# 阻尼谐振子 - 弹簧振子

弹簧-质量-阻尼系统的交互式模拟，实时可视化振动、位移曲线和相轨迹

> 标准页面: https://elysiatools.com/zh/visualizations/spring-oscillator

- **分类:** Physics

## 概述

Interactive simulation of damped spring-mass-damper harmonic oscillator with real-time visualization of spring animation, displacement-time curve with exponential envelope, and phase space trajectory. Features equation of motion: mx'' + cx' + kx = 0, with solution x(t) = Ae^(-γt)cos(ωt+φ). Displays natural frequency ω₀ = √(k/m), damping ratio ζ = c/(2√(km)), and damped frequency ωd = ω₀√(1-ζ²). Three preset damping modes: underdamped (ζ < 1) with oscillatory decay, critically damped (ζ = 1) for fastest return without oscillation, and overdamped (ζ > 1) with slow non-oscillatory return. Adjustable parameters: spring constant k (5-100 N/m), mass m (0.5-5 kg), damping coefficient c (0-20 Ns/m), initial position x₀, and initial velocity v₀. Visual displays include real-time spring-mass animation with coil stretching, displacement vs time chart showing envelope decay, phase space plot (x vs v) with inward spiral trajectory, and energy vs time chart showing kinetic, potential, and total energy dissipation. Multi-language support (zh, en, es, fr, de, ru, pt).

## 相关内容

- [零级反应 - Zero-Order Reaction](https://elysiatools.com/zh/visualizations/zero-order-reaction): 零级反应动力学和浓度随时间变化的交互式可视化
- [一级反应 - First-Order Reaction](https://elysiatools.com/zh/visualizations/first-order-reaction): 一级反应动力学和指数浓度衰减的交互式可视化
- [二级反应 - Second-Order Reaction](https://elysiatools.com/zh/visualizations/second-order-reaction): 二级反应动力学和双分子碰撞动力学的交互式可视化
- [阿伦尼乌斯方程 - Arrhenius Equation](https://elysiatools.com/zh/visualizations/arrhenius-equation): 温度对反应速率影响的交互式可视化 - 探索活化能、指数因子和速率常数的关系
- [可逆反应 - Reversible Reaction](https://elysiatools.com/zh/visualizations/reversible-reaction): A ⇌ B 可逆反应动力学的交互式可视化 - 探索正逆反应速率、平衡常数和浓度随时间的变化
- [连续反应 - Consecutive Reaction](https://elysiatools.com/zh/visualizations/consecutive-reaction): A → B → C 连续反应动力学的交互式可视化 - 探索中间物浓度峰值、决速步和所有物种的完整演化
- [链式反应 - Chain Reaction](https://elysiatools.com/zh/visualizations/chain-reaction): 自由基链式反应聚合的交互式可视化 - 探索引发、增长、终止步骤，链增长动画和分子量分布
- [勒夏特列原理 - Le Chatelier's Principle](https://elysiatools.com/zh/visualizations/le-chateliers-principle): 勒夏特列原理的交互式可视化 - 探索浓度、压强和温度变化如何影响化学平衡，配合分子动力学动画展示平衡移动过程