# Damped Driven Harmonic Oscillator.

• vkumar1403
In summary, the conversation discusses an oscillator with specific values for mass, stiffness, and mechanical resistance being driven by a sinusoidal force. The task is to plot the speed amplitude and phase angle between displacement and speed as a function of the driving frequency and determine the frequencies at which the phase angle is 45 degrees. The proposed equations for the solution are given, but there is a discrepancy in the interpretation of the question as the phase shift is typically between the driving force and position, not between speed and displacement. It is suggested that this is a misprint and the correct interpretation is likely between the driving force and position.
vkumar1403

## Homework Statement

An oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s is driven by a sinusoidal force of amplitude 2 N. Plot the speed amplitude and the phase angle between the displacement and speed as a function of the driving frequency and find the frequencies for which the phase angle for which the angle is 45 deg.

## The Attempt at a Solution

Using the general form of the solution:
x(t) = A(w) sin(wt-Φ)
where Φ=atan(2wp/(w_0^2-w^2))
A(w) = (F/m)/((w_0^2-w^2)^2 + (2wp)^2)^0.5

I am positive the above equations are correct and come from the differential equation for this case.

Now, u(t) [speed] = d x(t)/dt.
= w*A(w)*cos(wt-Φ)
=w*A(w)*sin(wt-Φ+pi/2)

My question: Now the speed amplitude, I believe, is wA(w). Won't the phase angle between the displacement and velocity always be pi/2 irrespective of w?[/B]

Yes, but this is not the phase shift. The phase shift is the phase between the driving force and the position, i.e., φ.

Orodruin said:
Yes, but this is not the phase shift. The phase shift is the phase between the driving force and the position, i.e., φ.
I agree but the question says the phase between the speed and the displacement. Am I interpreting this wrong?

vkumar1403 said:
I agree but the question says the phase between the speed and the displacement. Am I interpreting this wrong?
It is very likely a misprint.

Orodruin said:
It is very likely a misprint.
Thanks! I'm going to quote this as a misprint in my solution with the explanation of my rationale. Hopefully, that is good enough for my professor

## 1. What is a damped driven harmonic oscillator?

A damped driven harmonic oscillator is a physical system that exhibits periodic motion or oscillations in response to an external force, while also experiencing a resistive or damping force that decreases its amplitude over time. It can be modeled using a differential equation and is a common phenomenon in physics and engineering.

## 2. What is the equation of motion for a damped driven harmonic oscillator?

The equation of motion for a damped driven harmonic oscillator is given by: m*x'' + b*x' + k*x = F*cos(ω*t), where m is the mass of the oscillator, b is the damping coefficient, k is the spring constant, F is the amplitude of the driving force, ω is the angular frequency of the driving force, and t is time.

## 3. How does damping affect the motion of a driven harmonic oscillator?

Damping in a driven harmonic oscillator causes the amplitude of the oscillations to decrease over time. This is because the damping force acts in the opposite direction of the oscillations, reducing the energy of the system. As the damping coefficient increases, the oscillations become smaller and eventually the system reaches a steady-state motion with a constant amplitude.

## 4. How does the driving force affect the motion of a damped driven harmonic oscillator?

The driving force in a damped driven harmonic oscillator can affect the amplitude and frequency of the oscillations. If the driving force has a frequency close to the natural frequency of the oscillator, it can cause resonance and the amplitude of the oscillations can greatly increase. However, if the driving force is significantly different from the natural frequency, it can cause the system to exhibit more complex motion, such as beats or chaos.

## 5. What are some real-life examples of damped driven harmonic oscillators?

Damped driven harmonic oscillators can be found in many physical systems, such as a swing, a pendulum, or a mass-spring system. In engineering, they can be seen in structures or machines that experience vibrations, such as bridges, car suspensions, or guitar strings. They are also used in electronic circuits, such as in oscillators and filters.

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