Differentiate damped sinusoid

[akabyte]

Hi all,

I'm trying to sketch a unit response of a damped sinusoid an (unnderdamped system). I presume I'll have to differentiate the function such as the following (1 + 1.25e^(-6t)cos(8t+143))u(t) and then find the turning points of the function and then sketch it.

However I'm encountering problems in differentiating it properly and finding the two critical points, I was therefore hoping if any of you guys out there would know how to sketch the function either by inspection or any other method.

Cheers,

 

[mighty2000]

I would first like to clarify that a damped sinusoid is a type of response that occurs in a system that is undergoing damped oscillations. This means that the amplitude of the oscillations decreases over time due to the presence of a damping force. The equation you have provided, (1 + 1.25e^(-6t)cos(8t+143))u(t), is an example of a damped sinusoid, where u(t) is the unit step function.

To sketch the unit response of this damped sinusoid, you can follow these steps:

1. First, differentiate the equation with respect to time. This will give you the velocity equation, which represents the rate of change of the position of the system. In this case, the velocity equation will be: (-7.5e^(-6t)sin(8t+143)+10e^(-6t)cos(8t+143))u(t).

2. Next, find the turning points of the velocity equation by setting it equal to zero and solving for t. In this case, there will be two turning points at t = 0 and t = 0.4 seconds.

3. Using the turning points, you can sketch the velocity response by plotting the velocity values at different time points. For example, at t = 0, the velocity will be 10 units, and at t = 0.4 seconds, the velocity will be -7.5 units. You can connect these points to create a smooth curve.

4. To sketch the position response, you can integrate the velocity equation with respect to time. This will give you the position equation, which represents the displacement of the system from its equilibrium position. In this case, the position equation will be: (-1.25e^(-6t)sin(8t+143)+1.25e^(-6t)cos(8t+143))u(t).

5. Using the position equation, you can sketch the position response by plotting the position values at different time points. For example, at t = 0, the position will be 0 units, and at t = 0.4 seconds, the position will be approximately 0.08 units. You can connect these points to create a smooth curve.

6. Finally, you can combine the velocity and position responses to sketch the unit response of the damped sinusoid. The