Mass-Spring Oscillator Analysis

Consider now the mass-spring oscillator depicted physically in Fig.D.3, and in equivalent-circuit form in Fig.D.4.

By Newton's second law of motion, the force applied to a mass equals its mass times its acceleration:

By Hooke's law for ideal springs, the compression force applied to a spring is equal to the spring constant times the displacement :

By Newton's third law of motion (every action produces an equal and opposite reaction''), we have . That is, the compression force applied by the mass to the spring is equal and opposite to the accelerating force exerted in the negative- direction by the spring on the mass. In other words, the forces at the mass-spring contact-point sum to zero:

We have thus derived a second-order differential equation governing the motion of the mass and spring. (Note that in Fig.D.3 is both the position of the mass and compression of the spring at time .)

Taking the Laplace transform of both sides of this differential equation gives

To simplify notation, denote the initial position and velocity by and , respectively. Solving for gives

denoting the modulus and angle of the pole residue , respectively. From §D.1, the inverse Laplace transform of is , where is the Heaviside unit step function at time 0. Then by linearity, the solution for the motion of the mass is

If the initial velocity is zero (), the above formula reduces to and the mass simply oscillates sinusoidally at frequency , starting from its initial position . If instead the initial position is , we obtain

Mechanical Equivalent of a Capacitor is a Spring

The mechanical analog of a capacitor is the compliance of a spring. The voltage across a capacitor corresponds to the force used to displace a spring. The charge stored in the capacitor corresponds to the displacement of the spring. Thus, Eq.(E.2) corresponds to Hooke's law for ideal springs:

where is called the spring constant or spring stiffness. Note that Hooke's law is usually written as . The quantity is called the spring compliance.

Mechanical Equivalent of an Inductor is a Mass

The mechanical analog of an inductor is a mass. The voltage across an inductor corresponds to the force used to accelerate a mass . The current through in the inductor corresponds to the velocity of the mass. Thus, Eq.(E.4) corresponds to Newton's second law for an ideal mass:

where denotes the acceleration of the mass .

From the defining equation for an inductor [Eq.(E.3)], we see that the stored magnetic flux in an inductor is analogous to mass times velocity, or momentum. In other words, magnetic flux may be regarded as electric-charge momentum.

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Moving Mass