Spring response to sinusoidally varying force

• sridhar10chitta
In summary: The velocity lags the applied force by 90 degrees. This is because the velocity is proportional to the second harmonic of the applied force.
sridhar10chitta
Can one model how a spring (fixed at one end) responds to a sinusoidally varying applied force ? For example, in electric circuits, the reactance of a capacitor is modeled as 1/j(omega)C and is used to obtain the current when a sinusoidally varying voltage is applied.

The hyperphysics link referred by dlgoff shows a mass attached to the spring. If one were to model simply the spring without the mass, and apply a force that varies sinusoidally with respect to time then, can it be that the velocity leads the applied force by 90 degrees ? In which case will it be right to model the spring as k/j(omega) where k = spring constant ?

When a capacitor, modeled as 1/j(omega)C, is subjected to a sinusoidal input voltage, the current in the circuit leads the voltage (input voltage which is equal to the voltage across the capacitor) by 90 degrees. This is a consequence of the fringe field that develops across its plates as charges accumulate on its plate and which opposes the effect of the driving input voltage (field).

sridhar10chitta,

What you describe is a special case of complex harmonic motion.

In your analog equivalent of the reactance of a capacitor your spring establishes one input to the system (say; y-axis) with harmonic properties fixed by its physical characteristics while another input to the system (also in the y-axis) is your "sinusoidally varying applied force".

A more general expression which would apply in your case (where inputs to the system are coaxial, linear and time invariant) might be found within LTI system theory, see;

http://en.wikipedia.org/wiki/LTI_system

"Any system that can be modeled as a linear homogeneous differential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistors, inductors, and capacitors (RLC circuits). Ideal spring–mass–damper systems are also LTI systems, and are mathematically equivalent to RLC circuits."

So we can model the spring as k/j(omega), I take it. Then, as an extension of the inductance case, can one model the mass as j(omega)m where m is a mass ? In which case the velocity lags the applied sinusoidally varying (of course, again, because our notations for the impedance are valid only in the case of steady-state sinusoidal quantities) force by 90 degrees ?

A spring with a spring constant k (e.g., F = -kx) and an attached mass m at the end has a natural resonant frequency:
Fres = (1/2 pi) sqrt(k/m).
This means that you will need something in your circuit that has a resonant frequency; such as a capacitor AND an inductor.
Bob S

I think you are assuming the spring has mass. In the ideal case, the spring is taken to be massless and so the idea that the spring be modeled as a simple ideal capacitor. Resonance occurs only when there are two elements (mass and spring in the mechanical case and capacitor and inductor in the electric circuits case) with a natural frequency of vibration.
The mass is modeled separately as I indicated in my previous posting.

1. What is a spring response to sinusoidally varying force?

A spring response to sinusoidally varying force refers to the behavior of a spring when subjected to a force that varies in a sinusoidal pattern over time. This type of force can cause the spring to undergo oscillations, where it moves back and forth around its equilibrium position.

2. How does the amplitude of the force affect the spring response?

The amplitude of the force directly affects the amplitude or magnitude of the spring's oscillations. A larger force amplitude will result in larger oscillations, while a smaller force amplitude will result in smaller oscillations.

3. What is the relationship between the frequency of the force and the spring response?

The frequency of the force is directly proportional to the frequency of the spring's oscillations. This means that as the frequency of the force increases, the spring will oscillate at a higher frequency as well.

4. How does the stiffness of the spring affect its response to a sinusoidally varying force?

The stiffness of the spring, also known as its spring constant, determines how much force is required to stretch or compress the spring. A stiffer spring will require a larger force to produce the same amount of displacement as a less stiff spring. This means that a stiffer spring will have a smaller amplitude of oscillations for the same force amplitude compared to a less stiff spring.

5. Can the spring response to sinusoidally varying force be described mathematically?

Yes, the spring response to a sinusoidally varying force can be described by a mathematical equation known as Hooke's law. This law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. This equation can be used to calculate the magnitude and frequency of the spring's oscillations in response to a sinusoidally varying force.

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