Damped Harmonic Motion with a Sinusoidal Driving Force

In summary: The phase angle is the angle between the force and the velocity, and it also depends on ω. The problem is asking for a plot of both of these quantities as a function of ω, so you will have two curves on the same graph.In summary, the problem involves an oscillator with mass 0.5 kg, stiffness 100 N/m, and mechanical resistance 1.4 kg/s driven by a sinusoidal force of amplitude 2 N. The speed amplitude and phase angle between the force and speed are plotted as a function of the driving frequency ω, and the frequencies for which the phase angle is 45° are found. The equation of motion for the system is m\ddot{x} +Rm\
  • #1
roldy
237
2
1. 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 force and speed as a function of the driving frequency and find the frequencies for which the phase angle is 45°.



2. m[tex]\ddot{x}[/tex] +Rm[tex]\dot{x}[/tex]+kx=Fosin[tex]\omega[/tex]t

see attachment for rest of equations





3. m= 0.5 kg, s=100N/m, Rm=1.4 kg/s, Fo=2N

So my first question is this, is omega the independent variable in this case? Meaning, I solve everything that I am able to and leave omega alone. Also, is the differential equation in #2 the right form?

I am confused at how I obtain the equation of motion so that I can plot this.
 

Attachments

  • equations.doc
    51.5 KB · Views: 251
Physics news on Phys.org
  • #2
Yes, ω is the driving frequency, so it's the independent variable for what's being asked in the problem. I'm not sure what you mean by "leaving ω alone." You want to express the speed amplitude and the phase angle as a function of ω and the constant parameters of the system.

Yes, your differential equation is correct. It is the equation of motion for the system. Note that k is the spring constant, which you called also called s in the other equations.

I think you have a typo in your equation for the phase angle. Also, what does c represent in that formula?
 
  • #3
The equation for the phase angle is wrong it should be tan[tex]^{-1}[/tex](H)
where H=[tex]\frac{\omega*m-k/\omega}{R_{m}}[/tex].

I'm still a little confused about what they are asking for in regards to plotting.
 
  • #4
By "speed amplitude," I assume the problem is asking for the amplitude of v(t). It will depend on ω.
 
  • #5




I can provide a response to the above content by first clarifying the given information and then providing a possible approach to solving the problem.

Firstly, the given information states that we have an oscillator with a mass of 0.5 kg, stiffness of 100 N/m, and mechanical resistance of 1.4 kg/s. It is driven by a sinusoidal force with an amplitude of 2 N. The equation of motion for this system is given by m\ddot{x} + Rm\dot{x} + kx = Fosin(\omega t), where m is the mass, R is the mechanical resistance, k is the stiffness, and \omega is the angular frequency of the driving force.

To plot the speed amplitude and phase angle as a function of driving frequency, we need to first solve the equation of motion. This can be done by using methods such as Laplace transform or solving the differential equation directly. Once we have the solution, we can then plot the speed amplitude and phase angle as a function of driving frequency.

In this case, \omega is indeed the independent variable as it represents the frequency of the driving force. The differential equation given in #2 is in the correct form.

To find the frequencies for which the phase angle is 45°, we can set the phase angle (\phi) to 45° in the solution of the equation of motion and solve for \omega. This will give us the values of \omega at which the phase angle is 45°.

In conclusion, the approach to solving this problem would involve solving the equation of motion, obtaining a solution, and then using that solution to plot the speed amplitude and phase angle as a function of driving frequency. To find the frequencies for which the phase angle is 45°, we can set the phase angle to 45° in the solution and solve for \omega.
 

1. What is damped harmonic motion with a sinusoidal driving force?

Damped harmonic motion with a sinusoidal driving force is a type of motion in which a mass-spring system is subjected to a driving force that varies sinusoidally with time. The motion is damped due to the presence of a damping force, which causes the amplitude of the oscillations to decrease over time.

2. What is the formula for damped harmonic motion with a sinusoidal driving force?

The equation of motion for damped harmonic motion with a sinusoidal driving force is given by:
x(t) = A*cos(ωt + φ)*e^(-βt)
where x(t) is the displacement of the mass from its equilibrium position, A is the amplitude of the oscillations, ω is the angular frequency of the driving force, φ is the phase difference between the driving force and the displacement, and β is the damping coefficient.

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

The presence of damping in a damped harmonic oscillator causes the amplitude of the oscillations to decrease over time. This means that the oscillations become smaller and smaller until the system eventually comes to rest. Damping also affects the frequency of the oscillations, causing it to decrease as well.

4. What is the resonance frequency in damped harmonic motion with a sinusoidal driving force?

The resonance frequency in damped harmonic motion with a sinusoidal driving force is the frequency at which the amplitude of the oscillations is the highest. In other words, it is the frequency at which the driving force is in sync with the natural frequency of the system, causing the oscillations to build up rather than decrease over time.

5. How is damped harmonic motion with a sinusoidal driving force applied in real-life situations?

Damped harmonic motion with a sinusoidal driving force can be observed in many real-life situations, such as a swinging pendulum or a car suspension system. It is also used in engineering and technology, such as in the design of shock absorbers and vibration dampers to reduce the effects of vibrations and oscillations.

Similar threads

  • Introductory Physics Homework Help
Replies
17
Views
280
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
706
  • Introductory Physics Homework Help
Replies
2
Views
2K
Replies
1
Views
378
  • Introductory Physics Homework Help
Replies
4
Views
2K
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
831
  • Introductory Physics Homework Help
Replies
7
Views
1K
Back
Top