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Spring response to sinusoidally varying force

  1. Nov 30, 2009 #1
    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.
  2. jcsd
  3. Nov 30, 2009 #2


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    Last edited by a moderator: Apr 24, 2017
  4. Dec 1, 2009 #3
    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).
  5. Dec 1, 2009 #4

    dlgoff's post seems to address your question (as I understand your question).

    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;


    from above link;
    "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."
  6. Dec 1, 2009 #5
    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 ?
  7. Dec 2, 2009 #6
    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
  8. Dec 2, 2009 #7
    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.
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