Steady State Solution Mass Spring System
Lesson28.mw ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL Lesson 28 -- Application: Unforced Spring-Mass Oscillators Prof. Douglas B. Meade Industrial Mathematics Institute Department of Mathematics University of South Carolina Columbia, SC 29208 URL: http://www.math.sc.edu/~meade/ E-mail: meade@math.sc.edu Copyright 2001 by Douglas B. Meade All rights reserved ------------------------------------------------------------------- Outline of Lesson 28 28.A Unforced Oscillatory Motion - Undamped 28.A-1 Example 1 28.A-2 Example 2 28.B Unforced Oscillatory Motion - Damped 28.B-1 Case 1: Underdamped Motion ( 28.B-2 Case 2: Critically Damped Motion ( 28.B-3 Case 3: Overdamped Motion ( 28.C Resonance 28.C-1 Realistic Resonance 28.C-2 Unrealistic Resonance Initialization Warning, the name changecoords has been redefined 28.A Unforced Oscillatory Motion - Undamped A spring that obeys Hooke's law - the displacement is proportional to the applied force - is called a linear spring. Suppose that such a spring whose natural (unstretched or compressed) length is The force of gravity acting downward on the mass is According to Hooke's law, the force exerted by the spring on the mass is or Initial conditions would be the initial displacement of the mass from equilibrium , and any initial velocity given to the mass. Thus, the mass could be lifted initially, in which case If the mass is thrown upward initially, it would have a positive initial velocity, and hence, These considerations lead to the IVP for follows from the exponential guess Since the characteristic roots are then the general solution will be a linear combination of the members of the (real) fundamental set The trigonometric form can be transformed to where and the phase angle In this form, the general solution of the undamped spring-mass system clearly undergoes periodic motion with angular frequency 28.A-1 Example 1 Maple does not have a single command for converting the trigonometric form to the form Much of the management of this transition remains in the hands of the user. For example, the equation could be treated as an identity in It is typical to take the amplitude as positive, so the first solution would be the preferred one. Note also the usage of the two-argument arctangent function. The single-argument arctangent function, Most scientific programming languages have some form of the two-argument arctangent function, 28.A-2 Example 2 Convert the trigonometric form to the form Again treating the equation as an identity in Selecting the positive amplitude, the desired form is then The amplitude and phase angle are, respectively, 28.B Unforced Oscillatory Motion - Damped Linear damping is resistance proportional to where Surprisingly, the very similar DE governs the unforced RLC electric circuit (see Lesson 29 ) where The solution procedure for a second-order linear ODE with constant coefficients calls for a search for exponential solutions, so the exponential guess leads to and the characteristic equation The values of the characteristic roots are then Since the characteristic equation is quadratic, the discriminant determines the nature of the characteristic roots which, for These three cases are examined individually in the next three subsections. 28.B-1 Case 1: Underdamped Motion ( When the discriminant is negative, that is, in the case where A fundamental set contains the independent solutions which are both exponentially damped sinusoids . (The literature refers to both the sine and cosine functions as sinusoids.) The general solution, a linear combination of these functions, can be written either as or as where In this latter form, The decaying exponential term forces all solutions to decay to the zero function as A typical example of underdamped motion is generated by the differential equation whose solution for the initial conditions is Alternatively, this solution can be expressed in the form where Hence, the equivalent form of the solution is Figure 28.1 shows the solution in black, and the exponential envelopes + 28.B-2 Case 2: Critically Damped Motion ( When the discriminant is zero, that is, in the case so that the general solution is The motion that results from the case of the repeated characteristic root is called critically damped motion. It stands at the interface between underdamped motion and overdamped motion, the third case to be studied below. However, engineers consider that critically damped motion never occurs in actual practice. This is because the parameters in a physical system are never determined so precisely as to cause the discriminant to vanish. Even if the perfect system were built so that the oscillator experienced critically damped motion, the very motion of the spring would warm the metal and change the spring constant, and the system would no longer be truely critically damped. Thus, critical damping is really a mathematical concept. Initial values for whose solution is Surprisingly, the initial velocity The IVP governs the critically damped motion exhibited by the solution that is graphed in Figure 28.2. Subcase 1: So long as which is positive if and only if The IVP governs the critically damped motion exhibited by the solution and graphed in Figure 28.3. Lifting the mass and tossing it upward to set it into motion causes it to rise above the initial height, but then to descend exponentially to equilibrium. Subcase 2: Note that if a fundamental set of solutions is 28.B-3 Case 3: Overdamped Motion ( The third case is the most straightforward. If the discriminant is positive, that is, if where again but Writing these characteristic roots as a fundamental set of solutions is then so the general solution is The system giving rise to such a solution is called overdamped , and by a stretch of notation, the differential equation itself, its solution, or the resulting motion, can be modified by the adjective "overdamped." Observe that, because 0 < To determine when the critical point occurs with This time is positive precisely when 0 < To conclude, note that except for the trivial case with The driven or forced oscillator is governed by the nonhomogeneous ODE where where 28.C Resonance Anthropomorphically speaking, resonance can be described as a temper tantrum thrown by Mother Nature. It is an exaggerated response to a stimulus. Analytically speaking, resonance in a driven oscillator is the phenomenon whereby the steady-state response to a sinusoidal forcing term attains maximal amplitude. Thus, the amplitude of the steady-state solution of the driven damped oscillator described by the ODE where achieves a maximal value at some angular frequency Familiar occurrences of resonance might include the sound-box on a stringed musical instrument, the tuning circuit in an AM radio, the annoying rattle of a motor driven at certain speeds, or the collapse of the Tacoma Narrows Bridge. The body of an acoustic guitar is a resonator that amplifies the sound of the vibrating strings. The tuning circuit of an AM radio is essentially a driven RLC circuit whose resonant oscillations are used to oppose those of the carrier signal, leaving behind the broadcast information. Less useful instances of resonance include the irritation of a noisy motor whose resonance is causing nearby objects to vibrate noisily or destructively. The rythmic driving of a bridge under the footfalls of an army marching in step can cause destructive resonance. Supposedly, to avoid resonance, marching armies were instructed to break step when crossing bridges. The collapse in 1940 of the Tacoma Narrows Bridge over the Puget Sound requires more discussion. This enormous (almost 6000 feet long) engineering structure was designed to withstand winds of up to 120 mph. But, due to its tendency to undulate in light winds, the bridge was fondly known as Galloping Gertie. On November 7, 1940, only four months after being completed, the bridge was completely destroyed in 42 mph winds. The full incident is preserved on a famous four-minute film clip; a short excerpt is available at http://www.physics.bcit.ca/netshow/tacoma.avi . For many years the accepted explanation of this tragedy was resonance. In recent years, however, an alternate theory based on aerodynamically induced condition of self-excitation or "negative damping" has gained wide acceptance. Additional details can be found at many sites on the WWW, including http://www.urbanlegends.com/science/bridge_resonance.html http://www.ketchum.org/wind.html http://www.eng.uab.edu/cee/reu_nsf99/tacoma.htm This is not the place to debate these theories, but both require a solid understanding of resonance and damping. The following discussion of resonance proceeds in two parts. First, the realistic case of the driven damped oscillator is considered. Second, the unrealistic case of the undamped oscillator is discussed. Since all real systems have damping of some sort, undamped oscillators are actually only met as an approximation to real systems. 28.C-1 Realistic Resonance Without loss of generality, let the equation for the driven damped oscillator be where the driving (angular) frequency The steady-state response is essentially the particular solution since the homogeneous solution, containing exponentials whose real parts are negative, will be a transient of the system and become negligible after a certain time. This particular solution is The form is more convenient for determining the amplitude Again selecting the positive amplitude, and ignoring the phase angle, write This amplitude is a function of the driving frequency The positive solution is the resonant frequency, written as An alternate form for this expression is where Thus, for the damped oscillator, the natural frequency is slightly larger than the resonant frequency, whereas for the undamped oscillator, the natural and resonant frequencies will shortly be seen to be the same. If Choosing the parameter values gives the amplitude function whose graph is seen in Figure 28.4. Figure 28.4 supports the initial claim that resonance can be thought of as a temper tantrum by Mother Nature. The amplitude of the driving function is just The response to the driving term clearly depends on the damping parameter. Choose the system parameters to be so that the amplitude function is The dependence of the amplitude on both the damping coefficient and the driving frequency can be seen in Figure 28.5. As damping increases upward from 28.C-2Unrealistic Resonance Without damping, the driven oscillator equation would be in which case a particular solution would be Clearly, as Another approach to unrealistic resonance is to drive the system initially at the natural angular frequency. The differential equation for the system then becomes A particular solution is then where, in spite of the complicated form, it should be clear that one of the trigonometric terms is multiplied by To within a constant factor, the particular solution is actually Taking whose graph is then seen in Figure 28.7, along with the lines The oscillations would become unbounded were it not for the constraints of physical reality. [Back to ODE Powertool Table of Contents ]
< 0 )
= 0 )
> 0 )
is suspended vertically so that its upper end is attached to a fixed support, and its lower end is attached to a mass
. The attached mass will stretch the spring until it attains an equilibrium position. In a coordinate system
measured positive upward from the equilibrium position of the mass, measure displacements of the mass with the variable
. Again, the origin of this coordinate system is the equilibrium position of the suspended mass, and displacements upward are positive.
j , where
is the gravitational constant, is counterbalanced by the sag in the spring past its natural length
when the mass is first attached. This downward gravitational force is always in balance with the upward force generated by the spring as it sags from its natural length to the equilibrium position that establishes the origin
for the coordinate system. This gravitational force never appears in the equation of motion for the spring-mass system described above.
j , where
is the spring constant expressed in units of force divided by distance. The acceleration of the mass under the action of the spring is a =
j . Newton's second law, namely, F =
a , then gives the equation of motion as
. Alternatively, the mass could be pulled down below equilibrium so that
. Of course, if the mass is not displaced from equilibrium initially,
.
. Alternatively, the mass could be tossed downward initially, in which case
. If the mass is released from rest , then
.
> ode1 := m*diff(x(t),t,t)+k*x(t) = 0;
ic1 := x(0)=alpha, D(x)(0)=beta;
, the displacement of the mass from equilibrium. The solution of the differential equation itself, namely,
and the characteristic equation
= +
, where
is called the angular frequency .
is determined by the two equations
and
and amplitude
. Such motion is called harmonic motion , and without damping, would persist undiminished forever. Undamped spring-mass systems are therefore called natural harmonic oscillators .
> q := a*cos(omega*t) + b*sin(omega*t);
> Q := A*cos(omega*t - phi);
=
, so that Maple can compute the appropriate values of
and
as
> qq := solve(identity(q=Q,t), {A,phi});
, returns an angle in the range
, which is equivalent to the angle being in the first or fourth quadrants. This is the function one finds on the typical scientific calculator. Use of this function with a point
loses sign information when either
or
is negative.
, which returns an angle in the range (
]. This is equivalent to preserving the signs on the coordinates of points
so that the angle returned is in the proper one of the four quadrants.
> q := 3*cos(2*t) - 5*sin(2*t);
, Maple determines two solutions for the constants
and
via
> qq := solve(identity(q = Q,t), {A,phi});
> A = eval(A,qq[1]);
phi = eval(phi,qq[1]);
, the velocity, and opposing the motion. Air resistance against the moving mass would be an example of such a force. If linear damping is added to the spring-mass system of Section 28.A, the differential equation governing displacements from equilibrium becomes
> ode1 := m * diff( x(t), t$2 ) + b * diff( x(t), t ) + k * x(t) = 0;
is the coefficient of linear damping . The oscillatory motion experienced by such a system is called unforced, or free motion, and corresponds to the absence of a driving, or forcing term on the right-hand side of the equation. Hence, the DE for unforced motion is homogeneous.
represents the charge,
represents the current,
is the resistance,
is the inductance, and
is the capacitance (
is the elastance).
> q1 := factor( eval( ode1, x(t)=exp(lambda*t) ) );
> chareqn := q1 / exp( lambda*t );
> charvals := solve( chareqn, {lambda} );
> discrim(lhs(chareqn),lambda);
positive, and
and
nonnegative, can be negative, zero, or positive. In each case, the system experiences a distinctive motion called respective, underdamped, critically damped, or overdamped.
< 0 )
, the characteristic values are complex conjugates that can be written as
+
and
=
,
, and
.
is the damped amplitude ,
is the angular frequency,
is the quasi-frequency,
is the quasi-period, and
is the phase angle.
increases.
> q1 := eval(ode1, {m=1,b=4,k=16});
> ic1 := x(0)=2, D(x)(0)=2;
> X1 := collect(rhs(dsolve({q1,ic1},x(t))),exp);
> QQ := A*cos(2*sqrt(3)*t-phi)*exp(-2*t);
is the positive solution in
> params := solve(identity(X1=QQ,t), {A,phi});
> X2 := eval(QQ,params[1]);
in red.
> plot([X1, sqrt(7)*exp(-2*t), -sqrt(7)*exp(-2*t)], t=0..3, color=[black,red,red], title="Figure 28.1");
= 0 )
, the characteristic values are the repeated root
. In this case, two linearly independent solutions to the ODE are
and
> X := (c[1]+c[2]*t)*exp(-b/2/m*t);
and
lead to the equations
> eq1 := eval(X,t=0) = x(0);
eq2 := eval(diff(X,t),t=0) = `x'`(0);
> expand(solve({eq1,eq2},{c[1],c[2]}));
implies
.
> ode2 := eval(ode1, {m=1,b=8,k=16});
ic2 := x(0)=2, D(x)(0)=0;
> XC1 := rhs(dsolve({ode2, ic2}, x(t)));
> plot(XC1, t=0..3, title="Figure 28.2");
> 0
, the common exponential factor overwhelms the linear factor and forces all solutions to zero. The decay to zero is not necessarily monotonic. The general solution of the critically damped equation has a critical point at
> t[crit] = expand(solve( diff(X,t)=0, t));
and
<
.
> ode2;
ic3 := x(0)=2, D(x)(0)=5;
> XC2 := rhs(dsolve({ode2, ic3}, x(t)));
> plot(XC2, t=0..3, title="Figure 28.3");
= 0
= 0, then
= 0 and
imply that
. In this case of undamped motion , the ODE simplifies to
> eval( ode1, [m=1,b=0,k=0] );
, and the general solution is a linear function. Hence, solutions do not tend to zero as
increases. In fact, except in special cases where the solution is a constant function, the solutions become unbounded as
increases.
> 0 )
> 0, then there are two distinct, real characteristic roots, namely,
+
=
> lambda[1] := eval( lambda, charvals[1] );
lambda[2] := eval( lambda, charvals[2] );
> X[1] := exp( lambda[1]*t );
X[2] := exp( lambda[2]*t );
> X[h] := c[1]*X[1] + c[2]*X[2];
, both characteristic values are negative. In fact,
<
< 0. All solutions of an overdamped equation ultimately decay to zero as
increases. But, just as with the critically damped case, the solution can have one critical point:
> dX[h] := diff( X[h], t ):
> tcrit := solve( dX[h] = 0, t );
, observe that the location of the critical point can be expressed as
=
ln(
)
< 1.
=
= 0, all solutions to the ODE for unforced oscillatory motion are transient solutions that eventually "die out" as time passes. (The first generation of television sets were built with vacuum tubes that had to "warm up" to reach their operating points. During this warm-up phase, the TV screen would generally show a gradually brightening, wavering and distorted image that was the physical realization of the transients in the circuits.)
is called the forcing term. The general solution of this equation has the form
is the homogeneous solution built as a linear combination of the members of the fundamental set. This solution contains the decaying exponentials discussed in this worksheet. The particular solution
is determined by the driving term, and any part of it that persists over time constitutes the steady - state solution, the solution that is seen after all transients die out.
.
http://www.vibrationdata.com/Tacoma.htm
http://www.math.uconn.edu/~kmoore/tacoma.html
http://www.madsci.org/posts/archives/may98/892678504.Eg.r.html
http://www.nationmaster.com/encyclopedia/Tacoma-Narrows-Bridge
http://en2.wikipedia.org/wiki/Tacoma_Narrows_Bridge
> ODE := diff(x(t),t,t) + B*diff(x(t),t) + K*x(t) = A*cos(omega*t);
is as yet undetermined. Treat it as a parameter that can be adjusted, much like the speed on an electric fan can be changed by turning a knob. The standard equation for the damped oscillator has been divided by the mass
, so
and
.
> XP := dsolve(ODE,x(t),output=basis)[2];
> Q := a*cos(omega*t-phi);
, which is found via
> sol := solve(identity(XP=Q,t),{a,phi});
. Resonance occurs at the frequency that causes this function to be a maximum. Simple calculus reveals that this resonant frequency is
> SOL := solve(diff(AMP,omega)=0,omega);
> omega[R] = eval(SOL[2], {B=b/m, K=k/m});
is real and positive if
. Thus, if the damping coefficient
is nto less than
, no value of the driving frequency will cause the system to resonate. Alternatively, the threshold of damping for resonance is
, whereas the threshold of damping for the system to be underdamped is
. Loosely speaking, it might be said that half the systems that can oscillate can be made to resonate. An underdamped system that is made to resonate will also have a natural frequency
, the expression for the amplitude of the steady-state response is
> params1 := {A=1, B=1/10, K=1};
> F := eval(AMP, params1);
> plot(F,omega=0..3, title="Figure 28.4");
, but the magnitude of the response at steady-state is on the order of 10. For very little provocation, the oscillator has over-responded, much like a child having a temper tantrum.
and
> params2 := {A=1, B=b, K=1};
> G := eval(AMP, params2);
> plot3d(G, omega=0..2, b=0..1, axes=box, title="Figure 28.5");
, the magnitude of the steady-state amplitude diminishes. Thus, the greater the damping, the less the effect of resonance. As
decreases, the spike in amplitude at resonance clearly increases. The plane sections
shown in Figure 28.6 again capture the dependence of the resonant peak on damping. The tallest peak corresponds to the smallest value of
, and the smallest peak, to the largest.
> plot([seq(eval(G,b=j/10),j=1..9)],omega=0..2, title="Figure 28.6");
> dsolve(ODE1, x(t), output=basis)[2];
approaches
, the natural (angular) frequency of the system, the amplitude of the steady-state solution will become infinite. In reality, there is always some damping that prevents this solution from actually existing forever. In physical terms usually one or more system components break while the amplitude is still finite. Because these unbounded solutions are primarily of theoretical interest, this is called unrealistic resonance.
> ODE2 := eval(ODE1, {K=k/m, omega=sqrt(k/m)});
> XP := dsolve(ODE2, x(t), output=basis)[2];
. This is because, by the method of undetermined coefficients, the homogeneous solution and the driving term share a common term. Hence, the particular solution requires multiplication by
, and the trigonometric term in the particular solution without the factor
is actually redundant - it also appears in the homogeneous solution.
> Xp := t*sin(sqrt(k/m)*t);
= 1, this becomes
> xt := eval(Xp, {k=1,m=1});
= +
, the envelopes on the oscillations generated by
.
> plot([xt, t,-t], t=0..25, color=[black,red,red], title="Figure 28.7");
Steady State Solution Mass Spring System
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