# Does the transfer function exist for an unforced system?

• EastWindBreaks
In summary: Without further specifications given, there is one obvious target object in the image and that would be the mass-on-wheels. We might be interested in its position or velocity (or both) with respect to time when some driving force f(t) is applied.
EastWindBreaks

## Homework Statement

c is the coefficient of damping, k is the spring constant, and m is the mass.
if the cart is displaced from rest by distance X_0 and released at t=0 with initial velocity v_0=1.

## The Attempt at a Solution

Ok this might be a stupid question because I wasn't able to find a similar question like this anywhere, this question just pops into my mind when I was solving for transfer functions on a system with a force input. I think I am still a little confused on the definitions. So I found it's equation of motion in time and Laplace domain, but if I were asked to find the transfer function of the plant, does it exists? since G(s)= output/ input, as far as I see here, there is no input since it was released at t=0, if it has an input, I can just take the X(s)/ input(s). Can someone please confirm my understanding?

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EastWindBreaks said:
the transfer function of the plant, does it exist?
Of course it does. It exists independent of the fact that an actual input is applied or not.

EastWindBreaks
BvU said:
Of course it does. It exists independent of the fact that an actual input is applied or not.
Thank you, I thought the plant transfer function G_p(s) = output(s) / input(s)? how it is independent? are you saying that it always exists but cannot be calculated without some type of inputs?

You can calculate an amplitude and phase response for your system under a driving force of the type ##F_0\sin(\omega t)## even if it stands still in the lab. To validate your model, you'd need an actual input. But that's a different story.

EastWindBreaks said:
Thank you, I thought the plant transfer function G_p(s) = output(s) / input(s)? how it is independent? are you saying that it always exists but cannot be calculated without some type of inputs?
For the problem in your original post, the transfer function concept doesn't apply because there is no input to the system.

But you could take the same system and add a driving force to it, and then the transfer function idea makes sense. For a linear, time-invariant system, you can figure out the transfer function by looking at the differential equation that describes the (forced) system. The output y(t) of a system will obviously depend upon the input x(t), but the ratio Y(s)/X(s) is independent of the input and depends only on the system.

FactChecker
The transfer function describes the dynamics of a system (usually a linear system), without regard to whether it is excited at the present moment. i.e. How would the system respond if it were excited. In finding the transfer function you don't need any information about the input, just system properties like mass, inductance, density, etc.
So, for example, we can talk about the resonant frequency of a guitar string even when it is quiet.

rude man
IMHO, the problem is not well defined. One can hypothesize inputs of various types/locations, but none is specified in the problem statement.

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BvU
FactChecker said:
IMHO, the problem is not well defined. One can hypothesize inputs of various types, but none is specified in the problem statement.
The problem isn't well defined in the sense that there is nothing being asked in the problem statement.
But in the 'attempt at solution' the OP asks about a transfer function -- while mentally still busy with
EastWindBreaks said:
transfer functions on a system with a force input
and such a transfer function can definitely be determined for the system in the exercise.

rude man
BvU said:
The problem isn't well defined in the sense that there is nothing being asked in the problem statement.
But in the 'attempt at solution' the OP asks about a transfer function -- while mentally still busy with
and such a transfer function can definitely be determined for the system in the exercise.
This is not my specialty. What point is an input signal applied at and where is the output measured? Are there traditional input/output for this problem? I think that is necessary to determine the transfer function.

FactChecker said:
This is not my specialty. What point is an input signal applied at and where is the output measured? Are there traditional input/output for this problem? I think that is necessary to determine the transfer function.
Without further specifications given, there is one obvious target object in the image and that would be the mass-on-wheels. We might be interested in its position or velocity (or both) with respect to time when some driving force f(t) is applied.

f(t) itself can be left unspecified when finding the transfer function. Just represent it as F(s) in the Laplace domain, and then the transfer function might be obtained by finding, for example, ##H(s) = \frac{V(s)}{F(s)} = \{something\}##

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FactChecker
gneill said:
Without further specifications given, there is one obvious target object in the image and that would be the mass-on-wheels. We might be interested in its position or velocity (or both) with respect to time when some driving force f(t) is applied.
View attachment 236148
f(t) itself can be left unspecified when finding the transfer function. Just represent it as F(s) in the Laplace domain, and then the transfer function might be obtained by finding, for example, ##H(s) = \frac{V(s)}{F(s)} = \{something\}##
That certainly is what I would normally assume as an input. And I would assume the output is position, although there are other options.

FactChecker said:
That certainly is what I would normally assume as an input. And I would assume the output is position, although there are other options.
Yup, and it's simple enough to integrate or differentiate in the Laplace domain.

FactChecker
gneill said:
Yup, and it's simple enough to integrate or differentiate in the Laplace domain.
Good point.

## 1. What is a transfer function?

A transfer function is a mathematical representation of the relationship between the input and output of a system. It describes how an input signal is modified by the system to produce an output signal.

## 2. Can a transfer function exist for an unforced system?

Yes, a transfer function can exist for an unforced system. In this case, the input signal is zero, but the system still has an output signal. The transfer function will simply represent the inherent characteristics of the system.

## 3. How is a transfer function different from a frequency response?

A transfer function is a complex-valued function that describes the entire behavior of a system, while a frequency response is a real-valued function that only describes the output of a system for a specific input frequency.

## 4. Is it necessary to have a transfer function for an unforced system?

No, it is not necessary to have a transfer function for an unforced system. In some cases, a system may not have a transfer function or it may be difficult to obtain. Other methods, such as state-space analysis, can be used to analyze the behavior of unforced systems.

## 5. Can a transfer function change over time for an unforced system?

No, a transfer function for an unforced system will not change over time. The system's inherent characteristics will remain the same, regardless of the input signal. However, the response of the system to different inputs may vary over time.

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