Ideal Spring

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Impedance

Ideal Mass

One-Port Network Theory

Transients are short intervals during which the signal evolves quickly in some nontrivial or relatively unpredictable way (Bello, 2005). In computer music, the short attack portion of a sound is often treated as a transient signal. — Click for http://en.wikipedia.org/wiki/Transient_%28acoustics%29

An initial value problem consists of a differential equation and an initial condition. The solution to an initial value problem is a function y(t) solving the differential equation and satisfying the initial condition y(t₀)=y₀. — Click for http://en.wikipedia.org/wiki/Initial_value_problem

The wave impedance in a propagation medium relates the force-related component of the wave motion to the displacement-related component of the motion. — Click for http://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

Click for http://hyperphysics.phy-astr.gsu.edu/hbase/mass.html#mas

Click for https://ccrma.stanford.edu/CCRMA/Courses/152/SPL.html

The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.html

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

Click for https://ccrma.stanford.edu/~jos/filters/Laplace_Transform_Analysis.html

The displacement of an object describes how far away and in what direction it has been moved, or displaced, from its rest position. Displacement is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Plucked_String_Displacement.html

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

Click for https://ccrma.stanford.edu/~jos/pasp/Hooke_s_Law.html

According to Hooke's law, the force F exerted by a spring stretched a distance x from its rest position is F=-kx. — Click for http://en.wikipedia.org/wiki/Spring_%28device%29

Search JOS Website

JOS Home Page

JOS Online Publications

Index of terms in JOS Website

Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Impedance

Ideal Mass

One-Port Network Theory

Ideal Spring

Figure 7.4 depicts the ideal spring.

**Figure 7.4:** The ideal spring characterized by .
$\includegraphics[width=3in]{eps/lspring}$

From Hooke's law, we have that the applied force is proportional to the displacement of the spring:

$\displaystyle f(t) \eqsp k\, x(t) \isdefs k \int_0^t v(\tau)d\tau$

where it is assumed that

. The spring constant

is sometimes called the stiffness of the spring. Taking the Laplace transform gives

$\displaystyle F(s) = k V(s)/s$

so that the impedance of a spring is

$\displaystyle R_k(s) \isdef \frac{k}{s}$

and the admittance is

$\displaystyle \Gamma_k(s) \isdef \frac{s}{k}$

This is the transfer function of a differentiator. We can say that the ideal spring differentiates the applied force (divided by

) to produce the output velocity.

The frequency response of the ideal spring, given the applied force as input and resulting velocity as output, is

$\displaystyle \Gamma_k(j\omega) = \frac{j\omega}{k}$

In this case, the amplitude response grows

dB per octave, and the phase shift is $+\pi /2$ radians for all $\omega$ . Clearly, there is no such thing as an ideal spring which can produce arbitrarily large gain as frequency goes to infinity; there is always some mass in a real spring.

We call the compression velocity of the spring. In more complicated configurations, the compression velocity is defined as the difference between the velocity of the two spring endpoints, with positive velocity corresponding to spring compression.

In circuit theory, the element analogous to the spring is the capacitor, characterized by , or . In an equivalent analog circuit, we can use the value . The inverse of the spring stiffness is sometimes called the compliance of the spring.

Don't forget that the definition of impedance requires zero initial conditions for elements with ``memory'' (masses and springs). This means we can only use impedance descriptions for steady state analysis. For a complete analysis of a particular system, including the transient response, we must go back to full scale Laplace transform analysis.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Ideal Spring

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University