Sections 1 to 7: spring-mass model, free undamped motion, free damped motion, forced vibrations, resonance, the RLC circuit analogy, beats and practical resonance第1至7节：弹簧质量模型、自由无阻尼运动、自由有阻尼运动、受迫振动、共振、RLC电路类比、拍频与实际共振CALC IV
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PART I · CORE TECHNIQUES核心技法Computational fluency · 28 marks计算熟练度 · 28分
Free Motion and the Amplitude-Phase Form自由运动与幅相形式
Show all working. State the discriminant $c^{2}-4mk$ and classify the damping case before writing any solution. Report exact values; numerical approximations of $\pi$ or $\sqrt{\cdot}$ are not accepted.写出完整解题过程。在写出解之前，先计算判别式 $c^{2}-4mk$ 并判断阻尼类型。报告精确值，不接受对 $\pi$ 或 $\sqrt{\cdot}$ 的数值近似。
A mass of $2\ \text{kg}$ is attached to a spring with stiffness constant $k = 32\ \text{N/m}$. The system is undamped. Initial conditions are $x(0) = 3$ and $x'(0) = -8$.一个质量为 $2\ \text{kg}$ 的物体连接在刚度系数 $k = 32\ \text{N/m}$ 的弹簧上，系统无阻尼。初始条件为 $x(0) = 3$，$x'(0) = -8$。
(a)Write the IVP and identify the natural frequency $\omega_0$.写出初值问题并求自然频率 $\omega_0$。[2]
(b)Solve the IVP. Express the general solution in the form $x(t) = c_1\cos\omega_0 t + c_2\sin\omega_0 t$ and apply initial conditions.求解初值问题。将通解写成 $x(t) = c_1\cos\omega_0 t + c_2\sin\omega_0 t$ 的形式，并代入初始条件。[3]
(c)Convert the solution to amplitude-phase form $A\cos(\omega_0 t - \phi)$. State the amplitude $A$, the phase $\phi$, and the period $T$.将解转化为幅相形式 $A\cos(\omega_0 t - \phi)$，写出振幅 $A$、相位 $\phi$ 和周期 $T$。[3]
Q2MEDIUMCOREdamping classification by the discriminant利用判别式判断阻尼类型[6 marks]
For each spring-mass system below, compute the discriminant $\Delta = c^{2} - 4mk$, state the damping regime (underdamped, critically damped, or overdamped), and write the form of the general solution (do not apply initial conditions).对下列每个弹簧质量系统，计算判别式 $\Delta = c^{2} - 4mk$，判断阻尼状态（欠阻尼、临界阻尼或过阻尼），并写出通解形式（无需代入初始条件）。
(a)Find the characteristic roots. Compute $\Delta$ and confirm underdamped status. State the damping rate $\beta$ and the quasi-frequency $\omega_1$.求特征根。计算 $\Delta$ 并确认欠阻尼状态。写出阻尼率 $\beta$ 和准频率 $\omega_1$。[2]
(b)Write the general solution and apply the initial conditions to find $c_1$ and $c_2$.写出通解，代入初始条件求 $c_1$ 和 $c_2$。[3]
(c)Convert to amplitude-phase form $A e^{-\beta t}\cos(\omega_1 t - \phi)$. State $A$, $\phi$, and the quasi-period $T_1 = 2\pi/\omega_1$.转化为幅相形式 $A e^{-\beta t}\cos(\omega_1 t - \phi)$，写出 $A$、$\phi$ 和准周期 $T_1 = 2\pi/\omega_1$。[2]
(d)Explain in one sentence why $\omega_1 < \omega_0$ and what this means physically for the oscillation rate.用一句话解释为何 $\omega_1 < \omega_0$，以及这对振荡速率在物理上意味着什么。[1]
(a)Explain why the standard undetermined-coefficient guess $x_p = A\cos 3t + B\sin 3t$ fails, and state the correct modified guess.解释为何标准待定系数猜测 $x_p = A\cos 3t + B\sin 3t$ 失效，并写出正确的修正猜测。[2]
(b)Determine $x_p$ by substituting the correct guess into the ODE.将正确猜测代入常微分方程，求 $x_p$。[2]
(c)Apply initial conditions to find the complete solution and identify the factor that causes pure resonance.代入初始条件求完全解，并指出导致纯共振的因子。[2]
PART II · DEFINITIONS AND PROOF定义与证明Rigorous arguments · 26 marks严格论证 · 26分
Derivations, Conversions, and the Mechanical-Electrical Correspondence推导、转换与机电对应关系
These items are graded on the completeness of the argument. In derivations, start from the stated equation and name each algebraic step. For correspondence questions, state the explicit dictionary entry (mechanical quantity = electrical quantity) before drawing conclusions.本部分依据论证的完整性评分。推导时须从给定方程出发，并说明每一代数步骤。对应问题须先给出明确的对照条目（力学量 = 电学量），再得出结论。
Q5HARDPROOFderiving the amplitude-phase conversion推导幅相转换公式[8 marks]
Let $x(t) = c_1\cos\omega t + c_2\sin\omega t$ where $c_1, c_2$ are real constants and $\omega > 0$.设 $x(t) = c_1\cos\omega t + c_2\sin\omega t$，其中 $c_1, c_2$ 为实常数，$\omega > 0$。
(a)Define $A = \sqrt{c_1^{2} + c_2^{2}}$ and $\phi = \arctan(c_2/c_1)$ (choosing $\phi$ in the correct quadrant). Prove by expanding $A\cos(\omega t - \phi)$ using the cosine subtraction identity that $c_1\cos\omega t + c_2\sin\omega t = A\cos(\omega t - \phi)$.定义 $A = \sqrt{c_1^{2} + c_2^{2}}$，$\phi = \arctan(c_2/c_1)$（选取正确象限中的 $\phi$）。利用余弦差角公式展开 $A\cos(\omega t - \phi)$，证明 $c_1\cos\omega t + c_2\sin\omega t = A\cos(\omega t - \phi)$。[4]
(b)For the specific solution $x(t) = 3\cos 4t - 4\sin 4t$, carry out the conversion explicitly: compute $A$, determine $\phi$ (in radians, four significant figures), and write $x(t)$ in amplitude-phase form.对于特定解 $x(t) = 3\cos 4t - 4\sin 4t$，显式完成转换：计算 $A$，确定 $\phi$（弧度，四位有效数字），并将 $x(t)$ 写成幅相形式。[4]
Q6HARDPROOFsteady-state amplitude and the resonance condition稳态振幅与共振条件[10 marks]
The forced damped oscillator $mx'' + cx' + kx = F_0\cos\omega t$ ($c > 0$) has a particular (steady-state) solution of the form $x_p = A\cos(\omega t - \delta)$.受迫阻尼振荡器 $mx'' + cx' + kx = F_0\cos\omega t$（$c > 0$）的特解（稳态解）具有形式 $x_p = A\cos(\omega t - \delta)$。
(a)By substituting $x_p$ into the ODE and matching cosine and sine coefficients, derive the formula for the steady-state amplitude $$ C(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^{2}-\omega^{2})^{2}+4\beta^{2}\omega^{2}}} $$ where $\omega_0 = \sqrt{k/m}$ and $\beta = c/(2m)$.将 $x_p$ 代入常微分方程，比较余弦与正弦系数，推导稳态振幅公式 $$ C(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^{2}-\omega^{2})^{2}+4\beta^{2}\omega^{2}}} $$ 其中 $\omega_0 = \sqrt{k/m}$，$\beta = c/(2m)$。[5]
(b)Show by differentiating $C(\omega)^{-2}$ with respect to $\omega^{2}$ that $C$ is maximised at the practical resonance frequency $\omega_r = \sqrt{\omega_0^{2} - 2\beta^{2}}$, provided $2\beta^{2} < \omega_0^{2}$. State what happens to the peak amplitude as $\beta \to 0^{+}$.通过对 $\omega^{2}$ 求 $C(\omega)^{-2}$ 的导数，证明在实际共振频率 $\omega_r = \sqrt{\omega_0^{2} - 2\beta^{2}}$ 处（当 $2\beta^{2} < \omega_0^{2}$ 时）$C$ 取得最大值。说明当 $\beta \to 0^{+}$ 时峰值振幅的变化情况。[5]
Q7HARDPROOFRLC circuit analogy and mechanical-electrical correspondenceRLC电路类比与机电对应关系[8 marks]
A series RLC circuit obeys Kirchhoff's voltage law: $L\,q'' + R\,q' + \dfrac{1}{C}\,q = E(t)$, where $q(t)$ is the charge on the capacitor, $L$ is the inductance, $R$ is the resistance, and $C$ is the capacitance.串联RLC电路满足基尔霍夫电压定律：$L\,q'' + R\,q' + \dfrac{1}{C}\,q = E(t)$，其中 $q(t)$ 为电容上的电荷量，$L$ 为电感，$R$ 为电阻，$C$ 为电容。
(a)Write the full mechanical-electrical dictionary: state the electrical quantity that corresponds to each of $m$, $c$, $k$, $x$, and $F(t)$.写出完整的机电对照表：给出与 $m$、$c$、$k$、$x$ 和 $F(t)$ 各自对应的电学量。[3]
(b)For a circuit with $L = 1\ \text{H}$, $R = 4\ \Omega$, $C = \tfrac{1}{5}\ \text{F}$, and $E(t) = 0$, compute the discriminant $R^{2} - 4L/C$, classify the circuit, and state the quasi-frequency (for the underdamped case) or the reason why no quasi-frequency exists otherwise.对于 $L = 1\ \text{H}$，$R = 4\ \Omega$，$C = \tfrac{1}{5}\ \text{F}$，$E(t) = 0$ 的电路，计算判别式 $R^{2} - 4L/C$，判断电路类型，并给出准频率（欠阻尼情形）或说明不存在准频率的原因。[3]
(c)In one sentence, explain why inductance in a circuit plays the same role as mass in a mechanical system.用一句话解释为何电路中的电感与力学系统中的质量起相同作用。[2]
PART III · APPLICATIONS AND SYNTHESIS应用与综合Extended problems · 28 marks综合题 · 28分
Full IVPs, Forced Damped Response, and Resonance完整初值问题、受迫阻尼响应与共振
Set up each problem from physical data before solving. Carry exact values through all intermediate steps and simplify only at the end. Identify the transient and steady-state components by name.解题前先由物理数据建立方程。在所有中间步骤中保持精确值，仅在最后化简。须按名称区分暂态分量和稳态分量。
Q8HARDAPPLIEDcomplete damped spring-mass IVP from physical data由物理数据建立完整阻尼弹簧质量初值问题[10 marks]
A $4\ \text{kg}$ mass is attached to a spring with $k = 100\ \text{N/m}$. A dashpot provides a damping force of $40\ \text{N}$ when the velocity is $1\ \text{m/s}$, so $c = 40\ \text{N\,s/m}$. The mass is released from rest at $x(0) = 0.5\ \text{m}$ with $x'(0) = 0$.质量为 $4\ \text{kg}$ 的物体连接在 $k = 100\ \text{N/m}$ 的弹簧上。阻尼器在速度为 $1\ \text{m/s}$ 时提供 $40\ \text{N}$ 的阻尼力，故 $c = 40\ \text{N\,s/m}$。物体从 $x(0) = 0.5\ \text{m}$ 处由静止释放，$x'(0) = 0$。
(a)Write the IVP. Compute $\Delta = c^{2} - 4mk$ and classify the damping regime.写出初值问题。计算 $\Delta = c^{2} - 4mk$ 并判断阻尼状态。[2]
(b)Find the characteristic roots and write the general solution.求特征根并写出通解。[3]
(c)Apply initial conditions to find the specific solution. Verify your answer by substituting back into the ODE and checking that both initial conditions are satisfied.代入初始条件求特定解。将解代回常微分方程并验证两个初始条件均满足。[4]
(d)State the long-run behaviour of $x(t)$ as $t \to \infty$, explaining whether the mass crosses the equilibrium position at any time.说明 $x(t)$ 在 $t \to \infty$ 时的长期行为，并解释物体是否会越过平衡位置。[1]
Q9HARDAPPLIEDforced damped steady-state amplitude and practical resonance受迫阻尼稳态振幅与实际共振[10 marks]
(a)Identify $\omega_0$ and $\beta$. Confirm the free system is underdamped by checking the discriminant.确定 $\omega_0$ 和 $\beta$。通过检验判别式确认自由系统为欠阻尼状态。[2]
(b)Write the steady-state amplitude $C(\omega)$ as a function of $\omega$, and simplify the radicand as fully as possible.将稳态振幅 $C(\omega)$ 写成 $\omega$ 的函数，并尽量化简被开方式。[2]
(c)Find the practical resonance frequency $\omega_r$ that maximises $C(\omega)$. Compute the peak steady-state amplitude $C(\omega_r)$ when $F_0 = 1$.求使 $C(\omega)$ 最大的实际共振频率 $\omega_r$。当 $F_0 = 1$ 时计算峰值稳态振幅 $C(\omega_r)$。[4]
(d)Explain in one sentence how the peak amplitude $C(\omega_r)$ differs from the undamped resonance amplitude, and what happens to $\omega_r$ as $\beta \to 0^{+}$.用一句话说明峰值振幅 $C(\omega_r)$ 与无阻尼共振振幅的区别，以及当 $\beta \to 0^{+}$ 时 $\omega_r$ 的变化趋势。[2]
Q10HARDAPPLIEDRLC circuit with forcing: transient and steady-state受迫RLC电路：暂态与稳态[8 marks]
A series RLC circuit has $L = 1\ \text{H}$, $R = 4\ \Omega$, $C = \tfrac{1}{20}\ \text{F}$, and is driven by $E(t) = 10\cos 2t\ \text{V}$. Initially $q(0) = 0\ \text{C}$ and $q'(0) = i(0) = 0\ \text{A}$.串联RLC电路参数为 $L = 1\ \text{H}$，$R = 4\ \Omega$，$C = \tfrac{1}{20}\ \text{F}$，由 $E(t) = 10\cos 2t\ \text{V}$ 驱动。初始条件为 $q(0) = 0\ \text{C}$，$q'(0) = i(0) = 0\ \text{A}$。
(a)Write the governing ODE for $q(t)$. Classify the free ($E = 0$) circuit and find the quasi-frequency.写出 $q(t)$ 满足的常微分方程。对自由电路（$E = 0$）进行分类并求准频率。[3]
(c)Apply the initial conditions to determine the full solution $q(t)$, and state which part is the transient and which is the steady-state.代入初始条件求完全解 $q(t)$，并指出哪部分是暂态，哪部分是稳态。[2]