English translation
Vibration Analysis of a Two-Mass System
This document is an English translation of the original German-language application note “Schwingungsberechnung eines Zwei-Massen-Systems” published by AEG-Telefunken Datenverarbeitung.
1. Problem Statement
For the two-mass system shown in Figure 1, the deflections y₁(t) and y₂(t) are to be determined under 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 two masses m₁ and m₂ connected by spring c₁ and damper d₁ in parallel, with mass m₂ resting on spring c₂ which is grounded. A step disturbance y₃(t) is applied at the base.
- m₁, m₂ : masses
- c₁, c₂ : spring constants
- d₁ : damping constant
Figure 2 — Spring and Damper Characteristic Curves
Three cases (a, b, c) are considered for the damper characteristic (damping force vs. velocity ẏ₁ − ẏ₂) and the spring characteristic (spring force vs. deflection y₂ − y₃):
- Fall a: linear damper, linear spring
- Fall b: piecewise-linear (bilinear) damper, linear spring
- Fall c: piecewise-linear damper, piecewise-linear spring (with dead-band)
2. Equations of Motion
The equations of motion for the system are:
(1): d²y₁/dt² + (d₁/m₁)(dy₁/dt − dy₂/dt) + (c₁/m₁)(y₁ − y₂) = 0
(2): d²y₂/dt² + (d₁/m₂)(dy₂/dt − dy₁/dt) + (c₁/m₂)(y₂ − y₁) + (c₂/m₂)(y₂ − y₃) = 0
3. Constants
| Parameter | Value | Cases |
|---|---|---|
| m₁ | 20 kps²m⁻¹ | a, b, c |
| m₂ | 2 kps²m⁻¹ | a, b, c |
| c₁ | 500 kpm⁻¹ | a, b, c |
| d₁ | 80 kpsm⁻¹ | Case a |
| d₁ | 80 kpsm⁻¹ for ẏ₁ − ẏ₂ > 0; 8 kpsm⁻¹ for ẏ₁ − ẏ₂ < 0 | Cases b, c |
| c₂ | 2000 kpm⁻¹ | Cases a, b |
| c₂(y₂ − y₃) | 20(y₂ − y₃) kp for y₂ − y₃ > −0.37 cm; −20·0.37 kp for y₂ − y₃ < −0.37 cm | Case c |
Step disturbance: y₃(t) = 0 cm for t < 0; 6.5 cm for t ≥ 0
Initial conditions: y₁(0) = y₂(0) = 0
Estimated maximum values: y₁max = y₂max = y₃max = yₘ = 10 cm
4. Normalization
The voltages uₖ of the analog computer are conventionally normalized to the maximum computing voltage E:
u’ₖ = uₖ/E, so that −1 ≤ u’ₖ ≤ +1
The machine time t* is replaced by a dimensionless variable τ according to:
τ = k·t*, k [s⁻¹] …(3)
The problem quantities are handled correspondingly: the dependent variables and their derivatives are normalized to the analog voltage range of −1 to +1. The independent variable, here time t, is normalized according to the machine time scaling.
5. Amplitude and Time Scaling
The amplitude scale factors Yₖ = uₖ/yₖ (normalized variables) are determined from the estimated maximum values. For each variable the maximum signal is normalized to yₘ = 10 cm.
The time-scale factor λ allows the differential-equation coefficients to be adjusted. For the equations of motion the coefficients split into:
- Constant parts: d₀/(m₁λ), d₀/(m₂λ), c₀/(m₂λ²) — set by potentiometers
- Sign-dependent parts: d₁/d₀ and c₂/c₀ — realized by nonlinear circuits
The nonlinear damping and spring forces d₁(Y’₁ − Y’₂) and c₂/c₀(Y₂ − Y₃) are modeled by the nonlinear circuits shown in Figures 4 and 5.
Setting d₀ = 80 kpsm⁻¹ and c₀ = 2000 kpm⁻¹ (the linear-case values), the coefficients become (equation 8):
- d₀/(m₁λ) = 4/λ
- d₀/(m₂λ) = 40/λ
- c₁/(m₁λ²) = 25/λ²
- c₁/(m₂λ²) = 250/λ²
- c₀/(m₂λ²) = 1000/λ²
Choosing c = 1 and λ = 10 s⁻¹ (10× time expansion), the normalized scaled equations become:
(9): Y”₁ + 0.4·(d₁/d₀)(Y’₁ − Y’₂) + 0.25(Y₁ − Y₂) = 0
(10): Y”₂ + 4·(d₁/d₀)(Y’₂ − Y’₁) + 2.5(Y₂ − Y₁) + 10·(c₂/c₀)(Y₂ − Y₃) = 0
The nonlinear factors (equations 11, 12):
- For Case a: d₁/d₀ = 1
- For Cases b, c: d₁/d₀ = Y’₁ − Y’₂ for Y’₁ − Y’₂ > 0; 0.1(Y’₁ − Y’₂) for Y’₁ − Y’₂ < 0
- For Cases a, b: c₂/c₀ = 1
- For Case c: c₂/c₀·(Y₂ − Y₃) = Y₂ − Y₃ for Y₂ − Y₃ > −0.037; −0.037 for Y₂ − Y₃ < −0.037
The time-scale factor is ε = 1/k = 10/k₀. The simulation runs at 10× time expansion for integration factor k₀ = 1 s⁻¹, at real time for k₀ = 10 s⁻¹, and at 10× time compression for k₀ = 100 s⁻¹.
6. Analog Computer Circuit
Figure 3 — Computing Circuit for the System of Figure 1
The computing circuit implements both integrators for Y₁ and Y₂ with the associated summing amplifiers, and includes the nonlinear subcircuits for Cases b and c.
Figure 4 — Subcircuit 1 for Cases b, c
Implements the bilinear damper characteristic using diodes to switch between the two damping slopes.
Figure 5 — Subcircuit 2 for Case c
Implements the dead-band (bilinear) spring characteristic for the c₂ nonlinearity.
In the subcircuit “Pause” is the input to amplifier 3 when the computer is in Hold mode. The input and output signals are noted in the circuit. From the damper equation (11) and formula (14) it can be seen that when Y’₁ − Y’₂ < 0 the slope ratio is 0.1.
To create the circuit of Figure 3 with a slope ratio of 0.1, the slope is adjusted via potentiometers set to approximately 0.45 (potentiometer 5 in Table 1). The Begrenzung (limiting/clamping) function uses a ratio of approximately B/(B+1) ≈ 0.035.
7. Potentiometer List
Table 1 — Potentiometer List
| Pot. No. | Coefficient | Normalized Coefficient | Numerical Values (λ = 10 s⁻¹, yₘ = 10 cm) |
|---|---|---|---|
| 1 | c₁/m₁ | c₁/(m₁λ²) | 0.25 |
| 1 | c₁/m₂ | c₁/(m₂λ²) | 0.25·10 |
| 2 | d₁/m₁ | d₀/(m₁λ) | 0.4 |
| 2 | d₁/m₂ | d₀/(m₂λ) | 0.4·10 |
| 3 | y₃ | Y₃·y₃/yₘ | 0.65 |
| 4 | Diode offset compensation | ε | ≈ 0.05 |
| 5 | Characteristic slope | ≈ d₀/d₁−1 / 20 | ≈ 0.45 |
| 6 | Limiter | ≈ B/(B+1) | ≈ 0.035 |
8. Results
The time histories of deflections Y₁(t*) and Y₂(t*) for Cases a, b, c are recorded in Figures 6 through 12.
In Case b, amplifier 5 overdrives. The step from Y₃ = 0 to 0.65 is too large here. The amplitude normalization at yₘ = 10 cm for Cases a, b must be changed so that the amplifier no longer overloads.
- Figure 6 — Disturbance function (step input Y₃)
- Figure 7 — Deflection of mass m₂, Case a (damped oscillation settling to steady state)
- Figure 8 — Deflection of mass m₁, Case a (larger oscillation amplitude, slower settling)
- Figure 9 — Deflection of mass m₂, Case b (faster initial transient with bilinear damping)
- Figure 10 — Deflection of mass m₁, Case b (sustained oscillation due to lower negative-velocity damping)
- Figure 11 — Deflection of mass m₂, Case c (combined nonlinear effects)
- Figure 12 — Deflection of mass m₁, Case c (sustained oscillation with dead-band spring)
References
[1] Szabo, Einführung in die Technische Mechanik, Springer-Verlag 1961
[2] Giloi W., Herschel R., Rechenanleitung für Analogrechner, TELEFUNKEN-Fachbuch