.0 8.2759 22.4239 179.8742 87.0970 8.0646 86.8780 79.9805 8.2738 0.0000 120.0 12.4138 22.4239 179.8742 130.0011 8.0646 129.7820 119.9805 12.4118 0.0000 160.0 16.5517 22.4239 179.8742 172.4747 8.0646 172.2557 159.9805 16.5497 0.0000 200.0 20.6897 22.4239 179.8742 214.5867 8.0646 214.3676 199.9805 20.6876 0.0000 241.9 25.0265 22.4239 179.8742 258.4215 8.0646 258.2025 241.9231 25.0265 0.0000
3.86
圖 4.4 理想裝配條件下蝸桿 1 與蝸輪 2 之接觸齒數與運動誤差
表 4.5 理想裝配條件下蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -208.8 -21.6818 23.9915 177.3102 -243.7539 7.0487 -249.2445 -209.5909 -21.6818 0.0000
-200.0 -20.6897 23.9915 177.3102 -233.6341 7.0487 -239.1247 -200.7727 -20.7696 0.0000 -160.0 -16.5517 23.9915 177.3102 -187.7056 7.0487 -193.1962 -160.7727 -16.6317 0.0000 -120.0 -12.4138 23.9915 177.3102 -141.8459 7.0487 -147.3367 -120.7728 -12.4937 0.0000 -80.0 -8.2759 23.9915 177.3102 -96.2041 7.0487 -101.6948 -80.7727 -8.3558 0.0000 -40.0 -4.1379 23.9915 177.3102 -50.9128 7.0487 -56.4034 -40.7727 -4.2179 0.0000
0.0 0.0000 23.9915 177.3102 -6.0667 7.0487 -11.5573 -0.7727 -0.0799 0.0000 40.0 4.1379 23.9915 177.3102 38.2849 7.0487 32.7943 39.2273 4.0580 0.0000 80.0 8.2759 23.9915 177.3102 82.1355 7.0487 76.6448 79.2273 8.1959 0.0000 120.0 12.4138 23.9915 177.3102 125.5096 7.0487 120.0190 119.2273 12.3339 0.0000 160.0 16.5517 23.9915 177.3102 168.4506 7.0487 162.9600 159.2273 16.4718 0.0000 200.0 20.6897 23.9915 177.3102 211.0093 7.0487 205.5186 199.2273 20.6097 0.0000 240.0 24.8276 23.9915 177.3102 253.2369 7.0487 247.7462 239.2273 24.7477 0.0000 262.3 27.0504 23.9915 177.3102 276.6112 7.0487 271.1206 261.4877 27.0504 0.0000
3.92
圖 4.5 理想裝配條件下蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.6 理想裝配條件下蝸桿 1 與蝸輪 3 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -209.6 -21.6870 22.4239 179.8742 -279.1186 8.0646 -279.3376 -209.6412 -21.6870 0.0000
-160.0 -16.5517 22.4239 179.8742 -214.5829 8.0646 -214.8019 -160.0195 -16.5537 0.0000 -120.0 -12.4138 22.4238 179.8742 -159.7027 8.0646 -159.9216 -120.0194 -12.4158 0.0000 -80.0 -8.2759 22.4239 179.8742 -104.0043 8.0646 -104.2233 -80.0195 -8.2779 0.0000 -40.0 -4.1379 22.4239 179.8742 -50.4480 8.0646 -50.6671 -40.0195 -4.1400 0.0000 0.0 0.0000 22.4239 179.8742 -0.4170 8.0646 -0.6360 -0.0195 -0.0020 0.0000 40.0 4.1379 22.4239 179.8742 46.6238 8.0646 46.4047 39.9805 4.1359 0.0000 80.0 8.2759 22.4239 179.8742 91.6124 8.0646 91.3933 79.9805 8.2738 0.0000 120.0 12.4138 22.4239 179.8742 135.2540 8.0646 135.0350 119.9805 12.4118 0.0000 160.0 16.5517 22.4239 179.8742 177.9994 8.0646 177.7804 159.9805 16.5497 0.0000 200.0 20.6897 22.4239 179.8742 220.1304 8.0646 219.9113 199.9805 20.6876 0.0000 240.0 24.8276 22.4239 179.8742 261.8264 8.0646 261.6074 239.9805 24.8256 0.0000 280.0 28.9655 22.4239 179.8742 303.2053 8.0646 302.9863 279.9805 28.9635 0.0000 321.7 33.2755 22.4239 179.8742 346.0731 8.0646 345.8540 321.6631 33.2755 0.0000
4.42
圖 4.6 理想裝配條件下蝸桿 1 與蝸輪 3 之接觸齒數與運動誤差
表 4.7 理想裝配條件下蝸桿 2 與蝸輪 3 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -209.6 -21.7675 23.9915 177.3102 -281.9290 7.0487 -287.4196 -210.4189 -21.7675 0.0000
-160.0 -16.5517 23.9915 177.3102 -220.0536 7.0487 -225.5443 -160.7727 -16.6317 0.0000 -120.0 -12.4138 23.9915 177.3102 -167.8727 7.0487 -173.3632 -120.7726 -12.4937 0.0000 -80.0 -8.2759 23.9915 177.3102 -114.1971 7.0487 -119.6877 -80.7727 -8.3558 0.0000 -40.0 -4.1379 23.9915 177.3102 -60.7419 7.0487 -66.2326 -40.7727 -4.2179 0.0000 0.0 0.0000 23.9915 177.3102 -9.3305 7.0487 -14.8212 -0.7727 -0.0799 0.0000 40.0 4.1379 23.9915 177.3102 39.4411 7.0487 33.9505 39.2273 4.0580 0.0000 80.0 8.2759 23.9915 177.3102 85.9591 7.0487 80.4684 79.2273 8.1959 0.0000 120.0 12.4138 23.9915 177.3102 130.8244 7.0487 125.3337 119.2273 12.3339 0.0000 160.0 16.5517 23.9915 177.3102 174.5274 7.0487 169.0367 159.2273 16.4718 0.0000 200.0 20.6897 23.9915 177.3102 217.4111 7.0487 211.9204 199.2273 20.6097 0.0000 240.0 24.8276 23.9915 177.3102 259.7068 7.0487 254.2161 239.2273 24.7477 0.0000 280.0 28.9655 23.9915 177.3102 301.5709 7.0487 296.0802 279.2273 28.8856 0.0000 320.0 33.1034 23.9915 177.3102 343.1110 7.0487 337.6203 319.2273 33.0235 0.0000 345.3 35.6374 23.9915 177.3102 369.2195 7.0487 363.7288 344.4948 35.6374 0.0000
4.62
表 4.8 中心距誤差∆A=0.2 mm，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -212.5 -21.9412 23.6897 178.0115 -246.6302 7.1818 -249.7547 -211.0707 -21.8349 2.2110
-200.0 -20.6559 23.6786 178.0373 -232.1467 7.1894 -235.1308 -198.4059 -20.5247 2.1232 -160.0 -16.5230 23.6437 178.1190 -185.9562 7.2126 -188.4930 -158.0345 -16.3484 1.8017 -120.0 -12.3914 23.6123 178.1925 -139.8904 7.2321 -142.0140 -117.6971 -12.1756 1.4044 -80.0 -8.2607 23.5885 178.2483 -94.1397 7.2457 -95.9265 -77.4275 -8.0097 0.9538 -40.0 -4.1303 23.5748 178.2804 -48.8533 7.2522 -50.4083 -37.2477 -3.8532 0.4772 0.0 0.0000 23.5715 178.2881 -4.1133 7.2519 -5.5487 2.8379 0.2936 0.0000 40.0 4.1306 23.5772 178.2747 40.0636 7.2463 38.6475 42.8402 4.4317 -0.4580 80.0 8.2618 23.5895 178.2458 83.7064 7.2373 82.2315 82.7783 8.5633 -0.8845 120.0 12.3935 23.6061 178.2070 126.8681 7.2264 125.2802 122.6721 12.6902 -1.2736 160.0 16.5259 23.6249 178.1630 169.6096 7.2148 167.8752 162.5391 16.8144 -1.6238 200.0 20.6588 23.6444 178.1172 211.9903 7.2033 210.0920 202.3924 20.9371 -1.9367 240.0 24.7923 23.6639 178.0719 254.0641 7.1923 251.9954 242.2412 25.0594 -2.2151 256.8 26.5351 23.6718 178.0533 271.6072 7.1879 269.4672 258.9323 26.7861 -2.3155
3.91
圖 4.8 中心距誤差∆A=0.2 mm，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.9 中心距誤差∆A=−0.2 mm，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -205.3 -21.4488 24.2807 176.6450 -241.0591 6.9263 -248.8156 -208.2751 -21.5457 -2.0449
-160.0 -16.5792 24.3138 176.5691 -189.2591 6.9041 -197.4659 -163.3192 -16.8951 -1.7269 -120.0 -12.4355 24.3409 176.5073 -143.5789 6.8870 -152.1641 -123.6312 -12.7894 -1.3633 -80.0 -8.2908 24.3630 176.4567 -98.0466 6.8739 -106.9603 -83.8971 -8.6790 -0.9392 -40.0 -4.1455 24.3782 176.4221 -52.7772 6.8659 -61.9440 -44.0969 -4.5618 -0.4768 0.0 0.0000 24.3853 176.4059 -7.8674 6.8634 -17.1975 -4.2199 -0.4365 0.0000 40.0 4.1454 24.3845 176.4076 36.6149 6.8656 27.2124 35.7336 3.6966 0.4697 80.0 8.2904 24.3772 176.4244 80.6366 6.8715 71.2417 75.7538 7.8366 0.9161 120.0 12.4349 24.3649 176.4525 124.1970 6.8798 114.8727 115.8258 11.9820 1.3291 160.0 16.5788 24.3494 176.4878 167.3205 6.8895 158.1109 155.9338 16.1311 1.7041 200.0 20.7221 24.3323 176.5269 210.0468 6.8996 200.9792 196.0636 20.2824 2.0405 240.0 24.8648 24.3146 176.5673 252.4222 6.9095 243.5107 236.2042 24.4349 2.3400 267.6 27.5462 24.3025 176.5950 281.4616 6.9161 272.6609 263.8847 27.2984 2.5197
3.94
圖 4.9 中心距誤差∆A=−0.2 mm，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.10 水平軸向誤差∆γh =0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -209.3 -21.7503 23.9501 177.4059 -244.2753 7.0649 -249.3041 -209.7681 -21.7001 -0.9965
-200.0 -20.7056 23.9559 177.3927 -233.6550 7.0604 -238.7427 -200.5310 -20.7446 -0.9992 -160.0 -16.5669 23.9807 177.3352 -187.8941 7.0407 -193.2409 -160.7460 -16.6289 -0.9542 -120.0 -12.4269 24.0046 177.2800 -142.1990 7.0221 -147.8000 -120.9519 -12.5123 -0.8217 -80.0 -8.2856 24.0255 177.2315 -96.6961 7.0058 -102.5270 -81.1328 -8.3931 -0.6094 -40.0 -4.1432 24.0421 177.1933 -51.5005 6.9930 -57.5204 -41.2763 -4.2700 -0.3304 0.0 0.0000 24.0533 177.1675 -6.7005 6.9839 -12.8586 -1.3750 -0.1422 0.0000 40.0 4.1438 24.0590 177.1542 37.6501 6.9783 31.4069 38.5718 3.9902 0.3663 80.0 8.2879 24.0599 177.1522 81.5335 6.9759 75.2537 78.5595 8.1268 0.7555 120.0 12.4322 24.0568 177.1594 124.9617 6.9758 118.6852 118.5806 12.2670 1.1573 160.0 16.5766 24.0507 177.1735 167.9670 6.9774 161.7243 158.6266 16.4096 1.5647 200.0 20.7211 24.0424 177.1926 210.5927 6.9801 204.4054 198.6902 20.5542 1.9732 240.0 24.8655 24.0327 177.2150 252.8852 6.9835 246.7680 238.7654 24.6999 2.3801 263.8 27.2489 24.0265 177.2294 277.9267 6.9858 271.8557 262.6296 27.1686 2.6104
3.94
圖 4.10 水平軸向誤差∆γh =0.1°，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.11 水平軸向誤差∆γh =−0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -208.4 -21.6136 24.0327 177.2151 -243.2345 7.0327 -249.1862 -209.4162 -21.6637 1.0192
-200.0 -20.6734 24.0274 177.2272 -233.6148 7.0369 -239.5121 -201.0177 -20.7949 1.0224 -160.0 -16.5362 24.0017 177.2867 -187.5123 7.0573 -193.1415 -160.7951 -16.6340 0.9767 -120.0 -12.4004 23.9766 177.3447 -141.4789 7.0768 -146.8432 -120.5808 -12.4739 0.8401 -80.0 -8.2660 23.9544 177.3960 -95.6900 7.0939 -100.8147 -80.3926 -8.3165 0.6216 -40.0 -4.1326 23.9371 177.4362 -50.2987 7.1073 -55.2294 -40.2458 -4.1634 0.3358 0.0 0.0000 23.9257 177.4626 -5.4071 7.1166 -10.2001 -0.1482 -0.0153 0.0000 40.0 4.1321 23.9202 177.4753 38.9415 7.1218 34.2292 39.9008 4.1277 -0.3695 80.0 8.2638 23.9198 177.4760 82.7541 7.1239 78.0721 79.9077 8.2663 -0.7594 120.0 12.3953 23.9236 177.4674 126.0696 7.1236 121.3780 119.8816 12.4015 -1.1600 160.0 16.5268 23.9303 177.4518 168.9427 7.1216 164.2120 159.8314 16.5343 -1.5649 200.0 20.6583 23.9390 177.4316 211.4317 7.1186 206.6417 199.7649 20.6653 -1.9701 240.0 24.7898 23.9491 177.4083 253.5925 7.1149 248.7299 239.6879 24.7953 -2.3734 260.7 26.8487 23.9547 177.3954 275.2674 7.1128 270.3637 260.3158 26.9292 -2.5740
3.9
圖 4.11 水平軸向誤差∆γh =−0.1°，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.12 垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -204.3 -21.3007 24.2045 176.8194 -240.1542 6.8880 -248.6895 -207.8729 -21.5041 -3.1406
-200.0 -20.7384 24.2037 176.8213 -235.1656 6.8884 -243.6978 -203.5187 -21.0537 -3.0610 -160.0 -16.5889 24.1926 176.8468 -189.2101 6.8947 -197.6716 -163.4228 -16.9058 -2.3388 -120.0 -12.4401 24.1743 176.8888 -143.2446 6.9064 -151.5535 -123.2592 -12.7510 -1.6519 -80.0 -8.2922 24.1489 176.9472 -97.4225 6.9229 -105.4974 -83.0301 -8.5893 -1.0234 -40.0 -4.1454 24.1178 177.0187 -51.8990 6.9428 -59.6735 -42.7498 -4.4224 -0.4702 0.0 0.0000 24.0836 177.0977 -6.8029 6.9643 -14.2363 -2.4401 -0.2524 0.0000 40.0 4.1441 24.0488 177.1779 37.7850 6.9854 30.7045 37.8749 3.9181 0.3886 80.0 8.2871 24.0158 177.2541 81.8374 7.0048 75.0971 78.1752 8.0871 0.7033 120.0 12.4290 23.9859 177.3230 125.3715 7.0217 118.9427 118.4482 12.2533 0.9553 160.0 16.5701 23.9599 177.3833 168.4322 7.0360 162.2785 158.6883 16.4160 1.1559 200.0 20.7106 23.9376 177.4349 211.0766 7.0480 205.1605 198.8951 20.5754 1.3159 240.0 24.8506 23.9188 177.4785 253.3635 7.0577 247.6497 239.0708 24.7315 1.4438 262.1 27.0246 23.9098 177.4994 276.5758 7.0624 270.9601 261.2389 27.0247 1.5007
3.88
圖 4.12 垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.13 垂直軸向誤差∆γv =−0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -213.6 -22.0899 23.7638 177.8386 -247.5506 7.2203 -249.9106 -211.5076 -21.8801 3.1889
-200.0 -20.6428 23.7669 177.8315 -231.9999 7.2186 -234.3749 -197.9811 -20.4808 2.9436 -160.0 -16.5161 23.7812 177.7982 -186.1245 7.2094 -188.5999 -158.1013 -16.3553 2.2363 -120.0 -12.3888 23.8034 177.7464 -140.4046 7.1946 -143.0667 -118.2916 -12.2371 1.5725 -80.0 -8.2604 23.8326 177.6786 -94.9749 7.1753 -97.9004 -78.5424 -8.1251 0.9719 -40.0 -4.1308 23.8663 177.6002 -49.9363 7.1536 -53.1801 -38.8345 -4.0174 0.4465 0.0 0.0000 23.9019 177.5177 -5.3483 7.1314 -8.9378 0.8548 0.0884 0.0000 40.0 4.1320 23.9367 177.4369 38.7681 7.1103 34.8312 40.5466 4.1945 -0.3709 80.0 8.2651 23.9692 177.3617 82.4217 7.0914 78.1551 80.2565 8.3024 -0.6742 120.0 12.3992 23.9983 177.2946 125.6414 7.0750 121.0745 119.9937 12.4131 -0.9198 160.0 16.5339 24.0236 177.2361 168.4671 7.0611 163.6343 159.7619 16.5271 -1.1179 200.0 20.6693 24.0453 177.1860 210.9429 7.0496 205.8795 199.5612 20.6443 -1.2775 240.0 24.8052 24.0636 177.1437 253.1129 7.0401 247.8517 239.3893 24.7644 -1.4063 262.4 27.0747 24.0726 177.1229 276.6338 7.0356 271.2748 261.7270 27.0752 -1.4649
3.96
圖 4.13 垂直軸向誤差∆γv =−0.1°，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.14 中心距誤差∆A=0.2 mm，水平軸向誤差∆γh =−0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -212.1 -21.8742 23.7302 177.9170 -246.1208 7.1664 -249.6932 -210.8965 -21.8169 3.3296
-200.0 -20.6382 23.7110 177.9618 -232.1032 7.1805 -235.4534 -198.6158 -20.5465 3.2346 -160.0 -16.5064 23.6467 178.1121 -185.7104 7.2267 -188.3142 -157.9976 -16.3446 2.8484 -120.0 -12.3773 23.5865 178.2528 -139.4442 7.2684 -141.3407 -117.4240 -12.1473 2.2921 -80.0 -8.2504 23.5381 178.3665 -93.5321 7.3005 -94.8354 -76.9545 -7.9608 1.6015 -40.0 -4.1248 23.5065 178.4408 -48.1482 7.3205 -49.0258 -36.6300 -3.7893 0.8224 0.0 0.0000 23.4929 178.4727 -3.3775 7.3285 -4.0122 3.5401 0.3662 0.0000 40.0 4.1247 23.4951 178.4674 40.7774 7.3268 40.2211 43.5733 4.5076 -0.8299 80.0 8.2497 23.5094 178.4340 84.3648 7.3184 83.7579 83.5001 8.6379 -1.6431 120.0 12.3752 23.5317 178.3816 127.4544 7.3059 126.7060 123.3528 12.7606 -2.4262 160.0 16.5012 23.5588 178.3179 170.1186 7.2913 169.1703 163.1584 16.8785 -3.1729 200.0 20.6279 23.5884 178.2486 212.4236 7.2759 211.2418 202.9376 20.9935 -3.8821 240.0 24.7551 23.6188 178.1774 254.4267 7.2606 252.9952 242.7046 25.1074 -4.5551 255.1 26.3241 23.6302 178.1505 270.1846 7.2550 268.6573 257.6869 26.6573 -4.7826
3.89
圖 4.14 中心距誤差∆A=0.2 mm，水平軸向誤差∆γh =−0.1°，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.15 中心距誤差∆A=0.2 mm，垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -207.8 -21.5328 23.9164 177.4840 -242.8361 7.0102 -249.1057 -209.1729 -21.6386 -0.9942
-200.0 -20.7040 23.9085 177.5024 -233.8167 7.0154 -239.9958 -201.2716 -20.8212 -0.8997 -160.0 -16.5594 23.8632 177.6075 -187.6039 7.0449 -193.2650 -160.8015 -16.6346 -0.4852 -120.0 -12.4169 23.8118 177.7269 -141.4148 7.0780 -146.4878 -120.2800 -12.4428 -0.1921 -80.0 -8.2762 23.7586 177.8508 -95.4467 7.1113 -99.9068 -79.7460 -8.2496 -0.0231 -40.0 -4.1374 23.7086 177.9674 -49.8849 7.1415 -53.7631 -39.2454 -4.0599 0.0335 0.0 0.0000 23.6661 178.0666 -4.8589 7.1659 -8.2325 1.1844 0.1225 0.0000 40.0 4.1364 23.6335 178.1430 39.5763 7.1838 36.6046 41.5240 4.2956 -0.0971 80.0 8.2721 23.6107 178.1963 83.4305 7.1954 80.7547 81.7721 8.4592 -0.2339 120.0 12.4076 23.5964 178.2297 126.7538 7.2020 124.2795 121.9397 12.6144 -0.3914 160.0 16.5429 23.5889 178.2472 169.6127 7.2049 167.2635 162.0422 16.7630 -0.5563 200.0 20.6782 23.5865 178.2530 212.0758 7.2052 209.7934 202.0954 20.9064 -0.7201 240.0 24.8136 23.5874 178.2507 254.2056 7.2039 251.9476 242.1128 25.0462 -0.8780 256.5 26.5052 23.5886 178.2480 271.5435 7.2030 269.2862 258.6537 26.7573 -0.9368
3.86
圖 4.15 中心距誤差∆A=0.2 mm，垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.16 水平軸向誤差∆γh =0.1°，垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之 接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -204.8 -21.3690 24.1633 176.9142 -240.6731 6.9044 -248.7429 -208.0441 -21.5218 -4.1078
-200.0 -20.7537 24.1652 176.9099 -235.1670 6.9026 -243.2627 -203.2476 -21.0256 -4.0230 -160.0 -16.6038 24.1775 176.8816 -189.3672 6.8902 -197.6431 -163.3619 -16.8995 -3.2720 -120.0 -12.4530 24.1824 176.8702 -143.5615 6.8836 -151.9381 -123.4052 -12.7661 -2.4658 -80.0 -8.3019 24.1784 176.8795 -97.8821 6.8834 -106.2624 -83.3649 -8.6240 -1.6334 -40.0 -4.1507 24.1651 176.9101 -52.4646 6.8895 -60.7473 -43.2400 -4.4731 -0.8035 0.0 0.0000 24.1435 176.9597 -7.4263 6.9008 -15.5214 -3.0415 -0.3146 0.0000 40.0 4.1500 24.1159 177.0232 37.1511 6.9154 29.3113 37.2104 3.8493 0.7611 80.0 8.2993 24.0848 177.0949 81.2308 6.9316 73.6865 77.4925 8.0165 1.4725 120.0 12.4477 24.0524 177.1695 124.8167 6.9480 117.5837 117.7845 12.1846 2.1334 160.0 16.5955 24.0205 177.2433 167.9414 6.9638 161.0173 158.0712 16.3522 2.7477 200.0 20.7425 23.9899 177.3138 210.6536 6.9784 204.0243 198.3436 20.5183 3.3208 240.0 24.8890 23.9613 177.3800 253.0065 6.9916 246.6523 238.5972 24.6825 3.8590 263.7 27.2251 23.9454 177.4169 277.9055 6.9989 271.7035 262.3929 27.1441 4.1510
3.9
圖 4.16 水平軸向誤差∆γh =0.1°，垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之 接觸齒數與運動誤差
表 4.17 水平軸向誤差∆γh =−0.1°，垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -203.9 -21.2328 24.2456 176.7253 -239.6373 6.8719 -248.6372 -207.7040 -21.4866 -2.1506
-200.0 -20.7227 24.2426 176.7321 -235.1662 6.8740 -244.1391 -203.7936 -21.0821 -2.0766 -160.0 -16.5737 24.2074 176.8130 -189.0499 6.8995 -197.6939 -163.4812 -16.9118 -1.3834 -120.0 -12.4268 24.1647 176.9109 -142.9165 6.9304 -151.1444 -123.1025 -12.7347 -0.8192 -80.0 -8.2822 24.1167 177.0213 -96.9432 6.9645 -104.6895 -82.6772 -8.5528 -0.4008 -40.0 -4.1400 24.0670 177.1358 -51.3086 6.9989 -58.5453 -42.2370 -4.3693 -0.1312
0.0 0.0000 24.0196 177.2452 -6.1540 7.0308 -12.8952 -1.8161 -0.1879 0.0000 40.0 4.1381 23.9779 177.3418 38.4410 7.0582 32.1468 38.5584 3.9888 0.0124 80.0 8.2747 23.9434 177.4214 82.4613 7.0804 76.5457 78.8715 8.1591 -0.0705 120.0 12.4102 23.9168 177.4832 125.9389 7.0974 120.3284 119.1202 12.3228 -0.2266 160.0 16.5448 23.8972 177.5286 168.9318 7.1099 163.5571 159.3093 16.4803 -0.4370 200.0 20.6787 23.8836 177.5600 211.5058 7.1188 206.3072 199.4472 20.6325 -0.6870 240.0 24.8122 23.8750 177.5800 253.7246 7.1249 248.6530 239.5432 24.7803 -0.9656 260.5 26.8208 23.8722 177.5866 275.2169 7.1271 270.1944 260.0542 26.9022 -1.1131
3.87
圖 4.17 水平軸向誤差∆γh =−0.1°，垂直軸向誤差∆γv =0.1°，蝸桿 2 與蝸輪 2 之 接觸齒數與運動誤差
表 4.18 中心距誤差∆A=0.2 mm，水平軸向誤差∆γh =0.1°，垂直軸向誤差∆γv =0.1°， 蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -208.3 -21.6000 23.8757 177.5784 -243.3466 7.0261 -249.1624 -209.3446 -21.6563 -2.0654
-200.0 -20.7207 23.8726 177.5857 -233.8356 7.0274 -239.6105 -201.0291 -20.7961 -1.9480 -160.0 -16.5754 23.8535 177.6300 -187.8005 7.0361 -193.3332 -160.7887 -16.6333 -1.4880 -120.0 -12.4306 23.8284 177.6882 -141.7935 7.0487 -147.0172 -120.4937 -12.4649 -1.0554 -80.0 -8.2864 23.7990 177.7566 -95.9835 7.0638 -100.8505 -80.1617 -8.2926 -0.6617 -40.0 -4.1429 23.7679 177.8291 -50.5303 7.0796 -55.0237 -39.8192 -4.1192 -0.3111 0.0 0.0000 23.7378 177.8991 -5.5531 7.0942 -9.6869 0.5077 0.0525 0.0000 40.0 4.1424 23.7109 177.9620 38.8871 7.1066 35.0748 40.7993 4.2206 0.2797 80.0 8.2844 23.6881 178.0153 82.7844 7.1164 79.2436 81.0462 8.3841 0.5380 120.0 12.4263 23.6694 178.0588 126.1723 7.1238 122.8515 121.2477 12.5429 0.7839 160.0 16.5680 23.6545 178.0937 169.1046 7.1291 165.9572 161.4084 16.6974 1.0247 200.0 20.7098 23.6425 178.1218 211.6416 7.1329 208.6289 201.5354 20.8485 1.2653 240.0 24.8516 23.6328 178.1446 253.8415 7.1356 250.9330 241.6358 24.9968 1.5090 258.2 26.7148 23.6289 178.1537 272.9505 7.1366 270.0813 259.8795 26.8841 1.6326
3.88
圖 4.18 中心距誤差∆A=0.2 mm，水平軸向誤差∆γh =0.1°，垂直軸向誤差∆γv =0.1°
，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
表 4.19 中心距誤差∆A=−0.2mm，水平軸向誤差∆γh =−0.1°，垂直軸向誤差 ∆γv=0.1°
，蝸桿 2 與蝸輪 2 之接觸情形
φ′1(deg.
)
φ′2(deg.) u1(mm) θ(deg.) ϕ1(deg.) l(mm) ϕ2(deg.) φ1(deg.) φ2(deg.) K.E.
(arc-sec.)
接觸比 -200.6 -21.0217 24.5237 176.0908 -237.0971 6.7582 -248.2882 -206.5379 -21.3660 -4.2162
-160.0 -16.6029 24.5196 176.1002 -190.5217 6.7624 -201.8054 -165.9737 -17.1697 -3.2181 -120.0 -12.4500 24.5078 176.1271 -144.5998 6.7729 -155.8841 -125.9391 -13.0282 -2.2758 -80.0 -8.2983 24.4870 176.1743 -98.7768 6.7901 -109.9562 -85.8186 -8.8778 -1.4075 -40.0 -4.1482 24.4575 176.2414 -53.2013 6.8130 -64.1699 -45.6138 -4.7187 -0.6425 0.0 0.0000 24.4211 176.3242 -8.0066 6.8398 -18.6750 -5.3398 -0.5524 0.0000 40.0 4.1461 24.3807 176.4163 36.7114 6.8682 26.4036 34.9792 3.6185 0.5128 80.0 8.2902 24.3395 176.5105 80.9065 6.8961 70.9869 75.3161 7.7913 0.8996 120.0 12.4324 24.2999 176.6010 124.5796 6.9219 115.0471 115.6480 11.9636 1.1721 160.0 16.5731 24.2637 176.6838 167.7650 6.9448 158.5988 155.9591 16.1337 1.3452 200.0 20.7125 24.2318 176.7570 210.5154 6.9645 201.6831 196.2404 20.3007 1.4346 240.0 24.8507 24.2043 176.8200 252.8889 6.9812 244.3535 236.4888 24.4644 1.4550 265.9 27.3287 24.1888 176.8556 280.1637 6.9905 271.8006 262.5439 27.1597 1.4378
3.88
圖 4.19 中心距誤差∆A=−0.2 mm，水平軸向誤差∆γh =−0.1°，垂直軸向誤差 ∆γv=0.1°
，蝸桿 2 與蝸輪 2 之接觸齒數與運動誤差
圖 4.20 蝸桿與蝸輪在不同壓力角時之接觸點軌跡圖
表 4.20 裝配條件與運動誤差值
蝸桿 1(α1=20°) 蝸桿 2(α1=22.5°) 蝸輪 1
(γ=30°)
蝸輪 2 (γ=45°)
蝸輪 3 (γ=60°)
蝸輪 1 (γ=30°)
蝸輪 2 (γ=45°)
蝸輪 3 (γ=60°)
△A= 0.2 mm 2.413 4.531 6.09 2.419 4.526 6.203
△A=－0.2 mm 2.436 4.567 6.155 2.439 4.563 6.225
△γh= 0.1° 3.364 3.661 4.255 3.22 3.606 4.436
△γh=－0.1° 3.259 3.659 4.225 3.213 3.593 4.433
△γv= 0.1° 2.651 3.847 6.628 3.263 4.64 7.647
△γv=－0.1° 2.644 3.868 6.45 3.258 4.652 7.468
越大時，則可發現中心距誤差和垂直軸向誤差對蝸桿蝸輪組之運動誤 差的影響較為顯著。
在理想之裝配條件下，當蝸桿之法向壓力角改變時，接觸點落在 齒面之位置有上下移動的情形，一般而言，讓接觸點保持在齒面中央 處比較佳。茲改變蝸桿和蝸輪之法向壓力角，以探討其接觸點落在齒 面之位置的變化情形，如圖 4.20 所示；圖 4.20(a)顯示壓力角為 20°
之蝸輪與不同壓力角(20°及 22.5°)之蝸桿的接觸情形，圖 4.20(b)則為 壓力角 20°之蝸桿與不同壓力角(20°及 22.5°)之蝸輪的接觸情形。