Find the natural response of IVP-(VI).
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Find the forcing response of IVP-(VI).
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would there be a beat or resonance or neither?
The Correct Answer and Explanation is :
To analyze the natural and forced responses of the given initial value problem (IVP), we need to examine the system’s behavior under both free and forced conditions. Let’s break down the process step by step.
1. Natural Response of the System:
The natural response pertains to the system’s behavior when no external forces are applied, relying solely on its inherent properties. This response is governed by the homogeneous differential equation:
[ m \cdot x”(t) + k \cdot x(t) = 0 ]
where:
- ( m ) is the mass,
- ( k ) is the stiffness,
- ( x(t) ) is the displacement as a function of time.
The general solution to this equation is:
[ x(t) = A \cdot \cos(\omega_n t) + B \cdot \sin(\omega_n t) ]
where:
- ( \omega_n = \sqrt{\frac{k}{m}} ) is the natural frequency of the system,
- ( A ) and ( B ) are constants determined by initial conditions.
2. Forced Response of the System:
When an external force is applied, the system’s response includes both the natural response and the particular solution due to the forcing function. The equation of motion becomes:
[ m \cdot x”(t) + k \cdot x(t) = F(t) ]
where ( F(t) ) is the external forcing function. The particular solution depends on the form of ( F(t) ). For example, if ( F(t) = F_0 \cdot \cos(\omega t) ), the particular solution is:
[ x_p(t) = \frac{F_0}{m \cdot (\omega_n^2 – \omega^2)} \cdot \cos(\omega t) ]
The total response is the sum of the natural and particular solutions:
[ x(t) = x_h(t) + x_p(t) ]
3. Analysis of Beat and Resonance Phenomena:
- Resonance: Occurs when the forcing frequency ( \omega ) matches the system’s natural frequency ( \omega_n ). In this case, the amplitude of the forced response becomes unbounded, leading to large oscillations. This is because the system absorbs energy from the external force at its natural frequency, causing the amplitude to increase without bound.
- Beats: Arise when the forcing frequency ( \omega ) is close to, but not equal to, the natural frequency ( \omega_n ). This results in a modulated response where the amplitude oscillates at a frequency equal to the difference between ( \omega ) and ( \omega_n ). The system experiences periodic increases and decreases in amplitude, known as beats.
Conclusion:
To determine whether the system exhibits beats, resonance, or neither, we need to compare the forcing frequency ( \omega ) with the natural frequency ( \omega_n ). If ( \omega ) is equal to ( \omega_n ), resonance occurs. If ( \omega ) is close to ( \omega_n ) but not equal, beats are observed. If ( \omega ) is significantly different from ( \omega_n ), neither phenomenon occurs, and the system responds without these modulations.
For a more detailed analysis, including specific calculations and graphical representations, please refer to the provided link.