Driven Damped Mass-Spring System (2nd-Order IVP): 04-BS-1, December 2013, Q8

Question 8: Driven Damped Mass-Spring System (2nd-Order IVP) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

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The source paper's note says "$'$ denotes differentiation with respect to $x$" for this question, but the problem is a time-domain mass-spring equation of motion driven by $\cos(t)$ with $t$ as the only variable appearing — this is treated here as $'=d/dt$ (consistent with every other mass-spring/vibration problem of this type, and evidently a copy-paste artifact from a neighbouring question in the original).

Given. Linear, constant-coefficient, nonhomogeneous ODE $y''+2y'+2y=\cos t$ with $y(0)=1.2,\ y'(0)=1.4$.

Find. $y(t)$, the equation of motion.

Approach. Solve the homogeneous (underdamped) equation via its complex characteristic roots, find a particular solution by undetermined coefficients (no resonance, since the homogeneous modes are damped-oscillatory, not pure $\cos t$), then fix the two constants from the ICs.

Homogeneous solution. $r^2+2r+2=0\Rightarrow r=\dfrac{-2\pm\sqrt{4-8}}{2}=-1\pm i$, so $$y_h(t)=e^{-t}(C_1\cos t+C_2\sin t).$$
Particular solution. Try $y_p=A\cos t+B\sin t$. Substituting into the ODE: $$y_p''+2y_p'+2y_p=(A+2B)\cos t+(B-2A)\sin t.$$ Matching to $\cos t+0\sin t$: $A+2B=1,\ B-2A=0\Rightarrow B=2A$. Then $A+4A=1\Rightarrow A=0.2,\ B=0.4$, so $y_p=0.2\cos t+0.4\sin t$.
Apply the initial conditions. $y(t)=e^{-t}(C_1\cos t+C_2\sin t)+0.2\cos t+0.4\sin t$. $y(0)=C_1+0.2=1.2\Rightarrow C_1=1.0$. Differentiating, $y'(t)=e^{-t}[(-C_1+C_2)\cos t+(-C_1-C_2)\sin t]-0.2\sin t+0.4\cos t$, so $y'(0)=(-C_1+C_2)+0.4=1.4\Rightarrow -C_1+C_2=1.0\Rightarrow C_2=2.0$.

$$y(t)=\boxed{e^{-t}(\cos t+2\sin t)+0.2\cos t+0.4\sin t}$$

Quantity	Result
$y(t)$	$e^{-t}(\cos t+2\sin t)+0.2\cos t+0.4\sin t$

Concept & Theory

Topic. Underdamped forced mass-spring system (2nd-order linear ODE with complex characteristic roots).

Key relations. For $my''+cy'+ky=F(t)$, complex roots $\alpha\pm i\beta$ of the characteristic equation give a decaying oscillatory homogeneous solution $e^{\alpha t}(C_1\cos\beta t+C_2\sin\beta t)$; the steady-state response to $\cos\omega t$ forcing is $A\cos\omega t+B\sin\omega t$ found by direct substitution, valid whenever this trial is not itself a homogeneous solution.

Why this works. Here $\alpha=-1\neq0$, so the homogeneous solution decays to zero as $t\to\infty$ (transient) while $y_p$ persists (steady-state) — the trial $A\cos t+B\sin t$ is linearly independent of $y_h$ because $y_h$ carries the $e^{-t}$ envelope, so no resonance multiplier is needed.

Common pitfall. Sign errors differentiating the product $e^{-t}(C_1\cos t+C_2\sin t)$ (product rule on two oscillating/decaying factors); also mixing up which of $A,B$ pairs with $\cos t$ vs. $\sin t$ when matching coefficients.

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (§2.8 Modeling: Forced Oscillations, Resonance).

Question 8 of 8: Driven Damped Mass-Spring System (2nd-Order IVP)

Question 8: Driven Damped Mass-Spring System (2nd-Order IVP) (20 marks)