# Forced oscillations of a linear system

• watty
In summary, the system was being driven at frequencies not equal to its normal modes and had two small peaks at normal frequencies as well as a significant peak at 3 times the driving frequency. The two small peaks could be explained as imperfections in the system, while the significant peak could be explained by the system oscillating in multiple directions.
watty
Hello,

I am currently working on a lab in which we are studying the behaviour of chain of metal bars attached together with nylon wire in such a way as to to mimic the ability of solids or liquids to transmit a wave.

After studying the normal modes of the system as well as the quality factor, we excited the system at frequencies not equal to normal modes and analysed the frequencies present using an optical detector wired through a computer that can perform a Fourier analysis of the spectrum.

When the system was being driven, we expected to see only one frequency present in the Fourier spectrum: that of the driving oscillator. However we also saw two very small peaks at normal frequencies as well as a significant peak at what appeared to be 3 times the driving frequency.

The two small peaks I can understand as maybe imperfections of the system, ie the forced oscillation does not fully overcome the natural properties of the system but I don't understand why the system is also oscillating at 3 times the driving frequency.

NB. there is also a normal mode reasonable close to this significant unexplained peak.

When you theoretically analyze the propagation of waves in this chain, are you fully accounting for the rotational moment of inertia in the individual metal bars? The metal bars have both center-of-mass motion (linear inertia) and rotational inertia. Could this cause mode-conversion?
Bob S

no i have not taken this into account. you mean that they are oscillating not just on one axis but in several directions? this could be the case. each metal bar is attached at each end to two long nylon wires. the wires are at high tension but there could be some extension and contraction in the wires and i suppose the bars are not moving exclusively up and down but also left and right too. Is this what you mean?

ps. mode conversion? what does this mean?

## 1. What are forced oscillations of a linear system?

Forced oscillations of a linear system refer to the periodic motion of a system that is driven by an external force. This force causes the system to oscillate at a specific frequency, which is determined by the properties of the system and the magnitude of the external force.

## 2. How are forced oscillations different from free oscillations?

Forced oscillations are different from free oscillations in that they are driven by an external force, whereas free oscillations occur naturally without any external force. Additionally, forced oscillations tend to have a constant amplitude and frequency, while free oscillations may exhibit damping or changes in frequency over time.

## 3. What is the role of damping in forced oscillations?

Damping is a critical factor in forced oscillations as it affects the amplitude and frequency of the oscillations. In a system with high damping, the amplitude of the oscillations will decrease more quickly, resulting in a lower amplitude and shorter duration of oscillation. In contrast, a system with low damping will have a higher amplitude and longer duration of oscillation.

## 4. How do you calculate the resonance frequency of a forced oscillation system?

The resonance frequency of a forced oscillation system can be calculated by finding the frequency at which the external driving force is equal to the natural frequency of the system. This can be determined by solving the equation for the natural frequency, which is dependent on the properties of the system such as mass, stiffness, and damping.

## 5. What are some real-world applications of forced oscillations in linear systems?

Forced oscillations in linear systems have many applications in various fields, including engineering, physics, and biology. Some examples include the use of forced oscillations in building designs to reduce vibrations, in electronic circuits for signal processing, and in the study of heart and lung function in medical research.

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