commit 72d889f5ef2b6a7c1ab49928aa37f7a1c77d44d4 Author: hyungi Date: Wed Aug 13 07:24:06 2025 +0900 feat: local AI server scaffolding (FastAPI, RAG, embeddings). Port policy (>=26000), README/API docs, scripts. diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..42d5129 --- /dev/null +++ b/.gitignore @@ -0,0 +1,27 @@ +# Python +__pycache__/ +*.py[cod] +*$py.class + +# Virtual envs +.venv/ +venv/ +ENV/ + +# macOS +.DS_Store +.AppleDouble +.LSOverride + +# Editors/IDE +.vscode/ +.idea/ + +# Logs +*.log + +# Cache/Build +dist/ +build/ +.pytest_cache/ + diff --git a/README.md b/README.md new file mode 100644 index 0000000..582c6ce --- /dev/null +++ b/README.md @@ -0,0 +1,206 @@ +### 로컬 AI 서버 (Mac mini M4 Pro 64GB) + +이 저장소는 Apple Silicon(M4 Pro, RAM 64GB) 환경에서 로컬 AI 모델을 실행해 API 서버로 활용하기 위한 기본 구성과 가이드를 제공합니다. + +## 현재 설치 상태 + +- **러너**: Ollama 0.11.4 확인됨 +- **Ollama 로컬 모델**: + - qwen2.5:1.5b + - mistral:7b +- **LM Studio 로컬 모델**: + - gemma-3-4b-it + +## 하드웨어 요약 (권장 기준) + +- **칩셋**: Apple M4 Pro (Metal/ANE 가속 활용 가능) +- **메모리**: 64GB 통합 메모리 +- **권장 동시성**: 1–3 세션(프롬프트 길이에 따라 조절) + +## 권장 모델 (M4 Pro 64GB) + +- **일반 대화/업무** + - Llama 3.1 8B Instruct: 품질·속도 밸런스 좋음, 긴 문서 요약/대화에 적합 + - Qwen2.5 7B Instruct: 정보 회수/한글 대응 우수, 속도 양호 + - Mistral 7B Instruct: 경량/속도 지향, 기본 품질 안정적 + - Gemma 2 9B IT: 간결한 답변과 대화 품질 균형 + +- **코딩 보조** + - Qwen2.5-Coder 7B: 코드 생성/수정/해설에 실용적, 메모리 요구도 낮음 + - DeepSeek-Coder 6.7B 또는 16B(Lite): 코드 품질 강점, 16B는 속도·메모리 여유 필요 + +- **초경량** + - Phi-3.5/3.1 Mini(3–4B): 간단 질의응답/요약, 서버 부하가 낮음 + +- 참고: 14–32B급도 구동 가능하나(예: Qwen2.5 14B/32B), 긴 컨텍스트/동시성 시 메모리 여유가 적어질 수 있음. 70B급은 64GB 환경에서 가능하더라도 속도·안정성 상 비권장. + +## 설치 (Ollama) + +아래 명령으로 권장 모델을 내려받을 수 있습니다. 태그는 상황에 따라 업데이트될 수 있으니 `ollama run ` 시 안내를 확인하세요. + +```bash +# 일반 대화 +ollama pull llama3.1:8b-instruct +ollama pull qwen2.5:7b-instruct +ollama pull mistral:7b +ollama pull gemma2:9b-instruct + +# 코딩 보조 +ollama pull qwen2.5-coder:7b +ollama pull deepseek-coder:6.7b + +# 초경량 +ollama pull phi3:mini +``` + +이미 설치된 모델 확인: + +```bash +ollama list +``` + +모델 실행(대화형 테스트): + +```bash +ollama run qwen2.5:7b-instruct +``` + +## REST API로 바로 쓰기 (Ollama 내장 서버) + +Ollama는 기본적으로 `http://localhost:11434`에서 API를 제공합니다. + +```bash +# 단발성 텍스트 생성 +curl http://localhost:11434/api/generate \ + -H "Content-Type: application/json" \ + -d '{ + "model": "qwen2.5:7b-instruct", + "prompt": "한국어로 이 모델의 장점을 3가지로 요약해줘", + "stream": false + }' + +# Chat 형식 +curl http://localhost:11434/api/chat \ + -H "Content-Type: application/json" \ + -d '{ + "model": "llama3.1:8b-instruct", + "messages": [ + {"role": "user", "content": "로컬 LLM 서버 운영 팁을 알려줘"} + ], + "stream": false + }' +``` + +TIP: 긴 문서를 다루려면 `num_ctx`(컨텍스트 길이)와 `num_thread`를 모델/하드웨어에 맞춰 조정하세요. 과도하게 늘리면 속도와 메모리 사용량이 크게 증가합니다. + +## 포트 정책 + +- **AI 서버 표준 포트**: 26000 이상 사용 권장 (예: 26000) +- 환경 변수 `AI_SERVER_PORT`로 조정 가능. 기본값 26000. + +개발 서버 실행 스크립트(`scripts/dev_server.sh`)는 위 정책을 따릅니다. + +## API 개요 (Paperless/시놀로지 연동) + +- 기본 베이스 모델(24/7): `BASE_MODEL` (기본: `qwen2.5:7b-instruct`) +- 온디맨드 부스팅 모델: `BOOST_MODEL` (기본: `qwen2.5:14b-instruct`) +- 임베딩(RAG): `EMBEDDING_MODEL` (기본: `nomic-embed-text`), 인덱스 파일 `INDEX_PATH` (기본: `data/index.jsonl`) +- 문서화: `http://localhost:26000/docs` (FastAPI 자동 문서) + +### 헬스체크 + +```bash +curl -s http://localhost:26000/health +``` + +### 검색(Search, RAG용) + +```bash +curl -s -X POST http://localhost:26000/search \ + -H 'Content-Type: application/json' \ + -d '{ + "query": "질문 내용", + "top_k": 5 + }' +``` + +### 채팅(Chat, RAG/부스팅 자동) + +```bash +curl -s -X POST http://localhost:26000/chat \ + -H 'Content-Type: application/json' \ + -d '{ + "messages": [ + {"role": "user", "content": "문서 내용 기반으로 요약해줘"} + ], + "use_rag": true, + "top_k": 5, + "force_boost": false, + "options": {"num_ctx": 32768, "temperature": 0.3} + }' +``` + +필드 설명: +- `use_rag`: 인덱스(`INDEX_PATH`)에서 상위 청크를 검색해 시스템 프롬프트로 주입 +- `force_boost`: 강제로 부스팅 모델 사용(고난도/장문) +- `options`: Ollama 옵션(예: `num_ctx`, `temperature` 등) + +### 인덱스 갱신(Upsert) + +Paperless/시놀로지에서 추출한 본문 텍스트를 직접 인덱스에 추가합니다. + +```bash +curl -s -X POST http://localhost:26000/index/upsert \ + -H 'Content-Type: application/json' \ + -d '{ + "rows": [ + {"id": "paperless:123", "text": "문서 본문 텍스트", "source": "paperless"} + ], + "embed": true + }' +``` + +### 인덱스 리로드 + +```bash +curl -s -X POST http://localhost:26000/index/reload +``` + +### Paperless 훅(Webhook) 자리표시자 + +```bash +curl -s -X POST http://localhost:26000/paperless/hook \ + -H 'Content-Type: application/json' \ + -d '{"document_id": 123, "title": "문서제목", "tags": ["finance"]}' +``` + +해당 훅은 문서 도착을 통지받는 용도로 제공됩니다. 실제 본문 텍스트는 Paperless API로 조회해 `/index/upsert`로 추가하세요. + +## 시놀로지 메일/오피스 연동 가이드(요약) + +- **검색/QA 호출 엔드포인트**: `http://:26000/search`, `http://:26000/chat` +- **권장 흐름**: + - 메일/문서 본문 → `/index/upsert`로 인덱스 추가(임베딩 생성) + - 사용자 질의 → `/chat` 호출(`use_rag=true`) → 관련 청크 Top-k 주입 후 응답 +- **모델 라우팅**: + - 기본: 베이스 모델(7B/8B) + - 장문/고난도: `force_boost=true` 또는 메시지 길이에 따라 자동 부스팅(14B) + +## 환경 변수 + +- `AI_SERVER_PORT`(기본 26000): 서버 포트 +- `OLLAMA_HOST`(기본 `http://localhost:11434`): Ollama API 호스트 +- `BASE_MODEL`(기본 `qwen2.5:7b-instruct`) +- `BOOST_MODEL`(기본 `qwen2.5:14b-instruct`) +- `EMBEDDING_MODEL`(기본 `nomic-embed-text`) +- `INDEX_PATH`(기본 `data/index.jsonl`) +- `PAPERLESS_BASE_URL`, `PAPERLESS_TOKEN`(선택): Paperless API 연동 시 사용 + +## 이 저장소 사용 계획 + +1) Ollama API를 감싸는 경량 서버(Express 또는 FastAPI) 추가 +2) 표준화된 엔드포인트(`/v1/chat/completions`, `/v1/completions`) 제공 +3) 헬스체크/모델 선택/리밋/로깅 옵션 제공 + +우선 본 문서로 설치/선택 가이드를 정리했으며, 다음 단계에서 서버 스켈레톤과 샘플 클라이언트를 추가할 예정입니다. + diff --git a/data/index.jsonl b/data/index.jsonl new file mode 100644 index 0000000..6f964f1 --- /dev/null +++ b/data/index.jsonl @@ -0,0 +1,35 @@ +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:0", "text": "CHAPTER\nι\n10 유한 요소법 입문\n제6장에서, 우리는 구조물을 구조 요소들의 결합으로 생각함으로써 단순 프레임 구조의 강 \n성행렬을 구할 수 있었다. 구조이론으로부터 알려진 요소의 끝에서의 힘과 모멘트를 사용 \n하여, 요소들 사이의 결합부는 변위 구속(COmPatability)으로 연관되고, 결합부에서의 힘과 \n모멘트는 평형조건을 부과함으로써 구해진다.\n유한 요소법에서도 똑같은 과정을 따르지만 컴퓨터를 사용한 계산을 위하여 더욱 체 \n계적으로 되어진다. 비록 매우 적은 요소를 가진 구조에 대해서는 제6장의 기술한 방 \n법에 의하여 간단하게 해석될 수 있지만, 많은 요소로 된 대형 구조물에 대해서 기장 \n(bookkeeping)하는 것은 곧 해석자의 인내를 넘어서게 된다. 유한 요소법에서는, 요소좌표 \n와 힘은 전체 좌표로 변환되어지고 전체 구조물의 강성행렬은 공통의 방향을 가진 전체 좌 \n표에서 나타내어진다.\n유한 요소법에서 구해지는 정확도는 진동 모드 형태를 나타낼 수 있는가에 달려 있다. \n구조물의 결합부나 코너부 사이에 오직 하나의 유한 요소를 사용하면 정적 처짐곡선이 최 \n저차의 동적 모드 형상에 대한 좋은 근사가 되기 때문에 최저차 모드에 대하여 좋은 결과 \n를 낳는다. 고차 모드에 대해서는 구조 결합부 사이에 다수의 요소가 필요하다. 이것은 대 \n형 행렬을 낳게 되고 계의 고유값과 고유 벡터에 대하여 풀이하는 데 컴퓨터가 필수적으로 \n되도록 한다.\n본 장에서는 독자들에게 유한 요소법의 기본 개념을 소개하고, 동적 문제를 위한 운동 \n방정식을 완성하기 위하여 상당 질량 행렬식의 전개를 포함하고 있다.\n이 곳에서는 단지 축 요소와 보 요소와 같은 구조용 요소에 대해서만 논의 한다. 평판과 \n셸(ShenS)의 취급을 위해서는 독자들은 다른 참고서를 참조하기 바란다.\nIOl丄 요소강성 및 요소질량\n축 요소 단순지지 끝단으로 된 요소는 오직 축방향 힘만을 지지할 수 있기 때문에, 그러므 \n로 스프링과 같은 작용을 하게 된다. 그림 10丄1은 고정된 벽에 힘 F를 받으면서 단순지지 \n\n\n330 아IaPterlO 유한요소법 입문\nF = ku\nE시 t FI\nF = \\EA/l、U\n그림 10.1.1 그림 10.1.2\n그림 10.1.3\n된 스프링과 균일봉을 보여준다. 두 경우에 대한 힘-변위 관계식을 단순히 나타내면 다음 \n과 같다.\nM링 f=ku\nEA\n균일봉 F = U\n(10.1.1)\n일반적으로, 이들 축방향", "vector": [-0.22320415079593658, 0.8386243581771851, -2.4470958709716797, -0.511906087398529, 0.8749222755432129, -0.6825230121612549, 1.0908477306365967, 1.270628809928894, 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응용(제5판)_Chapter 10 유한 요소법 입문:1", "text": "에 힘 F를 받으면서 단순지지 \n\n\n330 아IaPterlO 유한요소법 입문\nF = ku\nE시 t FI\nF = \\EA/l、U\n그림 10.1.1 그림 10.1.2\n그림 10.1.3\n된 스프링과 균일봉을 보여준다. 두 경우에 대한 힘-변위 관계식을 단순히 나타내면 다음 \n과 같다.\nM링 f=ku\nEA\n균일봉 F = U\n(10.1.1)\n일반적으로, 이들 축방향 요소들은 양단의 변위를 가능하게 한 핀으로 연결된 구조의 일 \n부가 될 수 있다. 유한 요소법에서는 요소의 양단에서의 변위와 힘은 적절한 부호로서 간주 \n되어질 수 있다. 그림 10丄2에는 변위 싸, 均와 힘 F1, F2 모두 양의 방향으로 명시된 축방 \n향 요소를 보여준다. 만일 우리가 힘-변위 관계를 강성행렬로서 나타내면 방정식은 다음 \n과 같다.\n(10.1.2)\n강성행렬의 제 1 행의 요소는 그림 10.1.3에 보인 것과 같이 Wl = I과 w2 = 0일 때 양단에 \n서의 힘을 표시 한다. 그러므로 Fl — Aw1 이고 F2 — -이다.\n마찬가지로, W2 = 1 및 Wl=O으로 놓음으로써, 그림 10.1.4에서와 같이 Fl = -加2이고 \nF2 = ku2를 얻는다. 그러므로 식 (10.1.2)는 다음 식과 같이 된다.\n= k 1\n-1\n(10.1.2)\n그림 10.1.4\n\n\nιo.ι 요소강성 및 요소질량 331\n만일 스프링이 균일봉으로 바뀌면, k = AE∣lo∖ 되고 식은 다음과 같다.\nFlI _ EA∖ 1 \nf2J=-TL-I IJbZj (10.1.3)\n그러므로 이 식들은 축 요소에 대한 강성행렬을 축 요소의 방향에 관계없이 축 좌표 Wz와 \n축방향 힘 乃의 항으로 정의한 것이다.\n축 요소에 대한 모드 형상과 질량행렬 축요소의 두 끝이 Wl과 w2로 이동되면, 임의 점의 변 \n위 ξ = x∕/는 그림 10.1.5(a)에서 보는 것과 같이 직선의 형태로 가정되어진다. 그리하여 변 \n위는 그림 l(λl.5(b)에서 보여지는 두 개의 모드 형상의 중첩이 된다. 그러면 정규화된 모 \n드 형상은 다음과 같다.\nφ1 = (1 — ξ) 그리고 φ2 = ξ (10.1.4)\n질량행렬은 W를 두 모드 형상의 합으로서 표현한 후 운동 에너지를 위한 식을 쓰게 되면 \n구해진다.\nU = (I- ξ>1 + ξu2 (10.1.5)\n이 때 단위길이당 균일 질량 분포 끼을 가정한다.\nI f' IfI 2\nT= - U2 m dx = -m∖ [(1 - L)UI + ξiι^ldξ\nJO JO (", "vector": [-0.13675545156002045, 0.6610175371170044, -2.255779981613159, -1.1974515914916992, 0.41240471601486206, -0.5848233103752136, 0.4059920012950897, 0.5722971558570862, -1.2544077634811401, -0.48966890573501587, -0.6335870623588562, 1.8488949537277222, 1.1089563369750977, -0.08503951132297516, 0.2397901564836502, -0.48287492990493774, 0.17287763953208923, -0.7857689261436462, 0.21944041550159454, 0.31502583622932434, 0.04916687682271004, -0.6255623698234558, -0.7180657982826233, -1.6666080951690674, 0.4543558955192566, 0.6316214799880981, 0.4388263523578644, 0.30662742257118225, -1.3490052223205566, -0.35187819600105286, 0.43155670166015625, -0.9989564418792725, 0.724574089050293, 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대한 질량행렬을 다음과 같이 구할 수 있다.\n1 2\nml {2 1 \n^6\n또한 질량행렬의 각 항은 다음 식으로부터 구할 수 있다.\nmij\nJ φ↑φj dm\n(10.1.7)\nI 그림 10丄6에 보인 두 개의 단면적을 가진 봉의 길이방향 진동에 관한 운동 방정식을 I \n! 구하라.\n2 3\n그림 10.1.6\nI WS 결합부를 1,2 및 3으로번호를부여하면,두 개의축방향 요소1-2와2-3을 \n너 얻게 되고 변위는 G u2 및 內가 된다. Wl은 0이지만, 우선 우리는 그것을 구속시키지 \n서 않고나중에그것에 0의값을 부여한다.\n식 (10丄7)과 (10.1.3)으로부터 요소질량 및 요소강성 항들은 다음과 같다:\n\n\n10.1 요소강성 및 요소질량 333\nkb\n(10.1.8) -\n(10.1.9) -\n1\n2\n너 이 때 소 = EAa∕la9 kb = EAb∕lb, Ma = mala 및 Λζ = mzΛ이다.\n요소행렬들은 공통좌표 物를 가지고 그들을 중첩시킴으로써, 다음과 같은 3 × 3행 I \n! 렬로조합될수있다:\n질량행렬 }\nO\n1 -1\n-1 1요소 b：\n강성행렬\nW2 Ma 0% 2Λ쓰 + 씨\nMb\n0 Mb\n%\n— '(\n-ka 0\n-Eg ka + kb ~kb { Ul\n0 % kb u3\n우리는 이제 강성행렬은 특이(SingUlar)행렬이고, 역행렬이 없다는 데 주목하자. 이것 \n은 변위에 제한이 없었기 때문에 예상되었던 것이다. 강성행렬의 제 1 행과 제3행은 강 \n성행렬에서 나타나 있듯이 ka(ul - u2) = kb(u2 - u3) = 0이 된다. 이것은 좌표들 사이에 \n서 상대운동이 일어나지", "vector": [0.1517166793346405, 0.2527385950088501, -1.7375799417495728, -0.41894692182540894, 0.3700423240661621, 0.18621332943439484, 0.8736020922660828, 1.4357465505599976, -0.4077773988246918, -0.05146323889493942, -0.5845698714256287, 1.430216670036316, 1.2432353496551514, -0.5794823169708252, -0.5868003368377686, -0.3367791473865509, -0.8019363284111023, -0.8743544816970825, 0.578500509262085, 1.8649845123291016, 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없었기 때문에 예상되었던 것이다. 강성행렬의 제 1 행과 제3행은 강 \n성행렬에서 나타나 있듯이 ka(ul - u2) = kb(u2 - u3) = 0이 된다. 이것은 좌표들 사이에 \n서 상대운동이 일어나지 않는 것을 의미하고, 강체 이동에 해당되는 상태이다.\n만일 Ml=O이 되도록 1 인 점을 고정시키면, 행렬의 제 1 열은 없어질 수 있다. 두 개 \n단면을 가진 봉의 길이방향 진동에 관한 제2열 및 제3열은 다음 식으로 되어진다.\nIr 2(Ma + Mb)\n6 L Mb\nMql 이 +\n2Mb]{u3] 수\n(k1 + kb) -kh\n~kb kb\n특수한 경우 만일 Aa=Ab = A, ζ = ζ = IL이고 Ma = Mb = 이 면, 앞의 문제는 전체 길 \n이 L이고, 전체 질량 M인 균일봉 문제가 되고, 중간 지점에 좌표를 가지고 자유단을 가진 2 \n자유도계로 풀이될 수 있다. 그러면 앞의 문제의 방정식은 다음과 같이 된다.\nM∏4 llp2l + 2EAΓ 2 -l\"∣fw2l = fol\n지_1 2jU3∫ -L-L-I 1」Uj = IOj\n만일 우리가 λ = ω2ML∕24EA로 두면, 고유 진동수를 구하기 위한 특성 방정식은 다음과 \n같다.\n(2 — 4λ) -(1 + λ) _\n_(1 + λ) (1 - 2λ)\n\n\n334 ChaPterlo 유한 요소법 입문\n또는\nA2- 10 1 수\n■y λ - - = 0\n그 해는 다음과 같다.\n_ I 0.1082\nλ = 11.3204 ω —\n1.6115\n5.6293\nIEA\nMZ\nIEA\nML\n길이방향 진동에서 균일봉의 고유 진동수는 알려져 있고, 식 ωz = (2n+l)(√2)√EA/ML \n으로 주어져 있다. 처음의 두 모드에 대하여 이 방정식으로부터 계산한 결과는 다음과 같다.\n1.5708\nω —\n4.7124\n^EA\nIEA\nX7Z\n둘 사이를 비교하면 2자유도 유한 요소 모델의 결과와 연속 모델의 결과 사이의 일치는 \n1 차 모드에 대해서는 2.6% 높고 2차 모드에 대해서는 19.5% 높게 나타난다. 세 개 요소 모 \n델은 물론 더욱 가까운 일치된 값을 주리라고 기대되어 진다.\n변수의 성질 변수의 성질 문제에 대한 한 가지 단순한 접근은 짧은 길이를 가진 많은 요 \n소를 사용하는 것이다. 그러면 각각의 요소에 대한 질량과 강성의 차이는 매우", "vector": [0.43687447905540466, 0.4800492525100708, -2.213853597640991, -0.07480723410844803, 0.615935206413269, -0.8562817573547363, 1.2790515422821045, -0.08665214478969574, -0.7217844724655151, -0.1671793907880783, -0.42503103613853455, 1.4963643550872803, 1.6112574338912964, 0.09468381106853485, 0.003457044018432498, 0.18501387536525726, -0.6988611221313477, -0.7875200510025024, -0.35321253538131714, 0.9159902930259705, -0.4551399052143097, -0.28640520572662354, -0.764013409614563, -1.9057337045669556, 0.520989179611206, 0.567291259765625, -0.29961255192756653, 0.454805850982666, -0.5175756812095642, -0.7569369077682495, -0.8538672924041748, -1.5346013307571411, 0.45623794198036194, -0.22561189532279968, -2.065903663635254, -0.34046265482902527, 0.759392499923706, 0.41414546966552734, 0.9606385827064514, 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요소의 탄성계수 및 각 요소의 단면적 등을 입력하도록 요구한다. 그러면 모델에 대 \n한 질량행렬과 강성행렬을 구성하게 된다. 이 행렬이 3 X 3인 경우에 대해서는 식 (10.1.8) \n과 (10丄9)에 주어져 있다. 이들로부터 동행렬을 구성한다. 고유 진동수는 동행렬의 고유 \n값으로부터 구해진다. 이 문제에 대한 더 자세한 정보는 부록 F를 참고하라.\n\n\nιo.2 보요소에 대한 강성 및 질량 335\n10■기 보 요소에 대한 강성 및 질량\n보 강성 만일 요소의 끝단이 인접 구조에 단순지지되어 있지 않고 강하게 연결되어 있다 \n면, 요소는 결합부에서 모멘트와 축방향 힘이 작용하는 보와 같이 행동할 것이다. 일반적으 \n로, 상대 축방향 변위 U2 - WI은 보의 축방향 변위 V에 비하여 작게 될 것이고 0으로 가정할 \n수 있다. 보에 작용하는 힘과 모멘트뿐만 아니라 축방향 힘들도 고려되어야 할 경우에는 다 \n음에 보여주듯이 보 강성행렬에 더하는 것은 간단한 일이다.\n보 요소에 대한 국부 좌표계는 양단에서는 오직 축방향 변위와 회전이다. 우리는 이 토 \n론에서는 오직 평면구조만을 고려하고, 각각의 결합부는 축방향 변위 V와 회전 0를 하게 되 \n고 네 개의 좌표 v1, 01 과 v2, 仏를 가져온다. 이들 좌표계의 양의 의미는 임의이지만, 컴퓨터 \n의 계산을 목", "vector": [0.3512727618217468, -0.2951231598854065, -2.512925624847412, 0.5150349140167236, 1.1497023105621338, -0.35966601967811584, 0.8395098447799683, 0.9682655930519104, -1.3017048835754395, -0.6081584692001343, -0.816997230052948, 0.5497763752937317, 0.25030282139778137, -0.02159895747900009, 0.0652468279004097, 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변위 V와 회전 0를 하게 되 \n고 네 개의 좌표 v1, 01 과 v2, 仏를 가져온다. 이들 좌표계의 양의 의미는 임의이지만, 컴퓨터 \n의 계산을 목적으로 그림 10.2.1 의 선도가 대부분의 구조해석 공학자들에게 받아들여지는 \n것이다. 힘과 모멘트의 양의 의미 역시 같은 선도를 따른다.\n앞서의 변위들은 그림 10.2.2에 나타난 φ1(x), φ2(x), φ3W 및 少心)인 네 가지 모드 형상 \n의 중첩이라고 생각되어질 수 있다. 두 끝단에서 요구되는 힘과 모멘트는 제6장에서 구하\n보 변위 및 힘의 양의 방향\n그림 10.2.1\n그림 10.2.2\n\n\n336 아IaPterIO 유한요소법 입문\n였고, 그림 10.2.3에 계수 EIlf을 생략한 후 나타내었다. 이 그림으로부터 곧 힘-강성 관계 \n식을 구할 수 있다.\n(10.2.1)\n강성을 구하기 위한 식 (10.2.1)은 그림 10.2.3에 보인 것과 같은 주어진 힘과 모멘트에 \n서부터 구해진다. 질량행렬뿐만 아니라 강성행렬도 보의 형상함수 φ,(x)가 주어지면 포텐셜 \n에너지와 운동 에너지를 사용하여 구할 수 있다.\n보의 일반 방정식을 전개하면, 그것은 3차식이 되는데 처짐은 다음의 형태로 나타내어진다.\n沙U) = Pi 十 p2ξ + + (10.2.2)\n이 때\nξ= y 그리고Pz = 상수\n미분함으로써 기울기의 식을 구할 수 있다.\n∕∣9(X)=JP2 + 2p3ξ + 3p4ξ2 (10.2.3)\n\n\nιo.2 보 요소에 대한 강성 및 질량 337\n만일 경계조건으로 ξ = o과 ξ= 1 을 삽입하면, 경계 방정식은 다음의 행렬식으로 표시될 \n수 있다:\n”1 0 ! 0 O- \nOIlOO\nPI\nPi\nUl\nJg느 \nv2 己¾ >\n(10.2.4)T\"^i'i'ι\"\"T \n0 112 3\nP3\n< 乃4 >\n위에 보여진 것과 같은 구역지어진 행렬로부터, Pl과 P2는 단위행렬에 의하여 VI 및 ∕01 과 \n관련되는 것이 분명하다. 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PX = Vl과P2 = lθl을 대입하면 행렬의 마지막 두 열을 P3와P4로 쉽 \n게 풀이할 수 있다. 그렇게 하면 식 (10.2.4)의 구하고자 하는 역행렬은 다음과 같다.\n1 0 ! 0 0“PI\nP2_l\nP3 I\nPJ\nυ∖ \nlθλ0 1 : 0 0\n(10.2.5)\n-3-2∣3 -1\n2 1 ! -2 1\nυ2\n< 1어2 >\n이 방정식은 각각의 변위를 1 로 두고 다른 것을 0으로 둠으로써 A의 결정을 가능하게 한 \n다. 즉, 다른 모든 변위는 0으로 두고 Vl(X) = 1 인 경우에 대하여, 식 (10.2.5)의 제 1 행을 얻 \n을 수 있다.\nPl = 三 P2 = 0, P3 = —3, 그리고 p4 = 2\n이들을 식 (10.2.2)에 삽입함으로써 그림 10.2.2의 첫 번째 모양에 대한 형상함수를 구한다.\n 내적하면 다음 식 \n을 얻는다.\nu1(i∙i) + υ1(j∙i) = w1(i∙i) ÷ Vl(I i)\n즈\n+ 0 = COS α + 引 Sin a \n다음으로 드를 내적하면 다음 식을 얻는다.\n0 + υ1 = -UX Sin α + υ1 COS a \n그리하여 우리는 이들 결과를 다음의 행렬식으로 나타낼 수 있다", "vector": [0.374938040971756, -0.29427847266197205, -2.155832052230835, -0.4999278485774994, -0.20558196306228638, -1.0984606742858887, 0.9602518677711487, 0.35870465636253357, -0.388677179813385, -0.1431254744529724, -0.30193018913269043, 1.460297703742981, 1.0682638883590698, 0.22707094252109528, 0.02747572399675846, -0.24177296459674835, -0.4092448055744171, -0.15368299186229706, 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Sin α + υ1 COS a \n그리하여 우리는 이들 결과를 다음의 행렬식으로 나타낼 수 있다.\n(Q)\n그림 10.3.1\n(b)\n\n\n340 아IaPterlO 유한 요소법 입문\nSIn a\nCOS a\n1:}\n(10.3.1)\n앞의 식은 국부 좌표 w1, Vl을 전체 좌표 h1, 구의 항으로 표현하였다. 이들 결과는 그림 \n10.3.1(b)로부터 기하학적으로 손쉽게 확인되어진다.\n마찬가지로, 국부 좌표에서 결합부 ②의 변위는 동일한 변환식에 의하여 전체 좌표의 항 \n으로 표현되어질 수 있다. 두 좌표계에 대한 회전각은 물론 동일해야 만하고, Θ = 규로 된다. \n그리하여 우리는 변환행렬에서 0를 다음과 같이 포함시킬 수 있다.\n[ U COS a Sin a 0^ U\n► — -Sin a COS a 0 < V > z = 1, 2 (10.3.2)\ni 0 0 2_ i\n그리하여 수평에 대하여 반시계방향으로 측정하여 각도 α를 만듦으로써 임의 요소에 대한 \n변환행렬은 다음과 같게 된다.\n'w1' C S 0 r 示 >\n이 1 -S C 0 0\n( 어\n∖----- > =\n0 O 1\n0u2 C S 示 2\n이 2 0 I -S C 0\nW ! 0 0 1_\n(10.3.3)\n여기서 C = COS a 및 5ι = Sin 사!이다. 변위에 대하여 유도된 변환행렬은 마찬가지로 힘 벡터 \n에 대해서도 적용되어질 수 있다는 것을 쉽게 확인할 수 있다.\n더욱 단순한 표기로서, 우리는 국부에서 전체 좌표로 변환식을 다음과 같이 다시 쓸 수\n있다.\nr = Tr\n(10.3.4)\nF=TF\n여기서 T는 변환행렬, r, F 및 F, W 각각 국부 및 전체 좌표에서의 변위와 힘 이다. 우리는 \n이것을 r과 F사이의 관계에 더하게 되는데 이것은 강성행렬이다.\nF= kr (10.3.5)\n그리고 그것은 전체계에서는F= 方로 쓸 수 있다. 식 (10.3.4)에서 우리는 다음 식을 얻는다.\nP - τ~λF = TTF (10.3.6)\n\n\nιo.4 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O .0\n~τ -10:10\nO O ∙ O 0_\n_ ml \"2 Γ\nU) = ~6 _1 2_\n(10.4.2)\n그러면 이들 4X4 행렬은 그것을 전체 좌표계로 변환하기 위하여 식 (10.3.8)에 삽입되 \n어질 수 있다. 2\n\n\n342 ChaPterIO 유한요소법 입문\nCS ! -C2 -CS\nCS S\n-CS I C\n-S2 ； CS\n(10.4.3)\nm — TTmT — —\n6\n2c2\nIcs\n2cs ! c2 —I \n2s2 I CS_ _ S_\n(10.4.4)\nCS s2 I 2cs 2s2\n그림 10.4.1 의 힌지로 지지된 변길이가 3:4:5인 직삼각형 트러스에 대한 강성행렬을 \n구하라.\nWU 구조는 결합부 1, 2, 3을 가진 세 개의 단순지지 요소 a, b, C로 구성되어져 있 \n다. 각각의 결합부는 전체계에서 2자유도를 가지고, 여섯 개의 힘과 변위관계는 다음 \n식으로 나타낼 수 있다.\n아 \n푸 \n이 2\n＜石3；\n각각의 요소의 전체 강성은 특정 요소에 대하여 Sin α와 CoS α를 삽입함으로써 식 \n(10.4.3)으로부터 구해진다.\n요소 이(1 에서 2):\n\n\nιo.4 전체 좌표계에", "vector": [-0.17555753886699677, -0.1394663006067276, -2.1225357055664062, -0.4636605978012085, -0.14054478704929352, -0.6189761161804199, 1.3948723077774048, 0.07942894101142883, -1.152541160583496, -0.4235965311527252, -0.5288134813308716, 0.7248773574829102, 0.48925861716270447, -0.28897830843925476, 0.03989133983850479, -0.34270164370536804, -0.24767258763313293, -0.47550860047340393, 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2\n＜石3；\n각각의 요소의 전체 강성은 특정 요소에 대하여 Sin α와 CoS α를 삽입함으로써 식 \n(10.4.3)으로부터 구해진다.\n요소 이(1 에서 2):\n\n\nιo.4 전체 좌표계에서의 요소강성 및 요소질량 343\n4\n5\n3\n5\n요소 b（2에서 3）:\nC=-1, 5 = 0\n요소 c（3에서 1）:\nC = O, 5 = — 1\nVC\n이들은 이제 6X6 강성식으로 구성되어져야만 한다. 이와 으에 대한 행렬은 공통의 \n변위 ft］를 갖고, 그것은 공통의 변위와 관련된 단면이 서로 겹치는 것으로 쉽게 알 \n수 있다.\n16 12 -16 \"I -12\n/— 、\nUl\n! 12 9 -12 ； -9\n引\n으外 !-16 -12「16+ 쓰서 12 -V 0^! U2\n251） '-12 -9 : 12 J 9 0 0 ' >2\n： -ψ ’ O ψ 0 U3\nL 0 0 OOJ _\n\n\n344 아IaPterlO 유한요소법입문\nE에 대해서 적절한 위치를 찾기 위하여, 네 개의 2X2 행렬로 분리되어질 수 있고, \n그것은 다음과 같이 정리되어질 수 있다.\n—()\n0\n0 !\n125 I\n3 '\n!0\nIO\nO_ 끄\nT\n/— X\nA(≡\nI I\n우 ►\nO\nO\nO !\n125 '\n““I\n!θ\nIo\nO\n125\n示3\n패3>\n이들 세 개 행렬을 합침으로써, 우리는 트러스에 대한 강성행렬을 다음과 같이 구하게 \n된다.\nFIX\nΓ ! ! 1 / \\\nUI\n흐 _\n16 12 : -16 -12 ： 0 0\n12 9 + g引 -12 -9 ! O -ψ\n< F2x\n> = 씌 -16 -12 ; 16 + ψ 12 -ψ O\n< 示 2 I\n(25// -12 -9 : 12 9 ! O 0\n흐\n0 O ! O ψ 0\nO -ψ ! O Oio ψ\n_ 3 I I 3J\n<⅝>\n이것은 다음과 같이 변환되어 진다.\n이제 변위 O의 조건을 결합부 1 과 3에 적용시키면, 그것은 제 1, 2, 5 및 6열을 완전히 \n없애고 다음의 식으로 남게 된다.\n= 9 -12 0 1\n∖v2} ∖EA) 281.25 L —12 47.25J I-PJ\n\n\nιo.4 전체 좌표계에서의 요소강성 및 요소질량 345\n그러므로 결합부 2의 수직 및 수평방향 처짐은 다음과 같다.\n\"2 = （灰끄日4）（12月） = 1∙066 으\n¾ = （ 자끚司 ）（—47.25P） = -4.", "vector": [0.1335228830575943, 1.1895394325256348, -1.702836513519287, -1.4905263185501099, -0.48277974128723145, -0.07064646482467651, 0.6069087386131287, -0.633611798286438, -0.4051816761493683, -0.5178383588790894, -0.21103478968143463, 0.2993450164794922, -0.6421856880187988, -1.0285431146621704, 0.5731133818626404, -0.5999580025672913, 0.6707450747489929, -0.8183709383010864, 0.07227342575788498, 1.5154497623443604, 0.23280486464500427, 0.3117016851902008, -1.1915007829666138, -1.1249793767929077, 0.021586934104561806, 0.4885926842689514, -0.36206045746803284, -0.20120005309581757, -0.6701886653900146, -0.8525338768959045, 0.23305538296699524, -0.512428343296051, 0.8116117119789124, 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고정된 핀에 관한 모멘트를 취함으로써 쉽게 구할 수 있지만, 본 예 \n제는 더욱 복잡한 구조물인 경우에 따라해야 할 일반적인 절차를 나타내었다.\n0 0 O Ul \n사 \n아 \nU2 \n'으 \n#2\nml\n~6\n보 요소 보 요소에 대한 강성 및 질량행렬은 4 X 4차원이고, 이 때의 변환 행렬은 6 × 6이 \n다. 그리하여 이들 행렬을 전체 좌표로 변환하기 위하여, 축방향 성분을 더함으로써 다음과 \n같이 재구성하여 그들을 변형시킬 필요가 있다:\nEA\n느 1 0 ：-1\n0 O 0 I 0 0 0\n0 O O ! 0 0 0\n1 0 O ! 1 0 0\n0 0 0 : O 0 0\n0 0 O ! 0 0 0\n-2 0 O I 1 0 0\"\n0 O O ! O 0 0\n0 0 0： O 0 O\n1 0 O ! 2 0 O\n0 0 0 O 0 O\n_0 0 0 ! 0 0 0_\n그러면 변환에 사용되는 요소행렬은 다음과 같다.\n\n\n346 ChaPter 10 유한 요소법 입문\n느 R O O : —R O O 느\nO 12 6/ ! O -12 6/\n, EIk= 下 0 61 4/2 ； O -61\n의1__\n(10.4.5)\n-R O O ' R O 0\nO -12 -6/ ! 0 12 -61\n_ 0 61 2/2 : 0 -6/ 4匕 _\n여기서 ☆（을1）（⅛） = 쓰이다.\nN O OjPVo 0\n0 156 22/ '' O 54 -13/\nml 0 22/ 4/2 ： 0 13/ —3/2 (10.4.6)m =\n420 ∖N 0 O O\n0 54 13/ : O 156 -221\nO —131 -3l2∖ O —22/ 4Z2\n여기서 N= （쯔）（쯔） = 140이다.\n이들 6 × 6 요소행렬은 식 ;i = T7）cT와 m = T7mT에 의하여 전체 좌표（문자 위에", "vector": [0.079866424202919, 0.6693630218505859, -1.829393744468689, -0.5101503133773804, 0.1569349765777588, -0.6742144227027893, 0.8057575225830078, 0.4139045476913452, 0.14412032067775726, -0.4908224046230316, -0.054737892001867294, 0.25264957547187805, 1.0725936889648438, -0.011733783408999443, -0.4288097321987152, -1.0192502737045288, -0.1235065832734108, -0.5079165697097778, -0.45572879910469055, 0.8539738655090332, -0.8873613476753235, -0.3348623514175415, -0.9382326006889343, -0.8868852853775024, -0.737466037273407, 0.5963252782821655, 0.23169206082820892, 1.1640704870224, -0.18963558971881866, -0.3873831033706665, 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(―7? ÷ 12)c5 (-Rs2 - 12c2) GlC\nEl —6/5 Glc 4/2 6/5 -6lc\n__ 기2-\nTr (-Rc2 - 1252) (-R ÷ 12)α 6/5 [ (Rc2 + 1252j (R - 12)cs 6/5\n(-7? + 12)cs (-Rs2 — 12c2) —61C ； (R - 12)CS (Rs2 + 12c2) -6lc\n_ —61S 6lc 2l2 6/5 -6lc 4/2\nU\nV\nθ(10.4.7)\nm ml\n420\n-OVC흐 + 156유) OV - 156)CS —22IS \\! (PVC2 + 5¾2) (뉴 N - 54)α 1%\nOV - 156)c5 (M2 + 156c2) 22lc I! G/V —54)c5 GM그 + 54c2) -13/c\n22lc 시2 I — 13/5 13/c -3∕2\n(IyVC2 + 54^2) (PV- 54)cs -13/5 \\ (NC2 + 15652j (N - 156)CS 221S\n心 N - 54)C5 (IM2 + 54c2) 13/c ； (N — 156)CS (NS2 + 156c2) -22lc\n— 13/c -3∕2 I MlS -22∕c 4/2\n(10.4.8)\n10■디 보 요소를 포함하는 진동\n보에 대한 유한 요소법의 예로서, 제6장과 제7장에서 풀이한 몇몇 문제들을 고려하자. 이 \n곳에서의 목적은 첫째로 두 개의 요소를 사용하여 어떻게 계의 식을 조합하는가를 보이는 \n것이고, 둘째로 회전좌표를 제거함으로써 수식의 자유도를 줄이는 것이다.\n\n\nιo.5 보 요소를 포함하는 진동 347\nEHESk\n그림 10.5.1 에 보여준 보는 길이 ｛인 두 개의 동일한 요소로 간주되고, 그의 강성과 질 \n≡ 량행렬은 식 (10.2.1)과 (10.2.10)으로 주어진다. 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0.44422397017478943, 0.5588364005088806, 0.747902512550354, 0.8183513283729553, -0.05049830302596092, -0.190678671002388, -0.1647704839706421, -0.12072351574897766, -0.31045350432395935, -0.9985982775688171, -0.17403265833854675, -0.358766108751297], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:13", "text": "전좌표를 제거함으로써 수식의 자유도를 줄이는 것이다.\n\n\nιo.5 보 요소를 포함하는 진동 347\nEHESk\n그림 10.5.1 에 보여준 보는 길이 ｛인 두 개의 동일한 요소로 간주되고, 그의 강성과 질 \n≡ 량행렬은 식 (10.2.1)과 (10.2.10)으로 주어진다. I 대신 $을 삽입함으로써, 요소행렬은 \nI 다음과같다.\n： 요소 a:\n강성 (쪼)\n느 12\n_ 3/_\n3Z\nI2\n: -12\n! -3/\n3/ 느\n변위 벡터 ‘ 이 > \n的\n< ¾>\n-12\n_ 3/\n—3/ \n0.5/2\n: 12\n! -3/\n-3/\nI2 _\n질량 G\n끄斗\nr\n156\n11/\nIIZ \nZ2\n54\n[ 6.57\n—6.5/ \n-0.75/2\nZ40/ 54 \n-6.5/\n6.5/ \n-0.75Z2\n! 156\n! -11\n-11/\nI I2\n(α) (b)\nd=. 1)\n(1) (2)\nCi==d)\n(2) (3)\n그림 10.5.1\n요소 方: 변위 벡터를 제외하고는 요소 이와 동일하다. 변위 벡터는 다음과 같다.\n的\nl¾J\n보의 축과 일치하는 전체 좌표를 가지고, 계행렬의 조합은 단순히 요소 시와 그에 대한 \n이전의 행렬을 6X6 행렬로 중첩시키는 것이다. 그것은 강성행렬에 대해서는 다음과 \n같다.\n\n\n348 아IaPterlO 유한 요소법 입문\n요소 α\nL 요소占_!\n벽의 구속으로 인하여 y1 = 01 =O이기 때문에, 처음의 두 열은 무시될 수 있다. 또한 \n진동문제에서는 힘과 모멘트 Fl과 MI에 대해서도 모두 관심이 없다. 그러므로 처음 \n두 열뿐만 아니라 처음 두 행도 제외시킬 수 있어서 다음의 방정식으로 된다.\nml\n^ 312\n_ 0__\n0 !\n2l2 I\n54 \n6.5/\n-6.5/ ^\n-0.75/2\n840 54 \n-6.5/\n6.51 \\\n-0.75Z2 !\n156 \n-IlZ\n-IlZ\nZ2 b\n(10.5.1)\n보의 자유진동에 대하여 풀면, 힘 벡터는 0으로 되고 가속도 벡터는 -ω2에 변위를 곱 \n한 값으로 대치되어진다.\n^ 24 0 ! -12 3/ 더\n0 2Z2 ' -3/ 0.5/2 -¾->-12 -Z/ : 12 -Z/\n_ 31 0.5Z2 ! -3Z I2 _ l⅛J\n아 \n하 \nV2-\n名\"\n컴퓨터 프로그램 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0.03440294414758682, 0.38911712169647217, 1.584021806716919, -0.48519760370254517, 0.006370396353304386, -0.2739867568016052, 0.24476440250873566, 0.2955772280693054, -0.3806958794593811, -0.4602702558040619, -1.1686159372329712, -0.34715235233306885], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:14", "text": ": 12 -Z/\n_ 31 0.5Z2 ! -3Z I2 _ l⅛J\n아 \n하 \nV2-\n名\"\n컴퓨터 프로그램 beam.m 프로그램은 외팔보에 대하여 유한 요소 모델에 대해 결정되어 \n지는 고유 진동수를 계산하는 MATLAB®으로 쓰여진 파일이다. 사용자에게 보의 길이, 원하 \n는 요소의 수, 보의 질량, 보의 탄성계수 및 보의 관성 모멘트를 입력하라고 요구한 뒤, 프 \n로그램에서 모델에 대한 질량 및 강성행렬을 구성하게 된다. 두 개의 동일한 요소로 이루 \n어진 보에 대하여, 이들 행렬은 식 (10.5.1)과 같이 구해진다. 그러면 동행렬은 이들 두 행 \n렬에서부터 만들어진다. 동행렬의 고유값은 계산되어지고 모델의 고유 진동수를 구하는 데 \n사용되어진다. 프로그램에 관한 더욱 자세한 정보는 부록 F에 나타나 있다.\nEElEEBL 좍표저갈\nI 앞의 문제의 해를 구하는 데는 고유값-고유 벡터 관련 컴퓨터 프로그램이 필요하다.\nI 그러나 우리는 결합부 2와 3에서 균일 분포질량을 집중질량으로 대체함으로써 더욱 \nI 단순화된 문제로 만들 수 있다. 그러면 질량행렬은 요소 W2와 \"수를 제외하고는 모두 \n너 0인 값으로 된다. 이것은 변위 벡터를 정돈된 순서로 하기 위하여 앞의 식을 정리하는 \n! 것을의미한다.\n月\nAz2匕凶\n+\n\n\nιo.5 보 요소를 포함하는 진동 349\n이것은 단순히 제2열 및 제3열과 제2행 및 제3행을 서로 바꿈으로써 되어지고, 다음 \n과 같은 식으로 되어진다:\n이제 식은 다음과 같은 형태가 된다.\nP知 l0J∣∣-4 + [:K브丄⅛]μg = IOl \nL 0 ! 이 IdJ L尺21 ! ^22JbJ IOJ\n이는 다음과 같이 쓸 수 있다.\nMlIV ÷ KlIV+ Kγ2θ = 0\nTC21V + K22θ = 0\n두 번째 식에서 0는 V로 나타내어질 수 있다:\nθ = -TC221K21V\n첫 번째 식에 대입하면 다음으로 나타내어진다.\nΛf11V ÷ (K11 -K12TC221 K21)V = O\n원래의 항으로 표현하면 다음과 같다.\nm2 θ]∫⅛] (8EI∖ Γ 24 -12\nO λh3JI⅛J 十 ( Z3』_-12 12\n0 3/ Il2 0.5/2\n-3/ -3∕J∣0.5∕2 I2\nO\n3Z\n다음 항은 저감 강성(reduced StiffneSS)이고,\n尺11 _ 尺\\2尺』尺2↑\n곱해졌을 때 그 값은 다음과 같다.\n(10.5.2)\n(10.5.3)\n(10.5.4)\n\n\n350 ChaPterlo 유한요소법 입문\n8EI 96 -30", "vector": [0.9459292888641357, 0.9771109819412231, -2.31052827835083, -1.0079283714294434, 0.5766392946243286, -0.3289942145347595, 0.751593828201294, 0.7628238797187805, -0.3739086985588074, -0.7564211487770081, -0.25119394063949585, 0.6997271180152893, -0.013246648944914341, -0.281122624874115, 0.9054800271987915, -0.31337296962738037, 0.05480808764696121, 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강성(reduced StiffneSS)이고,\n尺11 _ 尺\\2尺』尺2↑\n곱해졌을 때 그 값은 다음과 같다.\n(10.5.2)\n(10.5.3)\n(10.5.4)\n\n\n350 ChaPterlo 유한요소법 입문\n8EI 96 -30 48£7 16 -5규\n-30 12 -5 2\n그러므로 원래의 4X4식은 2X2식으로 저감되어지고, 최종의 형태는 다음과 같다.\n7 ：1}+(쯔T： 기1：：}={：}\n수용가능한 이산질량 분포는 각 요소의 질량이 요소의 각 끝에 반씩 나누어지는 것이 \n다. 그리하여, 만일 길이 /인 균일보의 전체 질량은》이고, 그림 10.5.2에 보인 것과 \n같이 각 요소의 질량은 m∕∕2이고, m2 — 2(m∕∕4) — ml/2 및 g = m∕∕4이다\n그림 10.5.2\n균일 외팔보의 두 요소 이산질량 모델\n운동 방정식과 해는 다음과 같다.\n여기서\n시::］丄: 기］{::}={:}\nω1ml\n4\n7Z3\n48£7 ω⅛\nml4\nIi\nA1 = 0.3632 ω1 = 3.516 엄밀값 = 3.516\nA2 = 9.637 ω2 = 22.033 엄밀값= 22.034\n_ ∫0.3271 _ ∫-1.5271어 —(1.0Ooj ≠2 - I 1.0OOJ\n예제 10.5.3 \n동일 요소로 이루어진 문형 구조(Portal frame)의 자유 진동식을 구하라.\nMV 그림 10.5.3에 보인 것과 같이 결합부의 번호를 붙임으로써, 각 요소에 대한 강 \n성과 질량은 식 (10.4.7)과 (10.4.8)로 나타내어진다. 결합부 0와 3은 변위가 0 값을 가 \n지므로, 우리는 결합부 1 과 2에 대한 항만을 쓴다.\n\n\nιo.5 보 요소를 포함하는 진동 351\n요소 0—1, a — 90o, c = 0, 占 = 1:\n스0-1\nEI下\nI -12\nI 0 \n! 6/\n0\n- R\n0\n- 6/“ \n0\n2/2! 으 0 6/\n: 0 R 0! 이 0 4/2_\nml\n! 54\nI 0\n! -13/\n0\n0\n13厂 \n0 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N= 140⅜ 대입하여 자유진동에 대한 식을 세우면 다음과 같이 된다.\nml\n\"156 0 22/ I\nONOl\n221 0 _ 4/2_ ；_\n420 0\n이 행렬들을 조합하면 다음 식을 얻는다.\n요소 2—3, a — 270o, C = O, 占 = — 1:\nO\no\no\no\n=\n120\nlwl족\n -w2⅞\n^^296\n221 _\n221\n8/2\nI 70\n0\no’\n-3P_\n70\n_ 0\nO\n—3/2\nI 296\n； 22/\n22/\n8l2_\n브≈\n-wI石T -w2⅛\n^(\n12 ÷ 尺)\n6/\n6/\n8/2\n- -R\n0\n0^\n_ 기2_\n-R\n0\n0\n2l2\nj(12 +\n尺)\n61\n6/\n8/\\\n十\n으≈\n으≈\n-≡T\n - S r- Q r\n -w2½\n-\n⅛\n’(12 + \n0 \n6Z\n尺）\n(1 2\n0 \n+ \n6/\n尺） 6/\n6/\n8/2\nR \n0 \n0\n0\n12\n6/\n0 느 \n6/ \n_212__\nR \n0 \n0\n0\n12\n61\nO \n-6/ \n2Z2\nI (12 ÷ \n0 \n61\n尺）\n(12\n0 \n+ \n6/\n尺） 6/\n-6/\n8/2 _\n_£ _\n의\n그\n머\n⅛\n-\n½\n-\n¾\n‘(12 + \n0 \n6/\n尺）\n(12\n0 \n+ \n67\nR)\n61\n61\n8/2\nI - R \n0 \n0\n0\n12\n6/\n0 “ \n6/\n= _\nR\nO\n0\n0\n12\n6/\n0 \n-6/ \n2Z2\n!(12 + \n0 \n6/\nR)\n(12\n0\n+\n6/\nR)\n6/\n-6/\n8/2 _\n2\n \n2\n \n—2\n - \n3\n \n3\n \n)\n3\n \nW\n \n石》\n- 0\n -2\n \n石\n \n-\n0\n\n\n예제 10.5.4 \nιo.5 보요소를포함하는 진동 353", "vector": [-0.08480299264192581, 1.0308036804199219, 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-1.1819605827331543], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:17", "text": "61\n61\n8/2\nI - R \n0 \n0\n0\n12\n6/\n0 “ \n6/\n= _\nR\nO\n0\n0\n12\n6/\n0 \n-6/ \n2Z2\n!(12 + \n0 \n6/\nR)\n(12\n0\n+\n6/\nR)\n6/\n-6/\n8/2 _\n2\n \n2\n \n—2\n - \n3\n \n3\n \n)\n3\n \nW\n \n石》\n- 0\n -2\n \n石\n \n-\n0\n\n\n예제 10.5.4 \nιo.5 보요소를포함하는 진동 353\n그림 10.5.4는 문형 구조(POrtal frame)에 대한 자유진동의 최저차의 비대칭 및 최저차 \n의 대칭 모드들을 보여준다. 주어진 모드에 대한 고유 진동수를 구하라.\nWV 비대칭 모드 지점 1 과 2의 처짐 및 기울기는 處 = 必 및 @ = @2로 동일하다. 이 \n들 조건은 이전의 식에서 제3열을 제 1 열에 그리고 제4열을 제2열에 더함으로써 부과 되어질 수 있다. 이는 ｛스｝와 에 대하여 동일한 식으로 된다.\nω2ml Γ366 22/] £7 Γ12 6/ ^]][u↑∖ _ {01\n^420^ L 22/ 5∕2J + 7γL6∕ 10∕2Jjt⅛J = IOJ\nλ = ω2ml4∕420EI로 둠으로써, 이 식의 행렬식은3} 다음과 같다.\n(12 - 366A) (6 - 22λ)/ _\n(6 - 22Λ)/ (10 - 5λ)/2 = °\nA1 = 0.0245\nA2 = 2.543\n두 근을 구하면 다음과 같다.\nω1 = 3.21\nω2 = 32.68\n그림 10.5.4(a)에 보인 것과 같은 단순한 형상에 대응하는 최저차의 고유 진동수는 수 \n용할 만한 정확도를 가진다. 그러나 2차 비대칭 모드는 더욱 복잡한 형상을 갖게 되고, \n이 문제에서 사용한 몇 개의 지점으로 계산한 ω2는 정확하지 않을 것이다. 고차 모드 \n를 적절히 나타내는 데는 더욱 여러 개의 지점이 필요할 것이다.\nZ) 행렬식이 곱해질 때 Z2은 제거되어진다. 그리하여 λ1 및 X2의 값을 변화시키지 않고 주파수 방정식의 행렬에서 \nZ= 1.0을둘수 있다.\n\n\n그림 10.5.5의 외력을 전체계의 외력과 비교하면 다음과 같다.\nf 凡 + F2x]\n► — 녀\nr 야 '\n-Jzt1\nI J\n.—•사=\n354 아IaPterIO 유한요소법 입문\n- 그러면 λ와 ω는 다음과같다.\ni 대칭 모드 대칭 모드에 대해서는 Ml = W2 = 0 및 仏=—01 이다. 제1 열과 제 3 열을 지우 \nI 고, 제4열과 제2열을 제거하면, 別에 대한 오직 한 개의 식을 구하게 된다.\nEl\nω2ml Z X EI, ■\n— 즈5示 (IlZ)+", "vector": [0.21985791623592377, -0.06896553933620453, -1.9967114925384521, -0.3400980234146118, 0.45129045844078064, -0.4233298897743225, 0.5965010523796082, -0.05860976502299309, -1.3051912784576416, -0.04252035543322563, -0.5252266526222229, 1.6567597389221191, 1.427924394607544, 0.2793067991733551, 0.1652160882949829, -0.3301718235015869, 0.3695333003997803, -0.425748348236084, 0.05706166476011276, 0.09464696794748306, 0.6992003321647644, -0.608773410320282, -0.5983287692070007, -1.2774685621261597, 0.10571648925542831, 0.5993227362632751, 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경계조건을 ! 조사하고,강성행렬을 주어진좌표의항으로구하라.\n너 WD 요소의 신장이 없다는 조건 WI = U2⅛ 예제 10.5.3의 식 (C)에 제3열과 제 1 열을 \n서 더함으로써 만족되어진다. 이것은 신장의 항 R을 없앤다. 우리는 역시 제3행을 제 1행 \n너 에 더함으로써 강성행렬을 3X3행렬로 다시 쓸 수 있다:\n그림 10.5.5\n-W -Qr\n-¾ \n(\nI\nl\nl\n \n6\n2∕\n8∕ \n6\n8/\n2Z \n24\n6Z\n6/ \n끄\n ≈-\n、 \n>--- \n√ \n+ i\nλ\n∕1\nm2\n\n\nιo.6 구조물에서의 스프링 구속조건 355\n凡 = 0 및 日X = 야을 사용하면, 주어진 좌표와 주어진 하중의 항으로 나타낸 강성행렬 \n은다음과 같다.\n여\n—6/ —6/\n8/2 2/2\n2Z2 8/2\nU\n臥 스\nl¾J\n10■이 구조물에서의 스프링 구속조건\n제9장에서 스프링 구속조건들은 가상일에 의하여 일반화된 힘으로 취급되었다. 유한 요소 \n법의 경우에도 동일한 개념이 적용된다. 스프링의 작용점은 결합지점으로 선택한다. 그러 \n므로 전체 좌표에서 원래 구조에서의 하중은 스프링 힘으로 치환된다.\n스프링 힘은 항상 변위에 대하여 반대 방향이기 때문에, 결합부에서의 힘과 모멘트는 \n-kvi 또는 -Kθi로 줄어 들게 된다. 그러므로 방정식의 다른 변으로 이항되었을 때, 스프링 \n하중은 해당 강성항에 더해지게 된다.\n예제 10.6.1 \n그림 10.6.1(a)에 보인 선형인 회전 스프링을 갖고 균일보에 대한 강성행렬을 구하라.\nWV 우선 그림 10.6.1(b)에 있는 스프링이 없이 지점 2에 하중 P와 M이 작용하 \n는 보의 강성 행렬을 세우자. 각 단면 1-2 와 2-3에 대한 강성 행렬은 보요소 행렬식 \n(10.2.1)에서부터 세워질 수 있다. v1 = 01 = v3 = 03 = 0임을 주목하면, 우리는 좌표 v2 \n및 θ2와 관련된 행렬의 부분의", "vector": 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-0.5948840975761414, -0.5845783948898315, -0.10390608012676239, -0.775338351726532, -0.7136381268501282, -0.3132975697517395], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:19", "text": "대한 강성행렬을 구하라.\nWV 우선 그림 10.6.1(b)에 있는 스프링이 없이 지점 2에 하중 P와 M이 작용하 \n는 보의 강성 행렬을 세우자. 각 단면 1-2 와 2-3에 대한 강성 행렬은 보요소 행렬식 \n(10.2.1)에서부터 세워질 수 있다. v1 = 01 = v3 = 03 = 0임을 주목하면, 우리는 좌표 v2 \n및 θ2와 관련된 행렬의 부분의 값을 정할 필요가 있고, 그것은 다음과 같이 된다.\n그림 10.6.1\n\n\n지점 2에 작용하는 스프링들에 있어서, 힘 벡터는 다음 식으로 대치된다.\n스프링 힘을 식의 우측으로 보내면 다음 식을 구할 수 있다.\nPq _ EJl2(⅛ + ⅛) + ⅛ 쇠∕W)]∣히\n시- L -6GH) 1\n전체계에서 힘 호는 윗방향으로 양의 값을 갖고, 472는 반시계 방향으로 양의 값을 가 \n지므로, 앞의 식은 다음과 같이 정리되어진다.\n이는 스프링 구속조건을 갖는 보에 대한 강성행렬을 정의한다. 식으로부터 계는 \nIi = I2 = 1/2일 경우 비연성화가 되고, 그 경우 식은 다음과 같이 간단하게 된다.\n중심에서 처짐은 다음과 같다.\n_ _ (Pl3/Er) _ MliIEI\nVl = 192 + kl3∕EI 2 = 16/2 + Kl3/EI\n356 아IaPterlo 유한요소법 입문\n\n\nιo.6 구조물에서의 스프링 구속조건 357\n따라서 운동 방정식은 다음과 같다.\n0156\n20\n그리고 이 모드에 대한 고유 진동수는 다음과 같다.\n0.00521\n마찬가지로, 的에 대한 식은 다음과 같이 된다.\n- 그리고\nω1 = 22.37\n예제 10.6.2 \nkΓ\n156(Zl + Z2) \n—22(片 - ZD\n그리하여 유한 요소 접근에 있어서 1 차 모드에 대한 오차는 1.61%이고, 2차 모드에 대 \n한 오차는 33.9%이다. 보를 더욱 작은 요소로 나누면 이들 오차가 줄어들게 된다.\n또다시 좌표 v2 및 @2가 비 연성화된다. λ = ω2m∕4∕420E/로 둠으로써, 그에 대한 식은 \n다음과 같이 된다.\nkl3\nU\nω2 — 81.98\n-⅛(\"+2\n…+쯜\nEl\n+ π\nkp\n= 1.231 + 0.00641 —\n5 쓰 = 6167\n그러므로 두 고유 진동수는 구속 스프링에 의하여 증가된다. 만일 A = 尺=O이면, 고정 \n단을 가진 보의 정확한 고유 진동수는 다음과 같다.\nEI \nml4\nm\n420\n너 예제 10.6.1 에서 Z1 = ∕2 = Z/2인 경우 구속된 보의 고유 진동수를 구하라.\nI WV", "vector": [0.46023696660995483, 0.1656077653169632, -1.415460228919983, -1.1055184602737427, 0.6076514720916748, -0.23942168056964874, 0.7005326151847839, 0.0482882596552372, -0.48151394724845886, 0.18786579370498657, 0.12536492943763733, 1.1133838891983032, 1.4658993482589722, -0.40723779797554016, -0.3014642596244812, -0.6694104075431824, 0.7743754386901855, -1.1116911172866821, 0.474680632352829, 1.0804363489151, -0.7371329069137573, -0.7062344551086426, -1.3877043724060059, -1.5454612970352173, 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진동수를 구하라.\nI WV 이 값을 구하기 위하여 질량행렬이 필요하고, 식 (10.2.10)으로부터 다음과 같 \n≡ 이 표현할 수있다.\nω1ml\n-420^\nkl3\nEl\n0.0625\n0\n\n\n358 아IaPterlO 유한요소법 입문\nJq^기 일반화된 힘과분포하중\n제7장에서 논의한 바와 같이 일반화된 힘은 작용력의 가상일에서부터 구할 수 있다. 변위 \n가 다음과 같이 나타나 있을 때,\ny(x) — <∕>1(x)υ1 + <∕>2(x)^ι 十 φ3(x)υ2 十 ≠4(x)¾ (10.7.1)\n작용하는 분포력 P(X) 의 가상일은 다음과 같다.\nδW= [ P(X) δy(x) dx\nJO\n= δυ1 P(X)φ1(x) dx + δθ1 p(x)≠2(x) dx\nJO JO\n÷ δυ2 I p(x)≠3(x) dχ + 6Θ2 I p(x)ψ4(x) dx (10.7.2)\nJO JO\n식 (10.7.2)에서 적분표시된 것은 일반화된 힘이다.\n만일 끝단의 힘 F1, Ml, F2 및 M2에 동일 과정이 적용되면, 가상일은 다음과 같다.\nδw = Fl δυl + MI δθ1 + F2 δυ2 + MI δθ2 (10.7.3)\n앞의 두 경우에서의 가상일을 계산하면, 우리는 다음의 관계식을 얻을 수 있다.\nFI = P(X)ψ1(x) dx F2 - p(x)φ3(X) dx\njθ ⅛ (10-7.4)\nMi = P(X) <∕>2(x) dx M2 = P(X) ψ4(x) dx\nJO JO\n그리하여 분포하중에 대한 등가 유한요소하중은 지금 구한 일반화된 힘이다.\n예제 10.7.1 \n| 그림 10.7.1 에는 길이가 Z1 이고, 보의 바깥 반쪽 위에 균일 하중 P(X)=Plb/in를 받고 \n있는 외팔보를 나타내고 있다. 본 절의 방법을 사용하여 끝단에서의 처짐과 기울기를\nI 구하라.\nI WM 우리는 ①-②인 단일 요소를 사용하고, 강성행렬의 역행렬을 결정하자.\n! v1 = 02 = 0이므로, 식 (10.2.1)에서부터 강성수식은 다음과 같다.\n\n\nιo.7 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0.8399391770362854, -0.534338116645813, 0.005249244160950184, -0.6034844517707825, 0.17848028242588043, 0.16357465088367462, -0.2039090245962143, -0.5830338001251221, -0.4254564642906189, 0.014797031879425049], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:21", "text": "를 나타내고 있다. 본 절의 방법을 사용하여 끝단에서의 처짐과 기울기를\nI 구하라.\nI WM 우리는 ①-②인 단일 요소를 사용하고, 강성행렬의 역행렬을 결정하자.\n! v1 = 02 = 0이므로, 식 (10.2.1)에서부터 강성수식은 다음과 같다.\n\n\nιo.7 일반화된 힘과분포하중 359\n그림 10.7.1\n(이 _흐「12 -6∕1]∫띠\nIMj /?[-必 4/f JbJ\n수반연산 방법을 사용하면, 그 역은 다음과 같다.\n4zι 6zΓ∣{F2] \n¾∫ E∕12g[6∕ι 12_|IMJ\n식 (10.7.4)로부터 상당유한 요소 작용력은 다음과 같다.\nF2 = [ — PΦKx) dx= -p∖ Φ3(ξ)l1 dξ = -PIx f (3/ - 2ξ3) dξ= —\nJl2 J1/2 J1/2 JZ\n∫\n1\n1/2 - WfW = - 씨i/2(-f + 己 쌰 = 쁪此\n이 값들은 역식에 대입하면 다음과 같다.\nf 52 528 1 \nPK 32 1536 I \n12 EI\" 78 1056 |\n느 ~32Λ 十 I=J\n쑈＜\n48 EI\n5.125\n7.000\n이들 결과는 면적 모멘트법으로부터 구한 결과와 일치한다.\n\n\n360 ChaPterIo 유한요소법 입문\n10■이 변위에 비례하는 일반화된 힘\n일반화된 힘이 변위에 비례할 때, 자유진동을 위하여 강성행렬과 결합하기 위해 운동 방정 \n식의 좌측으로 옮겨질 수 있다. 본 절에서 제시되는 것은 다음의 두 경우이다.\n(1) 분포력이 보에 수직 일 때\n(2) 분포력이 보에 수평 일 때\n경우 1 식 (10.7.2)의 가상일의 항P(X)가Λx)y(x)에 의하여 대치되면, 다음의 식으로 된다.\n8W — f /(x)y(x) δy(x) dx (10.8.1)\nJO\n이 때 XX) = XZ≠⑷이다. 여기서 Φ는 보함수이고, 또 이는 식 (10.7.1)에서처럼 요소 끝단 처 \n짐이다.\nδW = 乞 乞 qj 6ql f f(x)φiφj dx (10.8.2)\nZ j Jo\n그리고 일반화된 힘은 다음과 같다.\nβ/ = 을Y= ∑ Qj f )WΦiΦj dx (10.8.3)\nj Jo\n이는 변위에 비례하는 것이다.\n예제 10.8.1 \n그림 10.8.1 은 보의 바깥 반쪽의 아랫부분에 탄성지지를 받고 있는 외팔보를 나타낸 \n다. 지지되는 강성은 —ky lbs∕in이고, 여기서 \") = —그로 일정하다. 이 때 운동 방정 \n식은 다음과 같다.\n그림 10.8.1\n\n\nιo.8 변위에 비례하는 일반화된 힘 361\n길이 /인 요소에 대하여 식 (10.8.", "vector": [-0.17929233610630035, 0.03934898599982262, -1.8330109119415283, -0.24470531940460205, 0.6243138313293457, -0.7065290808677673, 0.7196425199508667, -0.13031499087810516, 0.0814170092344284, -0.42877909541130066, -0.5642977952957153, 1.342734456062317, 0.8018938302993774, -0.3199140429496765, 0.5355207324028015, -0.002013757824897766, -0.29987749457359314, -0.10982385277748108, 0.5662893652915955, 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—ky lbs∕in이고, 여기서 \") = —그로 일정하다. 이 때 운동 방정 \n식은 다음과 같다.\n그림 10.8.1\n\n\nιo.8 변위에 비례하는 일반화된 힘 361\n길이 /인 요소에 대하여 식 (10.8.3)에 나타난 적분을 수행하면 다음의 식을 얻는다.\n~0.3714 0.524/ 0.1286 -0.03095/ '\n{Qi} = -kl 0.009524/2 0.03905/ -0.007143/2 H\n0.3714 -0.05238/ u3\n0.009524Z∖l⅛J\n/대신에 /∕2을 이 문제에 적용시키고 식의 좌측으로 옮기면, 보의 강성이 증가한다.\n경우 2 보에 평행한 분포력 P(X)JX는 P(X)dx ∙ δw(x)의 가상일을 하고, 이 때 M(X)는 \n변형 XX)에 의한 수평방향 변위이다. 변위 W(X)는 변형된 보의 수평 투영된 위치와 X\n축과의 차이와 같다.\nU(X) — J (ClS — dx) = j dx ∖∣1 •+■ j - dx — J ∖ya dr\n이 때 그은 X에 대한 가상변수이고 / = 砂/Jr이다. 그러므로 X의 가상변위는 다음과 같다.\nδw(x) = f {δy'2dr\nJO\n여기서 피적분함수는 다음과 같이 해석되어질 수 있다:\n오/2 = ；[(/ + 아')2 — /2]=/6/\n그러므로 분포력에 대한 가상일은 다음과 같다.\nδW = — [ P(X) f y,δy, drdx (10.8.4)\nJO JO\ny에 대하여 보함수의 항으로 대입하면 다음과 같다.\nδw \nQi = —\nP(X) φ'iφ,j drdx\nI Jo\nP(X) φ'φ- drdx\n) JO\nδw = - (10.8.5)\n(10.8.6)\ne\n 1\nβ\n2\nβ3β4\nΓ⅛⅜\n⅛\n⅞\n\n\n362 ChaPter 10 유한요소법 입문\n예제 10.8.2 \nI 회전 요소 여기서 관심이 있는 예제는 그림 10.8.2에 나타낸 각속도 ∩로 회전하는 \n≡ 헬리콥터 날개이다. 첫 번째 보 요소에 대하여 하중은 ∩2≡ 成이고, 식 (10.8.6)은 변 \n! 함없이 적용된다. 추가되는 요소에 대해서는, 보함수의 좌표와 확인하기 위하여 X좌표 \nI 는 새로운 요소의 시작지점부터 측정되어져야 한다. 요소에 작용하는 하중 은 단순히 \nI ∩2(Z,∙ + x)m 成이고, 여기서 I는 회전축에서부터 새로운 요소의 시작지점까지의 거리\nI 를나타낸다", "vector": [0.19502010941505432, 0.19514667987823486, -1.7533961534500122, -0.6270913481712341, -0.2405862957239151, -0.5023289918899536, 0.9844147562980652, 0.49141258001327515, -0.5658161640167236, -0.9778680205345154, -0.4919332265853882, 1.5203521251678467, 0.7918719053268433, -0.64851313829422, 0.0762743130326271, -0.010417365469038486, -0.22917711734771729, 0.10247458517551422, 0.27620729804039, 0.4046953022480011, -0.18085703253746033, -0.8045305013656616, -0.7850302457809448, -1.269372820854187, 0.8297248482704163, 0.6257882118225098, 0.01609181985259056, -0.38180842995643616, -0.9575079679489136, -0.7659577131271362, 0.38456982374191284, -0.365132212638855, 0.15457138419151306, -0.1609206199645996, -2.1216349601745605, -0.12678050994873047, 1.330959677696228, 0.15553414821624756, 0.6493310928344727, 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비례하는 일반화된 힘 363\nKV\n회전에 의한 항은 식 (10.8.6)으로 주어진 일반화된 힘 e로부터 찾을 수 있다. 그 평가 \n를 위하여 포함되어진 적분식은 다음과 같다.\n「/ [ φ,iφ,jldξ]ldξ\nOLJO -\nmΩ,2l I X I φ'iφ,j dr-dx — mΩ신 \nJo JO\n여기서\n少； = (—6ξ+6ξ2)}\n 수송할 수 있는 능력을 가진 큰 기종이 \n다. 모든 헬리콥터에서처럼 로터의 회전날개는 매우 유연하다. 그 회전속도는 날개 끝의 속 \n도가 음속보다 낮게 유지되는 요구조건에 의하여 제작된다.\n그림\n오일 승강장용 상용 헬리콥터(날개 폭 24 in, 길이 2", "vector": [0.202543705701828, 0.26754921674728394, -2.902200698852539, -0.30697718262672424, 0.03358953446149826, 0.10107557475566864, 0.6163579225540161, 0.11482098698616028, -1.7197763919830322, -0.38587892055511475, -1.2107956409454346, 0.4881288409233093, -0.34973734617233276, -0.06178019568324089, 0.6244078874588013, -0.8457543849945068, 0.4409564733505249, -1.677243709564209, 0.3639366626739502, 0.012869548052549362, 0.13709767162799835, 0.15526583790779114, -0.35860639810562134, -1.3707555532455444, 0.37671083211898804, 1.2633891105651855, 0.2227340191602707, 0.48465439677238464, -1.0517350435256958, -0.9753803610801697, 0.05519074946641922, 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0.5607739686965942, 0.05046740174293518, -0.6599812507629395, 0.46848762035369873, 0.2591957151889801, -0.8536597490310669, 0.13758589327335358, 0.39704641699790955, 0.30034980177879333, 0.04677649214863777, 0.12601880729198456, 0.6901310086250305, 0.0698540136218071, 1.0903949737548828, -0.14855004847049713, 0.25920310616493225, -1.0205984115600586, -0.2033350169658661, -0.9727770090103149, -0.6319360136985779, -0.5162640810012817], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:28", "text": "리콥터는 해변과 오일 작업대 사이에 재료와 작 \n업자를 운반하는 데 사용되는 최대 하중 6000 lbs> 수송할 수 있는 능력을 가진 큰 기종이 \n다. 모든 헬리콥터에서처럼 로터의 회전날개는 매우 유연하다. 그 회전속도는 날개 끝의 속 \n도가 음속보다 낮게 유지되는 요구조건에 의하여 제작된다.\n그림\n오일 승강장용 상용 헬리콥터(날개 폭 24 in, 길이 24 ft, 무게 각각 200 lb； 외팔보의 강성 \n은 선단에서 6 ft/100 Ib임; 헬리콥터의 총무게는 빈 상태일 때 7000 Ib이고 적재시 13,000 Ib임)\n11] COOK, R.D., COnCePtS and APPliCatiOnS Of Finite Element AnaIySis, NeW York: JOhn \nWiIey & Sons, 1974.\n[2] GALLAGHER, R.H., Finite Element AnalySiS FUndamentaI, EngIeWOOd Cliffs, NJ: \nPrentiCe-Ha11, 1975.\n[3] , K.C., EVANS, HR., GRlFFlTH, D.W., AND NETHEROOT, D.A., The Finite\nXul\n즈V / U O P\n3으UO\n PJo\n누XnP\n dp∣ :七Paj【3 OjOqJ\n\n\n370 아IaPterIo 유한요소법 입문\nEIement Method, NeW York: HaISted PreSS Book, JOhn WiIey & Sons, 1975.\n[4] YANG, T.Y., Finite EIement StrUCtUral AnalySis, EngIeWOOd Cliffs, W: PrentiCe-Ha11, \n1986.\n[5] WEAVER, W., AND JOHNSTON, P.R., StrUCtUral DynamiCS by Finite Elements, \nEngleWOOd Cliffs, NJ: PrentiCe-Ha11, 1987.\n[6] CLOUGH, R.W., AND PENZlEN, J” DynamiCS Of Structures, NeW York: McGraw- \nHill, 1975.\n[7] CRAIG, R.R. JR., StrUCtUral Dynamics, JOhn Wiley & Sons, 1981.\n10.1 한 쪽 끝단은 고정되고 다른 쪽은 자유이며, 사이지점이 고정단에서부터 Z/3인 \n지점에 있는 두 요소로 이루어진 균일봉에 대한 축방향 진동에서 두 고유 진동 \n수를 구하라. 그 결과를", "vector": [0.2551698088645935, -0.05154205858707428, -2.8952221870422363, -0.5189648866653442, 0.37736207246780396, 0.3553110361099243, 0.2790769040584564, 0.5232788920402527, -0.9366703033447266, -0.32645753026008606, -0.12198862433433533, 0.5350741744041443, 0.3612801432609558, 0.26070764660835266, 0.42074066400527954, -0.35065606236457825, -1.5199695825576782, -1.0823025703430176, -0.644412636756897, 1.0083115100860596, 0.42396116256713867, -0.5610737800598145, -0.7006188631057739, -1.5251951217651367, 1.1869843006134033, 0.2446696162223816, 0.34665173292160034, 0.25727394223213196, -1.221292495727539, -0.9616271257400513, -0.9358361959457397, -0.9229010343551636, 0.13526439666748047, -1.6659035682678223, -0.6841815710067749, 0.4375298321247101, -0.15748001635074615, 0.4662421941757202, 1.124903917312622, 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JR., StrUCtUral Dynamics, JOhn Wiley & Sons, 1981.\n10.1 한 쪽 끝단은 고정되고 다른 쪽은 자유이며, 사이지점이 고정단에서부터 Z/3인 \n지점에 있는 두 요소로 이루어진 균일봉에 대한 축방향 진동에서 두 고유 진동 \n수를 구하라. 그 결과를 사이지점이 중간에 선택되어진 경우와 비교하라. 지점의 \n위치의 선정과 관련하여 당신은 어떠한 결론에 도달하는가?\n10.2 그림 P10.2와 같이 테이퍼진 봉이 두 개의 균일 단면으로 모델링되었으며, 여기 \n서 E41 = 2EA2 및 m1 = 2m2이다. 길이방향 진동의 두 개의 고유 진동수를 구하라.\n서 W1,2771 α2, TnZ\nU.— t/2 —~∙L— 1/2 ———』\n그림 PIO.2\n10.3 각각 길이 /인 세 개의 축방향 요소를 사용하여, 길이 Z/3인 균일봉의 자유-자유 \n진동에 대한 식을 수립하라.\n10.4 균일축의 비틀림에 있어서 선형적 변화를 가정한 후, 비틀림 문제에 대한 유한 \n요소강성 및 질량행렬을 구하라. 이 문제는 축진동 문제와 동일하다.\n10.5 두 개의 동일한 요소를 사용하여, 비틀림 진동에서 고정-자유축의 초기 두 개의 \n고유 진동수를 구하라.\n10.6 비틀림 진동에서 두 개의 균일 단면인 경우에 대하여, 2자유도 집중질량 비틀림 \n계와의 유한 요소 관련식을 서술하라.\n10.7 그림 P10.7은 큰 쪽은 고정단이고, 다른 쪽은 자유단인 일정 두께를 가진 원뿔\n\n\n연습문제 371\n형의 관을 보여준다. 한 개의 요소를 사용하여 그의 길이방향 진동에 대한 식을 \n구하라.\n10.8 그림 P10.7의 관을 길이방향 진동에 있어서 동일 길이의 두 요소로 이루어진 문 \n제로 간주하여 식을 구하라.\n10.9 비틀림 진동에서 그림 P10.7의 관에 대하여 (a) 두 요소 (b) M단계의 균일 요소 \n를 사용하여 식을 구하라.\n10.10 그림 P10.10의 단순 구조는 단순지지된 결합부를 가지고 있다. 그 강성행렬을 \n구하라.\n그림 Pio?. 11\n10.11 단순지지된 트러스 그림 P10.ll에서 핀 ③은 고정되어 있다. ①의 핀은 수직통 \n로로 자유로이 움직일 수 있고, ②의 핀은 수평통로로만 움직일 수 있다. 만일 그 \n림에서와 같이 핀 ②에 힘 P가 작용한다면, W2와 Vl를 P의 항으로 구하라. 핀 ①, \n② 및 ③에서의 모든 반력을 계산하고", "vector": [1.0879185199737549, 0.6516323685646057, -2.146130323410034, -0.6738901734352112, 1.4157568216323853, -0.8546478152275085, 0.42789894342422485, 0.41460341215133667, -0.5207801461219788, 0.04881732165813446, -0.33652976155281067, 0.7332069873809814, 0.7398616671562195, 0.05849189683794975, -0.05806499347090721, -0.6790716648101807, 0.23656652867794037, -0.6424881219863892, -0.3455521762371063, 0.24761910736560822, 0.15444496273994446, -1.4215378761291504, -1.6389427185058594, -1.1201194524765015, 0.84775710105896, 1.1391040086746216, -0.17035731673240662, -0.046579692512750626, -1.161264419555664, -1.114126443862915, -0.2520742118358612, -0.7098807692527771, 0.22453723847866058, -1.1131693124771118, -1.5721355676651, -0.5680765509605408, 0.4491887390613556, 0.24741429090499878, 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-0.037443552166223526, 0.35517117381095886, -0.45399072766304016, -0.10354122519493103, 0.6497321128845215, 0.9343949556350708, 0.39748087525367737, 0.6417524814605713, -0.4759003520011902, 0.02456340193748474, -0.8907965421676636, 0.10295551270246506, -1.6952663660049438, -0.5225534439086914, -0.3121698796749115], "source": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.txt"} +{"id": "기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문:30", "text": "다. 그 강성행렬을 \n구하라.\n그림 Pio?. 11\n10.11 단순지지된 트러스 그림 P10.ll에서 핀 ③은 고정되어 있다. ①의 핀은 수직통 \n로로 자유로이 움직일 수 있고, ②의 핀은 수평통로로만 움직일 수 있다. 만일 그 \n림에서와 같이 핀 ②에 힘 P가 작용한다면, W2와 Vl를 P의 항으로 구하라. 핀 ①, \n② 및 ③에서의 모든 반력을 계산하고 평형조건이 만족되는지를 조사하라. 유한 \n요소법에 의하여 강성행렬을 인자 EAII로 유도하라.\n10.12 그림 P10.12에 보인 핀으로 연결된 사각 트러스에서, 전체 좌표로 요소강성과 \n질량 행렬을 결정하고, 전체 구조에 대하여 행렬들이 만들어지는 방법을 보여라. \n동행렬을 구성하라. 이것을 이용하여 이 구조에 대한 자유진동의 고유 진동수를 \n구하라.\n\n\n372 아IaPterIo 유한요소법 입문\n10.15 문제 10.14의 보에 대한 일관된 질량을 구하고 그 고유 진동수를 계산하라.\n10.16 그림 P10.16의 보에 대한 자유진동 방정식을 구하라.\n그림 P10.17\n10.17 한 개의 요소를 사용하여 그림 P10.17의 단순지지-자유보에 대한 운동 방정식, \n고유 진동수 및 모드 형상을 구하라. 참값과 비교하라.\n10.18 문제 10.17을 두 개의 요소를 사용하여 반복하라.\n그림 P10.13\n10.13 그림 P10.13의 단순지지된 트러스에서, 요소의 방향은 오직 세 개이다. 각 방향에 \n대한 강성행렬을 구하고, 전체계로 각각의 요소행렬들이 조립되는 빙법을 써라.\n10.14 두 개의 요소를 사용하여 양단이 고정되고 그림 P10.14와 같이 하중이 가해지는 \n균일보의 중간 지점에서의 처짐과 기울기를 구하라.\n½\n쇼\n쇼\n으\n、2//\n그림 P10.14\n\n\n연습문제 373\n10.19 문제 10.17을 여섯 개의 요소를 사용하여 반복하라. 참값과의 일치가 많은 수의 \n요소를 사용하면 향상되는가?\n10.20 그림 P10.20의 프레임에 대한 강성행렬을 구하라. 우측상부 끝단은 회전은 제한 \n되어 있지만 들어가고 나가는 것은 자유롭다.\n그림 P10.21\n10.21 그림 P10.21 의 프레임은 우측상부 끝단에서 회전과 이동이 자유롭다. 그 강성행 \n렬을구하라.\n10.22 그림 P10.22의 프레임에 대한 하중 작용점에서의 변형과 기울기를 구하라. 이 \n때 코너 각은 불변이라 생각하라\n10.23 문제 10.17의 단순지지-자유보가 그림 P10.23에서와 같이 단순지지", "vector": [0.03566126897931099, 0.45435646176338196, -2.1498913764953613, -0.9258386492729187, 1.2141332626342773, -0.4820806682109833, 1.2566546201705933, 0.6614838242530823, -0.5862761735916138, -0.42543283104896545, -0.37158843874931335, 0.7910647988319397, -0.07096617668867111, -0.3597128987312317, 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자유롭다. 그 강성행 \n렬을구하라.\n10.22 그림 P10.22의 프레임에 대한 하중 작용점에서의 변형과 기울기를 구하라. 이 \n때 코너 각은 불변이라 생각하라\n10.23 문제 10.17의 단순지지-자유보가 그림 P10.23에서와 같이 단순지지 지점에서 \n비틀림 강성 尺Ib ∙ in/rad인 스프링에 의하여 제한되어진다. 자체의 정지된 무게 \n에 의하여 보가 1/10회전하도록 수치적인 尺값을 구하고, 문제 10.18에 있는 것\n\n\n374 ChaPter 10 유한 요소법 입문\n과 같이 계산하라（이 때 한 개 및 두 개 요소를 사용하고 Smgii/EI= 1.0으로 한다）.\nEI\n그림 P10.23 그림 P10.24\n10.24 그림 P10.24에는 중간 지점에 선형 스프링 느를 가지고 우측 끝단에 비틀림 스프 \n링 尺를 가지는, 단순지지된 끝단을 가진 보가 나타나 있다. 두 요소 해석에 대한 \n강성행렬을 구하라.\n10.25 문제 10.24의 보에서 질량행렬을 구하고, 다음 식과 같을 때 모든 고유 진동수 \n및 모드 형상을 구하라.\nkl3\n~EΛ\n= ； 그리고 쓸; =》2\nk = 尺 =0이라둠으로써 문제 10.24의 해를 점검하라. 이 때 고유값은 단순지지- \n단순지지보의 고유값과 일치해야 한다.\n10.26 그림 P10.26 의 보에 대하여 유한 요소 운동 방정식을 구하라. kf∕*I= 1.0 및 \nKI3∕EI=2f 일 때의 보의 고유값과 고유 벡터를 구하라.\n10.27 그림 P10.27의 프레임에 대한 자유진동 방정식을 구하라.\n그림 PIOA27\n\n\n연습문제 375\n10.28 두 개의 요소를 사용하여, 그림 P10.28(a)에 보인 구간에 대하여 분포된 힘에 대 \n한 등가 결합부 하중을 구하라. 그림 P10.28(b)의 구간에 대하여 중간 지점에서 \n의 처짐과 기울기를 구하라.\n10.29 그림 Plo.29의 계에서 주어진 여섯 개의 좌표의 항으로 두 요소 식을 쓰고 고유 \n값과 고유 벡터를 구하라.\nIZl 化 匕\n사、 成木 =' +(10.2.4)T"^i'i'ι""T +0 112 3 +P3 +< 乃4 > +위에 보여진 것과 같은 구역지어진 행렬로부터, Pl과 P2는 단위행렬에 의하여 VI 및 ∕01 과 +관련되는 것이 분명하다. PX = Vl과P2 = lθl을 대입하면 행렬의 마지막 두 열을 P3와P4로 쉽 +게 풀이할 수 있다. 그렇게 하면 식 (10.2.4)의 구하고자 하는 역행렬은 다음과 같다. +1 0 ! 0 0“PI +P2_l +P3 I +PJ +υ∖ +lθλ0 1 : 0 0 +(10.2.5) +-3-2∣3 -1 +2 1 ! -2 1 +υ2 +< 1어2 > +이 방정식은 각각의 변위를 1 로 두고 다른 것을 0으로 둠으로써 A의 결정을 가능하게 한 +다. 즉, 다른 모든 변위는 0으로 두고 Vl(X) = 1 인 경우에 대하여, 식 (10.2.5)의 제 1 행을 얻 +을 수 있다. +Pl = 三 P2 = 0, P3 = —3, 그리고 p4 = 2 +이들을 식 (10.2.2)에 삽입함으로써 그림 10.2.2의 첫 번째 모양에 대한 형상함수를 구한다. + 내적하면 다음 식 +을 얻는다. +u1(i∙i) + υ1(j∙i) = w1(i∙i) ÷ Vl(I i) +즈 ++ 0 = COS α + 引 Sin a +다음으로 드를 내적하면 다음 식을 얻는다. +0 + υ1 = -UX Sin α + υ1 COS a +그리하여 우리는 이들 결과를 다음의 행렬식으로 나타낼 수 있다. +(Q) +그림 10.3.1 +(b) + + +340 아IaPterlO 유한 요소법 입문 +SIn a +COS a +1:} +(10.3.1) +앞의 식은 국부 좌표 w1, Vl을 전체 좌표 h1, 구의 항으로 표현하였다. 이들 결과는 그림 +10.3.1(b)로부터 기하학적으로 손쉽게 확인되어진다. +마찬가지로, 국부 좌표에서 결합부 ②의 변위는 동일한 변환식에 의하여 전체 좌표의 항 +으로 표현되어질 수 있다. 두 좌표계에 대한 회전각은 물론 동일해야 만하고, Θ = 규로 된다. +그리하여 우리는 변환행렬에서 0를 다음과 같이 포함시킬 수 있다. +[ U COS a Sin a 0^ U +► — -Sin a COS a 0 < V > z = 1, 2 (10.3.2) +i 0 0 2_ i +그리하여 수평에 대하여 반시계방향으로 측정하여 각도 α를 만듦으로써 임의 요소에 대한 +변환행렬은 다음과 같게 된다. +'w1' C S 0 r 示 > +이 1 -S C 0 0 +( 어 +∖----- > = +0 O 1 +0u2 C S 示 2 +이 2 0 I -S C 0 +W ! 0 0 1_ +(10.3.3) +여기서 C = COS a 및 5ι = Sin 사!이다. 변위에 대하여 유도된 변환행렬은 마찬가지로 힘 벡터 +에 대해서도 적용되어질 수 있다는 것을 쉽게 확인할 수 있다. +더욱 단순한 표기로서, 우리는 국부에서 전체 좌표로 변환식을 다음과 같이 다시 쓸 수 +있다. +r = Tr +(10.3.4) +F=TF +여기서 T는 변환행렬, r, F 및 F, W 각각 국부 및 전체 좌표에서의 변위와 힘 이다. 우리는 +이것을 r과 F사이의 관계에 더하게 되는데 이것은 강성행렬이다. +F= kr (10.3.5) +그리고 그것은 전체계에서는F= 方로 쓸 수 있다. 식 (10.3.4)에서 우리는 다음 식을 얻는다. +P - τ~λF = TTF (10.3.6) + + +ιo.4 전체 좌표계에서의 요소강성 및 요소질량 341 +여기서 우리는 변환행렬은 직교행렬이고 T-I = Tn)인 것에 주목하고자 한다. 강성식으로 +부터, F를 대치하고, r을 F의 항으로 바꿈으로써 다음 식을 얻는다. +2) 부록 C 참조 +(10.3.7) +그리하여 국부 좌표계에 대한 k는 다음의 식에 의하여 전체 좌표계에 대한『로 변환되어진다. +k = TTkT (10.3.8) +1(L41 전체 좌표계에서의 요소강성 및 요소질량 +축방향 요소 축방향 요소에 있어서, 요소 모멘트는 0이고, 끝단의 힘과 변위는 요소길이 +와 나란하다. 그러므로 오직 축방향 요소만을 포함한 계에 있어서는 6X6 변환행렬은 다음 +의 4 X 4 행렬로 축소되어진다. +(10.4.1) +우리는 축방향 요소에 대한 강성 및 질량행렬이 2 × 2차원이고, 그러므로 다음과 같이 +4X4 행렬로 다시 나타내어져야만 한다는 것을 주목한다. +W=M -11 UI +-1 1 U2 +0 I2 01 Wl +ml 0 0 I 0 0 이 1 +6 1 0 : 2 0 U2 +0 !0 00 Vl) +Ul +EA +1 0 : -1 o" +O O ! O .0 +~τ -10:10 +O O ∙ O 0_ +_ ml "2 Γ +U) = ~6 _1 2_ +(10.4.2) +그러면 이들 4X4 행렬은 그것을 전체 좌표계로 변환하기 위하여 식 (10.3.8)에 삽입되 +어질 수 있다. 2 + + +342 ChaPterIO 유한요소법 입문 +CS ! -C2 -CS +CS S +-CS I C +-S2 ； CS +(10.4.3) +m — TTmT — — +6 +2c2 +Ics +2cs ! c2 —I +2s2 I CS_ _ S_ +(10.4.4) +CS s2 I 2cs 2s2 +그림 10.4.1 의 힌지로 지지된 변길이가 3:4:5인 직삼각형 트러스에 대한 강성행렬을 +구하라. +WU 구조는 결합부 1, 2, 3을 가진 세 개의 단순지지 요소 a, b, C로 구성되어져 있 +다. 각각의 결합부는 전체계에서 2자유도를 가지고, 여섯 개의 힘과 변위관계는 다음 +식으로 나타낼 수 있다. +아 +푸 +이 2 +＜石3； +각각의 요소의 전체 강성은 특정 요소에 대하여 Sin α와 CoS α를 삽입함으로써 식 +(10.4.3)으로부터 구해진다. +요소 이(1 에서 2): + + +ιo.4 전체 좌표계에서의 요소강성 및 요소질량 343 +4 +5 +3 +5 +요소 b（2에서 3）: +C=-1, 5 = 0 +요소 c（3에서 1）: +C = O, 5 = — 1 +VC +이들은 이제 6X6 강성식으로 구성되어져야만 한다. 이와 으에 대한 행렬은 공통의 +변위 ft］를 갖고, 그것은 공통의 변위와 관련된 단면이 서로 겹치는 것으로 쉽게 알 +수 있다. +16 12 -16 "I -12 +/— 、 +Ul +! 12 9 -12 ； -9 +引 +으外 !-16 -12「16+ 쓰서 12 -V 0^! U2 +251） '-12 -9 : 12 J 9 0 0 ' >2 +： -ψ ’ O ψ 0 U3 +L 0 0 OOJ _ + + +344 아IaPterlO 유한요소법입문 +E에 대해서 적절한 위치를 찾기 위하여, 네 개의 2X2 행렬로 분리되어질 수 있고, +그것은 다음과 같이 정리되어질 수 있다. +—() +0 +0 ! +125 I +3 ' +!0 +IO +O_ 끄 +T +/— X +A(≡ +I I +우 ► +O +O +O ! +125 ' +““I +!θ +Io +O +125 +示3 +패3> +이들 세 개 행렬을 합침으로써, 우리는 트러스에 대한 강성행렬을 다음과 같이 구하게 +된다. +FIX +Γ ! ! 1 / \ +UI +흐 _ +16 12 : -16 -12 ： 0 0 +12 9 + g引 -12 -9 ! O -ψ +< F2x +> = 씌 -16 -12 ; 16 + ψ 12 -ψ O +< 示 2 I +(25// -12 -9 : 12 9 ! O 0 +흐 +0 O ! O ψ 0 +O -ψ ! O Oio ψ +_ 3 I I 3J +<⅝> +이것은 다음과 같이 변환되어 진다. +이제 변위 O의 조건을 결합부 1 과 3에 적용시키면, 그것은 제 1, 2, 5 및 6열을 완전히 +없애고 다음의 식으로 남게 된다. += 9 -12 0 1 +∖v2} ∖EA) 281.25 L —12 47.25J I-PJ + + +ιo.4 전체 좌표계에서의 요소강성 및 요소질량 345 +그러므로 결합부 2의 수직 및 수평방향 처짐은 다음과 같다. +"2 = （灰끄日4）（12月） = 1∙066 으 +¾ = （ 자끚司 ）（—47.25P） = -4.200 크 +∖ 281.25E√4 / EA +이들 값에서, 핀 1 과 3에서의 반력은 다음과 같다. +FyX — （ —— —16 X 1.066 - ÷ 12 × 4.200 —— — 1.333P +\ 251 ） L EΛ EA +FXy = 1.00OP +F3x = -1.333P +⅞ = 0 +물론 이들 반력은 고정된 핀에 관한 모멘트를 취함으로써 쉽게 구할 수 있지만, 본 예 +제는 더욱 복잡한 구조물인 경우에 따라해야 할 일반적인 절차를 나타내었다. +0 0 O Ul +사 +아 +U2 +'으 +#2 +ml +~6 +보 요소 보 요소에 대한 강성 및 질량행렬은 4 X 4차원이고, 이 때의 변환 행렬은 6 × 6이 +다. 그리하여 이들 행렬을 전체 좌표로 변환하기 위하여, 축방향 성분을 더함으로써 다음과 +같이 재구성하여 그들을 변형시킬 필요가 있다: +EA +느 1 0 ：-1 +0 O 0 I 0 0 0 +0 O O ! 0 0 0 +1 0 O ! 1 0 0 +0 0 0 : O 0 0 +0 0 O ! 0 0 0 +-2 0 O I 1 0 0" +0 O O ! O 0 0 +0 0 0： O 0 O +1 0 O ! 2 0 O +0 0 0 O 0 O +_0 0 0 ! 0 0 0_ +그러면 변환에 사용되는 요소행렬은 다음과 같다. + + +346 ChaPter 10 유한 요소법 입문 +느 R O O : —R O O 느 +O 12 6/ ! O -12 6/ +, EIk= 下 0 61 4/2 ； O -61 +의1__ +(10.4.5) +-R O O ' R O 0 +O -12 -6/ ! 0 12 -61 +_ 0 61 2/2 : 0 -6/ 4匕 _ +여기서 ☆（을1）（⅛） = 쓰이다. +N O OjPVo 0 +0 156 22/ '' O 54 -13/ +ml 0 22/ 4/2 ： 0 13/ —3/2 (10.4.6)m = +420 ∖N 0 O O +0 54 13/ : O 156 -221 +O —131 -3l2∖ O —22/ 4Z2 +여기서 N= （쯔）（쯔） = 140이다. +이들 6 × 6 요소행렬은 식 ;i = T7）cT와 m = T7mT에 의하여 전체 좌표（문자 위에 줄표시 +있는 것）로 변환되어진다. +' (Rc2+ 12s2) (R - 12)cs -6ls I (-Rc2 - 12s2) (-R + 12)CS -6ls' +(R - 12)cs (Rs2 ÷ Ylc2) 6lc ! (―7? ÷ 12)c5 (-Rs2 - 12c2) GlC +El —6/5 Glc 4/2 6/5 -6lc +__ 기2- +Tr (-Rc2 - 1252) (-R ÷ 12)α 6/5 [ (Rc2 + 1252j (R - 12)cs 6/5 +(-7? + 12)cs (-Rs2 — 12c2) —61C ； (R - 12)CS (Rs2 + 12c2) -6lc +_ —61S 6lc 2l2 6/5 -6lc 4/2 +U +V +θ(10.4.7) +m ml +420 +-OVC흐 + 156유) OV - 156)CS —22IS \! (PVC2 + 5¾2) (뉴 N - 54)α 1% +OV - 156)c5 (M2 + 156c2) 22lc I! G/V —54)c5 GM그 + 54c2) -13/c +22lc 시2 I — 13/5 13/c -3∕2 +(IyVC2 + 54^2) (PV- 54)cs -13/5 \ (NC2 + 15652j (N - 156)CS 221S +心 N - 54)C5 (IM2 + 54c2) 13/c ； (N — 156)CS (NS2 + 156c2) -22lc +— 13/c -3∕2 I MlS -22∕c 4/2 +(10.4.8) +10■디 보 요소를 포함하는 진동 +보에 대한 유한 요소법의 예로서, 제6장과 제7장에서 풀이한 몇몇 문제들을 고려하자. 이 +곳에서의 목적은 첫째로 두 개의 요소를 사용하여 어떻게 계의 식을 조합하는가를 보이는 +것이고, 둘째로 회전좌표를 제거함으로써 수식의 자유도를 줄이는 것이다. + + +ιo.5 보 요소를 포함하는 진동 347 +EHESk +그림 10.5.1 에 보여준 보는 길이 ｛인 두 개의 동일한 요소로 간주되고, 그의 강성과 질 +≡ 량행렬은 식 (10.2.1)과 (10.2.10)으로 주어진다. I 대신 $을 삽입함으로써, 요소행렬은 +I 다음과같다. +： 요소 a: +강성 (쪼) +느 12 +_ 3/_ +3Z +I2 +: -12 +! -3/ +3/ 느 +변위 벡터 ‘ 이 > +的 +< ¾> +-12 +_ 3/ +—3/ +0.5/2 +: 12 +! -3/ +-3/ +I2 _ +질량 G +끄斗 +r +156 +11/ +IIZ +Z2 +54 +[ 6.57 +—6.5/ +-0.75/2 +Z40/ 54 +-6.5/ +6.5/ +-0.75Z2 +! 156 +! -11 +-11/ +I I2 +(α) (b) +d=. 1) +(1) (2) +Ci==d) +(2) (3) +그림 10.5.1 +요소 方: 변위 벡터를 제외하고는 요소 이와 동일하다. 변위 벡터는 다음과 같다. +的 +l¾J +보의 축과 일치하는 전체 좌표를 가지고, 계행렬의 조합은 단순히 요소 시와 그에 대한 +이전의 행렬을 6X6 행렬로 중첩시키는 것이다. 그것은 강성행렬에 대해서는 다음과 +같다. + + +348 아IaPterlO 유한 요소법 입문 +요소 α +L 요소占_! +벽의 구속으로 인하여 y1 = 01 =O이기 때문에, 처음의 두 열은 무시될 수 있다. 또한 +진동문제에서는 힘과 모멘트 Fl과 MI에 대해서도 모두 관심이 없다. 그러므로 처음 +두 열뿐만 아니라 처음 두 행도 제외시킬 수 있어서 다음의 방정식으로 된다. +ml +^ 312 +_ 0__ +0 ! +2l2 I +54 +6.5/ +-6.5/ ^ +-0.75/2 +840 54 +-6.5/ +6.51 \ +-0.75Z2 ! +156 +-IlZ +-IlZ +Z2 b +(10.5.1) +보의 자유진동에 대하여 풀면, 힘 벡터는 0으로 되고 가속도 벡터는 -ω2에 변위를 곱 +한 값으로 대치되어진다. +^ 24 0 ! -12 3/ 더 +0 2Z2 ' -3/ 0.5/2 -¾->-12 -Z/ : 12 -Z/ +_ 31 0.5Z2 ! -3Z I2 _ l⅛J +아 +하 +V2- +名" +컴퓨터 프로그램 beam.m 프로그램은 외팔보에 대하여 유한 요소 모델에 대해 결정되어 +지는 고유 진동수를 계산하는 MATLAB®으로 쓰여진 파일이다. 사용자에게 보의 길이, 원하 +는 요소의 수, 보의 질량, 보의 탄성계수 및 보의 관성 모멘트를 입력하라고 요구한 뒤, 프 +로그램에서 모델에 대한 질량 및 강성행렬을 구성하게 된다. 두 개의 동일한 요소로 이루 +어진 보에 대하여, 이들 행렬은 식 (10.5.1)과 같이 구해진다. 그러면 동행렬은 이들 두 행 +렬에서부터 만들어진다. 동행렬의 고유값은 계산되어지고 모델의 고유 진동수를 구하는 데 +사용되어진다. 프로그램에 관한 더욱 자세한 정보는 부록 F에 나타나 있다. +EElEEBL 좍표저갈 +I 앞의 문제의 해를 구하는 데는 고유값-고유 벡터 관련 컴퓨터 프로그램이 필요하다. +I 그러나 우리는 결합부 2와 3에서 균일 분포질량을 집중질량으로 대체함으로써 더욱 +I 단순화된 문제로 만들 수 있다. 그러면 질량행렬은 요소 W2와 "수를 제외하고는 모두 +너 0인 값으로 된다. 이것은 변위 벡터를 정돈된 순서로 하기 위하여 앞의 식을 정리하는 +! 것을의미한다. +月 +Az2匕凶 ++ + + +ιo.5 보 요소를 포함하는 진동 349 +이것은 단순히 제2열 및 제3열과 제2행 및 제3행을 서로 바꿈으로써 되어지고, 다음 +과 같은 식으로 되어진다: +이제 식은 다음과 같은 형태가 된다. +P知 l0J∣∣-4 + [:K브丄⅛]μg = IOl +L 0 ! 이 IdJ L尺21 ! ^22JbJ IOJ +이는 다음과 같이 쓸 수 있다. +MlIV ÷ KlIV+ Kγ2θ = 0 +TC21V + K22θ = 0 +두 번째 식에서 0는 V로 나타내어질 수 있다: +θ = -TC221K21V +첫 번째 식에 대입하면 다음으로 나타내어진다. +Λf11V ÷ (K11 -K12TC221 K21)V = O +원래의 항으로 표현하면 다음과 같다. +m2 θ]∫⅛] (8EI∖ Γ 24 -12 +O λh3JI⅛J 十 ( Z3』_-12 12 +0 3/ Il2 0.5/2 +-3/ -3∕J∣0.5∕2 I2 +O +3Z +다음 항은 저감 강성(reduced StiffneSS)이고, +尺11 _ 尺\2尺』尺2↑ +곱해졌을 때 그 값은 다음과 같다. +(10.5.2) +(10.5.3) +(10.5.4) + + +350 ChaPterlo 유한요소법 입문 +8EI 96 -30 48£7 16 -5규 +-30 12 -5 2 +그러므로 원래의 4X4식은 2X2식으로 저감되어지고, 최종의 형태는 다음과 같다. +7 ：1}+(쯔T： 기1：：}={：} +수용가능한 이산질량 분포는 각 요소의 질량이 요소의 각 끝에 반씩 나누어지는 것이 +다. 그리하여, 만일 길이 /인 균일보의 전체 질량은》이고, 그림 10.5.2에 보인 것과 +같이 각 요소의 질량은 m∕∕2이고, m2 — 2(m∕∕4) — ml/2 및 g = m∕∕4이다 +그림 10.5.2 +균일 외팔보의 두 요소 이산질량 모델 +운동 방정식과 해는 다음과 같다. +여기서 +시::］丄: 기］{::}={:} +ω1ml +4 +7Z3 +48£7 ω⅛ +ml4 +Ii +A1 = 0.3632 ω1 = 3.516 엄밀값 = 3.516 +A2 = 9.637 ω2 = 22.033 엄밀값= 22.034 +_ ∫0.3271 _ ∫-1.5271어 —(1.0Ooj ≠2 - I 1.0OOJ +예제 10.5.3 +동일 요소로 이루어진 문형 구조(Portal frame)의 자유 진동식을 구하라. +MV 그림 10.5.3에 보인 것과 같이 결합부의 번호를 붙임으로써, 각 요소에 대한 강 +성과 질량은 식 (10.4.7)과 (10.4.8)로 나타내어진다. 결합부 0와 3은 변위가 0 값을 가 +지므로, 우리는 결합부 1 과 2에 대한 항만을 쓴다. + + +ιo.5 보 요소를 포함하는 진동 351 +요소 0—1, a — 90o, c = 0, 占 = 1: +스0-1 +EI下 +I -12 +I 0 +! 6/ +0 +- R +0 +- 6/“ +0 +2/2! 으 0 6/ +: 0 R 0! 이 0 4/2_ +ml +! 54 +I 0 +! -13/ +0 +0 +13厂 +0 +-3Z2 +420 ； 156 0 22/ +0 0 0 +• 22/ 0 4/2_ +요소 1—2, α = 0o, c= 1, 5 = 0: +_ R 0 0 ! -R 0 ()“ +0 12 6/ ; 0 -12 6/ 引 +EI 0 6/ 4Z2 ： 0 -61 2으 g += 7r —R 0 0 ' R 0 0 W2 +0 -12 —6/ ! 0 12 -6/ 石 2 +0 6/ +기2 ! +0 -6/ 4/2_ +꺼1-2 +느 N 0 0 U* 0 0“ +0 156 22/ : 0 54 -13/ +_ ml 0 22/ 4/2 ! 0 13/ -3/2 += 420 \ 시 +0 0 ! 川 0 0 +0 54 13/ I 0 156 —221 +_ 0 — 13Z -3Z2 : 0 -22/ 4/2 + + +352 ChaPterlo 유한요소법입문 +다음으로 VI=P2 = O임을 주목하면, 제2행과 제5행뿐만 아니 라 제2열과 제5 열 또한 +삭제되어진다. N= 140⅜ 대입하여 자유진동에 대한 식을 세우면 다음과 같이 된다. +ml +"156 0 22/ I +ONOl +221 0 _ 4/2_ ；_ +420 0 +이 행렬들을 조합하면 다음 식을 얻는다. +요소 2—3, a — 270o, C = O, 占 = — 1: +O +o +o +o += +120 +lwl족 + -w2⅞ +^^296 +221 _ +221 +8/2 +I 70 +0 +o’ +-3P_ +70 +_ 0 +O +—3/2 +I 296 +； 22/ +22/ +8l2_ +브≈ +-wI石T -w2⅛ +^( +12 ÷ 尺) +6/ +6/ +8/2 +- -R +0 +0^ +_ 기2_ +-R +0 +0 +2l2 +j(12 + +尺) +61 +6/ +8/\ +十 +으≈ +으≈ +-≡T + - S r- Q r + -w2½ +- +⅛ +’(12 + +0 +6Z +尺） +(1 2 +0 ++ +6/ +尺） 6/ +6/ +8/2 +R +0 +0 +0 +12 +6/ +0 느 +6/ +_212__ +R +0 +0 +0 +12 +61 +O +-6/ +2Z2 +I (12 ÷ +0 +61 +尺） +(12 +0 ++ +6/ +尺） 6/ +-6/ +8/2 _ +_£ _ +의 +그 +머 +⅛ +- +½ +- +¾ +‘(12 + +0 +6/ +尺） +(12 +0 ++ +67 +R) +61 +61 +8/2 +I - R +0 +0 +0 +12 +6/ +0 “ +6/ += _ +R +O +0 +0 +12 +6/ +0 +-6/ +2Z2 +!(12 + +0 +6/ +R) +(12 +0 ++ +6/ +R) +6/ +-6/ +8/2 _ +2 + +2 + +—2 + - +3 + +3 + +) +3 + +W + +石》 +- 0 + -2 + +石 + +- +0 + + +예제 10.5.4 +ιo.5 보요소를포함하는 진동 353 +그림 10.5.4는 문형 구조(POrtal frame)에 대한 자유진동의 최저차의 비대칭 및 최저차 +의 대칭 모드들을 보여준다. 주어진 모드에 대한 고유 진동수를 구하라. +WV 비대칭 모드 지점 1 과 2의 처짐 및 기울기는 處 = 必 및 @ = @2로 동일하다. 이 +들 조건은 이전의 식에서 제3열을 제 1 열에 그리고 제4열을 제2열에 더함으로써 부과 되어질 수 있다. 이는 ｛스｝와 에 대하여 동일한 식으로 된다. +ω2ml Γ366 22/] £7 Γ12 6/ ^]][u↑∖ _ {01 +^420^ L 22/ 5∕2J + 7γL6∕ 10∕2Jjt⅛J = IOJ +λ = ω2ml4∕420EI로 둠으로써, 이 식의 행렬식은3} 다음과 같다. +(12 - 366A) (6 - 22λ)/ _ +(6 - 22Λ)/ (10 - 5λ)/2 = ° +A1 = 0.0245 +A2 = 2.543 +두 근을 구하면 다음과 같다. +ω1 = 3.21 +ω2 = 32.68 +그림 10.5.4(a)에 보인 것과 같은 단순한 형상에 대응하는 최저차의 고유 진동수는 수 +용할 만한 정확도를 가진다. 그러나 2차 비대칭 모드는 더욱 복잡한 형상을 갖게 되고, +이 문제에서 사용한 몇 개의 지점으로 계산한 ω2는 정확하지 않을 것이다. 고차 모드 +를 적절히 나타내는 데는 더욱 여러 개의 지점이 필요할 것이다. +Z) 행렬식이 곱해질 때 Z2은 제거되어진다. 그리하여 λ1 및 X2의 값을 변화시키지 않고 주파수 방정식의 행렬에서 +Z= 1.0을둘수 있다. + + +그림 10.5.5의 외력을 전체계의 외력과 비교하면 다음과 같다. +f 凡 + F2x] +► — 녀 +r 야 ' +-Jzt1 +I J +.—•사= +354 아IaPterIO 유한요소법 입문 +- 그러면 λ와 ω는 다음과같다. +i 대칭 모드 대칭 모드에 대해서는 Ml = W2 = 0 및 仏=—01 이다. 제1 열과 제 3 열을 지우 +I 고, 제4열과 제2열을 제거하면, 別에 대한 오직 한 개의 식을 구하게 된다. +El +ω2ml Z X EI, ■ +— 즈5示 (IlZ)+ -P(6/)0ι = O +6 Λ=- ω = 15.14 +i 그림 10.5.5는 문형 구조(POital frame)에 외력이 작용하는 것을 나타낸다. 경계조건을 ! 조사하고,강성행렬을 주어진좌표의항으로구하라. +너 WD 요소의 신장이 없다는 조건 WI = U2⅛ 예제 10.5.3의 식 (C)에 제3열과 제 1 열을 +서 더함으로써 만족되어진다. 이것은 신장의 항 R을 없앤다. 우리는 역시 제3행을 제 1행 +너 에 더함으로써 강성행렬을 3X3행렬로 다시 쓸 수 있다: +그림 10.5.5 +-W -Qr +-¾ +( +I +l +l + +6 +2∕ +8∕ +6 +8/ +2Z +24 +6Z +6/ +끄 + ≈- +、 +>--- +√ ++ i +λ +∕1 +m2 + + +ιo.6 구조물에서의 스프링 구속조건 355 +凡 = 0 및 日X = 야을 사용하면, 주어진 좌표와 주어진 하중의 항으로 나타낸 강성행렬 +은다음과 같다. +여 +—6/ —6/ +8/2 2/2 +2Z2 8/2 +U +臥 스 +l¾J +10■이 구조물에서의 스프링 구속조건 +제9장에서 스프링 구속조건들은 가상일에 의하여 일반화된 힘으로 취급되었다. 유한 요소 +법의 경우에도 동일한 개념이 적용된다. 스프링의 작용점은 결합지점으로 선택한다. 그러 +므로 전체 좌표에서 원래 구조에서의 하중은 스프링 힘으로 치환된다. +스프링 힘은 항상 변위에 대하여 반대 방향이기 때문에, 결합부에서의 힘과 모멘트는 +-kvi 또는 -Kθi로 줄어 들게 된다. 그러므로 방정식의 다른 변으로 이항되었을 때, 스프링 +하중은 해당 강성항에 더해지게 된다. +예제 10.6.1 +그림 10.6.1(a)에 보인 선형인 회전 스프링을 갖고 균일보에 대한 강성행렬을 구하라. +WV 우선 그림 10.6.1(b)에 있는 스프링이 없이 지점 2에 하중 P와 M이 작용하 +는 보의 강성 행렬을 세우자. 각 단면 1-2 와 2-3에 대한 강성 행렬은 보요소 행렬식 +(10.2.1)에서부터 세워질 수 있다. v1 = 01 = v3 = 03 = 0임을 주목하면, 우리는 좌표 v2 +및 θ2와 관련된 행렬의 부분의 값을 정할 필요가 있고, 그것은 다음과 같이 된다. +그림 10.6.1 + + +지점 2에 작용하는 스프링들에 있어서, 힘 벡터는 다음 식으로 대치된다. +스프링 힘을 식의 우측으로 보내면 다음 식을 구할 수 있다. +Pq _ EJl2(⅛ + ⅛) + ⅛ 쇠∕W)]∣히 +시- L -6GH) 1 +전체계에서 힘 호는 윗방향으로 양의 값을 갖고, 472는 반시계 방향으로 양의 값을 가 +지므로, 앞의 식은 다음과 같이 정리되어진다. +이는 스프링 구속조건을 갖는 보에 대한 강성행렬을 정의한다. 식으로부터 계는 +Ii = I2 = 1/2일 경우 비연성화가 되고, 그 경우 식은 다음과 같이 간단하게 된다. +중심에서 처짐은 다음과 같다. +_ _ (Pl3/Er) _ MliIEI +Vl = 192 + kl3∕EI 2 = 16/2 + Kl3/EI +356 아IaPterlo 유한요소법 입문 + + +ιo.6 구조물에서의 스프링 구속조건 357 +따라서 운동 방정식은 다음과 같다. +0156 +20 +그리고 이 모드에 대한 고유 진동수는 다음과 같다. +0.00521 +마찬가지로, 的에 대한 식은 다음과 같이 된다. +- 그리고 +ω1 = 22.37 +예제 10.6.2 +kΓ +156(Zl + Z2) +—22(片 - ZD +그리하여 유한 요소 접근에 있어서 1 차 모드에 대한 오차는 1.61%이고, 2차 모드에 대 +한 오차는 33.9%이다. 보를 더욱 작은 요소로 나누면 이들 오차가 줄어들게 된다. +또다시 좌표 v2 및 @2가 비 연성화된다. λ = ω2m∕4∕420E/로 둠으로써, 그에 대한 식은 +다음과 같이 된다. +kl3 +U +ω2 — 81.98 +-⅛("+2 +…+쯜 +El ++ π +kp += 1.231 + 0.00641 — +5 쓰 = 6167 +그러므로 두 고유 진동수는 구속 스프링에 의하여 증가된다. 만일 A = 尺=O이면, 고정 +단을 가진 보의 정확한 고유 진동수는 다음과 같다. +EI +ml4 +m +420 +너 예제 10.6.1 에서 Z1 = ∕2 = Z/2인 경우 구속된 보의 고유 진동수를 구하라. +I WV 이 값을 구하기 위하여 질량행렬이 필요하고, 식 (10.2.10)으로부터 다음과 같 +≡ 이 표현할 수있다. +ω1ml +-420^ +kl3 +El +0.0625 +0 + + +358 아IaPterlO 유한요소법 입문 +Jq^기 일반화된 힘과분포하중 +제7장에서 논의한 바와 같이 일반화된 힘은 작용력의 가상일에서부터 구할 수 있다. 변위 +가 다음과 같이 나타나 있을 때, +y(x) — <∕>1(x)υ1 + <∕>2(x)^ι 十 φ3(x)υ2 十 ≠4(x)¾ (10.7.1) +작용하는 분포력 P(X) 의 가상일은 다음과 같다. +δW= [ P(X) δy(x) dx +JO += δυ1 P(X)φ1(x) dx + δθ1 p(x)≠2(x) dx +JO JO +÷ δυ2 I p(x)≠3(x) dχ + 6Θ2 I p(x)ψ4(x) dx (10.7.2) +JO JO +식 (10.7.2)에서 적분표시된 것은 일반화된 힘이다. +만일 끝단의 힘 F1, Ml, F2 및 M2에 동일 과정이 적용되면, 가상일은 다음과 같다. +δw = Fl δυl + MI δθ1 + F2 δυ2 + MI δθ2 (10.7.3) +앞의 두 경우에서의 가상일을 계산하면, 우리는 다음의 관계식을 얻을 수 있다. +FI = P(X)ψ1(x) dx F2 - p(x)φ3(X) dx +jθ ⅛ (10-7.4) +Mi = P(X) <∕>2(x) dx M2 = P(X) ψ4(x) dx +JO JO +그리하여 분포하중에 대한 등가 유한요소하중은 지금 구한 일반화된 힘이다. +예제 10.7.1 +| 그림 10.7.1 에는 길이가 Z1 이고, 보의 바깥 반쪽 위에 균일 하중 P(X)=Plb/in를 받고 +있는 외팔보를 나타내고 있다. 본 절의 방법을 사용하여 끝단에서의 처짐과 기울기를 +I 구하라. +I WM 우리는 ①-②인 단일 요소를 사용하고, 강성행렬의 역행렬을 결정하자. +! v1 = 02 = 0이므로, 식 (10.2.1)에서부터 강성수식은 다음과 같다. + + +ιo.7 일반화된 힘과분포하중 359 +그림 10.7.1 +(이 _흐「12 -6∕1]∫띠 +IMj /?[-必 4/f JbJ +수반연산 방법을 사용하면, 그 역은 다음과 같다. +4zι 6zΓ∣{F2] +¾∫ E∕12g[6∕ι 12_|IMJ +식 (10.7.4)로부터 상당유한 요소 작용력은 다음과 같다. +F2 = [ — PΦKx) dx= -p∖ Φ3(ξ)l1 dξ = -PIx f (3/ - 2ξ3) dξ= — +Jl2 J1/2 J1/2 JZ +∫ +1 +1/2 - WfW = - 씨i/2(-f + 己 쌰 = 쁪此 +이 값들은 역식에 대입하면 다음과 같다. +f 52 528 1 +PK 32 1536 I +12 EI" 78 1056 | +느 ~32Λ 十 I=J +쑈＜ +48 EI +5.125 +7.000 +이들 결과는 면적 모멘트법으로부터 구한 결과와 일치한다. + + +360 ChaPterIo 유한요소법 입문 +10■이 변위에 비례하는 일반화된 힘 +일반화된 힘이 변위에 비례할 때, 자유진동을 위하여 강성행렬과 결합하기 위해 운동 방정 +식의 좌측으로 옮겨질 수 있다. 본 절에서 제시되는 것은 다음의 두 경우이다. +(1) 분포력이 보에 수직 일 때 +(2) 분포력이 보에 수평 일 때 +경우 1 식 (10.7.2)의 가상일의 항P(X)가Λx)y(x)에 의하여 대치되면, 다음의 식으로 된다. +8W — f /(x)y(x) δy(x) dx (10.8.1) +JO +이 때 XX) = XZ≠⑷이다. 여기서 Φ는 보함수이고, 또 이는 식 (10.7.1)에서처럼 요소 끝단 처 +짐이다. +δW = 乞 乞 qj 6ql f f(x)φiφj dx (10.8.2) +Z j Jo +그리고 일반화된 힘은 다음과 같다. +β/ = 을Y= ∑ Qj f )WΦiΦj dx (10.8.3) +j Jo +이는 변위에 비례하는 것이다. +예제 10.8.1 +그림 10.8.1 은 보의 바깥 반쪽의 아랫부분에 탄성지지를 받고 있는 외팔보를 나타낸 +다. 지지되는 강성은 —ky lbs∕in이고, 여기서 ") = —그로 일정하다. 이 때 운동 방정 +식은 다음과 같다. +그림 10.8.1 + + +ιo.8 변위에 비례하는 일반화된 힘 361 +길이 /인 요소에 대하여 식 (10.8.3)에 나타난 적분을 수행하면 다음의 식을 얻는다. +~0.3714 0.524/ 0.1286 -0.03095/ ' +{Qi} = -kl 0.009524/2 0.03905/ -0.007143/2 H +0.3714 -0.05238/ u3 +0.009524Z∖l⅛J +/대신에 /∕2을 이 문제에 적용시키고 식의 좌측으로 옮기면, 보의 강성이 증가한다. +경우 2 보에 평행한 분포력 P(X)JX는 P(X)dx ∙ δw(x)의 가상일을 하고, 이 때 M(X)는 +변형 XX)에 의한 수평방향 변위이다. 변위 W(X)는 변형된 보의 수평 투영된 위치와 X +축과의 차이와 같다. +U(X) — J (ClS — dx) = j dx ∖∣1 •+■ j - dx — J ∖ya dr +이 때 그은 X에 대한 가상변수이고 / = 砂/Jr이다. 그러므로 X의 가상변위는 다음과 같다. +δw(x) = f {δy'2dr +JO +여기서 피적분함수는 다음과 같이 해석되어질 수 있다: +오/2 = ；[(/ + 아')2 — /2]=/6/ +그러므로 분포력에 대한 가상일은 다음과 같다. +δW = — [ P(X) f y,δy, drdx (10.8.4) +JO JO +y에 대하여 보함수의 항으로 대입하면 다음과 같다. +δw +Qi = — +P(X) φ'iφ,j drdx +I Jo +P(X) φ'φ- drdx +) JO +δw = - (10.8.5) +(10.8.6) +e + 1 +β +2 +β3β4 +Γ⅛⅜ +⅛ +⅞ + + +362 ChaPter 10 유한요소법 입문 +예제 10.8.2 +I 회전 요소 여기서 관심이 있는 예제는 그림 10.8.2에 나타낸 각속도 ∩로 회전하는 +≡ 헬리콥터 날개이다. 첫 번째 보 요소에 대하여 하중은 ∩2≡ 成이고, 식 (10.8.6)은 변 +! 함없이 적용된다. 추가되는 요소에 대해서는, 보함수의 좌표와 확인하기 위하여 X좌표 +I 는 새로운 요소의 시작지점부터 측정되어져야 한다. 요소에 작용하는 하중 은 단순히 +I ∩2(Z,∙ + x)m 成이고, 여기서 I는 회전축에서부터 새로운 요소의 시작지점까지의 거리 +I 를나타낸다. +이 곳에 나타낸 것은 하중 ∩2%m dx일 경우의 일반화된 힘이고, 그것은 첫 번째 요 ! 소에적용될수있다. +예제 10.8.3 +한 개의 요소를 사용하여, 길이 Z이고 회전속도 ∩로 회전하는 헬리콥터의 운동 방정식 +을 구하라. 날개는 회전축에 단단히 고정되어 있다고 가정한다. +OB 길이 /인 단일 요소에 대한 질량 및 강성항은 다음과 같다. +:vM/ - O +- +:; +어: +ς머 + ⅛ +¾ +3/ +3/ +-I- +2/ +4/ +2 +3/ +3/ +54 +13/ +56 +22/ +2Z +4/ +2 +4 +3/ +5 1 +56 +22// - O +끄 +42 + + +ιo.8 변위에 비례하는 일반화된 힘 363 +KV +회전에 의한 항은 식 (10.8.6)으로 주어진 일반화된 힘 e로부터 찾을 수 있다. 그 평가 +를 위하여 포함되어진 적분식은 다음과 같다. +「/ [ φ,iφ,jldξ]ldξ +OLJO - +mΩ,2l I X I φ'iφ,j dr-dx — mΩ신 +Jo JO +여기서 +少； = (—6ξ+6ξ2)} + List[str]: + chunks: List[str] = [] + start = 0 + n = len(text) + while start < n: + end = min(start + max_chars, n) + chunk = text[start:end].strip() + if chunk: + chunks.append(chunk) + if end == n: + break + start = max(0, end - overlap) + return chunks + + +def embed_texts_ollama(texts: List[str], model: str = "nomic-embed-text", host: str = "http://localhost:11434") -> List[List[float]]: + url = f"{host}/api/embeddings" + vectors: List[List[float]] = [] + for t in texts: + resp = requests.post(url, json={"model": model, "prompt": t}, timeout=120) + resp.raise_for_status() + data = resp.json() + vectors.append(data["embedding"]) # type: ignore[index] + return vectors + + +def main() -> None: + parser = argparse.ArgumentParser(description="Build simple vector index using Ollama embeddings") + parser.add_argument("--text", default=None, help="Path to extracted .txt; default = first in data/") + parser.add_argument("--model", default="nomic-embed-text", help="Ollama embedding model name") + parser.add_argument("--host", default="http://localhost:11434", help="Ollama host") + parser.add_argument("--out", default="data/index.jsonl", help="Output JSONL path") + parser.add_argument("--max-chars", type=int, default=1200, help="Max characters per chunk") + parser.add_argument("--overlap", type=int, default=200, help="Characters overlap between chunks") + args = parser.parse_args() + + data_dir = Path("data") + if args.text: + text_path = Path(args.text) + else: + txts = sorted(data_dir.glob("*.txt")) + if not txts: + raise SystemExit("data/*.txt가 없습니다. 먼저 scripts/pdf_stats.py로 PDF를 추출하세요.") + text_path = txts[0] + + text = text_path.read_text(encoding="utf-8") + chunks = chunk_text(text, max_chars=args.max_chars, overlap=args.overlap) + + vectors = embed_texts_ollama(chunks, model=args.model, host=args.host) + + out_path = Path(args.out) + out_path.parent.mkdir(parents=True, exist_ok=True) + with out_path.open("w", encoding="utf-8") as f: + for i, (chunk, vec) in enumerate(zip(chunks, vectors)): + row: Dict[str, Any] = { + "id": f"{text_path.stem}:{i}", + "text": chunk, + "vector": vec, + "source": text_path.name, + } + f.write(json.dumps(row, ensure_ascii=False) + "\n") + + meta = { + "source_text": str(text_path), + "embedding_model": args.model, + "host": args.host, + "chunks": len(chunks), + "index_path": str(out_path), + } + print(json.dumps(meta, ensure_ascii=False)) + + +if __name__ == "__main__": + main() + diff --git a/scripts/install_server.sh b/scripts/install_server.sh new file mode 100755 index 0000000..50afca6 --- /dev/null +++ b/scripts/install_server.sh @@ -0,0 +1,15 @@ +#!/usr/bin/env bash +set -euo pipefail + +VENV_DIR=".venv" + +if [ ! -d "$VENV_DIR" ]; then + python3 -m venv "$VENV_DIR" +fi + +source "$VENV_DIR/bin/activate" +python -m pip install --upgrade pip +pip install -r requirements.txt + +echo "[ok] server deps installed in $VENV_DIR" + diff --git a/scripts/pdf_stats.py b/scripts/pdf_stats.py new file mode 100644 index 0000000..d37b416 --- /dev/null +++ b/scripts/pdf_stats.py @@ -0,0 +1,99 @@ +#!/usr/bin/env python3 +import argparse +import json +import os +import re +from pathlib import Path + + +def detect_hangul_ratio(text: str) -> float: + han = len(re.findall(r"[\u3131-\u318E\uAC00-\uD7A3]", text)) + total = max(len(text), 1) + return han / total + + +def ensure_dir(path: Path) -> None: + if not path.exists(): + path.mkdir(parents=True, exist_ok=True) + + +def main() -> None: + parser = argparse.ArgumentParser(description="Extract full text from PDF and estimate token count") + parser.add_argument("pdf", nargs="?", help="Path to PDF; if omitted, first PDF in repo root is used") + parser.add_argument("--outdir", default="data", help="Output directory for extracted text") + args = parser.parse_args() + + repo_root = Path(os.getcwd()) + if args.pdf: + pdf_path = Path(args.pdf) + else: + # pick the first PDF in repo root + cands = sorted(repo_root.glob("*.pdf")) + if not cands: + print("{}") + return + pdf_path = cands[0] + + # Lazy import with helpful error if missing + try: + from pypdf import PdfReader + except Exception as e: + raise SystemExit( + "pypdf가 설치되어 있지 않습니다. 가상환경 생성 후 'pip install pypdf tiktoken'을 실행하세요." + ) + + # Tokenizer + try: + import tiktoken + enc = tiktoken.get_encoding("cl100k_base") + def count_tokens(s: str) -> int: + return len(enc.encode(s)) + tokenizer = "tiktoken(cl100k_base)" + except Exception: + def count_tokens(s: str) -> int: + # fallback heuristic + return int(len(s) / 3.3) + tokenizer = "heuristic_div_3.3" + + reader = PdfReader(str(pdf_path)) + num_pages = len(reader.pages) + + # Full extraction + all_text_parts = [] + for i in range(num_pages): + try: + page_text = reader.pages[i].extract_text() or "" + except Exception: + page_text = "" + all_text_parts.append(page_text) + full_text = "\n\n".join(all_text_parts).strip() + + # Stats + chars = len(full_text) + tokens = count_tokens(full_text) + hangul_ratio = detect_hangul_ratio(full_text) + size_bytes = pdf_path.stat().st_size + + # Save text + outdir = Path(args.outdir) + ensure_dir(outdir) + txt_name = pdf_path.stem + ".txt" + out_txt = outdir / txt_name + out_txt.write_text(full_text, encoding="utf-8") + + result = { + "pdf": str(pdf_path), + "pages": num_pages, + "size_bytes": size_bytes, + "chars": chars, + "tokens": tokens, + "hangul_ratio": round(hangul_ratio, 4), + "tokenizer": tokenizer, + "text_path": str(out_txt), + } + print(json.dumps(result, ensure_ascii=False)) + + +if __name__ == "__main__": + main() + diff --git a/scripts/venv_setup.sh b/scripts/venv_setup.sh new file mode 100755 index 0000000..d918712 --- /dev/null +++ b/scripts/venv_setup.sh @@ -0,0 +1,16 @@ +#!/usr/bin/env bash +set -euo pipefail + +VENV_DIR=".venv" + +if [ ! -d "$VENV_DIR" ]; then + python3 -m venv "$VENV_DIR" +fi + +source "$VENV_DIR/bin/activate" + +python -m pip install --upgrade pip +pip install pypdf tiktoken + +echo "[ok] venv ready at $VENV_DIR" + diff --git a/server/config.py b/server/config.py new file mode 100644 index 0000000..8f65a73 --- /dev/null +++ b/server/config.py @@ -0,0 +1,21 @@ +from __future__ import annotations + +import os +from dataclasses import dataclass + + +@dataclass(frozen=True) +class Settings: + ollama_host: str = os.getenv("OLLAMA_HOST", "http://localhost:11434") + base_model: str = os.getenv("BASE_MODEL", "qwen2.5:7b-instruct") + boost_model: str = os.getenv("BOOST_MODEL", "qwen2.5:14b-instruct") + embedding_model: str = os.getenv("EMBEDDING_MODEL", "nomic-embed-text") + index_path: str = os.getenv("INDEX_PATH", "data/index.jsonl") + + # Paperless (user will provide API details) + paperless_base_url: str = os.getenv("PAPERLESS_BASE_URL", "") + paperless_token: str = os.getenv("PAPERLESS_TOKEN", "") + + +settings = Settings() + diff --git a/server/index_store.py b/server/index_store.py new file mode 100644 index 0000000..d2b01e7 --- /dev/null +++ b/server/index_store.py @@ -0,0 +1,73 @@ +from __future__ import annotations + +import json +from dataclasses import dataclass +from pathlib import Path +from typing import List, Tuple +import math + + +def cosine_similarity(vec_a: List[float], vec_b: List[float]) -> float: + if not vec_a or not vec_b or len(vec_a) != len(vec_b): + return 0.0 + dot = sum(a * b for a, b in zip(vec_a, vec_b)) + na = math.sqrt(sum(a * a for a in vec_a)) + nb = math.sqrt(sum(b * b for b in vec_b)) + if na == 0.0 or nb == 0.0: + return 0.0 + return dot / (na * nb) + + +@dataclass +class IndexRow: + id: str + text: str + vector: List[float] + source: str + + +class JsonlIndex: + def __init__(self, path: str) -> None: + self.path = Path(path) + self.rows: List[IndexRow] = [] + self._load() + + def _load(self) -> None: + self.rows.clear() + if not self.path.exists(): + return + with self.path.open("r", encoding="utf-8") as f: + for line in f: + if not line.strip(): + continue + obj = json.loads(line) + self.rows.append(IndexRow( + id=obj["id"], + text=obj["text"], + vector=obj["vector"], + source=obj.get("source", "") + )) + + def search(self, query_vec: List[float], top_k: int = 5) -> List[Tuple[IndexRow, float]]: + scored: List[Tuple[IndexRow, float]] = [] + for row in self.rows: + score = cosine_similarity(query_vec, row.vector) + scored.append((row, score)) + scored.sort(key=lambda x: x[1], reverse=True) + return scored[:top_k] + + def append(self, new_rows: List[IndexRow]) -> int: + if not new_rows: + return 0 + self.path.parent.mkdir(parents=True, exist_ok=True) + with self.path.open("a", encoding="utf-8") as f: + for r in new_rows: + obj = {"id": r.id, "text": r.text, "vector": r.vector, "source": r.source} + f.write(json.dumps(obj, ensure_ascii=False) + "\n") + self.rows.extend(new_rows) + return len(new_rows) + + def reload(self) -> int: + self._load() + return len(self.rows) + diff --git a/server/main.py b/server/main.py new file mode 100644 index 0000000..1948427 --- /dev/null +++ b/server/main.py @@ -0,0 +1,144 @@ +from __future__ import annotations + +from fastapi import FastAPI, HTTPException +from pydantic import BaseModel +from typing import List, Dict, Any + +from .config import settings +from .ollama_client import OllamaClient +from .index_store import JsonlIndex + + +app = FastAPI(title="Local AI Server", version="0.1.0") +ollama = OllamaClient(settings.ollama_host) +index = JsonlIndex(settings.index_path) + + +class ChatRequest(BaseModel): + model: str | None = None + messages: List[Dict[str, str]] + use_rag: bool = True + top_k: int = 5 + force_boost: bool = False + options: Dict[str, Any] | None = None + + +class SearchRequest(BaseModel): + query: str + top_k: int = 5 + +class UpsertRow(BaseModel): + id: str + text: str + source: str | None = None + +class UpsertRequest(BaseModel): + rows: List[UpsertRow] + embed: bool = True + model: str | None = None + batch: int = 16 + + +@app.get("/health") +def health() -> Dict[str, Any]: + return { + "status": "ok", + "base_model": settings.base_model, + "boost_model": settings.boost_model, + "embedding_model": settings.embedding_model, + "index_loaded": len(index.rows) if index else 0, + } + + +@app.post("/search") +def search(req: SearchRequest) -> Dict[str, Any]: + if not index.rows: + return {"results": []} + qvec = ollama.embeddings(settings.embedding_model, req.query) + results = index.search(qvec, top_k=req.top_k) + return { + "results": [ + {"id": r.id, "score": float(score), "text": r.text[:400], "source": r.source} + for r, score in results + ] + } + + +@app.post("/chat") +def chat(req: ChatRequest) -> Dict[str, Any]: + model = req.model + if not model: + # 라우팅: 메시지 길이/force_boost 기준 간단 분기 + total_chars = sum(len(m.get("content", "")) for m in req.messages) + model = settings.boost_model if (req.force_boost or total_chars > 2000) else settings.base_model + + context_docs: List[str] = [] + if req.use_rag and index.rows: + q = "\n".join([m.get("content", "") for m in req.messages if m.get("role") == "user"]).strip() + if q: + qvec = ollama.embeddings(settings.embedding_model, q) + hits = index.search(qvec, top_k=req.top_k) + context_docs = [r.text for r, _ in hits] + + sys_prompt = "" + if context_docs: + sys_prompt = ( + "당신은 문서 기반 비서입니다. 제공된 컨텍스트만 신뢰하고, 모르면 모른다고 답하세요.\n\n" + + "\n\n".join(f"[DOC {i+1}]\n{t}" for i, t in enumerate(context_docs)) + ) + + messages: List[Dict[str, str]] = [] + if sys_prompt: + messages.append({"role": "system", "content": sys_prompt}) + messages.extend(req.messages) + + try: + resp = ollama.chat(model, messages, stream=False, options=req.options) + return {"model": model, "response": resp} + except Exception as e: + raise HTTPException(status_code=500, detail=str(e)) + + +@app.post("/index/upsert") +def index_upsert(req: UpsertRequest) -> Dict[str, Any]: + try: + if not req.rows: + return {"added": 0} + model = req.model or settings.embedding_model + new_rows = [] + for r in req.rows: + vec = ollama.embeddings(model, r.text) if req.embed else [] + new_rows.append({ + "id": r.id, + "text": r.text, + "vector": vec, + "source": r.source or "api", + }) + # convert to IndexRow and append + from .index_store import IndexRow + to_append = [IndexRow(**nr) for nr in new_rows] + added = index.append(to_append) + return {"added": added} + except Exception as e: + raise HTTPException(status_code=500, detail=f"index_upsert_error: {e}") + + +@app.post("/index/reload") +def index_reload() -> Dict[str, Any]: + total = index.reload() + return {"total": total} + + +# Paperless webhook placeholder (to be wired with user-provided details) +class PaperlessHook(BaseModel): + document_id: int + title: str | None = None + tags: List[str] | None = None + + +@app.post("/paperless/hook") +def paperless_hook(hook: PaperlessHook) -> Dict[str, Any]: + # NOTE: 확장 지점 - paperless API를 조회하여 문서 텍스트/메타데이터를 받아 + # scripts/embed_ollama.py와 동일 로직으로 인덱스를 업데이트할 수 있습니다. + return {"status": "ack", "document_id": hook.document_id} + diff --git a/server/ollama_client.py b/server/ollama_client.py new file mode 100644 index 0000000..cffee74 --- /dev/null +++ b/server/ollama_client.py @@ -0,0 +1,29 @@ +from __future__ import annotations + +import requests +from typing import List, Dict, Any + + +class OllamaClient: + def __init__(self, host: str) -> None: + host = host.strip() + if not host.startswith("http://") and not host.startswith("https://"): + host = "http://" + host + self.host = host.rstrip("/") + + def embeddings(self, model: str, text: str) -> List[float]: + url = f"{self.host}/api/embeddings" + resp = requests.post(url, json={"model": model, "prompt": text}, timeout=120) + resp.raise_for_status() + data = resp.json() + return data["embedding"] + + def chat(self, model: str, messages: List[Dict[str, str]], stream: bool = False, options: Dict[str, Any] | None = None) -> Dict[str, Any]: + url = f"{self.host}/api/chat" + payload: Dict[str, Any] = {"model": model, "messages": messages, "stream": stream} + if options: + payload["options"] = options + resp = requests.post(url, json=payload, timeout=600) + resp.raise_for_status() + return resp.json() + diff --git a/기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.pdf b/기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.pdf new file mode 100644 index 0000000..c24466f Binary files /dev/null and b/기계진동 이론과 응용(제5판)_Chapter 10 유한 요소법 입문.pdf differ