检查阻尼与受迫共振
通过有量纲参数连接自由衰减、稳态频率响应及时间积分。
目标与运行方法
本课适合具备 Python 基础的本科生。先阅读每一步的方法与图注，再修改参数。AI 练习为可选部分，不需要 AI 账号即可完成核心课程。数据写在 notebook metadata 中，不需要另下载数据文件。请保持 notebook 已保存到当前工作文件夹。
学习任务是理解本课的研究问题、执行透明的计算、校验结果，并说明结论的边界。
环境与参数
from pathlib import Path
import base64, gzip, hashlib, json, platform
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from IPython.display import display, HTML
from scipy.integrate import solve_ivp
from IPython.display import display as _display
def display(value):
if isinstance(value,pd.DataFrame):
table=value.to_html(max_rows=12,escape=True)
style="<style>.vibeit-readable-table{max-width:100%;overflow-x:auto}.vibeit-readable-table table{width:max-content;max-width:none;border-collapse:collapse}.vibeit-readable-table td,.vibeit-readable-table th{white-space:nowrap!important;word-break:normal!important;overflow-wrap:normal!important}</style>"
return _display(HTML(style+'<div class="vibeit-readable-table">'+table+'</div>'))
return _display(value)
RECIPE_ID='ph02-oscillator'
OUTPUT_DIR=Path.cwd()/"results"/RECIPE_ID
plt.rcParams.update({"figure.dpi":120,"font.size":10,"axes.spines.top":False,"axes.spines.right":False,
"axes.prop_cycle":plt.cycler(color=["#18785f","#416aa6","#bc6541","#9974a5"])})
print("Python:",platform.python_version(),"; NumPy:",np.__version__,"; Pandas:",pd.__version__)
print("Working folder:",Path.cwd())
print("Core data mode: embedded snapshot; no network requests")
Python: 3.13.13 ; NumPy: 2.5.3 ; Pandas: 3.0.6 Working folder: /Users/zhiluo/Developer/vibeit-cookbook/cookbook/downloads Core data mode: embedded snapshot; no network requests
读取内嵌快照
def load_snapshot(recipe_id, expected_sha256, snapshot_key):
"""Read embedded data from this notebook, including a renamed copy."""
for path in sorted(Path.cwd().glob("*.ipynb")):
try:
notebook = json.loads(path.read_text(encoding="utf-8"))
metadata = notebook.get("metadata", {}).get("vibeit_cookbook", {})
blob = metadata.get("snapshots", {}).get(snapshot_key)
if metadata.get("recipe_id") != recipe_id or not blob:
continue
if blob.get("sha256") != expected_sha256:
continue
raw = gzip.decompress(base64.b64decode(blob["gzip_base64"], validate=True))
if hashlib.sha256(raw).hexdigest() != expected_sha256:
raise ValueError("Embedded snapshot checksum mismatch: " + snapshot_key)
return json.loads(raw)
except (OSError, json.JSONDecodeError):
continue
raise FileNotFoundError(
"Save/open the complete .ipynb in the active working folder. "
"The embedded snapshot could not be located for " + recipe_id)
SNAPSHOT_HASHES={'models': '48fd2e6e706650fe2ee0c0ad80b6d794a546a1dec78dd9d370ac09db3cc3adbf'}
def snapshot(key):
return load_snapshot(RECIPE_ID,SNAPSHOT_HASHES[key],key)
print("Embedded snapshots:",list(SNAPSHOT_HASHES))
Embedded snapshots: ['models']
方法函数
def oscillator_solution(mass,stiffness,damping,force_amplitude=0,drive_frequency=0,duration_s=20,initial_displacement=1,samples=2001):
if mass<=0 or stiffness<=0 or damping<0 or duration_s<=0 or samples<3:
raise ValueError('Invalid oscillator parameters')
def dynamics(t,state):
x,v=state;return [v,(force_amplitude*np.cos(drive_frequency*t)-damping*v-stiffness*x)/mass]
t=np.linspace(0,duration_s,samples)
solved=solve_ivp(dynamics,[0,duration_s],[initial_displacement,0],t_eval=t,rtol=1e-8,atol=1e-10)
if not solved.success:raise RuntimeError(solved.message)
return dict(t=t,x=solved.y[0],velocity=solved.y[1])
print("Method ready: oscillator_solution")
Method ready: oscillator_solution
数据与模型边界
单摆、振子、抛体和温度轨迹是明确声明的数值模型，不是实验室测量。使用的 g0 为约定标准重力加速度。GW150914 使用真实冻结应变和已发表事件时间；滤波曲线不是独立的显著性或源参数分析。黑体曲线假定理想辐射体。
分步分析
1. 声明线性振子
方程为 m x″+c x′+k x=F cos(omega t)。质量 kg、刚度 N/m、阻尼 N·s/m。先检查从一米初始位移开始的自由衰减；该例是教学模型，不是装置实测。
p=snapshot("models")["oscillator"];mass=p["mass_kg"];stiffness=p["stiffness_N_per_m"];dampings=p["damping_N_s_per_m"]
natural=np.sqrt(stiffness/mass)
fig,ax=plt.subplots(figsize=(9,4))
for damping in dampings:
solution=oscillator_solution(mass,stiffness,damping)
ax.plot(solution["t"],solution["x"],label=f"c={damping} N·s/m")
ax.set(xlabel="Time (s)",ylabel="Displacement (m)",title="Free damped motion");ax.legend();plt.tight_layout();plt.show()
2. 解读稳态频率响应
正阻尼下振幅为 F/sqrt((k−m omega²)²+(c omega)²)。横轴为 rad/s 角频率，不是 Hz。阻尼后的振幅峰不一定等于无阻尼固有频率。
frequencies=np.linspace(.1,8,501);force=p["force_N"]
fig,ax=plt.subplots(figsize=(8,4))
for damping in dampings:
amplitude=force/np.sqrt((stiffness-mass*frequencies**2)**2+(damping*frequencies)**2)
ax.plot(frequencies,amplitude,label=f"c={damping}")
ax.axvline(natural,color="gray",linestyle="--",label="Undamped natural omega")
ax.set(xlabel="Drive angular frequency (rad/s)",ylabel="Steady displacement amplitude (m)",title="Damping changes peak location and width");ax.legend();plt.tight_layout();plt.show()
3. 将受迫轨迹与解析响应核对
从零初始状态积分足够长时间，使瞬态衰减。后期轨迹应与解析振幅和相位的正弦波匹配。不能将初期瞬态直接与稳态公式比较。
damping=1.6;drive=natural
forced=oscillator_solution(mass,stiffness,damping,force,drive,duration_s=80,initial_displacement=0,samples=8001)
amplitude=force/np.sqrt((stiffness-mass*drive**2)**2+(damping*drive)**2)
phase=np.arctan2(damping*drive,stiffness-mass*drive**2)
expected=amplitude*np.cos(drive*forced["t"]-phase);late=forced["t"]>70
fig,ax=plt.subplots(figsize=(9,4));ax.plot(forced["t"][late],forced["x"][late],label="ODE integration")
ax.plot(forced["t"][late],expected[late],"--",label="Analytic steady state")
ax.set(xlabel="Late model time (s)",ylabel="Displacement (m)",title="Time/frequency-domain consistency");ax.legend();plt.tight_layout();plt.show()
4. 核对并导出声明模型
使用独立闭式表达式核对后期振幅和相位。导出时保留固有角频率、阻尼单位及求解容差；本课不声称得到实验装置参数。
assert np.max(np.abs(forced["x"][late]-expected[late]))<1e-6
assert np.isclose(natural,4.0)
OUTPUT_DIR.mkdir(parents=True,exist_ok=True)
pd.DataFrame({"time_s":forced["t"],"ODE_displacement_m":forced["x"],"steady_displacement_m":expected}).to_csv(OUTPUT_DIR/"forced-oscillation.csv",index=False)
(OUTPUT_DIR/"parameters.json").write_text(json.dumps({"mass_kg":mass,"stiffness_N_per_m":stiffness,"damping_N_s_per_m":damping,"force_N":force,"drive_rad_per_s":drive,"natural_rad_per_s":natural},indent=2))
print("Steady-state cross-check passed; exports:",OUTPUT_DIR)
Steady-state cross-check passed; exports: /Users/zhiluo/Developer/vibeit-cookbook/cookbook/downloads/results/ph02-oscillator
自主练习与参考答案
- 区分 rad/s 与 Hz。
- 为何稳态响应只应在瞬态衰减后比较？
参考解释。 角频率等于每秒周期数乘 2pi。稳态正弦未包含由初始条件决定的瞬态。
先保留当前 notebook 副本，再修改参数。把新图与原图一起比较，并记录改变了哪些输入、哪些计算和哪些结论。
可选的 AI 编程练习
提示词 1
仅用 NumPy、SciPy、Matplotlib 加入阻尼 8 N·s/m，检查自由衰减及频响。
提示词 2
使用同样的包，用 atan2 导出解析相位曲线并说明弧度单位。
人工验收。 保留有量纲参数；正阻尼下后期振幅/相位核对通过。
检查修改后的源代码，再手动运行受影响单元及后续校验。要求助手解释修改的目的、使用的包和数据来源，并用实际输出核对解释。
可选联网扩展
RUN_ONLINE_UPDATE=False
if RUN_ONLINE_UPDATE:
import requests
try:
response=requests.get('https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html',timeout=20)
response.raise_for_status()
print("Source:",response.url)
print(response.text[:700])
except requests.RequestException as error:
print("Online extension unavailable:",type(error).__name__,str(error)[:160])
else:
print("Online extension skipped; embedded core data unchanged")
Online extension skipped; embedded core data unchanged
来源、快照与下一步
IAPWS saturation reference release — IAPWS SR1-86(1992): unrestricted publication allowed in all countries (cover page); retrieved 2026-10-09. SHA-256
a10d38d73339b0713c86cfac88f9adcba137202167ccdfa1cd58238c438361d7. Reference equations and numerical coefficients transcribed from release pages 2–3; pedagogical model values are not new experiments.Authored numerical teaching model parameters — Original teaching parameters/code MIT; reference equation coefficients attributed to IAPWS SR1-86(1992); retrieved 2026-10-09. SHA-256
4350c4305a83c772fa295a8be0ef86a247f2bfaae930a081720d3cd0c753abe0. Declared parameters and seed; generated trajectories and calibration observations must be labelled simulated.
下一步。 在保留本课的数据检查、参数记录和结果校验之后，将相同方法用于自己的公开或已获授权数据。记录研究设计与方法限制，不把示例结果当成新实验的验证。完整数据许可与生成说明见 cookbook 网页。