Analog Computers

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Anwendungsbeispiele für Analogrechner — Beispiel 3: Zwei-Massen-System

Complete English translation of the original German-language document (7 pages).

Application Examples for Analog Computers

Example 3                                                                                15 October 1963

TWO-MASS SYSTEM

1. Problem Statement

For the two-mass system shown in Figure 1, the displacements y₁(t) and y₂(t) are to be determined in response to a step-function disturbance y₃(t). The spring and damper characteristic curves are shown in Figure 2.

Figure 1 — Two-Mass System

The system consists of mass m₁ connected above mass m₂ by a spring c₁ in parallel with damper d₁. Mass m₂ is connected to the base excitation point via spring c₂. The displacements y₁, y₂, and y₃ are measured from their respective reference positions.

  • m₁, m₂ : masses
  • c₁, c₂ : spring constants
  • d₁ : damping constant

Figure 2 — Spring and Damper Characteristic Curves

The figure shows three cases (Case a, Case b, Case c) for each of the two nonlinear elements:

  • d₁ (damper): force as a function of (ẏ₁ − ẏ₂)

    • Case a: linear (symmetric, through origin)
    • Case b: linear with steeper positive slope (asymmetric — or bilinear)
    • Case c: zero for negative argument (one-sided)

  • c₂ (spring): force as a function of (y₂ − y₃)

    • Case a: linear (symmetric, through origin)
    • Case b: linear with steeper slope
    • Case c: zero for negative argument (one-sided, dead-band below zero)

2. Equations of Motion

The equations of motion for the two masses are:

$$\frac{d^2 y_1}{dt^2} + \frac{d_1}{m_1}!\left(\frac{dy_1}{dt} - \frac{dy_2}{dt}\right) + \frac{c_1}{m_1}(y_1 - y_2) = 0 \tag{1}$$

$$\frac{d^2 y_2}{dt^2} + \frac{d_1}{m_2}!\left(\frac{dy_2}{dt} - \frac{dy_1}{dt}\right) + \frac{c_1}{m_2}(y_2 - y_1) + \frac{c_2}{m_2}(y_2 - y_3) = 0 \tag{2}$$

3. Constants

  • m₁ = 20 Kp s² m⁻¹

  • c₁ = 1000 Kp m⁻¹

  • d₁ = 80 Kn s m⁻¹    for ẏ₁ − ẏ₂ > 0     Case b

  •      = 8 Kp cm⁻¹       for ẏ₁ − ẏ₂ < 0     Case c

  • m₂ = 2 Kp s² m⁻¹

  • c₂ = 4000 Kp m⁻¹   for y₂ − y₃ > −0.37 cm    Case c

  •      = 0                    for y₂ − y₃ < −0.37 cm

  • y₃(t) = 6.5 cm   for t ≥ 0

  •          = 0          for t < 0

  • y₁(0) = 0

  • y₂(0) = 0

Estimated maximum values: y₁max = y₂max = y₃max = yₘ = 10 cm

4. Normalization

Using the normalized amplitudes Yᵢ = yᵢ / yₘ, the time normalization τ = λt, and the abbreviations dY/dτ = Ẏ, d²Y/dτ² = Ÿ, the machine equations become:

$$\ddot{Y}_1 + \frac{d_1}{m_1 \lambda}(\dot{Y}_1 - \dot{Y}_2) + \frac{c_1}{m_1 \lambda^2}(Y_1 - Y_2) = 0 \tag{3}$$

$$\ddot{Y}_2 + \frac{d_1}{m_2 \lambda}(\dot{Y}_2 - \dot{Y}_1) + \frac{c_1}{m_2 \lambda^2}(Y_2 - Y_1) + \frac{c_2}{m_2 \lambda^2}(Y_2 - Y_3) = 0 \tag{4}$$

In order to make the coefficients settable on the machine, λ = 10 s⁻¹ is chosen. The process runs with integration constant k₀ = 1 s⁻¹ at a 10-fold time expansion; for k₀ = 10 s⁻¹ it runs in real time.

After substituting the coefficient values, the following equations result:

$$\ddot{Y}_1 = -0.4,(\dot{Y}_1 - \dot{Y}_2) - 0.25,(Y_1 - Y_2) \tag{5}$$

$$-\ddot{Y}_2 = -4.0,(\dot{Y}_1 - \dot{Y}_2) - 2.5,(Y_1 - Y_2) + 10,(Y_2 - Y_3) \tag{6}$$

5. Computing Circuit

Figure 3 — Computing Circuit for the System of Figure 1

The computing circuit implements equations (5) and (6) using integrators, summers, and coefficient potentiometers. Signal annotations in the circuit diagram:

  • −Ẏ₁, Y₁ (with output gain 1)
  • Y₂ (input gains 10, 10, 10)
  • |Ẏ₁ − Ẏ₂| (formed via an absolute-value or nonlinear subcircuit, coefficient 0.4)
  • −Y₂ (with output gain 1)
  • Y₃ (disturbance input, fed via potentiometer or summer)
  • Coefficient potentiometers set to 0.25 and 0.4

The dashed box labeled “1” in the circuit (Figure 3) is replaced by Circuit 1 (Figure 4) for Cases b and c. The dashed box labeled “2” is replaced by Circuit 2 (Figure 5) for Case c.

Figure 4 — Circuit 1 for Cases b and c

This circuit forms the signal (Ẏ₁ − Ẏ₂) with nonlinear (one-sided or bilinear) damping. An absolute-value or diode-based arrangement is used, with a threshold set at −1; the output represents the damping force with the appropriate asymmetric characteristic.

Figure 5 — Circuit 2 for Case c

This circuit implements the one-sided spring c₂. Inputs are −Y₂ and +1 along with Y₃. A diode-based limiter with threshold +1 produces the one-sided spring force (c₂ active only for y₂ − y₃ > −0.37 cm, zero below that threshold).

For Cases b and c, the dashed-boxed section labeled “Part 1” in the computing circuit (Figure 3) is to be replaced by Circuit 1 (Figure 4).

For Case c, the dashed-boxed section labeled “Part 2” is additionally to be replaced by Circuit 2 (Figure 5).

6. Results

The time histories of the displacements Y₁(τ) and Y₂(τ) are recorded for Cases a, b, and c in Figures 6 through 12.

Figure 6 — Disturbance Function

Y₃ steps from 0 to +1 at τ = 0 and remains constant thereafter (unit step). The horizontal axis is τ with maximum value 1; the vertical axis is Y₃.

Figure 7 — Displacement of Mass m₂, Case a

Y₂ rises rapidly after the step, overshoots slightly, then settles to a steady-state value close to +1. The response is lightly damped and monotonically approaches the final value after the initial overshoot. Horizontal axis: τ (0 to 1); vertical axis: Y₂.

Figure 8 — Displacement of Mass m₁, Case a

Y₁ rises, overshoots to a peak near +1, then undergoes a damped oscillation with two visible peaks before settling. The response exhibits more oscillation than Y₂ in Case a. Horizontal axis: τ (0 to 1); vertical axis: Y₁.

Figure 9 — Displacement of Mass m₂, Case b

With asymmetric damping (Case b), Y₂ shows a sharper initial rise with a distinct first overshoot followed by a rapid series of small oscillations (more highly damped for one direction of motion) before settling. The response is more oscillatory in the early phase compared to Case a. Horizontal axis: τ (0 to 1); vertical axis: Y₂.

Figure 10 — Displacement of Mass m₁, Case b

Y₁ in Case b exhibits sustained oscillation. The curve shows a smoothly varying, nearly sinusoidal shape over the time window, indicating that with asymmetric damping the first mass oscillates with relatively little net attenuation over the displayed interval. Horizontal axis: τ (0 to 1); vertical axis: Y₁.

Figure 11 — Displacement of Mass m₂, Case c

With both the one-sided damper and one-sided spring (Case c), Y₂ shows an initial rise similar to Case a but with a somewhat irregular intermediate region before settling near +1. The one-sided spring produces a slight plateau or inflection during the approach to steady state. Horizontal axis: τ (0 to 1); vertical axis: Y₂.

Figure 12 — Displacement of Mass m₁, Case c

Y₁ in Case c undergoes a sustained, nearly undamped sinusoidal oscillation around a mean value near +1. The amplitude remains essentially constant over the time window, in contrast to Case a (where oscillations decay) and Case b (where asymmetric damping modifies the waveform). Horizontal axis: τ (0 to 1); vertical axis: Y₁.

References

[1] Szabo, Einführung in die Technische Mechanik [Introduction to Engineering Mechanics], Springer-Verlag, 196–

[2] Giloi W., Herschel R., Rechenanleitung für Analogrechner [Computing Guide for Analog Computers], TELEFUNKEN-Fachbuch

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