Predicting

### Pendulum Rate Change vs Temperature Change

Refers to the Electronically-Driven Pendulum (read first),
but applies to any pendulum.

Max Carter

#### Rationale

A pendulum has a natural tendency to speed up or slow down as its shaft shortens or lengthens; its period varies as the square root of it's length. Here we are interested in how much the pendulum lengthens or shortens due to temperature change and how it affects the pendulum's rate.

Predicting how much temperature change should affect a pendulum's rate requires knowing the shaft material's coefficient of thermal expansion, that is, how much a given material expands or contracts with a given temperature change. Fortunately, the coefficient of thermal expansion, expressed as meters per meter-degree Celsius (m/m°C), has been determined for virtually every material known. Some common materials are shown in this chart. The shaft of the electronically-driven pendulum clock is 6061 alloy aluminum. It's coefficient of thermal expansion is .0000232 m/m°C.

Comparing the rate variation (ΔT) predicted by the pendulum's coefficient of thermal expansion over a span of time to the observed rate variation over the same span will demonstrate the effect of temperature change on the 1-meter electronically-driven pendulum's temporal stability, and conversely verify the shaft material's coefficient of thermal expansion.

#### Running the Numbers

Rate variation of a 1-meter pendulum due to thermal expansion is predicted by the equation:

ΔT = √(1 + α) - 1     (see derivation)

Where:

• ΔT is the change in pendulum period in seconds
• α is coefficient of thermal expansion (m/m°C)
Substituting the coefficient of thermal expansion for 6061 aluminum (.0000232m/m°C), the predicted effect of a 1°C temperature change on the aluminum 1-meter aluminum pendulum is:

ΔT = √(1 + .0000232) - 1 = .0000116 seconds/°C

Which is to say that on every pendulum cycle (two beats) the pendulum would gain or lose .0000116 seconds per degree C of temperature change. Twenty-four hours is a convenient time span to test the proposition. Given that there are 43200 2-second periods in a day, expected time gained or lost per °C in 24 hours would be:

43200 x 0.0000116 = 0.5 seconds/°C/day

Given that the ambient temperature change in the room the as-built pendulum occupies is typically 5 to 10°C over 24 hours, the daily rate change would be expected to be within the range of:

[5,10]°C x 0.5 = 2.5 to 5.0 seconds/day

The graphs below show phase and temperature data collected in real time by the phase comparator that monitors the pendulum. It covers the time span from 72 hours ago to the present moment. The red trace shows ΔT (seconds per day) to be slightly less than the numbers predicted, suggesting there is some temperature compensation being provided by the driver circuit, possibly by the silicon diodes shunting the energy storage capacitor.

Graphs

#### Electronically-Driven Pendulum

Rate Error vs Temperature - 72-Hour Run  Derivation

A pendulum's period varies as the square root of it's length per this relationship:

T ≈ 2π√L/g

Where:

• T is pendulum period in seconds,
• L is pendulum length in meters,
• g is acceleration of gravity (m/sec2).
Temperature change lengthens or shortens the pendulum. Adding the coefficient of thermal expansion term (α) to the length term (L), the relationship becomes:

T ≈ 2π√(L + α)/g

Subtracting the relationship without the expansion coefficient term from the relationship with the expansion coefficient term determines the difference (ΔT):

ΔT = 2π√(L + α)/g - 2π√L/g

The equation simplifies to:

ΔT = √(L + α) - √L

Setting the value of L to 1 reduces the equation to:

ΔT = √(1 + α) - 1

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### Related Pages

#### Phase Comparator 