I've got a weight-driven pendulum clock, I increased the driving weight by 10% and the period decreased by 1%. Why?
What's the equation?
The pendulum is length = 32cm; bob weight = 7g; rod weight = 7g. So it's a compound pendulum
The amplitude of the swing is 0.05rad.
The Q factor is
Q = 2pi(energy at start of swing)/(energy lost during swing) = 200
Apparently a clock's pendulum mostly loses energy through viscous drag; i.e. assume drag is proportional to velocity.
The energy is replenished by the escapement. It's a deadbeat escapement so (I think) it can be modelled by adding a constant torque to the pendulum. The torque is in the direction of the swing and only operates when the angle of the pendulum is in the range
-0.03 < angle < 0.03
Outside that range, the escapement applies no torque to the pendulum.
On the web, I've found equations for a compound pendulum with drag, and a compound pendulum driven by an external sine wave but none of those describe a real pendulum clock.
You can assume that angle=sin(angle). Horologists call the error that introduces "circular error" and it's tiny for the sort of angles I've got. And it would increase the period by, say, 50ppm whereas I see a 1% decrease in the period.
1% is a big number. I want to understand it but the maths is beyond me.
I would simply accept that your initial estimates are off by a small percentage, and recalibrate by moving the bob-weight. –
I'm wondering if I can build electronics to alter the force in the driving chain to regulate the clock. So I quite like the 1% change in period that I observed - it's big enough to be useful. My original question was "what's the equation". I want to know whether the effect I've seen is common to all pendulum clocks with a deadbeat escapement. I think the math involved is the sort of thing engineers do every day.
you can easily calibrate the clock after building it.
I already have the clock. I'm wanting to regulate it.
How would you alter the driving force using electronics?
I have built an auto-winder, https://www.instructables.com/An-Auto-winder-for-a-Weight-driven-Clock/
You'll see that the weight moves slightly side-to-side. In other words, the tension due to the weight shifts from the left-hand chain (the taut chain) to the right-hand chain (the loose chain). So with a little cleverness inside the winder I can change the "weight" or, more precisely, the torque on the chain wheel inside the clock. Will the clock mind? No. The weight is 550g and the chain is 50g so originally as the weight descended, the tension increased by 20%. The clock is happy with a varying weight. How will the clock react? I think it will change it's speed.
The winder will know whether the clock is running slow or fast by measuring how far the chain-wheel on the winder has turned. It will adjust the speed to compensate by moving the weight to the left or right (on average).
I know how to do the electronics and I should end up with a clock that's as maintenance-free and accurate as a quartz clock I can buy for £1.
... if you want your result to be so precise, you would need to know all of the parameters in the equations very precisely as well.
The result of the calculation does not need to be precise. The winder/regulator measures what the clock is doing and adjusts its speed up/down. It's a feedback system. The mechanics and electronics will be extremely simple and the software is straightforward.
I want to know what the math is given the information in the original question. I want to know because if I come up with a successful design, will it apply to a wide range of clocks or is my clock idiosyncratic in some way? If the math explains what I measured then I'll know how generally my solution can be applied.