本文由20级学生整理，包括实验目的和仪器、实验原理、实验步骤三个部分。主要是想节约一下大家手机拍照扫描、语音输入或手打的时间。（可能有些任课老师要求手写，那就爱莫能助了）
【5.4 缺乏实验原理
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注：有些实验报告的排版有点难看，可以根据需要改。
5.1 研究受迫振动
5.2 霍尔效应
5.3 PN结特性
5.4 测量液体变温粘滞系数PID使用
5.6 迈克尔逊干涉仪
5.7 全息照相
5.8 用光栅测量光波波长
5.9 使用光栅光谱仪
5.10 光电效应
5.11 测定电子电荷e值
5.12 弗兰克-赫兹实验
7.4 超声波在物质中的传播和超声成像
薄膜-toc" style="margin-left:80px;">7.6 金属薄膜采用低真空获取、测量和直流溅射法制备
太阳能电池-toc" style="margin-left:80px;">5.17 太阳能电池
补充实验1-金属丝线膨胀系数测定
补充实验2-偏振光的观察和研究
3-拉曼光谱实验
实验目的和仪器
<p><label>实验目的</label></p> <p>（1）研究波尔共振仪中弹性摆轮受迫振动的幅频特性和相频特性。</p> <p>(2)研究不同阻尼扭矩对受迫振动的影响，观察共振现象。</p> <p>(3)学习用频闪法测定差异的方法。</p> <p>(4)纠正系统误差的方法。</p> <p><label>实验仪器</label></p> <p>BG-2型波尔共振器</p>实验原理
<p> 系统不仅受到驱动力的影响，还受到回复力和阻尼力的影响。因此，在稳定振动状态下，物体的位移、速度和驱动力的变化不同。当驱动频率接近系统的固有频率时，振幅会增加。当物体振幅达到最大时，称为位移共振。位移共振频率接近振动物固有频率，但低于振动物固有频率。阻尼越小，位移共振频率越接近振动物的固有频率。当驱动频率与振动物体固有频率相同时，强制振动的速度范围达到最大，产生速度共振时，物体的振动位移落后于驱动力90°。</p> <p> 波尔共振中的摆轮可以在弹性扭矩的作用下自由摆动。如果同时添加阻尼力矩和驱动力矩，摆轮可以进行强制振动。它可以直观地显示机械振动中的一些物理现象，用于研究被迫振动的特性。</p> <p> 当摆轮周期驱动外力矩M=M<sub>0</sub>cosw<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="3">ω</var></span></span> t的作用，并在有空气阻尼和电磁尼的介质中运动(阻尼扭矩设置为-<span class="mq-math-mode" latex-data="\gamma\frac{d\Theta}{dt}" style="font-size: 29.9px;"><span class="mq-texarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 64.3646px;"><var mathquill-command-id="5">γ</var><span class="mq-fraction mq-non-leaf" mathquill-command-id="7" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="8"><var mathquill-command-id="11">d</var><span mathquill-command-id="12">Θ</span></span><span class="mq-denominator" mathquill-block-id="9" style="width: 37.8229px;"><var mathquill-command-id="15">d</var><var mathquill-command-id="23">t</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> ,<span class="mq-math-mode" latex-data="\gamma" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 15.7917px;"><var mathquill-command-id="5">γ</var></span></span> 为尼力矩系数)，其运动方程为J<span class="mq-math-mode" latex-data="\frac{d^2\Theta}{dt^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 64.9062px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="24" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="25"><var mathquill-command-id="33">d</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="30" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="31"><span mathquill-command-id="34">2</span></span></span><span mathquill-command-id="35">Θ</span></span><span class="mq-denominator" mathquill-block-id="26" style="width: 50.1562px;"><var mathquill-command-id="37">d</var><var mathquill-command-id="43">t</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="40" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="41"><span mathquill-command-id="44">2</span></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =-k<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="45">Θ</span></span></span> -<span class="mq-math-mode" latex-data="\gamma\frac{d\Theta}{dt}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 64.3646px;"><var mathquill-command-id="47">γ</var><span class="mq-fraction mq-non-leaf" mathquill-command-id="49" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="50"><var mathquill-command-id="53">d</var><span mathquill-command-id="54">Θ</span></span><span class="mq-denominator" mathquill-block-id="51" style="width: 37.8229px;"><var mathquill-command-id="56">d</var><var mathquill-command-id="57">t</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> +M<sub>0</sub>cos<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="58">ω</var></span></span> t(5.1-1)式中。J为摆轮的转动惯量:-k<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="60">Θ</span></span></span> 为弹性力矩;k为弹簧的劲度系数;M<sub>0</sub>为驱动力矩的幅值；<span style="font-family: "Times New Roman", Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">ω</span>为驱动力的角频率。令<span class="mq-math-mode" latex-data="\omega_0^2" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="70">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="67" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="65"><span mathquill-command-id="72">2</span></span><span class="mq-sub" mathquill-block-id="68"><span mathquill-command-id="73">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> <span style="font-family: Symbola, "Times New Roman", serif; font-size: 16.56px; text-align: center; white-space: nowrap;">Θ</span>=<span class="mq-math-mode" latex-data="\frac{k}{J}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 32.0729px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="74" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="75"><var mathquill-command-id="78">k</var></span><span class="mq-denominator" mathquill-block-id="76" style="width: 17.3229px;"><var mathquill-command-id="79">J</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> ,2β=<span class="mq-math-mode" latex-data="\frac{\gamma}{J}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 32.0729px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="74" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="75"><var mathquill-command-id="80">γ</var></span><span class="mq-denominator" mathquill-block-id="76" style="width: 17.3229px;"><var mathquill-command-id="79">J</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> ，m=<span class="mq-math-mode" latex-data="\frac{M_0}{J}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 54.875px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="74" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="75"><var mathquill-command-id="93">M</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="90" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="91"><span mathquill-command-id="94">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span><span class="mq-denominator" mathquill-block-id="76" style="width: 40.125px;"><var mathquill-command-id="79">J</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> 则式(5.1-1)变为 当<span style="font-family: Symbola, "Times New Roman", serif; font-size: 16.56px; text-align: center; white-space: nowrap;">mcos</span><span style="font-family: "Times New Roman", Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">ωt</span>=0时，即在无周期性驱动外力矩作用时，式(5.1-2)即为阻尼振动方程；且当阻尼系数β=0,即无阻尼时，式(5.1-2)变为简谐振动方程，<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="121">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="118" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="119"><span mathquill-command-id="123">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> 即为振动系统的固有角频率。 方程(5.1-2)的通解为</p>
<p><span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="124">Θ</span></span></span> =<span style="font-family: Symbola, "Times New Roman", serif; font-size: 18.4px; white-space: nowrap;">Θ<sub>1</sub></span>e<sup>-βt</sup>cos(<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="126">ω</var></span></span> <sub>1</sub>t+<span class="mq-math-mode" latex-data="\alpha" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 19.6979px;"><var mathquill-command-id="128">α</var></span></span> )+<span style="font-family: Symbola, "Times New Roman", serif; font-size: 18.4px; white-space: nowrap;">Θ</span><sub>2</sub>cos(<span style="font-family: "Times New Roman", Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">ω</span>t+φ) (5.1-3）</p>
<p>由式(5.1-3）可见，受迫振动可分成两部分:第一部分，<span style="font-family: Symbola, "Times New Roman", serif; font-size: 18.4px; white-space: nowrap;">Θ<sub>1</sub></span>e<sup>-βt</sup>cos(<span class="mq-math-mode" latex-data="\omega" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 17px;"><var mathquill-command-id="126">ω</var></span></span> <sub>1</sub>t+<span class="mq-math-mode" latex-data="\alpha" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 13.6625px;"><var mathquill-command-id="128">α</var></span></span> )表示阻尼振动，经过一定时间后振动衰减至可忽略不计。第二部分，因驱动力矩对摆轮做功，向振动系统传送能量，使系统最终达到稳定的动状态。此时振幅不变，其值为</p>
<p><span class="mq-math-mode" latex-data="\Theta_2" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8125px;"><span mathquill-command-id="137">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="134" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="135"><span mathquill-command-id="139">2</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =<span class="mq-math-mode" latex-data="\frac{m}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+4\beta^2\omega^2}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 292.417px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="140" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="141"><var mathquill-command-id="144">m</var></span><span class="mq-denominator" mathquill-block-id="142" style="width: 277.667px;"><span class="mq-non-leaf" mathquill-command-id="145"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 1.95556);">√</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="146"><span class="mq-non-leaf" mathquill-command-id="207"><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">(</span><span class="mq-non-leaf" mathquill-block-id="189"><var mathquill-command-id="188">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="194" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="196"><span mathquill-command-id="195">2</span></span><span class="mq-sub" mathquill-block-id="192"><span mathquill-command-id="191">0</span></span><span style="display: inline-block; width: 0px;"></span></span><span class="mq-binary-operator" mathquill-command-id="198">−</span><var mathquill-command-id="200">ω</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="202" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="204"><span mathquill-command-id="203">2</span></span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="184" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="185"><span mathquill-command-id="209">2</span></span></span><span class="mq-binary-operator" mathquill-command-id="210">+</span><span mathquill-command-id="211">4</span><var mathquill-command-id="217">β</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="214" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="215"><span mathquill-command-id="219">2</span></span></span><var mathquill-command-id="232">ω</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="229" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="230"><span mathquill-command-id="234">2</span></span></span></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> (5.1-4)</p>
<p>它与驱动力矩之间的相差<span class="mq-math-mode" latex-data="\varphi" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 20.5417px;"><var mathquill-command-id="235">φ</var></span></span> 为<span class="mq-math-mode" latex-data="\varphi" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 14.1875px;"><var mathquill-command-id="235">φ</var></span></span> =arctan<span class="mq-math-mode" latex-data="\frac{2\beta\omega}{\omega_0^2-\omega^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 114.448px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="237" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="238"><span mathquill-command-id="241">2</span><var mathquill-command-id="242">β</var><var mathquill-command-id="244">ω</var></span><span class="mq-denominator" mathquill-block-id="239" style="width: 99.6979px;"><var mathquill-command-id="254">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="251" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="249"><span mathquill-command-id="256">2</span></span><span class="mq-sub" mathquill-block-id="252"><span mathquill-command-id="257">0</span></span><span style="display: inline-block; width: 0px;"></span></span><span class="mq-binary-operator" mathquill-command-id="258">−</span><var mathquill-command-id="264">ω</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="261" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="262"><span mathquill-command-id="266">2</span></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =arctan<span class="mq-math-mode" latex-data="\frac{\beta T_0^2T}{\pi\left(T^2-T_0^2\right)}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 148.667px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="237" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="238"><var mathquill-command-id="242">β</var><var mathquill-command-id="275">T</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="272" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="270"><span mathquill-command-id="276">2</span></span><span class="mq-sub" mathquill-block-id="273"><span mathquill-command-id="277">0</span></span><span style="display: inline-block; width: 0px;"></span></span><var mathquill-command-id="278">T</var></span><span class="mq-denominator" mathquill-block-id="239" style="width: 133.917px;"><span class="mq-nonSymbola" mathquill-command-id="281">π</span><span class="mq-non-leaf" mathquill-command-id="283"><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">(</span><span class="mq-non-leaf" mathquill-block-id="284"><var mathquill-command-id="292">T</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="289" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="290"><span mathquill-command-id="293">2</span></span></span><span class="mq-binary-operator" mathquill-command-id="294">−</span><var mathquill-command-id="286">T</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="251" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="249"><span mathquill-command-id="256">2</span></span><span class="mq-sub" mathquill-block-id="252"><span mathquill-command-id="257">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">)</span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> (5.1-5)由式(5.1-4)和式(5.1-5)可看出，稳定振动状态的振幅<span class="mq-textarea" style="font-size: 18.4px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: text; box-sizing: border-box; position: relative; font-family: Symbola, "Times New Roman", serif;"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: text; box-sizing: border-box; position: absolute; clip: rect(1em, 1em, 1em, 1em); transform: scale(0); resize: none; width: 1px; height: 1px;" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.425px; font-size: 18.4px; line-height: inherit; margin: 0px; border-color: black; user-select: none; font-family: Symbola, "Times New Roman", serif;"><span mathquill-command-id="137" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="134" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em;"><span class="mq-sub" mathquill-block-id="135" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="139" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">2</span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;"></span></span></span>与相差的数值<span class="mq-math-mode" latex-data="\varphi" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 14.1875px;"><var mathquill-command-id="235">φ</var></span></span> 取决于驱动力矩的幅值M<sub>0</sub>、驱动力的频率<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="295">ω</var></span></span> 、系统的固有频率<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="302">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="299" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="300"><span mathquill-command-id="304">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> 和阻尼系数β四个因素，而与振动的起始状态无关。</p>
<p>受迫振动的振幅与驱动力频率有关，由极大值条件<span class="mq-math-mode" latex-data="\frac{\delta\Theta}{\delta\omega}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 51.6354px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="305" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="306"><var mathquill-command-id="311">δ</var><span mathquill-command-id="313">Θ</span></span><span class="mq-denominator" mathquill-block-id="307" style="width: 36.8854px;"><var mathquill-command-id="317">δ</var><var mathquill-command-id="315">ω</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =0可知，当驱动力角频率为</p>
<p><span class="mq-math-mode" latex-data="\omega_r" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.5938px;"><var mathquill-command-id="324">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="321" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="322"><var mathquill-command-id="326">r</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =<span class="mq-math-mode" latex-data="\sqrt{\omega_0^2-2\beta^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 156.208px;"><span class="mq-non-leaf" mathquill-command-id="327"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 1.75);">√</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="328"><var mathquill-command-id="338">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="335" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="333"><span mathquill-command-id="340">2</span></span><span class="mq-sub" mathquill-block-id="336"><span mathquill-command-id="341">0</span></span><span style="display: inline-block; width: 0px;"></span></span><span class="mq-binary-operator" mathquill-command-id="342">−</span><span mathquill-command-id="343">2</span><var mathquill-command-id="349">β</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="346" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="347"><span mathquill-command-id="351">2</span></span></span></span></span></span></span> (5.1-6)</p>
<p><span class="mq-math-mode" latex-data="\Theta_r" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.5833px;"><span mathquill-command-id="357">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="354" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="355"><var mathquill-command-id="359">r</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =<span class="mq-math-mode" latex-data="\frac{m}{2\beta\sqrt{\omega_0^2-\beta^2}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 170.479px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="360" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="361"><var mathquill-command-id="364">m</var></span><span class="mq-denominator" mathquill-block-id="362" style="width: 155.729px;"><span mathquill-command-id="365">2</span><var mathquill-command-id="366">β</var><span class="mq-non-leaf" mathquill-command-id="368"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 1.73333);">√</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="369"><var mathquill-command-id="379">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="376" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="374"><span mathquill-command-id="381">2</span></span><span class="mq-sub" mathquill-block-id="377"><span mathquill-command-id="382">0</span></span><span style="display: inline-block; width: 0px;"></span></span><span class="mq-binary-operator" mathquill-command-id="383">−</span><var mathquill-command-id="390">β</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="386" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="387"><span mathquill-command-id="389">2</span></span></span></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> (5.1-7)</p>
<p> 此时，系统产生共振。阻尼系数β越小，共振时驱动力角频率越接近系统固有角频率，振幅就越大。</p>
<p><img alt="" height="512" src="/files/testpaper/106/2022/03-31/201701d10575308923.jpg" width="400" /></p>
实验步骤
<p><label>1. 定性观察摆轮的自由振动和阻尼振动</label></p>
<p>选中进入“自由振荡”(或“阻尼振荡”，再选中“阻尼选择”)，在“测量”置于默认的“关”状态下，可以直接从屏幕上读出每次振动的振幅和周期，注意观察振幅变化的特点. 数据的自动记录与查询功能:当测量界面中的“测量”处于被选择时，可以更改(通过上下按键，以下同)它的状态，将其置于“开”状态，则控制箱可以在摆轮摆动时自动记录屏上所显示的数据。当“测量”回到“关”状态时，可以选中“回查”，进入查询界面，通过上下按键查看所有记录的数据，按“确认”按钮可以退出回查状态。</p>
<p><label> 2. 测定受迫动的幅频特性和相频特性曲线(至少完成一种阻尼下的测量)</label></p>
<p> 选中“阻尼振荡”，将“阻尼选择”选中于适当的档位(如“阻尼2”)，再由测量界面返回到“实验步骤”界面后，才能选中“强迫振荡”进行测量。注意在实验过程中“阻尼选择”不能任意改变，或将整机电源切断，否则由于电磁铁剩磁现象将引起β值变化，只有在某一阻尼系数β的所有实验数据测量完毕，需要改变β值时，才可改变“阻尼选择”。进人“强迫振荡”的测量界面(默认选择“电机”)后，更改“电机”状态令其置于“开”，则启动电动机，当保持“周期”为“1”时，屏上可以直接显示摆轮和电动机的周期、振幅值。改变电动机的转速.即改变驱动力矩的频率w。设定某一电动机转速，当受迫振动稳定后，才可以开始准备测量(此时摆轮和电动机的周期必须趋向一致)。选择“周期”，把“周期”更改为“10”，再选择“测量”，更改其状态为“开”，控制箱开始自动记录数据。一次测量完成，测量”状态显示“关”，读出摆轮的振幅值0。记录驱动力矩10次振动周期10T、按住“闪光灯”按钮，利用闪光灯测定受迫振动位移与驱动力间的相差。将所测电动机转速刻度值、驱动力周期10T、振幅、相差等数据记录。</p>
<p> 每次改变强迫力周期钮的刻度(即改变电动机转速)进行测量前，均需返回“周期”为1的测量界面，等待系统稳定，之后再进行相应的测量. 强迫振动测量完毕、选中“返回”，回到“实验步骤”选择界面</p>
<p><label> 3. 测定尼系数β</label></p>
<p> 在“实验步骤”界面选中“阻尼振荡”，将“阻尼选择”置于与“强迫振荡”测量时相同的档位，确认进入阻尼测量界面。将有机玻璃盘上零度标志放在0位置，用手将摆轮转动140°~150°左右。松手后将“测量”状态更改为“开”。控制箱开始自动连续记录摆轮做阻尼振动10次的振幅数值<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="3">Θ</span></span></span><sub> 0</sub>、<span style="font-family: Symbola, "Times New Roman", serif; font-size: 18.4px; white-space: nowrap;">Θ<sub>1</sub>、Θ<sub>2</sub>、……、Θ<sub>n</sub></span>。及周期。在“测量”回到“关”时，可以利用回查功能查询记录的数据。测量数据记录，利用公式ln<span class="mq-math-mode" latex-data="\frac{\Theta_0e^{-\beta t}}{\Theta_0e^{-\beta\left(t+nT\right)}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 176.042px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="6"><span mathquill-command-id="14">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="11" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="12"><span mathquill-command-id="16">0</span></span><span style="display: inline-block; width: 0px;"></span></span><var mathquill-command-id="23">e</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="20" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="21"><span mathquill-command-id="24">−</span><var mathquill-command-id="25">β</var><var mathquill-command-id="27">t</var></span></span></span><span class="mq-denominator" mathquill-block-id="7" style="width: 161.292px;"><span mathquill-command-id="28">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="30" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="32"><span mathquill-command-id="31">0</span></span><span style="display: inline-block; width: 0px;"></span></span><var mathquill-command-id="34">e</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="36" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="38"><span mathquill-command-id="37">−</span><var mathquill-command-id="39">β</var><span class="mq-non-leaf" mathquill-command-id="44"><span class="mq-scaled mq-paren" style="transform: scale(0.997531, 1.18519);">(</span><span class="mq-non-leaf" mathquill-block-id="45"><var mathquill-command-id="47">t</var><span class="mq-binary-operator" mathquill-command-id="48">+</span><var mathquill-command-id="49">n</var><var mathquill-command-id="50">T</var></span><span class="mq-scaled mq-paren" style="transform: scale(0.997531, 1.18519);">)</span></span></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> =nβt=ln<span class="mq-math-mode" latex-data="\frac{\Theta_0}{\Theta_n}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 51.4479px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="6"><span mathquill-command-id="14">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="11" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="12"><span mathquill-command-id="16">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span><span class="mq-denominator" mathquill-block-id="7" style="width: 36.6979px;"><span mathquill-command-id="28">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="30" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="32"><var mathquill-command-id="51">n</var></span><span style="display: inline-block; width: 0px;"></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> (5.1-8)求出β式中，n为阻尼振动的周期次数;<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.5625px;"><span mathquill-command-id="58">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><var mathquill-command-id="60">n</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> 为第n次振动的振幅，T为阻尼振动周期的平均值(可以测出10个摆轮振动周期值，取其平均值)。重复2、3次。</p>
<p><label> 4. 测定振幅<span style="font-family: Symbola, "Times New Roman", serif; font-size: 18.4px; white-space: nowrap;">Θ</span>与固有频率<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="61">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><span mathquill-command-id="63">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> 的对应关系</label></p>
<p>将有机玻璃盘上零度标志线保持在“0”处，选中“自由振荡”测量。用手将摆轮提动到较大偏转处(约140°~150°)后放手，可以直接从屏上读出每次振幅值及其相应的摆动周期。若振幅变小时，周期不变，则可不必记录，也即只记录周期值变化时对应的幅值。重复几次即可作出<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.5625px;"><span mathquill-command-id="64">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><var mathquill-command-id="66">n</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> 与T<sub>0n</sub>的对应关系。请注意、记录振幅幅值范围应该涵盖上表所测量的振幅值。</p>
<p> 如果选择控制箱自动记录数据的测量功能(此时只记录摆轮周期值变化时对应的振幅值)，则可以利用回查功能对振幅和周期值进行查询</p>
<p>在“实验步骤”界面，持续按住复位钮几秒钟，仪器自动复位，实验数据全部清除，关闭电源，实验结束。</p>
<p><label>5.数据记录和处理</label></p>
<p>1)绘制幅频特性和相频特性测量曲线:分别求出阻尼系数β和各个振幅所对应的固有振动周期T<sub>0</sub>(频率<var mathquill-command-id="61" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: "Times New Roman", Symbola, serif; white-space: nowrap;">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, "Times New Roman", serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="63" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;"></span></span>)。作出幅频特性和相频特性曲线(至少作出一种阻尼下的特性曲线)。</p>
<p>2)计算阻尼系数β</p>
<p> ①利用作图法或直线拟合最小二乘法求出β值:根据式(5.1-8)、利用表中的数据，采用作图法或直线拟合最小二乘法得到线性关系In(<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span></span><span class="mq-math-mode" latex-data="\Theta_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8125px;"><span mathquill-command-id="64">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><span mathquill-command-id="67">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> /<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 38.5625px;"><span mathquill-command-id="64">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><var mathquill-command-id="66">n</var></span><span style="display: inline-block; width: 0px;"></span></span></span></span> )~n,由该直线的斜率求出β值。</p>
<p> ②利用共振曲线确定β值:当受迫振动达到稳定状态时，阻尼振动部分可以认为已衰减至零，因而振动位移只需要考虑强迫振动部分、即式(5.1-3)的第二项。阻尼系数较小(满足β<sup>2</sup>≤<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="68">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><span mathquill-command-id="67">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span></span> <sup>2</sup>)时，在共振位置附近(<span style="font-family: "Times New Roman", Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">ω</span>=<var mathquill-command-id="68" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: "Times New Roman", Symbola, serif; white-space: nowrap;">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, "Times New Roman", serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="67" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span></span>)，由于<span style="font-family: "Times New Roman", Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">ω+</span><var mathquill-command-id="68" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: "Times New Roman", Symbola, serif; white-space: nowrap;">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, "Times New Roman", serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="67" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span></span>=2<var mathquill-command-id="68" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: "Times New Roman", Symbola, serif; white-space: nowrap;">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, "Times New Roman", serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="67" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span></span>，从式(5.1-4)和式(5.1-7)可得出<span class="mq-math-mode" latex-data="\left(\frac{\Theta}{\Theta_r}\right)^2" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 94.125px;"><span class="mq-non-leaf" mathquill-command-id="105"><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.82);">(</span><span class="mq-non-leaf" mathquill-block-id="103"><span class="mq-fraction mq-non-leaf" mathquill-command-id="94" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="96"><span mathquill-command-id="95">Θ</span></span><span class="mq-denominator" mathquill-block-id="98" style="width: 33.7917px;"><span mathquill-command-id="97">Θ</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="99" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="101"><var mathquill-command-id="100">r</var></span><span style="display: inline-block; width: 0px;"></span></span></span><span style="display: inline-block; width: 0px;"></span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.82);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="90" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="91"><span mathquill-command-id="107">2</span></span></span></span></span> =<span class="mq-math-mode" latex-data="\frac{4\beta^2\omega_0^2}{4\omega_0^2\left(\omega-\omega_0\right)^2+4\beta^2\omega_0^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 290.615px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="109" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="110"><span mathquill-command-id="113">4</span><var mathquill-command-id="119">β</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="116" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="117"><span mathquill-command-id="121">2</span></span></span><var mathquill-command-id="130">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="127" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="125"><span mathquill-command-id="132">2</span></span><span class="mq-sub" mathquill-block-id="128"><span mathquill-command-id="133">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span><span class="mq-denominator" mathquill-block-id="111" style="width: 275.865px;"><span mathquill-command-id="134">4</span><var mathquill-command-id="151">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="148" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="146"><span mathquill-command-id="153">2</span></span><span class="mq-sub" mathquill-block-id="149"><span mathquill-command-id="154">0</span></span><span style="display: inline-block; width: 0px;"></span></span><span class="mq-non-leaf" mathquill-command-id="186"><span class="mq-scaled mq-paren" style="transform: scale(1.08889, 1.73333);">(</span><span class="mq-non-leaf" mathquill-block-id="176"><var mathquill-command-id="175">ω</var><span class="mq-binary-operator" mathquill-command-id="177">−</span><var mathquill-command-id="179">ω</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="181" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="183"><span mathquill-command-id="182">0</span></span><span style="display: inline-block; width: 0px;"></span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.08889, 1.73333);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="171" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="172"><span mathquill-command-id="18