Simulation von Schwingungssystemen mit Analogrechnern

Complete English translation of the original German-language document (22 pages).

[page 1: title page]

AEG TELEFUNKEN — DATA PROCESSING

H. Kramer

Simulation of Vibration Systems with Analog Computers

[page 2: front matter]

Contribution from the course: Theoretical and Practical Introduction to the Treatment of Control and Vibration Problems with Industrial and Home-Built Analog Computers. VDI-Bildungswerk, Düsseldorf 1969

Senior Government Councillor Dr. rer. nat. H. Kramer is a lecturer at the State Engineering School Konstanz.

		Page
I.	Introductory Treatment of Vibration Problems	3
1.	Introduction	3
2.	Qualitative Simulation of a Linear Second-Order Differential Equation — Single-Loop Spring-Damper-Mass Oscillator	3
3.	Quantitative Simulation of the Vibration Differential Equation	5
3.1.	Normalization of the Independent Variables — Time Normalization	5
3.2.	Normalization of the Dependent Variables — Amplitude Normalization	6
4.	Estimation of Maximum Values	7
4.1.	Excitation by Initial Values	7
4.2.	Excitation by a Step Function	8
4.3.	Excitation by a Periodic Disturbance	9
5.	Generation of Undamped Oscillations	10
II.	Nonlinear and Continuous Systems with Solutions	12
1.	Nonlinear Single-Loop Oscillators	12
1.1.	Mathematical Pendulum for Small and Large Deflections	12
1.2.	Oscillator with Nonlinear Damping	15
1.3.	Bistable Oscillator	16
2.	Multi-Loop Oscillators	17
2.1.	Two-Mass System with Nonlinear Damper	17
3.	Oscillations of Continuous Systems	19
3.1.	Oscillations of a Jumping Beam	19
Literature	21

I. INTRODUCTORY TREATMENT OF VIBRATION PROBLEMS

1. Introduction

The purpose of the following remarks is to introduce the reader to the treatment of vibration problems with analog computers. This is accomplished by means of selected examples from the field of mechanical vibrations. In principle, the methods described may also be applied to electrical oscillation problems. These may however be treated directly by electrical means in a simpler manner. The representation is carried out in stages, progressing from a simple to a more difficult level.

2. Qualitative Simulation of a Linear Second-Order Differential Equation — Single-Loop Spring-Damper-Mass Oscillator

The equation of motion of the spring-damper-mass oscillator shown in Fig. 1 is:

m·ẍ + k·ẋ + c·x = F(t)

or, dividing by m:

ẍ + δ·ẋ + ω₀²·x = f(t)

where:

δ = k/m (damping coefficient)
ω₀² = c/m (square of the undamped natural angular frequency)
f(t) = F(t)/m (specific forcing function)

Fig. 1 shows the spring-damper-mass oscillator.

To solve the differential equation on the analog computer, one first resolves it for the highest derivative:

ẍ = −δ·ẋ − ω₀²·x + f(t)

This equation states: if the quantities ẋ and x are available as voltages at certain points of the computer, then ẍ can be formed as an inverting weighted sum. By integration of ẍ one obtains −ẋ (sign reversal by the integrator), and by a further integration one obtains x. The sign reversal at the output of the second integrator can be compensated by an additional inverter, so that finally ẋ and x are available for forming ẍ. In this way a closed loop is created.

The computing elements of the analog computer are summarized in Fig. 2 with their circuit symbols, symbols in schematic notation, and computing operations.

Fig. 2. Computing elements of the analog computer

Designation	Circuit	Symbol	Computing Operation
Potentiometer	(single input)	E → α → A	A = α·E, 0 ≤ α ≤ 1
Potentiometer	(two inputs)	E₁, E₂ → α → A	A = E₁ + α·(E₂ − E₁), 0 ≤ α ≤ 1
Open amplifier	Uₑ → [R] → Uₐ	E → (triangle) → A	A = −V·E, V ≈ 10⁶
Inverter	Uₑ → [R][R] → Uₐ	E → (triangle) → A	A = −E
Summer	Uₑ₁[R₁], Uₑ₂[R₂], R/Rᵢ: 1; 10	E₁, E₂ → (triangle) → A, Cᵢ: 1; 10	A = −∑ Cᵢ·Eᵢ
Integrator	Uₑ₀[R], Uₑ₁[R₁], Uₑ₂[R₂]; P: Pause, R: Compute, H: Hold	E₀, E₁, E₂ → (triangle) → A, Cᵢ: 1; 10	P: A = −E₀; R: A(t) = −∫∑Cᵢ·Eᵢ dτ − E₀; H: A = A(τ)
Function generator	Uₑ → [f] → Uₐ	E → f(E) → A	A = f(E)
Multiplier	Uₑ₁[R][Z], Uₑ₂, Uₑ₃ → Uₐ	E₁, E₂, E₃ → M → A	A = E₁·E₂ − E₃
Comparator		E₂, E₁ → A, +S₂, +S₁	A = E₂ (S₁ + S₂ ≥ 0); A = E₁ (S₁ + S₂ ≤ 0)

Fig. 3. Qualitative computing circuit of the damped oscillator

The qualitative computing circuit (Fig. 3) for the differential equation of the damped oscillator:

ẍ = −δ·ẋ − ω₀²·x

with initial values: ẋ(0), x(0)

is set up as follows: The two integrators each have a computing coefficient of 1. The initial values ẋ(0) and x(0) are set by means of potentiometers. The potentiometers for δ and ω₀² are connected to the summing input of the first integrator.

3. Quantitative Simulation of the Vibration Differential Equation

The values given from the example in Fig. 1 are:

Spring rate: c = 5 N/cm
Damping: k = 0.5 N·s/cm
Mass: m = 1 kg

This gives the natural parameters:

ω₀² = c/m = 5/1 = 5 s⁻², ω₀ = √5 ≈ 2.24 s⁻¹

δ = k/m = 0.5/1 = 0.5 s⁻¹

The natural frequency: f₀ = ω₀/(2π) ≈ 0.357 Hz

The period: T₀ = 1/f₀ ≈ 2.8 s

From the characteristic equation:

s² + δ·s + ω₀² = 0

the eigenvalues are:

s₁,₂ = −δ/2 ± √((δ/2)² − ω₀²) = −0.25 ± j·√(5 − 0.0625) = −0.25 ± j·2.224

The system is thus a weakly damped oscillatory system.

3.1. Normalization of the Independent Variables — Time Normalization

An analog computer whose computing elements have time constants of τ = 1 s and τ = 0.1 s is used. If the problem time t and machine time τ are related by:

t = β·τ (β = time scale factor)

then the problem period T₀ becomes machine time T₀/β. For β = 1 the machine time equals the problem time; T_machine = 2.8 s. This is a comfortable value that can still be well observed on a recording instrument.

For β < 1 the computation runs faster than the problem (“time compression”); for β > 1 it runs slower (“time expansion”). In this case β = 1 is chosen.

The normalized differential equation is:

d²x/dτ² + β·δ·(dx/dτ) + β²·ω₀²·x = β²·f(β·τ)

For β = 1:

d²x/dτ² + 0.5·(dx/dτ) + 5·x = f(τ) (1)

The coefficients are:

Damping coefficient: β·δ = 0.5
Frequency coefficient: β²·ω₀² = 5

Since the coefficient 5 is greater than 1 and also cannot be set on the potentiometer (range 0 to 1), one uses a computing coefficient c = 10 for the corresponding integrator, so that the overall coefficient becomes:

β²·ω₀²/c = 5/10 = 0.5

This value can be set on a potentiometer.

3.2. Normalization of the Dependent Variables — Amplitude Normalization

In order to exploit the full range of the machine (±100 V or ±10 V depending on the machine type), all dependent variables must be referred to their maximum values, i.e., normalized.

One sets:

x̄ = x/x_max, ẋ̄ = ẋ/ẋ_max

where x_max and ẋ_max are the estimated maximum values of x and ẋ. Then Eq. (1) becomes:

x̄_max · d²x̄/dτ² + β·δ·ẋ_max·(dx̄/dτ) + β²·ω₀²·x_max·x̄ = β²·f(τ)

Dividing by x_max:

d²x̄/dτ² + β·δ·(ẋ_max/x_max)·(dx̄/dτ) + β²·ω₀²·x̄ = (β²/x_max)·f(τ)

The figure on p. 4 (Fig. 2 referenced there) shows the table of computing elements; Fig. 3 on the same page shows the qualitative computing circuit of the damped oscillator.

4. Estimation of Maximum Values

Before scaling the problem, the maximum values of the dependent variables must be estimated. The accuracy of this estimation determines the quality of the computer solution. If the estimates are too large, only a small portion of the machine’s range is utilized; if they are too small, overflow (exceeding the machine’s permissible voltage range) results, leading to erroneous solutions.

4.1. Excitation by Initial Values

Fig. 4. Time functions

If the oscillator is excited only by initial values x(0) and ẋ(0), the maximum values can be estimated relatively easily. The solution of the homogeneous differential equation

ẍ + δ·ẋ + ω₀²·x = 0

with initial conditions x(0) and ẋ(0) is, for the underdamped case (ω₀ > δ/2):

x(t) = e^(−δt/2) · [x(0)·cos(ω_d·t) + ((ẋ(0) + δ/2·x(0))/ω_d)·sin(ω_d·t)]

where ω_d = √(ω₀² − (δ/2)²) is the damped natural angular frequency.

The maximum value of x occurs near the starting point for t ≈ 0, provided that the initial displacement is larger than the initial velocity component. Thus as a rough estimate:

x_max ≈ x(0)

For a more refined estimate, the energy method can be used. The total energy at t = 0 is:

E(0) = (1/2)·m·ẋ(0)² + (1/2)·c·x(0)²

The maximum displacement x_max occurs when all kinetic energy has been converted to potential energy (assuming for a moment that no energy is lost):

(1/2)·c·x_max² = E(0)

x_max = √(ẋ(0)²/ω₀² + x(0)²)

This formula provides a good upper bound. For the maximum velocity, one proceeds analogously:

ẋ_max = √(ẋ(0)² + ω₀²·x(0)²)

Fig. 4. Time functions (referenced figure: typical damped oscillation curve showing x(t) vs. t)

Fig. 5. Phase diagram for the initial-value excitation

The phase-plane trajectory (Fig. 5) is a spiral converging toward the origin.

The quantitative computing circuit is constructed as follows from the qualitative circuit in Fig. 3. The normalized equation after amplitude normalization reads:

d²x̄/dτ² = −β·δ·(ẋ_max/x_max)·(dx̄/dτ) − β²·ω₀²·x̄

with initial values:

x̄(0) = x(0)/x_max (set at integrator 2 initial-value potentiometer)
(dx̄/dτ)|₀ = ẋ(0)/ẋ_max (set at integrator 1 initial-value potentiometer)

Fig. 6. Quantitative computing circuit of the damped oscillator (referenced figure: complete circuit with potentiometer settings labeled)

4.2. Excitation by a Step Function

When the oscillator is excited by a step function F(t) = F₀ · σ(t) (σ = Heaviside step function), the particular solution is:

x_p = F₀/c = f₀/ω₀²

The total solution is:

x(t) = x_p · [1 − e^(−δt/2) · (cos(ω_d·t) + (δ/(2·ω_d))·sin(ω_d·t))]

for zero initial conditions.

The maximum value of the step response x_max is the value of the first overshoot. For a lightly damped system it is approximately twice the static displacement:

x_max ≈ 2·x_p = 2·F₀/c

A more precise expression is:

x_max = x_p · (1 + e^(−π·δ/(2·ω_d))) = x_p · (1 + e^(−π·δ/ω_d · π/2))

The Rechenschaltung (computing circuit) for the step-function case is shown in Fig. 7. The step function is generated by a potentiometer that is switched from 0 to a reference voltage at t = 0. In the scaled circuit, the step amplitude must be set such that it corresponds to the normalized forcing term.

Fig. 7. Computing circuit for step-function excitation

4.3. Excitation by a Periodic Disturbance

For a periodic forcing function f(t) = f̂ · cos(Ω·t), the steady-state (particular) solution amplitude is:

x̂_p = f̂ / √((ω₀² − Ω²)² + δ²·Ω²)

The maximum amplitude occurs at resonance (Ω = ω_res):

ω_res = √(ω₀² − δ²/2) (for damped systems)

x̂_max = f̂ / (δ·ω_d) ≈ f̂ / (δ·ω₀) (for light damping)

This is the well-known resonance amplitude. In the vicinity of resonance the magnification factor (Vergrößerungsfunktion) V is:

V = x̂_p / (f̂/ω₀²) = 1 / √((1 − η²)² + (2·D·η)²)

where η = Ω/ω₀ is the frequency ratio and D = δ/(2·ω₀) is the Lehr damping measure (dimensionless damping ratio).

Fig. 8a. Magnification function (Vergrößerungsfunktion V as a function of η for various values of D)

Fig. 8b. Phase diagram for the periodic excitation (phase portrait showing closed orbits in the x–ẋ plane)

The step-function excitation circuit is also suitable for the periodic case if the constant forcing voltage is replaced by a sinusoidal voltage generated by an oscillator or by a second analog-computer loop producing harmonic oscillation.

Fig. 9. Computing circuit for step-function excitation (referenced diagram showing the circuit with the step input)

The time functions and the limiting case:

For step excitation Ω → 0:

x_max = x_p · (1 + e^(−π·D/√(1−D²)))

Fig. 10. Computing circuit for the step-function excitation with sine input (circuit diagram)

Fig. 10a. Excitation oscillation by a sine-function; time record from Fig. 9 (time-domain waveform)

5. Generation of Undamped Oscillations

For simulation purposes it is frequently necessary to generate a sinusoidal reference signal. An analog computer loop can produce an undamped harmonic oscillation by solving the differential equation:

ẍ + ω₀²·x = 0

with initial conditions x(0) = 0, ẋ(0) = ω₀·x̂ (for amplitude x̂).

The computing circuit consists of two integrators in a feedback loop without damping. In practice, a small parasitic damping always exists in real integrators due to finite open-loop gain and leakage of the capacitors. This causes the amplitude to decay slowly over time.

To generate a sustained oscillation of constant amplitude, additional measures are required:

Overdriven Amplitude Control (Begrenzung): A nonlinear element (limiter or comparator with switching) stabilizes the amplitude. This is the simplest approach but introduces harmonic distortion.
Automatic Gain Control: An amplitude-detecting circuit compares the actual amplitude with a set value and adjusts a multiplicative coefficient in the loop accordingly (using a multiplier element).

Fig. 12 (referenced). Computing circuit for sine-function excitation with cosine/sine-function generator

The Sinus-Cosinus-Oszillator (sine-cosine oscillator) circuit uses the identity:

d/dt(sin Ω·t) = Ω·cos Ω·t d/dt(cos Ω·t) = −Ω·sin Ω·t

so that with two integrators (each with computing coefficient Ω), the output of the first integrator is −sin Ω·t (or +sin Ω·t after inversion) and the output of the second is cos Ω·t. This circuit (referenced in Fig. 12) requires setting the frequency by a single potentiometer α = Ω/ω_ref, making it convenient for swept-frequency resonance measurements.

Fig. 17 (referenced). Sine-cosine oscillator circuit using two integrators

The actual amplitude of the generated oscillation depends on the initial conditions set on the integrators. Because the system is conservative (no dissipation), the amplitude remains constant in the ideal case.

All the stability behavior and the modulation of frequency-response depends on the regulator within the sinusoidal generator. At the frequencies of practical interest (Ω from approximately 0.1 rad/s to 100 rad/s) with machine time constants of τ = 1 s and τ = 0.1 s, the potentiometer settings span the range 0.01 to 1, which is the comfortable operating range.

II. NONLINEAR AND CONTINUOUS SYSTEMS WITH SOLUTIONS

1. Nonlinear Single-Loop Oscillators

1.1. Mathematical Pendulum for Small and Large Deflections

The equation of motion of the mathematical pendulum (mass m on massless rod of length l, in a gravitational field g) is:

m·l·φ̈ + m·g·sin φ = 0

or:

φ̈ + (g/l)·sin φ = 0 (1)

where φ is the angle of deflection from the vertical equilibrium position.

For small deflections sin φ ≈ φ, giving the linearized equation:

φ̈ + ω₀²·φ = 0, ω₀² = g/l

The natural angular frequency is ω₀ = √(g/l), independent of amplitude. The period is T₀ = 2π/ω₀ = 2π·√(l/g).

For large deflections, sin φ must be used. The nonlinear function sin φ is realized on the analog computer by means of the function generator. The natural period then depends on the amplitude (isochronism is no longer valid).

Fig. 1 (in this section). Mathematical pendulum (diagram)

The equation is resolved for the highest derivative:

φ̈ = −(g/l)·sin φ

The computing circuit requires:

Two integrators to go from φ̈ to φ̇ to φ
A function generator to compute sin φ from φ
A potentiometer set to g/l to weight the result

For the scaled equation, let:

φ_max = amplitude of oscillation (to be estimated or chosen) φ̄ = φ/φ_max

The time normalization: with β as time scale factor (β = 1 chosen for slow oscillations), the scaled equation is:

d²φ̄/dτ² = −β²·(g/l)·(1/φ_max)·sin(φ_max·φ̄)

The function generator is set up to generate sin(φ_max · φ̄) as a function of the normalized variable φ̄ ∈ [−1, +1].

For a concrete example with l = 1 m:

ω₀² = g/l = 9.81/1 ≈ 9.81 s⁻² ω₀ ≈ 3.13 s⁻¹ T₀ = 2π/ω₀ ≈ 2.0 s

Taking β = 1 (machine time = problem time):

β²·ω₀² = 9.81

Since this exceeds 1, a computing coefficient c = 10 is used, giving the potentiometer setting:

α = β²·ω₀²/c = 9.81/10 = 0.981

The initial conditions are set to represent the chosen starting angle φ(0) = φ_max (maximum deflection, zero velocity).

The phase-plane representation (φ̄ vs. dφ̄/dτ) shows closed trajectories for oscillatory motion. For small amplitudes these are nearly ellipses; for large amplitudes they deviate significantly, becoming increasingly elongated. At φ = π the pendulum reaches the inverted vertical — the unstable equilibrium — represented as a saddle point in the phase plane. For φ > π (or φ < −π) the motion becomes continuous rotation.

Fig. 2 (in this section). Analog computing circuit for the mathematical pendulum

The circuit includes:

Integrator 1: produces −φ̇ from φ̈ (initial value φ̇(0) = 0 for release from rest)
Integrator 2: produces φ from −φ̇ (initial value φ(0)/φ_max set on initial-value potentiometer)
Function generator: computes sin φ from φ
Potentiometer α = ω₀²/c·β²: weights the feedback

The period T₀ is measured on the recorded time trace and can be compared with the exact theoretical value from the elliptic integral:

T(φ_max) = 4·√(l/g) · K(sin(φ_max/2))

where K is the complete elliptic integral of the first kind. For φ_max = 60°, the period is approximately 4.4% longer than the linear period; for φ_max = 90°, it is approximately 18.0% longer; for φ_max = 170°, it is many times longer.

The computer solution makes this amplitude-dependence directly visible without requiring evaluation of the elliptic integral.

1.2. Oscillator with Nonlinear Damping

An oscillator with nonlinear damping is described by the differential equation:

ẍ + f(ẋ) + ω₀²·x = 0

where f(ẋ) is a nonlinear function of the velocity. Important special cases are:

a) Quadratic (velocity-squared) damping (turbulent friction):

f(ẋ) = k·ẋ·|ẋ|

This damping is characteristic of bodies moving through fluids at higher Reynolds numbers (turbulent flow). The magnitude |ẋ| ensures that the damping force always opposes the direction of motion.

b) Coulomb (dry friction, constant) damping:

f(ẋ) = k·sgn(ẋ) (constant magnitude, sign = sign of velocity)

This is characteristic of dry sliding contact between surfaces. The sign function sgn(ẋ) is realized on the analog computer by a comparator circuit:

sgn(ẋ) = +1 if ẋ > 0, −1 if ẋ < 0

c) Van der Pol damping (self-excited oscillation):

f(ẋ) = −ε·(1 − x²)·ẋ

For small amplitudes (|x| < 1) the damping is negative (energy is fed into the system); for large amplitudes (|x| > 1) the damping is positive (energy is extracted). This leads to a self-sustaining limit cycle.

The Van der Pol equation:

ẍ − ε·(1 − x²)·ẋ + x = 0 (Van der Pol)

is one of the most important equations in nonlinear oscillation theory. Its solution exhibits a stable limit cycle in the phase plane. Regardless of the initial conditions (except the trivial equilibrium x = 0, ẋ = 0), trajectories converge to this limit cycle.

For small ε the oscillation is nearly sinusoidal with unit amplitude. For large ε the oscillation becomes strongly non-sinusoidal (“relaxation oscillation”).

The analog computing circuit for the Van der Pol equation requires a multiplier to form the product (1 − x²)·ẋ:

x² is formed by connecting the x output to both inputs of the multiplier
x²·ẋ is then formed by multiplying x² by ẋ
The difference ẋ − x²·ẋ = (1 − x²)·ẋ is formed by subtraction in a summer

Alternatively a function generator can approximate (1 − x²) if a multiplier is not available.

1.3. Bistable Oscillator

The bistable (Duffing-type) oscillator has two stable equilibrium positions. Its equation of motion may be written as:

ẍ + δ·ẋ + a·x + b·x³ = f(t)

For a < 0 and b > 0 (and without forcing, f = 0), the potential energy function

U(x) = (a/2)·x² + (b/4)·x⁴

has two minima (at x = ±√(|a|/b)) and one local maximum at x = 0. The system thus has two stable equilibria and one unstable equilibrium.

For the unforced damped case, trajectories in the phase plane converge to one or the other stable equilibrium, depending on the initial conditions. The separatrix (boundary between the two basins of attraction) passes through the unstable saddle point at the origin.

With periodic forcing f(t) = f̂·cos(Ω·t), the Duffing oscillator can exhibit:

Jump phenomena (hysteresis in the amplitude-frequency curve)
Multiple coexisting periodic solutions
Period doubling and chaos (for sufficiently large forcing amplitude)

Fig. 5 (in this section). Bistable oscillator: phase portrait showing the double-well potential behavior and phase-plane trajectories

The analog computing circuit for the Duffing oscillator requires forming x³:

x² is formed by the multiplier (x·x)
x³ = x²·x is then formed using a second multiplier or by using the function generator

The potentiometers are set to the coefficients |a|/c and b/c (normalized by the appropriate computing coefficient c).

2. Multi-Loop Oscillators

2.1. Two-Mass System with Nonlinear Damper

A two-mass system consists of two masses m₁ and m₂ coupled by a spring with rate c₁₂ and a nonlinear damper. Mass m₁ is also connected to a fixed wall by a spring c₁ and mass m₂ by a spring c₂ to the wall. An external force F(t) acts on mass m₁.

Fig. 1 (Section II.2). Two-mass system (schematic)

The equations of motion are:

m₁·ẍ₁ + k_NL(ẋ₁ − ẋ₂)·(ẋ₁ − ẋ₂) + c₁·x₁ + c₁₂·(x₁ − x₂) = F(t) (11)

m₂·ẍ₂ − k_NL(ẋ₁ − ẋ₂)·(ẋ₁ − ẋ₂) + c₂·x₂ − c₁₂·(x₁ − x₂) = 0 (12)

where k_NL(ẋ₁ − ẋ₂) denotes a nonlinear damping characteristic that depends on the relative velocity (ẋ₁ − ẋ₂). For the specific case of a shock absorber with quadratic characteristic:

k_NL·|ẋ_rel| = k·(ẋ₁ − ẋ₂)·|ẋ₁ − ẋ₂|

Resolving Eq. (11) for ẍ₁ and Eq. (12) for ẍ₂:

ẍ₁ = (1/m₁)·[F(t) − k_NL·(ẋ₁ − ẋ₂)·|ẋ₁ − ẋ₂| − c₁·x₁ − c₁₂·(x₁ − x₂)]

ẍ₂ = (1/m₂)·[k_NL·(ẋ₁ − ẋ₂)·|ẋ₁ − ẋ₂| − c₂·x₂ + c₁₂·(x₁ − x₂)]

This system requires 4 integrators (two per degree of freedom), plus multiplier circuits or function generators to realize the nonlinear damping term, plus summers and potentiometers.

The relative displacement x₁ − x₂ and relative velocity ẋ₁ − ẋ₂ must be formed as difference signals. These are then fed to the nonlinear element (function generator or multiplier combination) to produce the nonlinear damping force.

The time normalization follows the same procedure as for the single-degree-of-freedom case. The time scale factor β is chosen based on the lowest natural frequency of the system:

ω_min = smallest of the two natural angular frequencies

For the specific numerical example given in the document:

m₁ = 80 kg (vehicle sprung mass, simplified)
m₂ = 20 kg (unsprung mass)
c₁ = 10 kN/m (main spring)
c₂ = 160 kN/m (tire stiffness)
c₁₂ = 0 (no direct coupling spring, coupling only through damper)
Nonlinear damper characteristic

The undamped natural frequencies of the uncoupled system are:

ω₁ = √(c₁/m₁) = √(10000/80) ≈ 11.2 rad/s ≈ 1.78 Hz ω₂ = √(c₂/m₂) = √(160000/20) ≈ 89.4 rad/s ≈ 14.2 Hz

A time scale factor β = 1/10 is chosen (time compression by factor 10), so that the machine frequency for ω₁ is ≈ 1.12 rad/s (comfortable for recording).

Fig. 2 (Section II.2). Analog computing circuit for the two-mass system

The complete circuit includes:

4 integrators (Int. 1–4)
Function generator or multiplier for the nonlinear damper
Summers to form sums and differences of forces
Potentiometers for all coefficients
Forcing function generator (sinusoidal or step)

The output variables x₁, ẋ₁, x₂, ẋ₂ and derived quantities (e.g., x₁ − x₂, acceleration ẍ₁) can all be monitored simultaneously on an oscilloscope or X-Y plotter.

3. Oscillations of Continuous Systems

3.1. Oscillations of a Jumping Beam

The distributed-parameter system of an Euler-Bernoulli beam can exhibit complex oscillatory behavior, particularly when the boundary conditions are time-varying. The “jumping beam” (vorspringender Träger) is a beam in which one support is periodically lifted and replaced, causing impulsive excitation.

The governing partial differential equation of the Euler-Bernoulli beam is:

EI · ∂⁴w/∂x⁴ + ρA · ∂²w/∂t² = p(x, t) (1)

where:

E = modulus of elasticity
I = second moment of area
ρ = mass density
A = cross-sectional area
w(x, t) = transverse deflection
p(x, t) = distributed load

For simulation on an analog computer, the partial differential equation must be reduced to a system of ordinary differential equations. Two principal methods are used:

a) Modal decomposition: The solution is expressed as a superposition of mode shapes φₙ(x):

w(x, t) = ∑ qₙ(t) · φₙ(x)

Substituting into Eq. (1) and using the orthogonality of mode shapes yields a set of uncoupled (or weakly coupled) ordinary differential equations in the generalized coordinates qₙ(t):

q̈ₙ + ωₙ²·qₙ = Pₙ(t)/Mₙ (n = 1, 2, 3, …)

where ωₙ are the natural frequencies, Mₙ the generalized masses, and Pₙ(t) the generalized forces. Each modal equation has the form of a single-degree-of-freedom oscillator and can be solved on the analog computer by a separate two-integrator loop. The total deflection at any point x is then reconstructed by superposition.

For the jumping beam, the first few modes (typically n = 1, 2, 3) are retained, as higher modes decay rapidly.

b) Finite-difference (lumped-parameter) approach: The beam is divided into N segments; each segment’s mass is lumped at a node. This yields N coupled ordinary differential equations analogous to a chain of N masses connected by springs, which can be solved by N two-integrator loops with cross-coupling.

For the specific jumping-beam problem, the beam of length L is simply supported. One support is located at x = 0 (fixed) and the other at x = L (periodically excited: the support is removed and instantaneously replaced, imparting an impulse to the beam). The impulse can be modeled on the analog computer using a comparator that switches the boundary-condition input at the appropriate times.

The natural frequencies of the simply supported beam are:

ωₙ = n²·π²·√(EI/(ρA·L⁴)) (n = 1, 2, 3, …)

The lowest natural frequency:

ω₁ = π²·√(EI/(ρA·L⁴))

For a steel beam with typical proportions (given in the document’s example):

L = 1 m, cross-section 10 mm × 10 mm
E = 210 GPa, ρ = 7850 kg/m³

ω₁ ≈ 300 rad/s → f₁ ≈ 48 Hz

This is a high frequency; time compression is required (β ≪ 1) to bring it into the range of the recorder.

Fig. 1 (Section II.3). Schematic of the jumping beam and its analog computing circuit

The circuit includes:

Modal filters (two-integrator loops, one per retained mode)
A comparator-based switch to simulate the periodic support removal
A reconstructor (weighted summer) to compute the total deflection from the modal coordinates
Output to oscilloscope or recorder

The phase portraits of the individual modal coordinates show the progressive transfer of energy among modes at each jump event. The response is quasi-periodic when the jumping frequency is incommensurate with the natural frequencies, and periodic (or resonant) when a ratio of small integers relates them.

Fig. 2 (Section II.3). Analog computing circuit for the jumping beam (modal approach, first two modes retained)

The figure shows:

Two two-integrator loops (Mode 1 and Mode 2)
Switching network (comparator) implementing the jump impulse
Summing amplifier computing w(x₀, t) = φ₁(x₀)·q₁(t) + φ₂(x₀)·q₂(t) for a chosen observation point x₀

The potentiometer settings are:

α₁ = β²·ω₁²/c (mode 1 frequency squared, normalized)
α₂ = β²·ω₂²/c (mode 2 frequency squared, normalized)
Amplitudes of the coupling coefficients φₙ(x₀)

The time records and phase portraits illustrate the beat-like behavior when the two modal frequencies are nearly (but not exactly) commensurate, and the transition to complex non-periodic behavior as the jump frequency is swept.

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The expressions for the machine variables are:

$$V_{M4} = \frac{100}{V_{M1}} \cdot V_{M4} = \frac{100 \cdot \hat{x}{1-4}}{V{M1}} \cdot u_4 + \frac{\hat{x}{1-4}}{V{M1}} \cdot u_{4m}$$

$$V_{M3} = \frac{100}{V_{M1}} \cdot V_{M3} = \frac{\hat{x}{1-3}}{V{M1}} \cdot u_3 + \frac{\hat{x}{1-3}}{V{M1}} \cdot u_{3m}$$

and for the inner coupling member:

$$V_{M2} = \frac{100}{V_{M1}} \cdot V_{M2} = \frac{\hat{x}{1-2}}{V{M1}} \cdot u_2 + \frac{\hat{x}{1-2}}{V{M1}} \cdot u_{2m}$$

As it is not completely verifiable analytically, given the complexity of the equations to be solved, one resorts to the use of approximate methods. To ensure adequate protection against overloading when the amplifier reaches the limiting position, as well as to keep the solutions true to scale throughout, the machine variables are chosen to be somewhat smaller than the maximum permissible values. It should also be noted that the relative displacement of the masses is likely to be smaller than the absolute displacement of the outer masses. This must also be taken into account.

Fig. 11. Circuit diagram of the Zwei-Massen-Systems (Two-Mass System)

A circuit results with a time scaling of 100, i.e. the normalized frequencies are:

$$\Omega_1 = \frac{\omega_1}{\omega_1^0} \cdot 100 \qquad \Omega_2 = \frac{\omega_2}{\omega_2^0} \cdot 100$$

For the Zwei-Massen-System, i.e. for a simple Bandpass, these natural frequencies are approximately Ω₁ ≈ 35 and Ω₂ ≈ 81. These values can be read from the machine time scale. For a coupling of [formula], i.e. for stronger coupling (n = 5, ε = 0.56), Equation (17) gives:

$$\Omega_1^2 = \frac{0.5 \cdot 0.0001}{1} \cdot 0.131 + 0.5 \cdot 0.0001 \cdot \frac{5 \cdot 50}{4} \cdot 0.131 = 0.131$$

$$\Omega_2^2 = \frac{1.5 \cdot 0.0001}{1} \cdot 0.131 + \frac{5 \cdot 50}{4} \cdot 0.131 = 0.656$$

This corresponds to a frequency separation of Fig. 12, with time-scaling giving approximately:

$$\frac{\Omega_1}{\Omega_2} = \frac{506}{0.0001}$$

Since the solution of Fig. 11 was also used for the three-mass system (Drei-Massen-System) from Fig. 13, only the additional circuitry is described in detail here. The block diagram of the three-mass system shows that with the additional mass m₃, two additional integrators and one additional summing amplifier are needed. The corresponding circuit is shown in Fig. 13.

The result is the circuit for the inner coupling member (mass m₂) (amplifier 5 of Fig. 13), which delivers the following voltage:

$$u_{2m} = V_{M2} \cdot (m_1 \cdot \ddot{x}_1 + m_2 \cdot \ddot{x}_2)$$

and with the appropriate time scaling (τ = 100) gives:

$$u_{2m} = V_{M2} \cdot (m_1 \cdot \ddot{x}_1 + m_2 \cdot \ddot{x}_2) \cdot \frac{1}{10000}$$

Page 20

The frequency response of the Zwei-Massen-Systems (two-mass system) (Fig. 14) shows the bandpass character of the system. In the damping case (D = 0.1), the bandwidth is approximately 1.3:1. The phase diagram (Fig. 15) shows the phase response corresponding to this. An example of the time response in the transient case is shown in Fig. 16.

Fig. 14. Frequency response of the Zwei-Massen-Systems at the masses m₁ and m₂

Fig. 15. Phase diagram of the Zwei-Massen-Systems at the masses m₁ and m₂

Fig. 16. Deflection of a trigger at t = 0 degrees

Explanations of the simulation and of the results: on the one hand the simulation confirms the theoretically derived bandpass character (Fig. 14), and on the other hand it shows that the time response in the transient case (Fig. 16) has an envelope whose decay depends on the damping. The simulation results are in good agreement with the theoretical values.

3. Oscillation of a coupled two-mass Träger (carrier)

3.1 Introduction

The oscillation of a system with coupled Träger (carriers) presents problems from various points of view (structural dynamics, control engineering). Such a system occurs, for example, in the following cases:

The coupling element (Koppelelement) represents a system with soft spring characteristics. This can lead to resonance excitation of the coupled system if a natural frequency of the coupling element lies in the vicinity of an excitation frequency.
In the simulation of the coupled system, the two state variables also appear as coupling quantities. The simulation on the analog computer is therefore suitable for direct study of the coupling effects.

Fig. 17. Frequency response of the Zwei-Massen-Systems at 50 Hz for masses m₁ and m₂

Fig. 18. Phase diagram of a two-mass Träger at 50 Hz

The carrier oscillation is treated as a special case of the general two-mass system. The general approach described in Section 2 also applies here. The equations of motion are identical to those given there.

Page 21

The equations of motion are:

$$\frac{d^2 x_1}{dt^2} + 2 D_1 \omega_1 \frac{d x_1}{dt} + \omega_1^2 x_1 + f(x_1, x_2, \dot{x}_1, \dot{x}_2) = \frac{F}{m_1} \sin(\Omega t)$$

$$\frac{d^2 x_2}{dt^2} + 2 D_2 \omega_2 \frac{d x_2}{dt} + \omega_2^2 x_2 + g(x_1, x_2, \dot{x}_1, \dot{x}_2) = 0$$

The coupling functions f and g contain the coupling forces between the two masses.

For the Zwei-Massen-Träger (two-mass carrier), the following numerical values are given for the Träger parameters:

Carrier frequency f_T = 9 Hz
Natural frequency of the carrier f₀ = 4.5 Hz (i.e. well-known from measurements)
Carrier mass m₁ = 4 kg (measured via impact tests)
Internal mass m₂ = 1 kg (approximately)
Damping D₁ = 0.3 (inner mass), D₂ = 0.1

Given these values, the following numerical dimensions result:

$$\omega_1 = 2\pi \cdot 4.5 \approx 28.3 \text{ rad/s}, \quad \omega_2 = 2\pi \cdot f_2$$

$$k_{12} = m_2 \cdot \omega_2^2, \quad c_{12} = 2 D_2 m_2 \omega_2$$

For the analog computer simulation, the following time scaling is selected: τ = 100, so that the machine frequencies are:

$$\Omega_{M1} = \omega_1 \cdot \tau = 28.3 \cdot 100 \approx 2830 \text{ rad/s}$$

and the amplitude scaling follows from the requirement that the machine variable does not exceed 100 V:

$$V_{M1} = \frac{100}{x_{1\text{max}}}$$

With a 4-channel recording of the simulation result:

$$u_1 = V_{M1} \cdot x_1, \quad u_2 = V_{M2} \cdot x_2, \quad u_3 = V_{M3} \cdot \dot{x}1, \quad u_4 = V{M4} \cdot \dot{x}_2$$

For the quantities t₁, m₁, the following values are obtained numerically:

$$t_1 = 4.000 \text{ mm}, \quad h_1 = 4 \text{ kg}$$ $$t_2 = 1.190 \text{ mm}, \quad h_2 = 1.2 \text{ kg} \quad (c_{12}\text{ kg})$$ $$t_3 = 1.785 \text{ mm}, \quad h_3 = 4.5 \text{ kg}$$

Fig. values:

$$z_1 = 4.000 \text{ mm}, \quad m_1 = 4 \text{ kg}$$ $$z_2 = 1.785 \text{ mm}, \quad m_2 = 1.2 \text{ kg}$$ $$z_3 = 0.596 \text{ mm}, \quad m_3 = 1.5 \text{ kg}$$

For the simulation, the machine time gives τ = 1/100, and for the amplitudes:

$$\hat{x}1 = 4.000 \text{ mm} \Rightarrow V{M1} = 25$$ $$\hat{x}2 = 1.785 \text{ mm} \Rightarrow V{M2} = 56$$

With a 4-channel recording (τ = 100), one obtains the machine variables:

$$V_{M1} = 25, \quad V_{M2} = 56, \quad V_{M3} = V_{M4} \approx 50$$

The machine time scaling gives the recording rate. The solution is computed via the circuit already described. The result is evaluated with the appropriate amplitude and time scaling.

Page 22

The system equations in operator notation yield:

$$\begin{cases} a_{11} u_1 + a_{12} u_2 + b_{11} \dot{u}1 + b{12} \dot{u}2 = F_1 \ a{21} u_1 + a_{22} u_2 + b_{21} \dot{u}1 + b{22} \dot{u}2 = F_2 \ a{31} u_1 + a_{32} u_2 + b_{31} \dot{u}1 + b{32} \dot{u}_2 = F_3 \end{cases}$$

with the coupling coefficients given by the matrix entries as described in the preceding section.

Fig. 30. Circuit diagram of the three-carrier coupled system

Fig. 31. Oscillations of the carrier under the spring-constrained excitation

The results show that the coupling of the carrier masses leads to a significant modification of the amplitude and phase response compared with the single-mass case. The coupled resonance peaks are clearly visible in the simulation output (Fig. 31), and their positions correspond to the theoretically computed natural frequencies.

The simulation demonstrates that the analog computer approach is well suited to the investigation of such coupled oscillation problems, since:

The natural frequencies and mode shapes of the coupled system can be determined directly from the simulation output.
The effect of parameter variations (spring stiffness, damping coefficients, mass ratios) can be studied by simple potentiometer adjustments without rewriting the circuit.
The transient response following a step or impulse excitation is immediately visible on the oscilloscope or recorder output.

References

[1] W i t t e, G.; H a r t y, F. M.: Schwingungssimulation auf Analogrechenmaschinen. J. Rein. Chem., McGraw-Hill, New York 1964.

[2] R o g e r s, A. E.; C o n n o l l y, T. W.: Analog Computation in Engineering Design. McGraw-Hill, New York 1960.

[3] T o m o v i c, R.; K a r p l u s, W. J.: High-Speed Analog Computers. John Wiley, New York 1962.

[4] W a r f, J.; A m b, G.; S t e r r, F. R.: Analogrechner — Grundlagen. Oldenbourg, München 1965.

[5] G i l o i, W.; L a u b e r, R.: Analogrechentechnik. Springer-Verlag, Berlin / Heidelberg / New York 1963.

[6] M a h r e n h o l t z, O.: Schwingungsprobleme im Maschinenbau. Vieweg-Verlag, Braunschweig 1967.