The Raspberry Pi is generally powered from a 5V wall-wart type of DC power supply. The secondary is galvanically isolated from the mains voltage for reasons of personal safety (a fault will not expose the end user to the mains voltage).
The DC return of your bridge rectifier circuit is most certainly not isolated from the mains. The Raspberry Pi ground is 'floating' with respect to the bridge rectifier ground - there is no galvanic connection between them, hence your voltage measurement.
If you were to connect the DC return of the bridge rectifier circuit to the Raspberry Pi ground, you bypass the galvanic isolation that the DC power supply gives you. This means your Raspberry PI is now mains-referenced, and any fault could potentially expose you you to lethal voltages. I wouldn't do this.
A further complication comes if you also hook up an earth-referenced return to the Raspberry Pi, like a connection to a PC, with the mains-reference return connected. When you mix a mains-referenced return like your rectifier circuit with earth, things are going to explode (you essentially short out your bridge through the earthed return, which is often a flimsy wire that gets really hot and melts/catches fire while blowing up everything connected to it). Another reason not to do this.
You would be much better off with a small line frequency transformer to (1) step down the mains voltage to a lower level ahead of your resistive attenuator, and (2) provide galvanic isolation from the mains. Put your bridge and attenuator in the secondary of the transformer. With this, you can safely connect the low voltage isolated rectifier return to the Raspberry Pi return.
(You also must include a fuse in the line to isolate the rectifier circuit from the mains if there is a severe fault like a transformer fault or a short circuit.)
While using a one-shot timer circuit will work, I think an easier solution can be used. Take a look at this circuit.
For clarification, "VBAT" is a 12V source that is always on as long as the battery is connected. However, "ACC" is a 12V source that is only on when the ignition is on or the key is set to "accessory." Rather than using a 5V relay just to control the power to the RPi, why not use a standard 12V auto relay as shown. This way, there is no wasted power (except for the coil current while the power is on) because everything will be disconnected from the battery.
One side of the coil is always connected to 12V. The opposite side is connected to ground (chassis) through an N-Channel FET (Q1). While a MOSFET is used in the diagram, any FET capable of sinking the coil current can be used. When "ACC" is powered ON, Q1 will switch ON, connecting the coil to ground and actuating the switch. This will in turn power whatever 5V regulation circuit you plan to use (a simple 7805 regulator with heat sink, a switching DC-DC converter, the USB supplies mentioned, etc).
The diode D2 is there to ensure the capacitor can only discharge into Q1 and can be regular or Shottky. Other methods should probably be used for over voltage and current protection from the battery.
The "ACC" voltage can be put through a voltage divider to create a 3.3V signal for the RPi. Be careful with this voltage level, considering a 12V auto battery can really be more like 14V DC. As long as this signal is HI, the RPi knows that the power is on. Obviously, this GPIO pin should be set as an input with any internal pullups disabled. When "ACC" is turned off, the RPi should see the LO signal on the pin and begin its shutdown.
When the "ACC" voltage is turned off, the capacitor C1 will retain the charge for so long, discharging through the resistor R1. Once the capacitor voltage drops below the gate threshold of Q1, it will switch OFF, disconnecting the relay coil from ground and removing power from the peripheral circuit. If a "logic level MOSFET" is used for Q1, it will remain switched ON until C1 voltage is fairly low. I tested this circuit using an NTD4960 (Datasheet), and it remained on for around 15 seconds - until C1 was around 2V. To increase the time, increase the capacitance value.
Best Answer
The usual procedure I use for motors is through a H-Bridge. But since a vibration motor isn't bidirectional I believe we can use just one switch in the following configuration. (I'm assuming its a 3V motor).
Instead of the D3 pin used on the arduino you can use the GPIO pin from the RPi. Furthermore you can also use the 5V output from the Raspberry pi for testing, otherwise i usually like to use an external battery for this.
Source for further reading:
http://learningaboutelectronics.com/Articles/Vibration-motor-circuit.php