Allow me to jump in and make a few observations. If the slow cranking is intermittent, clean and tighten the battery cable connections first to eliminate that possibility. If you can get it to act up more often, pull out the automatic shutdown (ASD) relay so it won't start, have a helper handy to turn the ignition switch, then use a digital voltmeter to measure the voltage drops across both battery cable connections. If you want to try that, I'll suggest a page that gives the complete description.
Also measure the battery voltage with the engine off first. It should be near 12.6 volts if it's fully charged. If you find closer to 12.0 volts, it's good but discharged and the engine will crank slowly. Next, measure the voltage again with the engine running. It must be between 13.75 and 14.75 volts. If it's low, measure the voltages on the two smaller wires on the back of the alternator. That must be done with the engine running.
Charging problems are often intermittent and you have to catch them and do any testing when they're acting up. Watch for a low volts reading on the dash gauge, dim head lights, or lights that dim noticeably when you turn on the heater fan. You have the little silver Nippendenso alternator that commonly develops worn brushes. That causes intermittent operation and can go on for weeks before the failure is permanent. Those brushes can be replaced for less than 20 bucks.
If the battery voltage got drawn down far enough due to a poor connection, or if it is disconnected for any reason or run dead, the Engine Computer will have to relearn "minimum throttle" before it will know when it has to be in control of idle speed. Until that relearn takes place you also may not get the nice idle flare-up to 1500 rpm when you start the engine. The other clue is you have to hold the gas pedal down 1/4" to get it to start and stay running. The relearn takes place when you drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.
The Transmission Computer also goes through a learning cycle, then it constantly updates its shift schedules and harshness to mask any wear that has taken place. It may shift sloppy but it usually shifts rather hard until the relearn is complete. You do that by driving and just letting it shift. The relearn typically takes a few miles or a few shift cycles.
You're probably hearing the solenoids buzzing when you shift into reverse or drive. That is done to modulate, or soften the engagement of the clutches to give a more comfortable feel instead of a hard clunk into gear. If you listen closely, you should also hear that ratcheting sound when you come to a stop and it downshifts into first gear.
Thursday, June 14th, 2012 AT 2:55 AM