Mathematical Appendix

The spring pendulum is characterized by the spring constant D, the mass m and the constant of attenuation Γ. (Γ is a measure of the friction force assumed as proportional to the velocity.)

The top of the spring pendulum is moved to and fro according to the formula
y_{E} = A_{E} cos (ωt).

y_{E} means the exciter's elongation compared with the mid-position; A_{E} is the amplitude of the exciter's oscillation, ω means the corresponding angular frequency and t the time.

It is a question of finding the size of the resonator's elongation y (compared with its mid-position) at the time t. Using
ω_{0} = (D/m)^{1/2}
this problem is described by the following differential equation:

y''(t) = ω_{0}^{2}
(A_{E} cos (ωt) − y(t))
− Γ y'(t)Initial conditions: y(0) = 0; y'(0) = 0 |

If you want to solve this differential equation, you have to distinguish between several cases:

Case 1: Γ <
2 ω
_{0} |

Case 1.1:
Γ < 2 ω
_{0};
Γ ≠ 0 or ω ≠ ω_{0} |

y(t) = A_{abs} sin (ωt)
+ A_{el} cos (ωt)
+ e^{−Γt/2}
[A_{1} sin (ω_{1}t)
+ B_{1} cos (ω_{1}t)]

ω_{1} =
(ω_{0}^{2}
− Γ^{2}/4)^{1/2}

A_{abs} = A_{E}
ω_{0}^{2}
Γ ω
/ [(ω_{0}^{2}
− ω^{2})^{2}
+ Γ^{2} ω^{2}]

A_{el} = A_{E}
ω_{0}^{2}
(ω_{0}^{2}
− ω^{2})
/ [(ω_{0}^{2}
− ω^{2})^{2}
+ Γ^{2} ω^{2}]

A_{1} = − (A_{abs} ω
+ (Γ/2) A_{el})
/ ω_{1}

B_{1} = − A_{el}

Case 1.2:
Γ < 2 ω
_{0};
Γ = 0 and
ω = ω_{0} |

y(t) = (A_{E} ω t / 2)
sin (ωt)

Case 2: Γ =
2 ω
_{0} |

y(t) = A_{abs} sin (ωt)
+ A_{el} cos (ωt)
+ e^{−Γt/2}
(A_{1} t + B_{1})

A_{abs} = A_{E}
ω_{0}^{2}
Γ ω
/ (ω_{0}^{2}
+ ω^{2})^{2}

A_{el} = A_{E}
ω_{0}^{2}
(ω_{0}^{2}
− ω^{2})
/ (ω_{0}^{2}
+ ω^{2})^{2}

A_{1} = − (A_{abs} ω
+ (Γ/2) A_{el})

B_{1} = − A_{el}

Case 3:
Γ >
2 ω
_{0} |

y(t) = A_{abs} sin (ωt)
+ A_{el} cos (ωt)
+ e^{−Γt/2}
[A_{1} sinh (ω_{1}t)
+ B_{1} cosh (ω_{1}t)]

ω_{1} =
(Γ^{2}/4
− ω_{0}^{2})^{1/2}

A_{abs} = A_{E}
ω_{0}^{2}
Γ ω
/ [(ω_{0}^{2}
− ω^{2})^{2}
+ Γ^{2} ω^{2}]

A_{el} = A_{E}
ω_{0}^{2}
(ω_{0}^{2}
− ω^{2})
/ [(ω_{0}^{2}
− ω^{2})^{2}
+ Γ^{2} ω^{2}]

A_{1} = − (A_{abs} ω
+ (Γ/2) A_{el})
/ ω_{1}

B_{1} = − A_{el}

URL: http://www.walter-fendt.de/ph14e/resmath_e.htm

© Walter Fendt, September 9, 1998

Last modification: February 3, 2010