机械动力学

绪论

机械动力学的研究内容

• 动力学分析

研究现有机械在力作用下的运动,机械在运动中的力

• 动力学综合
  从力和运动的相互作用角度下设计新机械或改进现有机械

• 动力学反问题

已知运动求力

• 动力学正问题
  已知力求运动

机械动力学分析方法

名称	方法原理	适用范围	基本假设
静力分析	静力学方法	低速机械	运动中产生的惯性力可以忽略不计,刚性物件
动态静力分析	达朗贝尔原理	速度较高	原动构件等速回转,刚性物件
动力分析		动力学正问题	刚性物件
弹性动力分析		机械轻量化

动态静力学分析

• 将惯性力计入静力平衡方程

• 惯性力−mi​si​¨​

• 惯性矩−Ji​φi​¨​

• 特殊情况

  1. 往复直线运动--惯性矩为0
  2. 绕质心回转运动--惯性力为0

• 构件自由度

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

pL​低副,pH​高副

平面机构的平衡

平衡的分类

• 惯性载荷三种危害,分为

  1. 机构在机座上的平衡
  2. 机构输入转矩的平衡
  3. 运动副中动压力的平衡

• 根据采取措施不同,分为

  1. 加配重---较通用
  2. 合理布局,设置附加机构---不具普遍性

• 惯性载荷被平衡程度,分为

  1. 部分平衡
  2. 完全平衡
  3. 优化综合平衡

摆动力(矩)平衡条件

• 机构摆动力完全平衡的条件

1. 机构总质心静止不动

xs​=m1​∑i=1n−1​mi​xi​

ys​=m1​∑i=1n−1​mi​yi​

2. 定义∑i=1n−1​mi​xi​ , ∑i=1n−1​mi​yi​为 质量矩,则可表述为

   • 机构的质量矩为常数

3. H0​=∑i=1n−1​[mi​(xi​yi​˙​−yi​xi​˙​)+Ji​φi​˙​]为 动量矩,则机构摆动力矩完全平衡的条件

   • 机构动量矩为常数

完全平衡的局限性

1. 不是任何机构都可以通过施加配重来实现完全平衡,需满足通路定理:

   • 机构内任何一个构件都有一条通到固定件的途径,在此途径上只含转动副而无移动副

2. 完全平衡导致机械结构复杂化

3. 完全平衡是机械质量大为增加

质量代换的条件

1. 各代换质量的综合应等于原来构件的质量
2. 各质量代换的总质心应与原来的质心相重合
3. 各代换质量对坐标原点的转动惯量之和应等于原构件对坐标点的转动惯量

单自由度机械系统动力学

机构运转阶段

1. 启动阶段
2. 稳定运转阶段
3. 停车阶段

动力分析中主要的力

力	描述	一个循环做功
驱动力	原动机发出传给驱动构件的力	正功
生产阻力	完成有用功时作用于机械的阻力	负功
重力		零功
摩擦力	运动副表面的摩擦	负功

力	常数	位移	速度	时间
生产阻力	o	o	o	o
驱动力	o	o	o	x

等效构件

1. 作回转运动的主动构件的转角选成系统的广义坐标,称为 等效构件

   1. 转动惯量称为 等效转动惯量
   2. 系统总动能Jqi​˙​2

2. 等效构件上作用的力矩为 等效力矩 Me​

   1. 整个系统各构件全部的力和力矩折算到等效构件上
   2. 等效力矩所做的功等于系统所受全部的力和力矩所做的功

3. 对直线运动

   1. 等效惯量-->等效质量
   2. 等效力矩-->等效力

等效转动惯量的计算

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>J</mi><mi>e</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>l</mi></munderover><mo>[</mo><msub><mi>m</mi><mi>i</mi></msub><mo>(</mo><mfrac><mrow><msub><mi>v</mi><mrow><mi>S</mi><mi>i</mi></mrow></msub></mrow><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover></mrow></mfrac><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><msub><mi>J</mi><mrow><mi>S</mi><mi>i</mi></mrow></msub><mo>(</mo><mfrac><mrow><msub><mi>ω</mi><mi>i</mi></msub></mrow><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover></mrow></mfrac><msup><mo>)</mo><mn>2</mn></msup><mo>]</mo></mrow><annotation encoding="application/x-tex">J_e=\sum_{i=1}^l[m_i(\frac{v_{Si}}{\dot{q}})^2+J_{Si}(\frac{\omega_{i}}{\dot{q}})^2]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.8361130000000003em;"></span><span class="strut bottom" style="height:3.1137820000000005em;vertical-align:-1.277669em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span><span class="mord mathit mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span><span class="mord mathit mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span>

等效力矩

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mi>e</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo>(</mo><mfrac><mrow><msub><mi>F</mi><mi>k</mi></msub><msub><mi>v</mi><mi>k</mi></msub><mi>c</mi><mi>o</mi><mi>s</mi><msub><mi>α</mi><mi>k</mi></msub></mrow><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover></mrow></mfrac><mo>)</mo><mo>+</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo>(</mo><mo>±</mo><msub><mi>M</mi><mi>j</mi></msub><mfrac><mrow><msub><mi>ω</mi><mi>j</mi></msub></mrow><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover></mrow></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">M_e=\sum_{k=1}^m(\frac{F_kv_kcos\alpha_k}{\dot{q}})+\sum_{j=1}^m(\pm M_j\frac{\omega_j}{\dot{q}})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.6513970000000007em;"></span><span class="strut bottom" style="height:3.0651740000000007em;vertical-align:-1.4137769999999998em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord mathit">c</span><span class="mord mathit">o</span><span class="mord mathit">s</span><span class="mord"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"></span></span></span></span><span class="mopen">(</span><span class="mord">±</span><span class="mord"><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span>

动力学方程

1. 力矩形式运动学方程

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>J</mi><mi>e</mi></msub><mover accent="true"><mrow><mi>q</mi></mrow><mo>¨</mo></mover><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>J</mi><mi>e</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><mi>q</mi></mrow></mfrac><msup><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover><mn>2</mn></msup><mo>=</mo><msub><mi>M</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">J_e\ddot{q}+\frac12\frac{\partial J_e}{\partial q}\dot{q}^2=M_e
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.37144em;"></span><span class="strut bottom" style="height:2.25188em;vertical-align:-0.8804400000000001em;"></span><span class="base"><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">¨</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">1</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span></span>

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>m</mi><mi>e</mi></msub><mover accent="true"><mrow><mi>q</mi></mrow><mo>¨</mo></mover><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>m</mi><mi>e</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><mi>q</mi></mrow></mfrac><msup><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover><mn>2</mn></msup><mo>=</mo><msub><mi>F</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">m_e\ddot{q}+\frac12\frac{\partial m_e}{\partial q}\dot{q}^2=F_e
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.37144em;"></span><span class="strut bottom" style="height:2.25188em;vertical-align:-0.8804400000000001em;"></span><span class="base"><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.16666em;">¨</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">1</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span></span>

2. 能量形式动力学方程

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>J</mi><mrow><mi>e</mi><mn>2</mn></mrow></msub><msubsup><mi>ω</mi><mn>2</mn><mn>2</mn></msubsup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>J</mi><mrow><mi>e</mi><mn>1</mn></mrow></msub><msubsup><mi>ω</mi><mn>1</mn><mn>2</mn></msubsup><mo>=</mo><msubsup><mo>∫</mo><msub><mi>φ</mi><mn>1</mn></msub><msub><mi>φ</mi><mn>2</mn></msub></msubsup><msub><mi>M</mi><mi>e</mi></msub><mi>d</mi><mi>φ</mi></mrow><annotation encoding="application/x-tex">\frac12 J_{e2}\omega_2^2-\frac12 J_{e1}\omega_1^2=\int_{\varphi_1}^{\varphi_2}M_ed\varphi
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.4142919999999999em;"></span><span class="strut bottom" style="height:2.46235em;vertical-align:-1.048058em;"></span><span class="base"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">1</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">e</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">1</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">e</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4142919999999999em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"></span></span></span></span></span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">φ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.048058em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord mathit">d</span><span class="mord mathit">φ</span></span></span></span></span>

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>m</mi><mrow><mi>e</mi><mn>2</mn></mrow></msub><msubsup><mi>v</mi><mn>2</mn><mn>2</mn></msubsup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>m</mi><mrow><mi>e</mi><mn>1</mn></mrow></msub><msubsup><mi>v</mi><mn>1</mn><mn>2</mn></msubsup><mo>=</mo><msubsup><mo>∫</mo><msub><mi>s</mi><mn>1</mn></msub><msub><mi>s</mi><mn>2</mn></msub></msubsup><msub><mi>F</mi><mi>e</mi></msub><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">\frac12 m_{e2}v_2^2-\frac12 m_{e1}v_1^2=\int_{s_1}^{s_2}F_eds
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.4142920000000003em;"></span><span class="strut bottom" style="height:2.426342em;vertical-align:-1.01205em;"></span><span class="base"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">1</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">e</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">1</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">e</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4142920000000003em;"><span style="top:-1.7880500000000004em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"></span></span></span></span></span></span></span></span><span style="top:-3.8129000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathit mtight">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.01205em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mord mathit">d</span><span class="mord mathit">s</span></span></span></span></span>

等效转动惯量的导数

<span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>d</mi><msub><mi>J</mi><mi>e</mi></msub></mrow><mrow><mi>d</mi><mi>q</mi></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mrow><msup><mover accent="true"><mrow><mi>q</mi></mrow><mo>˙</mo></mover><mn>3</mn></msup></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>l</mi></munderover><mo>[</mo><msub><mi>m</mi><mi>j</mi></msub><mo>(</mo><msub><mi>v</mi><mrow><mi>S</mi><mi>j</mi><mi>x</mi></mrow></msub><msub><mi>a</mi><mrow><mi>S</mi><mi>j</mi><mi>x</mi></mrow></msub><mo>+</mo><msub><mi>v</mi><mrow><mi>S</mi><mi>j</mi><mi>y</mi></mrow></msub><msub><mi>a</mi><mrow><mi>S</mi><mi>j</mi><mi>y</mi></mrow></msub><mo>)</mo><mo>+</mo><msub><mi>J</mi><mi>j</mi></msub><msub><mi>ω</mi><mi>j</mi></msub><msub><mi>α</mi><mi>j</mi></msub><mo>]</mo></mrow><annotation encoding="application/x-tex">\frac{dJ_e}{dq}=\frac2{\dot{q}^3}\sum_{j=1}^l[m_j(v_{Sjx}a_{Sjx}+v_{Sjy}a_{Sjy})+J_j\omega_j\alpha_j]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.8361130000000006em;"></span><span class="strut bottom" style="height:3.2498900000000006em;vertical-align:-1.4137769999999998em;"></span><span class="base"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">d</span><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit">d</span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mord rule" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.66786em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.055550000000000016em;">˙</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy" style="height:0.2em;"><svg width="400em" height="0.2em" viewBox="0 0 400000 200" preserveAspectRatio="xMinYMin slice"><path d="M0 80H400000 v40H0z M0 80H400000 v40H0z"></path></svg></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord">2</span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000006em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathit">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mord mathit mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mord mathit mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord"><span class="mord mathit">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight" style="margin-right:0.05764em;">S</span><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span><span class="mord mathit mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mclose">)</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mord rule" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.09618em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mord"><span class="mord mathit" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathit mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span><span class="mclose">]</span></span></span></span></span>

盈功和亏功

1. 输入功大于输出功--盈功
2. 输出功大于输入功--亏功

不均匀系数

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

单自由度系统的振动

单自由度系统小阻尼自由振动

1. 周期性振动,固有频率略小于无阻尼自由振动
2. 振幅随时间成指数衰减

狄里克雷条件

能进行傅里叶变换的充分非必要条件:
在一个周期内

1. 只存在有限个第一类不连续点
2. 极大值极小值的数目有限
3. 绝对可积

振动的分类

1. 振动产生原因:

   1. 自由振动
   2. 强迫振动
   3. 自激振动

2. 结构参数特性

   1. 线性振动
   2. 非线性振动

3. 系统自由度

   1. 单自由度系统振动
   2. 多自由度系统振动
   3. 连续体振动

4. 振动规律

   1. 简谐振动
   2. 周期振动
   3. 瞬态振动
   4. 随机振动

简谐振动的表示方法

1. 正弦余弦表示法
2. 旋转向量表示法
3. 复数表示法

简谐振动的基本性质

1. 同方向同高频率合成仍未简谐运动

2. 不同频率合成

   1. 频率比为有理数--周期振动
   2. 频率比为无理数--非周期振动

3. 两个频率十分接近的简谐振动合成周期性的拍振

机械振动三要素

惯性,弹性,阻尼
产生振动--惯性,弹性

1. 等效质量--等效前后系统动能相等
2. 等效刚度--等效前后系统弹性势能相等
3. 等效粘性阻尼--振动周期内非线性粘性阻尼消耗能量等于等效粘性阻尼消耗能量

• 阻尼对单自由度系统自由振动的影响为
  1. 改变了振动频率
  2. 使振幅衰减

二自由度系统的振动

振型矢量的正交性

A(1)TMA(2)=0
A(2)TMA(1)=0
A(1)TKA(2)=0
A(2)TKA(1)=0

主振动

正定系统只能出现形如x=uacos(ωt−φ)的同步运动,称为主振动