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Simple Harmonic Motion Coupled Oscillators Eigenvalue Problem

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Coupled Oscillators Concepts of primary interest: Simple Harmonic Motion Coupled Oscillators Eigenvalue problem Developing the equations of Motion: Intuitive Newton’s Law Method Lagrangian Mechanics Hamiltonian Mechanics Work-Energy for Mechanical Engineers Sample calculations: A matched equal mass pair of coupled oscillator – with variable coupling An oscillator pair with unequal masses Initial condition matching for near degenerate modes Comparison of two choices for generalized coordinates Applications Tools of the Trade Cramer’s Rule Coupled Differential Equation Principles Underlying the Coupled Oscillator Procedures Engineers can use this handout by replacing -2 by . A brief discussion of forced couple oscillations will be added. dd example with damping The Plan: A system of coupled linear oscillators is to be studied. To anchor the discussion, the solution to the single simple harmonic oscillator is reviewed. Next a system of two masses coupled by a linear spring is analyzed by guessing the solution. The result is a pair of two-particle sinusoidal oscillation patterns each with its characteristic frequency. More general systems are analyzed by assuming that sinusoidal solution modes exist and applying a diagonalization approach to identify them. The method can be made more formal by noting that it is a particular example of the matrix solution to systems of coupled linear differential equations. At several points, physical interpretations of results are presented that enhance our confidence in the procedure. Contact: [email protected] Just the single simple harmonic oscillator: The motion of a mass about its equilibrium position while subject to a linear restoring force is well- studied. The model equation is: ma mx kx where x is measured relative to the equilibrium position. The solutions are known to be: xt( ) C cos( t ) D sin( t ) A cos( t ) Re Aeit where 2 k . m The mass hanging on a vertical spring reduces to the same equation when the displacement y is measured relative to the equilibrium position for the mass. ma my k y That is: The Hooke’s Law model for the force due to a spring Fx kx for displacements measured relative to equilibrium can be used for all springs. In this model k is the spring or Hooke’s constant. A simple example: An equal mass system of coupled oscillators x1 x2 k k m m The springs are linear Hooke’s law springs; there is no friction; x1 and x2 are measured from the equilibrium positions of the masses. (The springs exert forces on the masses at the points of contact.) As the spring are linear, the net force on each mass can be deduced by computing the force when (a) mass one is displaced while the mass two remains at its equilibrium position and adding (b) the force deduced when mass one is held fixed at equilibrium and mass two is displaced. (a) Mass one displaced and mass two fixed: Contribution to net force on mass one: F1a = - k x1 - x1 Contribution to net force on mass two: F2a = + x1 (b) Mass one fixed and mass two displaced: Contribution to net force on mass one: F1a = + x2 Contribution to net force on mass two: F2a = - k x2 - x2 Newton’s Law for the motions of the masses in the x direction yields: 12/13/2013 Physics Handout Series.Tank: Coupled Oscillator’s CO-2 ma11 mx() k x 12 x mx 1 kx 112 ( x x ) ma22 mx x 1() k x 2 mx 2 kx 221 ( x x ) The equations are symmetric as is the problem. The situation begs for a symmetric solution. Add the two equations together. mx()()12 x kx 12 x That strategy was so successful that as a next try, the two equations are to be differenced. mx()(2)()21 x k x 21 x k 1/2 The system has two modes. The (x1+x2) symmetric mode oscillates at S = ( /m) and the (x2–x1) k+2 1/2 anti-symmetric mode that oscillates at S = ( /m) . As a general rule, a symmetric problem is expected to have symmetric and anti-symmetric solutions. A debate can arise over which solution is symmetric and which is anti-symmetric. Convention is to define the lowest frequency mode to be a symmetric mode. Identifying the oscillation pattern: If the two masses move the same distance in the same direction, the length of the spring joining them does not change. The forces due to the spring witch constant drop out as they maintain their values at equilibrium. Effectively each mass works k 1/2 against the one spring with constant k that it contacts. The frequency of the mode is: S = ( /m) . If the two masses move with equal magnitude displacements that are oppositely directed, the coupling spring undergoes a length change double the magnitude of the length changes in the end springs. Each mass is subject to a force from one end spring (k) and from the coupling spring () that is being k+2 1/2 double stretched. The frequency of the mode is therefore: S = ( /m) . As the problems become more complex, a formal procedure is needed to cover the cases that our intuition fails to untangle. A multi-step plan of attack is to be introduced and applied to the symmetric oscillator. The Formal Method: Step One - Find the equations of Motion for the Masses: When mass one is displaced by x1 to the right while mass two is held fixed, forces of – k x1 and – x1 are applied to it by the two springs in contact with it. When mass one is held fixed at equilibrium while mass two is displaced by x2 to the 12/13/2013 Physics Handout Series.Tank: Coupled Oscillator’s CO-3 right, the center spring exerts a force of + x2 on mass one. The equation of motion for mass one is: mx112 k x x. Following the same procedure, the forces on mass two are identified when mass two is held fixed and mass one is displaced and then added to those that arise when mass one is fixed and mass two is displaced. mx21 x() k x 2 Equations of Motion: mx112 k x x and mx21 x() k x 2 An alternative approach uses the Lagrangian approach to generate the equations of motion. Execute Step-One using your method of choice. Step Two – Recast the equations of motion in matrix form and identify the Mass and Spring Constant Matrices x x 1 1 Equation of motion summary: = - or x = - x . x2 x2 The signs are chosen to resemble the single oscillator equation. is the mass matrix, and is the spring constant matrix. m 0 x1 k x1 m 0 k = - where = and = . 0 m x2 k x2 0 m k Step Three – Assume Harmonic Solutions in which the masses execute motion at the same frequency. A harmonic solution is one that varies sinusoidally in time. These solutions can be represented using complex exponentials in which case it must be understood that only the real part is physically meaningful. xt1() a it xt1() 2 a it e and e yielding xt2 () b xt2 () b ()km2 a0 eit 2 ()kmb0 ()km2 a0 For engineers: et 2 ()kmb0 As an alternative, the real sinusoids are to be used. xt1() a xt1() 2 a cos t and cost yielding xt2 () b xt2 () b 12/13/2013 Physics Handout Series.Tank: Coupled Oscillator’s CO-4 ()km2 a0 cos t 2 ()kmb0 The important point is that in either case the second time derivative is equivalent to multiplication by 2 it - . Neither e nor cos( t +) is identically zero, so the equation for the coefficients must have a non-trivial solution. A trivial solution is one in which both a and b are zero. Step Four – Solve the modified eigenvalue problem. Cramer’s rule specifies that a homogeneous set of n linear equations for n unknowns can have a non- trivial solution for a and b (a solution in which both a and b are not zero) only if the determinant of the matrix of coefficients is zero. ()km2 a0 2 ()kmb0 ()km2 0 or kmkm222 0 ()km2 This matrix equation is called the eigenvalue equation, and the values of that result in a zero determinant are the eigenvalues for the problem. As in so many cases, a little notation is helpful. The equation is rewritten in terms of the reference or scale value for the frequency squared: 2 k and the relative spring constant = / . Taking the 0 m k 2 across and taking the square root. km 2 2 22 Dividing by m : 0 1 . 2 22 2 2 2 Solving for : SAS1and1212() 000 k What rule was used to identify the symmetric (S) and the anti-symmetric (AS) mode? Step Five – Identify the patterns of motion To identify the symmetric mode, substitute S into the eigenvalue equation. First try 22k . S 0 m 12/13/2013 Physics Handout Series.Tank: Coupled Oscillator’s CO-5 ()km2 a0 a 0 ab 2 ()kmb0 b 0 As is the usual case, the two row equations redundantly state a single relation which for this case is: xt1() 1 it a = b. The pattern of motion is: Ae 0 . The prose description of this mode is that xt2 () 1 the two masses oscillate in phase (in the same direction at the same time) with equal amplitude. Note that these prose descriptions of the motion are crucial. You need to describe the pattern of the motion without reference to your choice of x1 and x2. Express your answers in coordinate independent form! Complex motions are not the order of the day so the real part of the complex exponential is taken.
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