Mathematical Appendix

The spring pendulum is characterized by the spring constant D, the mass m and the constant of attenuation G. (G 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 (wt).

y_{E} means the exciter's elongation compared with the mid-position; A_{E} is the amplitude of the exciter's oscillation, w 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
w_{0} = (D/m)^{1/2}
this problem is described by the following differential equation:

y''(t) = w_{0}^{2}
(A_{E} cos (wt) – y(t))
– G 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: G <
2 w
_{0} |

Case 1.1:
G < 2 w
_{0};
G ¹ 0 or
w ¹
w_{0} |

y(t) = A_{abs} sin (wt)
+ A_{el} cos (wt)
+ e^{–Gt/2}
[A_{1} sin (w_{1}t)
+ B_{1} cos (w_{1}t)]

w_{1} =
(w_{0}^{2}
– G^{2}/4)^{1/2}

A_{abs} = A_{E}
w_{0}^{2}
G w
/ [(w_{0}^{2}
– w^{2})^{2}
+ G^{2} w^{2}]

A_{el} = A_{E}
w_{0}^{2}
(w_{0}^{2}
– w^{2})
/ [(w_{0}^{2}
– w^{2})^{2}
+ G^{2} w^{2}]

A_{1} = – (A_{abs} w
+ (G/2) A_{el})
/ w_{1}

B_{1} = – A_{el}

Case 1.2:
G < 2 w
_{0};
G = 0 and
w = w_{0} |

y(t) = (A_{E} w t / 2)
sin (wt)

Case 2: G =
2 w
_{0} |

y(t) = A_{abs} sin (wt)
+ A_{el} cos (wt)
+ e^{–Gt/2}
(A_{1} t + B_{1})

A_{abs} = A_{E}
w_{0}^{2}
G w
/ (w_{0}^{2}
+ w^{2})^{2}

A_{el} = A_{E}
w_{0}^{2}
(w_{0}^{2}
– w^{2})
/ (w_{0}^{2}
+ w^{2})^{2}

A_{1} = – (A_{abs} w
+ (G/2) A_{el})

B_{1} = – A_{el}

Case 3:
G >
2 w
_{0} |

y(t) = A_{abs} sin (wt)
+ A_{el} cos (wt)
+ e^{–Gt/2}
[A_{1} sinh (w_{1}t)
+ B_{1} cosh (w_{1}t)]

w_{1} =
(G^{2}/4
– w_{0}^{2})^{1/2}

A_{abs} = A_{E}
w_{0}^{2}
G w
/ [(w_{0}^{2}
– w^{2})^{2}
+ G^{2} w^{2}]

A_{el} = A_{E}
w_{0}^{2}
(w_{0}^{2}
– w^{2})
/ [(w_{0}^{2}
– w^{2})^{2}
+ G^{2} w^{2}]

A_{1} = – (A_{abs} w
+ (G/2) A_{el})
/ w_{1}

B_{1} = – A_{el}

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

© Walter Fendt, September 9, 1998

Last modification: January 18, 2003