I want to know if I used a lower voltage than what was labeled on the relay was OK. The relay was a 24V relay. I check that the relay turns on when the voltage when the voltage was 13.5v. If I apply that voltage is it reliable?. EDIT: I do not have a datasheet but here is the part number: HTCA024V
Electromechanical relays are really current operated devices. The relay pulls in when the current is sufficient. The current is determined by the operating voltage and the coil resistance. But the plot thickens - the coil resistance is a function of temperature- the coil temperature- increasing by about 0.4% for every degree C. The coil is heated by the current through it, as well as by the ambient conditions. It is also heated by resistive losses when the contacts are carrying current. The temperature rise above ambient might be 35 C so if the temperature outside your box is 45 C, and the temperature inside is 70 C the coil could easily be running above 100 C. That means the pull in voltage will increase by 1/3. In fact high temperature versions of relays have less tolerance than the usual ones.
Bottom line is that the relay engineers designed the relay to be reliable at the nominal voltage (with a bit of tolerance for unregulated supply voltage) over a reasonable temperature range, and with rated current flowing, and to account for manufacturing variations (friction, spring constant, magnetic properties).
If you know that those assumptions are somehow conservative in your situation (say you have a tightly regulated supply) you may be able to shave a bit off, but you if you are wrong the relay operation may not be assured or contact life may be compromised due to more arcing from sluggish operation. For example the relay may not pull back in on a hot day immediately after it has been on for a while.
Holding current is less than operating current (and contact pressure is not compromised in most relay designs) so it's sometimes possible to give a higher voltage pulse to operate the relay then back off to a lower voltage, but that's seldom worth the trouble.
It depends on the application.
You will find that the relay switches much more quickly when powered at the rated voltage. This results in clean switching with minimum arcing and contact bounce. For a low current application it may be fine. For a high-current application you will probably burn out the contacts fairly quickly.
Note that as the relay armature pulls in that the magnetic circuit air-gap reduces and this increases the magnetic field strength even further, etc. You should find that if you can pick from fully open at 13.5 V that it should hold on until the voltage drops to maybe 2/3 of that.
When the armature gap is closing it is working against the springiness of the contacts. If you can get the cover off the relay you could energise the relay and then press the armature with your finger to confirm that it is fully closed. If not then you might find that it stops when the first contact closes and the second contact doesn't make.
The voltage value that is specified for a relay is the value at which the manufacturer guarantees the operation.
From experience, I know that usually the relay is activated with lower voltage on and off with a much less than the activation voltage.
However, to ensure the functioning of the circuit, you must respect the nominal voltage, as specified by the manufacturer.
Most relays have a range of voltage specified in their datasheet in which the relay can operate. The minimum voltage of that range is only needed to reliably switch the relay to its new state, and is sometimes called the "must operate voltage".
After switching, a voltage significantly lower than the "must operate voltage" could be applied to keep the relay activated. Take a look at a snippet from this this answer:
12 V relay at 5 V
Here we get to the Real Savings™. First we'll have to explain the 5 V operation. We already saw that we can operate the relay at 9 V, since the "must operate voltage" was 8.4 V. But 5 V is considerably lower than that, so it won't activate the relay. It appears, however, that the "must operate voltage" is only needed to activate the relay; once it's activated it will remain active even at much lower voltages. You can easily try this. Open the relay and place 5 V across the coil, and you'll see it doesn't activate. Now close the contact with the tip of a pencil and you'll see that it remains closed. Great.