damped harmanic oscillator, haSdamping constanta= 2 We, that is acted upon by a driving forceF= Fo sin wt, The system Starts from rest With anandinitial displacement ofXo lie,xLO) = Xo10)=0), Find the equation of motion and itscorres panding salution xlt), determine all of thecoefficients le.g., Ai,A2, B,, Bzretc)%3DBe,Bz,et c)

damped harmanic oscillator, has damping aonstant a B = 2 Wo, that is acted upon bya driving force F Fo sin wt The System Starts from rest with an initial displacement of 10)=0), Find the equation of motian and itS corres panding solution x(t), determine all Of the coefficients (e.g. AL,A2, B, Bzretc) Xe li.e, xL0) = Xo and

damped harmanic oscillator, has damping aonstant a B = 2 Wo, that is acted upon bya driving force F Fo sin wt The System Starts from rest with an initial displacement of 10)=0), Find the equation of motian and itS corres panding solution x(t), determine all Of the coefficients (e.g. AL,A2, B, Bzretc) Xe li.e, xL0) = Xo and

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Question

damped harmanic oscillator, haS
damping constant
a
= 2 We, that is acted upon by a driving force
F
= Fo sin wt, The system Starts from rest With an
and
initial displacement of
Xo lie,xLO) = Xo
10)=0), Find the equation of motion and its
corres panding salution xlt), determine all of the
coefficients le.g., Ai,A2, B,, Bzretc)
%3D
Be,Bz,et c)

Expert Solution

Step 1

Given:

Driven force F = F_o sin $ωt$

Damping Constant $β=2ω$

Fid the equation of motion and its corresponding solution x(t)

Step 2

Driven Force F=F_o $sinωt$

Here, $F_{o}$ = Amplitude of force

$ω$ = Angular frequency

Let restoring force be -cx and damping force - $β \frac{dx}{dt}$ . Then the equation of motion for oscillator of mass 'm' can be expressed as

$\frac{{md}^{2} x}{{dt}^{2}} = - β \frac{dx}{dt} - cx + F_{o} sinωt$

$\frac{d^{2} x}{{dt}^{2}} + \frac{β}{m} (\frac{dx}{dt}) + \frac{cx}{m} = \frac{F_{o}}{m} sinωt$ ----- Equation (1)

Where, $\frac{β}{m}$ =2k ; $\frac{c}{m}$ = $p_{o}^{2}$ ; $\frac{F_{o}}{m} = f_{o}$

In Steady-state the oscillator will execute oscillations of frequency $\frac{ω}{2 π}$ . We get

$x (t) = A \sin (ωt - θ)$

Where A = Amplitude, $θ$ = Phase angle

Differentiating equation (2), with respect to we have

$\frac{dx}{dt} = Aω \cos (pt - θ)$

Step by step

Solved in 3 steps

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