BALTIMORE -- If you sign one reliever, you have the chance to sign two. And if you sign two, you have a chance to sign four.

That's the lesson the Orioles learned this offseason, as their relief staff has sought safety in numbers. Baltimore announced two finalized deals for relievers on Thursday, when veterans Chad Bradford and Scott Williamson were both officially added to the fold.

Bradford and Williamson were the third and fourth free-agent relievers signed to contracts by Baltimore this offseason, though there had been concern that they may be turned off by the team's turbulent pursuit of a better bullpen. Instead, shortly after the O's signed Danys Baez, they learned that they opposite was true.

"For me, it was all positive when [Baez] signed," said Bradford, who inked a three-year deal worth $10.5 million. "If you want a good bullpen, you've got to have more than one good guy. You've got to have more than just a closer. In New York we had a lot of good relievers, and I thought that made the bullpen better."

"Some of the feedback we got almost instantly with getting Baez onboard," said Mike Flanagan, the team's executive vice president of baseball operations. "I pitched out of a bullpen for a couple of years, and you'd really like the guy that's getting the ball after you to be good. It becomes a chain reaction in the bullpen where you pick each other up.

"I probably didn't think, at the time, [that signing Baez] was going to make that much of a difference. But all the people that we talked to -- and certainly the relievers -- noticed what was happening in the bullpen."

It doesn't take a seasoned analyst to notice what Baltimore has done. The Orioles have spent more than $40 million on four veteran relievers -- all 29 or older -- and in the process, they've completely transformed the weakest part of their ballclub. Last year the bullpen was packed with inexperienced arms.

Baltimore's relievers allowed 86 home runs in 2006, and only one other relief staff (Kansas City) was even within 10 homers of that total. The Orioles also had the second-highest bullpen ERA (5.27) in baseball, which explains the preoccupation with relief help. Now the only question is whether they've chosen the right relievers.

"They made a lot of moves already with the bullpen, and I think there's more to come with other parts of the team," said Bradford, who has spent most of his career in the American League. "It just felt like they were going in the right direction [and] trying to win. It felt like a good fit, and they were kind of anxious to get something done."

Bradford notched a 2.90 ERA last season and has a 3.40 mark for his career. He's worked in at least 65 games in four of the last five years, and he's given up just two home runs since the start of the 2005 season. Part of that is due to his submarine windup, which takes his hand within an inch or two of the mound before he releases the ball.

"I had a coach in high school that dropped me down," he said, explaining the origin of his delivery. "He dropped me down from over the top, and over the years, I've gotten lower and lower. I can't really go any further down from where I am, so I guess I've got to stop there. It started when I was 15 years old, really, and over the years it's changed."

Bradford topped out at $1.4 million in 2006, and his average salary over the next three years will be more than double that amount. He said that his last team, the Mets, were interested in re-signing him, but didn't come close to matching the winning offer in terms of length or annual salary.

Williamson, meanwhile, was an easier deal. The fireballing right-hander has had two reconstructive surgeries on his pitching elbow, and this winter he had to have bone chips removed from the same joint. Jim Duquette, Baltimore's vice president of baseball operations, expects Williamson to make the team and be a key reliever.

"We're obviously happy he's healthy," Duquette said of Williamson, who signed a one-year deal for $900,000. "He had a bone chip removed at the end of the season, so that's one thing we have to be cautious about. ... He hasn't started his throwing program, but he's on schedule, and we expect him to be ready for the beginning of the season."

Bradford has been more durable, though he had to have back surgery in 2005. He said that the operation didn't leave many lingering effects and doesn't force him to do anything extra. Judging by his career history, the Orioles are getting a pitcher who will likely pitch in 65 to 70 games and serve in a variety of roles.

In fact, Bradford doesn't care when he pitches -- as long as he does so on a regular basis.

"It doesn't matter one bit. Fifth inning. Ninth inning. Eighth inning. Sixth or seventh," he said. "I've come in and faced a hitter or two, and I've come in and pitched a couple of innings. And everything in between. It doesn't matter.

"I kind of look at it as, 'If the phone rings, they need you for a situation. If you get through it, they may need you for the next inning.' Either way, it doesn't matter."

"It's refreshing," said Flanagan. "Jim and I were sitting there, noting that fact. He didn't throw in there that he was going to start. I was a little disappointed, because we could probably use him as a starter, [too]. ... We heard the same thing about Jamie Walker, saying 'Any role.' When we go through this, we ask those questions, too."

The Orioles also have two more agreements that aren't official yet. Baltimore has reached terms on a one-year deal worth $900,000 with backup catcher Paul Bako, who is expected to take a physical next week. In addition, the O's have agreed to terms with first baseman Kevin Millar on a one-year deal worth $2.5 million.