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3. ( 20 points) For a single degree of freedom damped system under harmonic for...
Sep 26, 2024
Solution
a
The magnification factor M M is given by the equation: M=1(1−r2)2+(2ζr)2M = \frac{1}{\sqrt{(1 - r^2)^2 + (2 \zeta r)^2}} where r=ωωn r = \frac{\omega}{\omega_n} . To find the maximum value of M M , we need to analyze the expression for M M in terms of ζ \zeta and r r
b
We substitute r=1−2ζ2 r = \sqrt{1 - 2\zeta^2} into the equation for M M : M=1(1−(1−2ζ2))2+(2ζ1−2ζ2)2=1(2ζ2)2+(2ζ1−2ζ2)2M = \frac{1}{\sqrt{(1 - (1 - 2\zeta^2))^2 + (2\zeta\sqrt{1 - 2\zeta^2})^2}} = \frac{1}{\sqrt{(2\zeta^2)^2 + (2\zeta\sqrt{1 - 2\zeta^2})^2}}. Simplifying this will help us find the maximum value
c
To find the maximum value of M M for 0 < \zeta < \frac{1}{\sqrt{2}} , we differentiate M M with respect to ζ \zeta and set the derivative to zero to find critical points. The maximum value is given by the equation: Mmax=12ζ1−ζ2M_{max} = \frac{1}{2\zeta\sqrt{1 - \zeta^2}}
Answer
The maximum value of M M is given by Mmax=12ζ1−ζ2 M_{max} = \frac{1}{2\zeta\sqrt{1 - \zeta^2}} for 0 < \zeta < \frac{1}{\sqrt{2}} .
Key Concept
The magnification factor M M in a damped harmonic oscillator is influenced by the damping ratio ζ \zeta and the frequency ratio r r . The maximum value of M M can be derived from the relationship between these parameters.
Explanation
By substituting the expression for r r and differentiating, we can find the conditions under which M M reaches its maximum value, confirming the given formula for Mmax M_{max} .
Solution
a
The given differential equation for a damped harmonic oscillator is mx¨+cx˙+kx=0m \ddot{x} + c \dot{x} + k x = 0. We can rewrite it in the standard form as x¨+2ζωnx˙+ωn2x=0\ddot{x} + 2 \zeta \omega_n \dot{x} + \omega_n^2 x = 0, where ωn=km\omega_n = \sqrt{\frac{k}{m}} and ζ=c2mωn\zeta = \frac{c}{2m\omega_n}. This form helps us analyze the system based on the damping ratio ζ\zeta
b
For the case of undamped motion (ζ=0\zeta = 0), the solution is given by x(t)=Acos⁡(ωnt−ϕ)x(t) = A \cos(\omega_n t - \phi), where A=x02+(v0ωn)2A = \sqrt{x_0^2 + \left(\frac{v_0}{\omega_n}\right)^2} and ϕ=tan⁡−1(v0x0ωn)\phi = \tan^{-1}\left(\frac{v_0}{x_0 \omega_n}\right). This represents simple harmonic motion with no damping
c
For the underdamped case (0 < \zeta < 1), the solution is x(t)=Ae−ζωdtcos⁡(ωdt−ϕ)x(t) = A e^{-\zeta \omega_d t} \cos(\omega_d t - \phi), where ωd=1−ζ2ωn\omega_d = \sqrt{1 - \zeta^2} \omega_n. The amplitude decays exponentially due to the damping factor, and the expressions for AA and ϕ\phi are given as: A=x02+(ζωnx0+v0ωd)2A = \sqrt{x_0^2 + \left(\frac{\zeta \omega_n x_0 + v_0}{\omega_d}\right)^2} and ϕ=tan⁡−1(ζωnx0+v0x0ωd)\phi = \tan^{-1}\left(\frac{\zeta \omega_n x_0 + v_0}{x_0 \omega_d}\right)
d
For the critically damped case (ζ=1\zeta = 1), the solution is x(t)=[x0+(v0+ωnx0)t]e−ωntx(t) = \left[x_0 + \left(v_0 + \omega_n x_0\right) t\right] e^{-\omega_n t}. This case represents the fastest return to equilibrium without oscillation
Answer
The solutions for the damped harmonic oscillator are derived based on the damping ratio, leading to distinct behaviors for undamped, underdamped, and critically damped cases.
Key Concept
Damped Harmonic Oscillator: The behavior of a damped harmonic oscillator is characterized by the damping ratio ζ\zeta, which influences the system's response. The solutions vary based on whether the system is undamped (ζ=0\zeta = 0), underdamped (0 < \zeta < 1), or critically damped (ζ=1\zeta = 1). The general form of the equation is mx¨+cx˙+kx=0m \ddot{x} + c \dot{x} + k x = 0.
Explanation
The solutions for the damped harmonic oscillator illustrate how the system's response changes with different damping conditions, affecting the amplitude and oscillation behavior over time.
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