Spring pendulum

A spring pendulum or spring oscillator is a harmonic oscillator, which consists of a coil spring and an attached mass piece which can move linearly along the direction lengthened or shortened in the spring itself. If the mass moves in the perpendicular direction, gravity affects the rest.

Upon release of the deflected from its rest position spring oscillator is a harmonic oscillation, which no longer decays in the absence of damping begins.

Operation

An ideal spring exerts on the mass of a force composed of the force in the rest position and a share proportional to the distance from the rest position. The force in the rest position compensates for the weight and has no effect on the vibration behavior. The proportion proportional to the deflection always acts resetting. A deflected spring oscillator always has therefore a tendency to return to the rest position. Its mass is accelerated toward the rest position and oscillating due to the principle of inertia again beyond.

The energy stored in the spring potential energy is converted into kinetic energy of the mass. In the absence of damping of the system is removed, no energy, such that the process is repeated periodically with a constant amplitude.

If the spring oscillator periodically excited by an external force, the amplitude can be very large and lead to disaster response.

Derivation of the wave equation

The force acting on the mass of spring force according to Hooke's law proportional to the deflection y.

The proportionality factor D is the spring constant or constant Directorate.

The spring force caused by the action principle, an acceleration of the mass is now opposite to the deflection. The acceleration can also be expressed as the second derivative of displacement with respect to time.

After forming the equation one obtains finally

A linear homogeneous differential equation that can be solved with a Exponentialansatz.

Is called the undamped natural angular frequency.

The natural angular frequency is generally, change the position after the period T gives

The period is the time required for a complete oscillation.

Solving the wave equation

The deflection is an exponential function of the form. The second derivative of the function is, according to the chain rule

Provides the insertion of y in the equation of oscillation

Because the constant c can not be zero (otherwise the approach useless ) and the exponential function for all values ​​of t can be non-zero, the so-called characteristic equation must be satisfied.

For there are two complex solutions:

And

The two solutions, and can be added. For the deflection y of the spring oscillator is obtained thus:

The constants and must be determined. Are at the beginning of the vibration and. After quarter of a period T of the oscillator has reached its maximum deflection.

The exponential function with complex numbers can be converted into sine and cosine functions with the help of Euler's formula.

Provides insertion of

This gives and. The constants can now be used in the trigonometric form of Auslenkungsfunktion, which is then converted in accordance with the quadrant and relationships.

The wave equation for the ideal spring chair without deflection at the beginning of the oscillation ()

Energy of a spring oscillator

The kinetic energy of a spring oscillator with mass m can be calculated with.

After insertion velocity v is obtained

For the natural angular frequency. Therefore, the kinetic energy can also be expressed by:

The potential energy is generally

Since the spring force is applies

The total spring energy EF is composed of the potential and kinetic energy.

Due to the trigonometric Pythagoras applies, the total energy simplifies to:

Fused mass spring

The equations of motion for an ideal spring oscillators are only valid for massless springs. When the elastic spring is assumed to be mass -prone and the mass is homogeneously distributed, the period of oscillation results to

The parameters m and mF correspond to the mass of the oscillator and the mass of the spring.

The total length of the spring is L, S is the distance between the suspension spring of the oscillator and an arbitrary point on the spring. A portion of the spring with the length ds then the mass. The speed of the sprung portion, because it will increase linearly with increasing distance from the suspension. As a result of the kinetic energy of the spring section

The total kinetic energy of the spring is obtained by integrating:

The kinetic energy of a spring vibrator in consideration of the mass spring lossy

It can be seen that one-third of the spring mass behaves as if it were a part of the mass of the body. It follows the period of a mass- spring -prone described above.