Two Spring-Coupled Masses

Here, we have made use of the fact that a mass attached to the left end of a spring of extension and spring constant experiences a horizontal force , whereas a mass attached to the right end of the same spring experiences an equal and opposite force .

Equations (149)-(150) can be rewritten in the form

where . Let us search for a solution in which the two masses oscillate in phase at the same angular frequency, . In other words,

where , , and are constants. Equations (151) and (152) yield

(155) | ||

(156) |

or

where . By searching for a solution of the form (153)-(154), we have effectively converted the system of two coupled linear differential equations (151)-(152) into the much simpler system of two coupled linear algebraic equations (157)-(158). The latter equations have the trivial solutions , but also yield

Hence, the condition for a nontrivial solution is

In fact, if we write Equations (157)-(158) in the form of a homogenous (i.e., with a null right-hand side) matrix equation, so that

(161) |

then it is apparent that the criterion (160) can also be obtained by setting the determinant of the associated matrix to zero (Riley 1974).

Equation (160) can be rewritten

It follows that

or | (163) |

Here, we have neglected the two negative frequency roots of (162)--that is, and --because a negative frequency oscillation is equivalent to an oscillation with an equal and opposite positive frequency, and an equal and opposite phase. In other words, . It is thus apparent that the dynamical system pictured in Figure 149 has two unique frequencies of oscillation: namely, and . These are called the

The patterns of motion associated with the two normal frequencies can be deduced from Equation (159). Thus, for (i.e., ), we get , so that

where and are constants. This first pattern of motion corresponds to the two masses executing simple harmonic oscillation with the same amplitude and phase. Such an oscillation does not stretch the middle spring. On the other hand, for (i.e., ), we get , so that

where and are constants. This second pattern of motion corresponds to the two masses executing simple harmonic oscillation with the same amplitude but in anti-phase: that is, with a phase shift of radians. Such oscillations do stretch the middle spring, implying that the restoring force associated with similar amplitude displacements is greater for the second pattern of motion than for the first. This accounts for the higher oscillation frequency in the second case. (The inertia is the same in both cases, so the oscillation frequency is proportional to the square root of the restoring force associated with similar amplitude displacements.) The two distinctive patterns of motion that we have found are called the

The most general motion of the system is a linear combination of the two normal modes. This immediately follows because Equations (149) and (150) are linear equations. [In other words, if and are solutions then so are and , where is an arbitrary constant.] Thus, we can write

We can be sure that this represents the most general solution to Equations (149) and (150) because it contains four arbitrary constants: namely, , , , and . (In general, we expect the solution of a second-order ordinary differential equation to contain two arbitrary constants. It, thus, follows that the solution of a system of two coupled, second-order, ordinary differential equations should contain four arbitrary constants.) These constants are determined by the initial conditions.

For instance, suppose that , , , and at . It follows, from Equations (168) and (169), that

(170) | ||

0 | (171) | |

0 | (172) | |

0 | (173) |

which implies that and . Thus, the system evolves in time as

(174) | ||

(175) |

where , and use has been made of the trigonometric identities and . (See Appendix B.) This evolution is illustrated in Figure 16. (Here, . The solid curve corresponds to , and the dashed curve to .)

Finally, let us define the so-called *normal coordinates*, which (in the present case) take the
form

(176) | ||

(177) |

It follows from Equations (168) and (169) that, in the presence of both normal modes,

Thus, in general, the two normal coordinates oscillate sinusoidally with unique frequencies, unlike the regular coordinates, and . (See Figure 16.) This suggests that the equations of motion of the system should look particularly simple when expressed in terms of the normal coordinates. In fact, it can be seen that the sum of Equations (151) and (152) reduces to

whereas the difference gives

Thus, when expressed in terms of the normal coordinates, the equations of motion of the system reduce to two uncoupled simple harmonic oscillator equations. The most general solution to Equation (180) is (178), whereas the most general solution to Equation (181) is (179). Hence, if we can guess the normal coordinates of a coupled oscillatory system then the determination of the normal modes of oscillation is considerably simplified.