Moving Mass

Figure D.1 depicts a free mass driven by an external force along an ideal frictionless surface in one dimension. Figure D.2 shows the electrical equivalent circuit for this scenario in which the external force is represented by a voltage source emitting volts, and the mass is modeled by an inductor having the value Henrys.

**Figure D.1:** Physical diagram of an external force driving a mass along a frictionless surface.
$\begin{figure}\input fig/forcemass.pstex_t \end{figure}$

**Figure:** Electrical equivalent circuit of the force-driven mass in Fig.D.1.
$\begin{figure}\input fig/forcemassec.pstex_t \end{figure}$

From Newton's second law of motion ``'', we have

$\displaystyle f(t) = m\,a(t) \isdef m\,{\dot v}(t) \isdef m\,{\ddot x}(t).$

Taking the unilateral Laplace transform and applying the differentiation theorem twice yields

$\begin{eqnarray*} F(s) &=& m\,{\cal L}_s\{{\ddot x}\}\\ &=& m\left[\,s {\cal L... ...right\}\\ &=& m\left[s^2\,X(s) - s\,x(0) - {\dot x}(0)\right]. \end{eqnarray*}$

Thus, given

Laplace transform of the driving force ,
initial mass position, and
${\dot x}(0)\isdeftext v(0) =$ initial mass velocity,

we can solve algebraically for

, the Laplace transform of the mass position for all $t\ge 0$ . This Laplace transform can then be inverted to obtain the mass position

for all $t\ge 0$ . This is the general outline of how Laplace-transform analysis goes for all linear, time-invariant (LTI) systems. For nonlinear and/or time-varying systems, Laplace-transform analysis cannot, strictly speaking, be used at all.

If the applied external force is zero, then, by linearity of the Laplace transform, so is , and we readily obtain

$\displaystyle X(s) = \frac{x(0)}{s} + \frac{{\dot x}(0)}{s^2} = \frac{x(0)}{s} + \frac{v(0)}{s^2}.$

Since

is the Laplace transform of the Heaviside unit-step function

$\displaystyle u(t)\isdef \left\{\begin{array}{ll} 0, & t<0 \\ [5pt] 1, & t\ge 0 \\ \end{array}\right.,$

we find that the position of the mass

is given for all time by

$\displaystyle x(t) = x(0)\,u(t) + v(0)\,t\,u(t).$

Thus, for example, a nonzero initial position

and zero initial velocity

results in

for all $t\ge 0$ ; that is, the mass ``just sits there''.^D.3 Similarly, any initial velocity

is integrated with respect to time, meaning that the mass moves forever at the initial velocity.

To summarize, this simple example illustrated use the Laplace transform to solve for the motion of a simple physical system (an ideal mass) in response to initial conditions (no external driving forces). The system was described by a differential equation which was converted to an algebraic equation by the Laplace transform.

Order a Hardcopy of Introduction to Digital Filters

Previous: Laplace Analysis of Linear Systems
Next: Mass-Spring Oscillator Analysis

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Sponsor

Chapters

Introduction to Digital Filters
    Introduction to Laplace Transform Analysis
       Laplace Analysis of Linear Systems
          Moving Mass

Moving Mass

Order a Hardcopy of Introduction to Digital Filters

Sign in

Search Online Books

Free Online Books

Sponsor

Chapters

Introduction to Digital Filters Introduction to Laplace Transform Analysis Laplace Analysis of Linear Systems Moving Mass

Moving Mass

Order a Hardcopy of Introduction to Digital Filters

Introduction to Digital Filters
Introduction to Laplace Transform Analysis
Laplace Analysis of Linear Systems
Moving Mass