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# 物理代写|理论力学代写Theoretical Mechanics代考|MATH4022 Free Damped Linear Oscillator

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## 物理代写|理论力学代写Theoretical Mechanics代考| Free Damped Linear Oscillator

Each real oscillator eventually comes to stop because of the unavoidable frictional forces. We therefore want to now include them into our considerations where, however, we will restrict ourselves to the simplest case of the Stokes’s friction. Then the extended equation of motion reads:
$$m \ddot{x}=-k x-\alpha \dot{x} .$$
One can realize this situation by a ‘tongue’, dipping into a liquid and being fixed to the mass $m$ (Fig. 2.27). While the frictional term in Eq. (2.167) in general represents a certain approximation, there exists an exact non-mechanical realization of the damped harmonic oscillator by the electrical oscillator circuit. The sum of the partial voltages in the circuit sketched in Fig. $2.28$ must be zero. The electrical current therefore obeys the following differential equation:
$$\ddot{L}+R \dot{I}+\frac{1}{C} I=0 .$$
The ohmic resistance $R$ simulates the frictional term.

After division by $m$ we get from (2.167) the following homogeneous differential equation of second order:
$$\ddot{x}+2 \beta \dot{x}+\omega_0^2 x=0 ; \quad \beta=\frac{\alpha}{2 m} .$$
As ansatz an exponential function appears again plausible:
$$x(t)=e^{\lambda t} .$$
It is exactly then a solution if $\lambda$ fulfills the following relation:
$$\lambda^2+2 \beta \lambda+\omega_0^2=0 .$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Damped Linear Oscillator Under the Inﬂﬂuence of an External Force

Because of the unavoidable friction every oscillating process is exponentially damped unless an additional external force acts. We will now include the latter in our considerations. The equation of motion (2.169) is then to be replaced by
$$\ddot{x}+2 \beta \dot{x}+\omega_0^2 x=\frac{1}{m} F(t) .$$
We choose the same denotations as in the last section and restrict ourselves to the important special case of a periodic force:
$$F(t)=f \cos \bar{\omega} t .$$
One can realize the periodic force by, for instance, a wheel spinning with constant angular velocity and being connected via a drive rod to the oscillating body (Fig. 2.31).

Here again we have an exact non-mechanical realization (Fig. 2.32) by the electrical oscillator circuit if one applies to it a periodic alternating voltage $U_0 \sin \bar{\omega} t$ :
$$\ddot{L}+R \dot{I}+\frac{1}{C} I=U_0 \bar{\omega} \cos \bar{\omega} t$$

## 物理代写|理论力学代写Theoretical Mechanics代考| Free Damped Linear Oscillator

$$m \ddot{x}=-k x-\alpha \dot{x} .$$

$$\ddot{L}+R \dot{I}+\frac{1}{C} I=0 .$$

$$\ddot{x}+2 \beta \dot{x}+\omega_0^2 x=0 ; \quad \beta=\frac{\alpha}{2 m} .$$

$$x(t)=e^{\lambda t}$$

$$\lambda^2+2 \beta \lambda+\omega_0^2=0 .$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Damped Linear Oscillator Under the Inflfluence of an External Force

$$\ddot{x}+2 \beta \dot{x}+\omega_0^2 x=\frac{1}{m} F(t) .$$

$$F(t)=f \cos \bar{\omega} t .$$

$$\ddot{L}+R \dot{I}+\frac{1}{C} I=U_0 \bar{\omega} \cos \bar{\omega} t$$

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