火箭轨迹三维模拟（低空）

“Failure is an option here. If things are not failing, you are not innovating enough.”——Elon Musk

过程概述

发射阶段（Liftoff）

• 火箭继续以倾角 $\theta_0$ 上升，速度逐渐增加。
• 空气阻力与速度方向相反，需要考虑空气阻力的影响。
• 火箭的姿态可能会随时间调整（例如通过控制舵面或推力矢量控制）。

相关公式：

• 推力：
  $
  F = \dot{m} v_e + A_e (P_e - P_0)
  $
  其中，$ \dot{m} $ 为推进剂流量，$ v_e $ 为喷射气体速度，$ A_e $ 为喷口面积，$ P_e $ 和 $ P_0 $ 分别为喷口出口压力和大气压力。

• 质量变化：
  $
  m(t) = m_0 - \dot{m} t
  $
  其中，$ m_0 $ 为火箭的初始质量。

• 加速度：
  $
  a = \frac{F}{m} - g - \frac{1}{2} C_D \rho v^2 A / m
  $
  其中，$ C_D $ 为阻力系数，$ \rho $ 为空气密度，$ v $ 为火箭的速度，$ A $ 为火箭的迎风面积，$ g $ 为重力加速度。

• 速度：
  $
  v(t) = \int a \, dt
  $

• 位置：
  $
  h(t) = \int v \, dt
  $

---

倾斜上升阶段（Inclined Ascent）

• 火箭继续以倾角 $\theta_0$ 上升，速度逐渐增加。
• 空气阻力与速度方向相反，需要考虑空气阻力的影响。
• 火箭的姿态可能会随时间调整（例如通过控制舵面或推力矢量控制）。

相关公式：

• 推力：
  $
  F = \dot{m} v_e + A_e (P_e - P_0)
  $

• 质量变化：
  $
  m(t) = m_0 - \dot{m} t
  $

• 加速度：
  $
  a = \frac{F}{m} - g - \frac{1}{2} C_D \rho v^2 A / m
  $

• 速度：
  $
  v(t) = \int a \, dt
  $

• 位置：
  $
  h(t) = \int v \, dt
  $

---

重力转弯阶段（Gravity Turn）

• 火箭在倾斜上升过程中，重力会逐渐改变火箭的飞行方向，使其轨迹向水平方向弯曲。
• 这一过程通常不需要额外的控制输入，而是由重力和火箭的初始倾角自然形成。

相关公式：

• 重力转弯角度变化：
  $
  \frac{d\gamma}{dt} = \frac{F_T \sin(\theta)}{m v} - g \cos(\gamma)
  $
  其中，$\gamma$ 为俯仰角，$ F_T $ 为推力，$ m $ 为火箭质量，$ v $ 为火箭速度，$ \theta $ 为倾角，$ g $ 为重力加速度。

• 速度变化：
  $
  \mathbf{v} = \int a \, dt
  $
  此处 $a$ 为重力转弯阶段的加速度。

---

空速下降阶段（Airspeed Descent）

• 火箭在重力转弯后，开始以空速下降，速度逐渐减小。
• 空气阻力与速度方向相反，需要考虑空气阻力的影响。
• 火箭的姿态可能会随时间调整（例如通过控制舵面或推力矢量控制）。

相关公式：

• 空气阻力：
  $
  F_D = \frac{1}{2} C_D \rho v^2 A
  $
  其中，$ v $ 为火箭的空速，$ C_D $ 为阻力系数，$ \rho $ 为空气密度，$ A $ 为迎风面积。

• 速度变化：
  $
  v(t) = \int \left( \frac{F_T}{m} - g - \frac{1}{2} C_D \rho v^2 A / m \right) dt
  $

• 位置变化：
  $
  h(t) = \int v \, dt
  $

---

空速着陆阶段（Airspeed Landing）

• 火箭在空速下降后，开始以空速着陆，速度逐渐减小。
• 空气阻力与速度方向相反，需要考虑空气阻力的影响。
• 火箭的姿态可能会随时间调整（例如通过控制舵面或推力矢量控制）。

相关公式：

• 空气阻力：
  $
  F_D = \frac{1}{2} C_D \rho v^2 A
  $

• 速度变化：
  $
  v(t) = \int \left( - g - \frac{1}{2} C_D \rho v^2 A / m \right) dt
  $

• 位置变化：
  $
  h(t) = \int v \, dt
  $

---

着陆阶段（Landing）

• 火箭在空速着陆后，开始以着陆速度着陆，速度逐渐减小。
• 空气阻力与速度方向相反，需要考虑空气阻力的影响。

相关公式：

• 空气阻力：
  $
  F_D = \frac{1}{2} C_D \rho v^2 A
  $

• 速度变化：
  $
  v(t) = \int \left( - g - \frac{1}{2} C_D \rho v^2 A / m \right) dt
  $

• 位置变化：
  $
  h(t) = \int v \, dt
  $

---

火箭六阶段运动仿真方程（刚体物理 + 空气动力学）

火箭模型假设

• 几何形状： 圆柱 + 圆锥组合
• 质心变化： 由于燃料燃烧，质心沿纵轴下降
• 空气动力学： 考虑空气阻力、风力、重力、推力
• 旋转动力学： 考虑火箭的角动量变化（绕质心旋转）

火箭模型参数

• 质量： $ m(t) = m_0 - \dot{m} t $（线性减少，假设均匀燃烧）
• 质心位置： $ r_c(t) $ 逐渐向下（圆锥部分燃尽后下降）
• 惯性矩： $ I = I{cone} + I{cylinder} $（绕质心计算）
  • $ I{cone} = \frac{3}{10} m{cone} R^2 $
  • $ I{cylinder} = \frac{1}{2} m{cylinder} R^2 $
• 空气阻力（沿速度方向）： $ F_D = \frac{1}{2} C_D \rho v^2 A $
• 风力作用： $ F_W = \frac{1}{2} C_L \rho v_w^2 A $，其中 $ C_L $ 是升力系数， $ v_w $ 是风速
• 推力 $ F_T $ 的方向随火箭姿态变化： $ F_T = \dot{m} v_e $，其中 $ v_e $ 是喷射气体速度

---

六阶段运动方程

1. 发射阶段（Liftoff）

• 火箭从地面垂直升起，初始倾角 $ \theta_0 $
• 受推力、重力、空气阻力影响

平动方程（牛顿第二定律）：

$$m \frac{d\mathbf{v}}{dt} = F_T \hat{e}_T - mg \hat{e}_z - F_D \hat{e}_v + F_W \hat{e}_w$$

角动量方程（Euler方程）：

$$I \frac{d\omega}{dt} = M_{\text{wind}} + M_{\text{thrust}} + M_{\text{drag}}$$

2. 倾斜上升阶段（Inclined Ascent）

• 火箭倾角 $ \theta $ 逐渐增大，风力影响显著
• 受推力、重力、空气阻力、风力作用

平动方程：

$$m \frac{d\mathbf{v}}{dt} = F_T \hat{e}_T - mg \hat{e}_z - F_D \hat{e}_v + F_W \hat{e}_w$$

角动量方程：

$$I \frac{d\omega}{dt} = M_{\text{wind}} + M_{\text{thrust}} + M_{\text{drag}}$$

其中：

• 风力矩： $ M_{\text{wind}} = r_c \times F_W $
• 推力矩： $ M_{\text{thrust}} = d \times F_T $（假设推力偏离质心）

3. 重力转弯阶段（Gravity Turn）

• 火箭逐渐由垂直转向水平
• 主要受重力和空气阻力影响

角加速度方程：

$$\frac{d\gamma}{dt} = \frac{1}{v} \left( \frac{F_T \sin\theta}{m} - g \cos\gamma \right)$$

4. 推进剂耗尽阶段（Burnout）

• 推力消失，仅受重力和空气阻力影响

平动方程：

$$m \frac{d\mathbf{v}}{dt} = -mg \hat{e}_z - F_D \hat{e}_v + F_W \hat{e}_w$$

5. 自由飞行阶段（Free Flight）

• 进入弹道飞行或轨道
• 仅受重力和空气阻力作用

平动方程：

$$m \frac{d\mathbf{v}}{dt} = -mg \hat{e}_z - F_D \hat{e}_v + F_W \hat{e}_w$$

6. 降落阶段（Descent）

• 考虑风力和空气阻力

平动方程：

$$m \frac{d\mathbf{v}}{dt} = -mg \hat{e}_z - F_D \hat{e}_v + F_W \hat{e}_w$$

---

XYZ 三轴运动方程

1. 三轴加速度

$$\begin{array}{l} a_x = -\frac{1}{2} C_D \rho v^2 A \frac{v_x}{v} + a_{\text{wind},x} \\ a_y = -\frac{1}{2} C_D \rho v^2 A \frac{v_y}{v} + a_{\text{wind},y} \\ a_z = -g - \frac{1}{2} C_D \rho v^2 A \frac{v_z}{v} + a_{\text{wind},z} \end{array}$$

2. 三轴速度

$$\begin{array}{l} v_x = \int a_x \, dt \\ v_y = \int a_y \, dt \\ v_z = \int a_z \, dt \\ \end{array}$$

3. 三轴位矢

$$\begin{array}{l} r_x = \int v_x \, dt \\ r_y = \int v_y \, dt \\ r_z = \int v_z \, dt \\ \end{array}$$

程序初步仿真

本人还不是挺懂这个物理模型，所以就没有用具体的流体力学的粒子模型，而是用简单的空气阻力模型，所以可能和实际有误差。

这里很推荐一个plotly的库：
这个库可以将三维数据可视化，并且可以动态展示。
接口也很简单，可以参考这个链接。
它还有其他很好玩的功能，我认为是现代matlab可视化的一个很好的替代品。

具体模拟程序如下：

利用了scipy的solve_ivp函数，这个函数可以用来求解常微分方程，还有plotly库来可视化结果。

import numpy as np
from scipy.integrate import solve_ivp
import plotly.graph_objects as go
from plotly.subplots import make_subplots

# ----------------------------
# 常量与参数设置
# ----------------------------
g = 9.81       # 重力加速度 (m/s^2)
rho = 1.225    # 海平面空气密度 (kg/m^3)
m0 = 20.0
m_final = 5.0 # 火箭初始/燃尽质量 (kg)
burn_time = 3.0 # 燃烧时间 (s)
m_dot_val = (m0 - m_final) / burn_time # 质量流失率 (kg/s)
v_e = 1500.0 # 有效排气速度 (m/s)
F_T_const = m_dot_val * v_e # 推力 (N) = m_dot * v_e
C_D = 0.75
diameter = 0.2 # 空气阻力系数与迎风面积
A = np.pi * (diameter/2)**2  # 迎风面积
v_w = np.array([-3.0, 4.0, 0.0]) # 风速 (m/s)，例如：[-3, 4, 0] 表示在 x 负方向和 y 正方向上

# 发射角度设置：与正北夹角 87.4°，仰角 85°
azimuth_deg = 87.4   
elevation_deg = 85.0
azimuth = np.deg2rad(azimuth_deg)
elevation = np.deg2rad(elevation_deg)

# 推力方向 (单位矢量)，坐标：x正东、y正北、z正上
thrust_dir = np.array([
    np.cos(elevation) * np.sin(azimuth),
    np.cos(elevation) * np.cos(azimuth),
    np.sin(elevation)
])
thrust_dir = thrust_dir / np.linalg.norm(thrust_dir)

# ----------------------------
# 定义火箭动力学模型（ODE）
# 状态变量： [x, y, z, vx, vy, vz, m]
# ----------------------------
def rocket_dynamics(t, state):
    x, y, z, vx, vy, vz, m = state
    vel = np.array([vx, vy, vz])
    if t < burn_time:
        F_thrust = F_T_const * thrust_dir
        dm_dt = -m_dot_val
    else:
        F_thrust = np.array([0.0, 0.0, 0.0])
        dm_dt = 0.0
    v_rel = vel - v_w
    v_rel_norm = np.linalg.norm(v_rel)
    if v_rel_norm < 1e-6:
        F_drag = np.array([0.0, 0.0, 0.0])
    else:
        F_drag = -0.5 * C_D * rho * v_rel_norm**2 * A * (v_rel / v_rel_norm)
    F_gravity = np.array([0.0, 0.0, -m*g])
    F_total = F_thrust + F_drag + F_gravity
    accel = F_total / m

    return [vx, vy, vz, accel[0], accel[1], accel[2], dm_dt]

# ----------------------------
# 定义事件函数：火箭落地（z <= 0）时终止积分
# ----------------------------
def hit_ground(t, state):
    return state[2]  # z 分量

hit_ground.terminal = True
hit_ground.direction = -1 

# ----------------------------
# 初始条件与数值积分
# ----------------------------
initial_state = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, m0]
t_span = (0, 60)

sol = solve_ivp(rocket_dynamics, t_span, initial_state, 
                events=hit_ground, max_step=0.01, rtol=1e-8, atol=1e-8)

t_vals = sol.t
x_vals = sol.y[0]
y_vals = sol.y[1]
z_vals = sol.y[2]
vx_vals = sol.y[3]
vy_vals = sol.y[4]
vz_vals = sol.y[5]
m_vals  = sol.y[6]

# 计算速度大小 speed(t)
speed_vals = np.sqrt(vx_vals**2 + vy_vals**2 + vz_vals**2)

# 计算加速度 (ax, ay, az) 以及加速度大小
acc_array = np.zeros((3, len(t_vals)))
for i in range(len(t_vals)):
    dxdt = rocket_dynamics(t_vals[i], sol.y[:, i])
    # dxdt = [vx, vy, vz, ax, ay, az, dm_dt]
    acc_array[0, i] = dxdt[3]
    acc_array[1, i] = dxdt[4]
    acc_array[2, i] = dxdt[5]
acc_mag = np.sqrt(acc_array[0]**2 + acc_array[1]**2 + acc_array[2]**2)

print(f"仿真结束时间：{t_vals[-1]:.2f} s")
print(f"落地时火箭位置：x={x_vals[-1]:.2f} m, y={y_vals[-1]:.2f} m")
print(f"最远水平距离：{np.sqrt(x_vals[-1]**2 + y_vals[-1]**2):.2f} m")

# ----------------------------
# 准备 3D 轨迹图的 hover 信息 (位置、速度、加速度)
# ----------------------------
hover_texts = []
for i in range(len(t_vals)):
    hover_texts.append(
        f"time = {t_vals[i]:.2f} s<br>"
        f"x = {x_vals[i]:.2f} m, y = {y_vals[i]:.2f} m, z = {z_vals[i]:.2f} m<br>"
        f"vx = {vx_vals[i]:.2f}, vy = {vy_vals[i]:.2f}, vz = {vz_vals[i]:.2f} (m/s)<br>"
        f"ax = {acc_array[0, i]:.2f}, ay = {acc_array[1, i]:.2f}, az = {acc_array[2, i]:.2f} (m/s^2)"
    )

# ----------------------------
# 1) 绘制火箭 3D 轨迹图
# ----------------------------
fig3d = go.Figure(data=[go.Scatter3d(
    x=x_vals, y=y_vals, z=z_vals,
    mode='lines',
    line=dict(
        width=4,
        color=speed_vals,        
        colorscale='Turbo',   
        cmin=0,
        cmax=speed_vals.max(),
        colorbar=dict(title='Speed (m/s)'),
        showscale=True        
    ),
    text=hover_texts,
    hoverinfo='text',
    name='火箭轨迹'
)])
fig3d.update_layout(
    title="火箭三维运动轨迹图",
    scene=dict(
        xaxis_title='x (正东方向)',
        yaxis_title='y (正北方向)',
        zaxis_title='z (竖直向上)'
    ),
    width=800,
    height=800
)

# ----------------------------
# 2) 绘制 x-t, y-t, z-t
# ----------------------------
fig_xyz = make_subplots(rows=3, cols=1,
                        subplot_titles=["x(t)", "y(t)", "z(t)"])
fig_xyz.add_trace(go.Scatter(x=t_vals, y=x_vals, mode='lines', name='x(t)'), row=1, col=1)
fig_xyz.add_trace(go.Scatter(x=t_vals, y=y_vals, mode='lines', name='y(t)'), row=2, col=1)
fig_xyz.add_trace(go.Scatter(x=t_vals, y=z_vals, mode='lines', name='z(t)'), row=3, col=1)

fig_xyz.update_layout(height=900, width=800, title_text="火箭位置随时间变化")
fig_xyz.update_xaxes(title_text="时间 (s)", row=3, col=1)
fig_xyz.update_yaxes(title_text="x (m)", row=1, col=1)
fig_xyz.update_yaxes(title_text="y (m)", row=2, col=1)
fig_xyz.update_yaxes(title_text="z (m)", row=3, col=1)

# ----------------------------
# 3) 绘制速度 (v-t) 图
# ----------------------------
fig_v = make_subplots(rows=4, cols=1,
                      subplot_titles=["vx(t)", "vy(t)", "vz(t)", "speed(t)"])
fig_v.add_trace(go.Scatter(x=t_vals, y=vx_vals, mode='lines', name='vx'), row=1, col=1)
fig_v.add_trace(go.Scatter(x=t_vals, y=vy_vals, mode='lines', name='vy'), row=2, col=1)
fig_v.add_trace(go.Scatter(x=t_vals, y=vz_vals, mode='lines', name='vz'), row=3, col=1)
fig_v.add_trace(go.Scatter(x=t_vals, y=speed_vals, mode='lines', name='|v|'), row=4, col=1)

fig_v.update_layout(height=900, width=800, title_text="火箭速度随时间变化")
fig_v.update_xaxes(title_text="时间 (s)", row=4, col=1)
fig_v.update_yaxes(title_text="速度 (m/s)", row=1, col=1)
fig_v.update_yaxes(title_text="速度 (m/s)", row=2, col=1)
fig_v.update_yaxes(title_text="速度 (m/s)", row=3, col=1)
fig_v.update_yaxes(title_text="速度 (m/s)", row=4, col=1)

# ----------------------------
# 4) 绘制加速度 (a-t) 图
# ----------------------------
fig_a = make_subplots(rows=4, cols=1,
                      subplot_titles=["ax(t)", "ay(t)", "az(t)", "acc magnitude(t)"])
fig_a.add_trace(go.Scatter(x=t_vals, y=acc_array[0], mode='lines', name='ax'), row=1, col=1)
fig_a.add_trace(go.Scatter(x=t_vals, y=acc_array[1], mode='lines', name='ay'), row=2, col=1)
fig_a.add_trace(go.Scatter(x=t_vals, y=acc_array[2], mode='lines', name='az'), row=3, col=1)
fig_a.add_trace(go.Scatter(x=t_vals, y=acc_mag, mode='lines', name='|a|'), row=4, col=1)

fig_a.update_layout(height=900, width=800, title_text="火箭加速度随时间变化")
fig_a.update_xaxes(title_text="时间 (s)", row=4, col=1)
fig_a.update_yaxes(title_text="加速度 (m/s^2)", row=1, col=1)
fig_a.update_yaxes(title_text="加速度 (m/s^2)", row=2, col=1)
fig_a.update_yaxes(title_text="加速度 (m/s^2)", row=3, col=1)
fig_a.update_yaxes(title_text="加速度 (m/s^2)", row=4, col=1)

# ----------------------------
# 显示图像
# ----------------------------
fig3d.show()
fig_xyz.show()
fig_v.show()
fig_a.show()

其流程图如下：

flowchart TD
    B[定义常量与参数] --> C[定义火箭动力学模型]
    C --> D[定义事件函数]
    D --> E[设置初始条件]
    E --> F[数值积分求解]
    F --> G[提取仿真结果]
    G --> H[计算速度和加速度]
    H --> I[绘制图表]
    I --> J[显示图像]

该程序的缺点：

1. 没有考虑火箭的旋转，所以火箭的飞行轨迹是直线，没有倾斜。
2. 落地事件未考虑地形或其他障碍物的影响。如果火箭在仿真时间内未落地，程序会继续运行到最大时间，可能导致不必要的计算。
3. 3D 轨迹图仅显示火箭的路径，未显示其他信息（如推力方向、风速等）
4. 未提供动画功能，无法直观展示火箭的运动过程。
5. 未考虑火箭开伞时的降落过程。

提升空间：

1. 添加火箭旋转模型，使火箭的飞行轨迹更加真实，并考虑火箭开伞时的降落过程，使仿真更加准确。
2. 添加其他信息（如推力方向、风速等）到 3D 轨迹图中，使图表更加直观。
3. 添加动画功能，使火箭的运动过程更加直观。
4. 加入RK45积分方法，提高积分精度，并加入实时轨迹更新功能，以及加入统计学的预测模型，对火箭的飞行轨迹进行预测。

2025

03-11

更新了基于node.js的前端页面，让用户输入更可视化。(至于为什么之前我不写到blog上或者发布到GitHub上，是因为没时间啊！今天(3月30号)才想起这事)

更新的代码

结构：
rock-sim/
├── static/
│ ├── index.css
│ ├── index.html
│ └── simulation.js
├── package-lock.json
├── package.json
├── rocket_sim.py
└── server.js

下面是具体的代码：

server.js

/**
 * server.js
 * 使用 Express 提供静态页面，同时定义 /simulate 接口调用 Python 脚本，
 * 将火箭仿真数据（JSON 格式）返回给前端。
 */
const express = require('express');
const { spawn } = require('child_process');
const path = require('path');

const app = express();
app.use(express.json());

// 提供 static 目录下的静态文件
app.use(express.static(path.join(__dirname, 'static')));

app.post('/simulate', (req, res) => {
  // 从请求中获取参数
  const {
    m0,
    m_final,
    burn_time,
    v_e,
    C_D,
    diameter,
    azimuth_deg,
    elevation_deg,
    wind_x,
    wind_y,
    wind_z,
    total_time
  } = req.body;

  // 将参数依次传入 Python 脚本
  const pyArgs = [
    String(m0),
    String(m_final),
    String(burn_time),
    String(v_e),
    String(C_D),
    String(diameter),
    String(azimuth_deg),
    String(elevation_deg),
    String(wind_x),
    String(wind_y),
    String(wind_z),
    String(total_time)
  ];

  // 调用 Python 脚本 rocket_sim.py
  const pyProcess = spawn('python', [path.join(__dirname, 'rocket_sim.py'), ...pyArgs]);

  let outputData = '';
  let errorData = '';

  pyProcess.stdout.on('data', (data) => {
    outputData += data.toString();
  });
  pyProcess.stderr.on('data', (data) => {
    errorData += data.toString();
  });
  pyProcess.on('close', (code) => {
    if (code !== 0) {
      console.error('Python script error:', errorData);
      return res.status(500).json({ error: errorData || 'Python script failed' });
    }
    try {
      const result = JSON.parse(outputData);
      res.json(result);
    } catch (err) {
      console.error('JSON parse error:', err);
      res.status(500).json({ error: 'Invalid JSON output from Python' });
    }
  });
});

// 启动服务器
const PORT = 3000;
app.listen(PORT, () => {
  console.log(`Server started on http://localhost:${PORT}`);
});

rocket_sim.py

#!/usr/bin/env python
# -*- coding: utf-8 -*-

"""
rocket_sim.py
-------------
命令行参数顺序：
  m0, m_final, burn_time, v_e, C_D, diameter, azimuth_deg, elevation_deg, wind_x, wind_y, wind_z, total_time
如果未传入参数，则使用默认值：
  20.0, 5.0, 3.0, 1500.0, 0.75, 0.2, 87.4, 85.0, -3.0, 4.0, 0.0, 60.0
输出 JSON 数据，包含：
  t_vals, x_vals, y_vals, z_vals, vx_vals, vy_vals, vz_vals, speed_vals,
  acc_array (二维列表: [ax_vals, ay_vals, az_vals]), acc_mag
"""

import sys
import json
import numpy as np
from scipy.integrate import solve_ivp

def simulate_rocket(m0, m_final, burn_time, v_e, C_D, diameter,
                    azimuth_deg, elevation_deg, wind_vector, total_time):
    g = 9.81        # 重力加速度 (m/s^2)
    rho = 1.225     # 空气密度 (kg/m^3)
    
    # 计算燃料流失率与推力（简化公式 F = m_dot * v_e）
    m_dot_val = (m0 - m_final) / burn_time
    F_T_const = m_dot_val * v_e
    
    # 迎风面积（火箭截面面积）
    A = np.pi * (diameter / 2)**2
    
    # 推力方向：由发射角度（水平投影与正北夹角）和仰角决定
    azimuth = np.deg2rad(azimuth_deg)
    elevation = np.deg2rad(elevation_deg)
    thrust_dir = np.array([
        np.cos(elevation) * np.sin(azimuth),
        np.cos(elevation) * np.cos(azimuth),
        np.sin(elevation)
    ])
    thrust_dir = thrust_dir / np.linalg.norm(thrust_dir)
    
    v_w = np.array(wind_vector)
    
    def rocket_dynamics(t, state):
        x, y, z, vx, vy, vz, m = state
        vel = np.array([vx, vy, vz])
        if t < burn_time:
            F_thrust = F_T_const * thrust_dir
            dm_dt = -m_dot_val
        else:
            F_thrust = np.array([0.0, 0.0, 0.0])
            dm_dt = 0.0
        v_rel = vel - v_w
        v_rel_norm = np.linalg.norm(v_rel)
        if v_rel_norm < 1e-6:
            F_drag = np.array([0.0, 0.0, 0.0])
        else:
            F_drag = -0.5 * C_D * rho * (v_rel_norm**2) * A * (v_rel / v_rel_norm)
        F_gravity = np.array([0.0, 0.0, -m * g])
        F_total = F_thrust + F_drag + F_gravity
        accel = F_total / m
        return [vx, vy, vz, accel[0], accel[1], accel[2], dm_dt]

    # 落地事件：当 z 从正变负时触发
    def hit_ground(t, state):
        return state[2]
    hit_ground.terminal = True
    hit_ground.direction = -1

    initial_state = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, m0]
    sol = solve_ivp(rocket_dynamics, (0, total_time), initial_state,
                    events=hit_ground, max_step=0.01, rtol=1e-8, atol=1e-8)

    t_vals = sol.t.tolist()
    x_vals = sol.y[0].tolist()
    y_vals = sol.y[1].tolist()
    z_vals = sol.y[2].tolist()
    vx_vals = sol.y[3].tolist()
    vy_vals = sol.y[4].tolist()
    vz_vals = sol.y[5].tolist()

    # 合速度
    speed_vals = (np.sqrt(sol.y[3]**2 + sol.y[4]**2 + sol.y[5]**2)).tolist()

    # 计算加速度：遍历每个时刻
    n = len(t_vals)
    ax_vals = []
    ay_vals = []
    az_vals = []
    for i in range(n):
        deriv = rocket_dynamics(t_vals[i], sol.y[:, i])
        ax_vals.append(deriv[3])
        ay_vals.append(deriv[4])
        az_vals.append(deriv[5])
    acc_mag = (np.sqrt(np.array(ax_vals)**2 + np.array(ay_vals)**2 + np.array(az_vals)**2)).tolist()
    
    # 返回字典，注意把 numpy 数组转为普通列表
    return {
        "t_vals": t_vals,
        "x_vals": x_vals,
        "y_vals": y_vals,
        "z_vals": z_vals,
        "vx_vals": vx_vals,
        "vy_vals": vy_vals,
        "vz_vals": vz_vals,
        "speed_vals": speed_vals,
        "acc_array": [ax_vals, ay_vals, az_vals],
        "acc_mag": acc_mag
    }

def main():
    # 如果命令行参数不足，则使用默认值
    if len(sys.argv) < 13:
        m0 = 20.0
        m_final = 5.0
        burn_time = 3.0
        v_e = 1500.0
        C_D = 0.75
        diameter = 0.2
        azimuth_deg = 87.4
        elevation_deg = 85.0
        wind_vector = [-3.0, 4.0, 0.0]
        total_time = 60.0
    else:
        m0 = float(sys.argv[1])
        m_final = float(sys.argv[2])
        burn_time = float(sys.argv[3])
        v_e = float(sys.argv[4])
        C_D = float(sys.argv[5])
        diameter = float(sys.argv[6])
        azimuth_deg = float(sys.argv[7])
        elevation_deg = float(sys.argv[8])
        wind_x = float(sys.argv[9])
        wind_y = float(sys.argv[10])
        wind_z = float(sys.argv[11])
        total_time = float(sys.argv[12])
        wind_vector = [wind_x, wind_y, wind_z]

    sim_data = simulate_rocket(m0, m_final, burn_time, v_e, C_D, diameter,
                               azimuth_deg, elevation_deg, wind_vector, total_time)
    # 输出 JSON 数据
    print(json.dumps(sim_data))

if __name__ == "__main__":
    main()

package.json

{
    "name": "rocket-sim",
    "version": "1.0.0",
    "description": "Rocket simulation using Node.js + Python + Plotly",
    "main": "server.js",
    "scripts": {
      "start": "node server.js"
    },
    "dependencies": {
      "express": "^4.18.2"
    }
  }

package-lock.json

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    "requires": true,
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        "node_modules/vary": {
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}

index.css

.navbar {
    box-shadow: 0 2px 4px rgba(0, 0, 0, 0.1);
  }
  
  .section {
    padding: 3rem 1.5rem;
  }
  
  .box {
    box-shadow: 0 2px 4px rgba(0, 0, 0, 0.1);
    transition: box-shadow 0.3s ease;
  }
  
  .box:hover {
    box-shadow: 0 4px 8px rgba(0, 0, 0, 0.2);
  }
  
  .button.is-primary {
    transition: background-color 0.3s ease;
  }
  
  .button.is-primary:hover {
    background-color: #3273dc;
  }
  
  .plot-container {
    margin-top: 2rem;
  }
  
/* 仿真结果部分样式 */
.section {
    background-color: #f8f9fa; 
    padding: 4rem 1.5rem; 
  }
  
  .section .title.is-2 {
    color: #363636; 
    margin-bottom: 2rem; 
    text-align: center; 
    font-weight: bold; 
  }
  
  .plot-container {
    background-color: white; 
    border-radius: 10px; 
    box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); 
    margin-bottom: 2rem; 
    padding: 1.5rem; 
    transition: transform 0.3s ease, box-shadow 0.3s ease; 
  }
  
  .plot-container:hover {
    transform: translateY(-5px); /* 悬停时轻微上移 */
    box-shadow: 0 8px 16px rgba(0, 0, 0, 0.2); /* 悬停时阴影加深 */
  }
  
  /* 图表容器的内联样式保留，确保图表尺寸不变 */
  .plot-container > div {
    width: 100% !important; 
    height: auto; 
  }
  
  @media (max-width: 768px) {
    .section {
      padding: 2rem 1rem; /* 小屏幕下减少内边距 */
    }
  
    .plot-container {
      padding: 1rem; 
    }
  
    #plot3d {
      height: 500px !important; 
    }
  
    #plotXYZ,
    #plotV,
    #plotA {
      height: 400px !important; 
    }
  }

index.html

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1">
  <title>火箭仿真</title>
  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/bulma@0.9.4/css/bulma.min.css">
  <link rel="stylesheet" href="index.css">
  <script src="https://cdn.plot.ly/plotly-2.18.2.min.js"></script>
</head>
<body>
<nav class="navbar is-primary" role="navigation" aria-label="main navigation">
  <div class="container">
    <div class="navbar-brand">
      <a class="navbar-item">
        <h1 class="title is-3 has-text-white">火箭仿真演示</h1>
      </a>
    </div>
  </div>
</nav>

<section class="section">
  <div class="container">
    <h1 class="title">输入参数，点击「开始仿真」</h1>
    <form id="simForm" class="box">
      <div class="columns">
        <div class="column">
          <div class="field">
            <label class="label">初始质量 (kg)</label>
            <div class="control">
              <input class="input" type="number" id="m0" value="20.0" required>
            </div>
          </div>
          <div class="field">
            <label class="label">燃烧后质量 (kg)</label>
            <div class="control">
              <input class="input" type="number" id="m_final" value="5.0" required>
            </div>
          </div>
          <div class="field">
            <label class="label">燃烧时间 (s)</label>
            <div class="control">
              <input class="input" type="number" id="burn_time" value="3.0" required>
            </div>
          </div>
          <div class="field">
            <label class="label">排气速度 (m/s)</label>
            <div class="control">
              <input class="input" type="number" id="v_e" value="1500" required>
            </div>
          </div>
        </div>
        <div class="column">
          <div class="field">
            <label class="label">阻力系数 C_D</label>
            <div class="control">
              <input class="input" type="number" step="0.01" id="C_D" value="0.75" required>
            </div>
          </div>
          <div class="field">
            <label class="label">火箭直径 (m)</label>
            <div class="control">
              <input class="input" type="number" step="0.01" id="diameter" value="0.2" required>
            </div>
          </div>
          <div class="field">
            <label class="label">方位角 (°) <small>(与正北夹角)</small></label>
            <div class="control">
              <input class="input" type="number" id="azimuth_deg" value="87.4" required>
            </div>
          </div>
          <div class="field">
            <label class="label">仰角 (°)</label>
            <div class="control">
              <input class="input" type="number" id="elevation_deg" value="85" required>
            </div>
          </div>
        </div>
        <div class="column">
          <div class="field">
            <label class="label">xyz方向风速 (m/s)</label>
            <label class="label">x正东、y正北、z正上</label>
            <div class="field has-addons">
              <div class="control">
                <input class="input" type="number" id="wind_x" value="-3.0" required>
              </div>
              <div class="control">
                <input class="input" type="number" id="wind_y" value="4.0" required>
              </div>
              <div class="control">
                <input class="input" type="number" id="wind_z" value="0.0" required>
              </div>
            </div>
          </div>
          <div class="field">
            <label class="label">仿真总时长 (s)</label>
            <div class="control">
              <input class="input" type="number" id="total_time" value="60" required>
            </div>
          </div>
          <div class="field">
            <div class="control">
              <button class="button is-primary is-fullwidth" type="submit">开始仿真</button>
            </div>
          </div>
        </div>
      </div>
    </form>
  </div>
</section>

<section class="section">
    <div class="container">
      <h2 class="title is-2">仿真结果</h2>
      <div class="plot-container">
        <div id="plot3d" style="width: 100%; height: 800px;"></div>
      </div>
      <div class="plot-container">
        <div id="plotXYZ" style="width: 100%; height: 600px;"></div>
      </div>
      <div class="plot-container">
        <div id="plotV" style="width: 100%; height: 600px;"></div>
      </div>
      <div class="plot-container">
        <div id="plotA" style="width: 100%; height: 600px;"></div>
      </div>
    </div>
  </section>

<script src="simulation.js"></script>
</body>
</html>

simulation.js

document.getElementById('simForm').addEventListener('submit', async function (e) {
    e.preventDefault();
  
    // 获取输入参数
    const m0 = parseFloat(document.getElementById('m0').value);
    const m_final = parseFloat(document.getElementById('m_final').value);
    const burn_time = parseFloat(document.getElementById('burn_time').value);
    const v_e = parseFloat(document.getElementById('v_e').value);
    const C_D = parseFloat(document.getElementById('C_D').value);
    const diameter = parseFloat(document.getElementById('diameter').value);
    const azimuth_deg = parseFloat(document.getElementById('azimuth_deg').value);
    const elevation_deg = parseFloat(document.getElementById('elevation_deg').value);
    const wind_x = parseFloat(document.getElementById('wind_x').value);
    const wind_y = parseFloat(document.getElementById('wind_y').value);
    const wind_z = parseFloat(document.getElementById('wind_z').value);
    const total_time = parseFloat(document.getElementById('total_time').value);
  
    try {
      const resp = await fetch('/simulate', {
        method: 'POST',
        headers: { 'Content-Type': 'application/json' },
        body: JSON.stringify({
          m0, m_final, burn_time, v_e, C_D, diameter,
          azimuth_deg, elevation_deg,
          wind_x, wind_y, wind_z,
          total_time
        })
      });
      const data = await resp.json();
      const t_vals = data.t_vals;
      const x_vals = data.x_vals;
      const y_vals = data.y_vals;
      const z_vals = data.z_vals;
      const vx_vals = data.vx_vals;
      const vy_vals = data.vy_vals;
      const vz_vals = data.vz_vals;
      const speed_vals = data.speed_vals;
      const acc_array = data.acc_array; // acc_array[0]: ax, [1]: ay, [2]: az
      const acc_mag = data.acc_mag;
  
      // 1) 绘制 3D 轨迹图（颜色按速度大小着色）
      const trace3d = {
        x: x_vals,
        y: y_vals,
        z: z_vals,
        mode: 'lines',
        type: 'scatter3d',
        line: {
          width: 4,
          color: speed_vals,
          colorscale: 'Jet', // 可改成 'Viridis', 'Turbo' 等
          cmin: 0,
          cmax: Math.max(...speed_vals),
          colorbar: { title: 'Speed (m/s)' },
          showscale: true
        },
        text: x_vals.map((x, i) =>
          `t=${t_vals[i].toFixed(2)} s<br>x=${x.toFixed(2)} m, y=${y_vals[i].toFixed(2)} m, z=${z_vals[i].toFixed(2)} m<br>vx=${vx_vals[i].toFixed(2)}, vy=${vy_vals[i].toFixed(2)}, vz=${vz_vals[i].toFixed(2)}`
        ),
        hoverinfo: 'text',
        name: '火箭轨迹'
      };
      const layout3d = {
        title: '火箭三维轨迹图',
        scene: {
          xaxis: { title: 'x East(m)' },
          yaxis: { title: 'y North(m)' },
          zaxis: { title: 'z Vertical upwards(m)' }
        }
      };
      Plotly.newPlot('plot3d', [trace3d], layout3d);
  
      // 2) 绘制 x-t, y-t, z-t 图（独立显示）
      const traceX = { x: t_vals, y: x_vals, mode: 'lines', name: 'x(t)' };
      const traceY = { x: t_vals, y: y_vals, mode: 'lines', name: 'y(t)' };
      const traceZ = { x: t_vals, y: z_vals, mode: 'lines', name: 'z(t)' };
      const layoutXYZ = {
        title: '位置随时间变化',
        height: 600,
        xaxis: { title: 'Time (s)' },
        yaxis: { title: 'Position (m)' }
      };
      Plotly.newPlot('plotXYZ', [traceX, traceY, traceZ], layoutXYZ);
  
      // 3) 绘制速度图：vx, vy, vz, |v|（独立显示）
      const traceVx = { x: t_vals, y: vx_vals, mode: 'lines', name: 'vx(t)' };
      const traceVy = { x: t_vals, y: vy_vals, mode: 'lines', name: 'vy(t)' };
      const traceVz = { x: t_vals, y: vz_vals, mode: 'lines', name: 'vz(t)' };
      const traceSpeed = { x: t_vals, y: speed_vals, mode: 'lines', name: '|v|(t)' };
      const layoutV = {
        title: '速度随时间变化',
        height: 600,
        xaxis: { title: 'Time (s)' },
        yaxis: { title: 'Speed (m/s)' }
      };
      Plotly.newPlot('plotV', [traceVx, traceVy, traceVz, traceSpeed], layoutV);
  
      // 4) 绘制加速度图：ax, ay, az, |a|（独立显示）
      const traceAx = { x: t_vals, y: acc_array[0], mode: 'lines', name: 'ax(t)' };
      const traceAy = { x: t_vals, y: acc_array[1], mode: 'lines', name: 'ay(t)' };
      const traceAz = { x: t_vals, y: acc_array[2], mode: 'lines', name: 'az(t)' };
      const traceA = { x: t_vals, y: acc_mag, mode: 'lines', name: '|a|(t)' };
      const layoutA = {
        title: '加速度随时间变化',
        height: 600,
        xaxis: { title: 'Time (s)' },
        yaxis: { title: 'Acceleration (m/s²)' }
      };
      Plotly.newPlot('plotA', [traceAx, traceAy, traceAz, traceA], layoutA);
    } catch (err) {
      console.error('仿真出错:', err);
      alert('仿真失败，请查看控制台日志。');
    }
  });

提示：

• 首先在终端中运行 node server.js 启动服务器。具体步骤如下：
  1. 在终端中进入项目目录。
  2. 运行 npm install 安装依赖。
  3. 运行 npm start 启动服务器。
• 然后在浏览器中打开npm给你的本地服务器地址（通常是 http://localhost:3000）。

需要注意的是，需要安装以下py的库：

• numpy
• scipy
• matplotlib
• plotly