Date: 201807
Location: ZZZ.ARTCC, US
Flight conditions: VMC
Aircraft: B757 Undifferentiated or Other Model
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Aircraft component: Turbine Engine
Component problem: Malfunctioning
Reporter organization: Air Carrier
Reporter function: Pilot Not Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Time Pressure; Troubleshooting; Distraction
Second reporter function: Pilot Flying; First Officer
Anomaly: Aircraft Equipment Problem Critical; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Security
Detector: Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Provided Assistance; Flight Crew Diverted; Flight Crew Landed in Emergency Condition; Flight Crew Inflight Shutdown
Contributing factors: Aircraft
Primary problem: Aircraft

Narrative: While climbing out to 35;000 feet cruising altitude everything was normal and air was smooth. The first officer left the cockpit for a bathroom break. Flight Attendant was in the cockpit with me when the aircraft began to shudder and shake very strongly. I quickly assessed the situation having no idea what it was. I looked at the engine gauges and saw that the right engine EGT was in the red over 850 C. I pulled the right throttle back about halfway and the temperature quickly fell back into the green but the aircraft shudder remained. I looked at the engine vibration reading and saw 5.0 N1 on the right engine. The First Officer quickly knocked on the door and entered the cockpit and the flight attendant left. I asked her to fly the aircraft and she took control. The autopilot was still engaged. She adjusted the rudder to compensate for the asymmetrical thrust and we stabilized the jet. Then I pulled the QRH out and thought about which checklist was appropriate. I settled on the engine fire checklist because I saw airframe vibration as a condition plus the engine EGT had already gone into the red.We had stopped climbing at 29;200 feet and asked ATC for 29;000 feet to assess the problem. Once I decided on the checklist we [advised ATC]. I was thinking of heading back to [the departure airport] when the FO (First Officer) told me we were only 56 miles from ZZZ; so we decided to go there. As I read the checklist it told me to shutdown the right engine; I was hesitant to do it at first because the engine was still running well except for the high vibration indication on the gauge and the shuddering in the airframe. We secured the engine and shut it down while stabilizing the aircraft with rudder trim.About this time we let deadheading [a check airman] enter the cockpit to assist us. He had tried to enter the cockpit by using the emergency door access twice but we were too busy shutting down the engine to let him in. It was an unnecessary distraction; him knocking on the door and trying to get in; to what we were trying to do and a little annoying but I knew he only wanted to help us.After we had secured the engine and the airplane was stable did I get up out of my seat and let him in. I quickly told him of our situation and what our intentions were. He suggested him talking to the flight attendants and passengers and I agreed. We told the flight attendants (who were well aware of what was going on) to do a cabin advisory but no evacuation prep and of our intentions to land normally in ZZZ and the approximate time to landing.I sent a message to dispatch letting them know we were an emergency aircraft with an engine shutdown. He replied that he knew we were diverting to ZZZ. I didn't feel a need to communicate further with dispatch as we could see ZZZ in the window and ATC let us know that weather conditions were fine with calm winds. We asked ATC to have the fire trucks available for us on landing. We now turned our attention to landing in ZZZ. FO was still flying the aircraft as we descended with the autopilot still engaged. After I had read the engine shutdown checklist down to the approach and landing checklist; I took control of the aircraft and FO became the pilot monitoring. We accomplished the briefings for Runway XXL even though it was shorter than Runway XXR because it had an ILS. We planned a 20 flaps landing with target speed of 156 and flaps 5 for go-around if necessary with autobrakes set to 2.ATC vectored us for a base turn and cleared us for the visual approach. I intended to fly the autopilot to around 500 feet AGL and told FO of my intentions. As I turned final; about 9-12 miles out approximately; the aircraft overshot the localizer and banked steeply to the left to intercept. I disengaged the autopilot and turned the aircraft onto final approach manually. I saw that I was too high to intercept the glide slope and stated so verbally. I asked for gear down flaps 20 and pulled the speed brakelever to get down. Speed was at 180. I think we were about 5-7 miles out on final when this occurred but am not sure; my eyes were outside at this point looking at the runway. I asked for flaps 25 to assist intercepting the glide slope and once on glide slope returned the flaps to 20 and stowed the speedbrakes and armed them. All remaining checklists were accomplished and at 1;000 feet; I announced cleared to land and at 500 feet; announced stable.I landed the aircraft in the landing zone and engaged the thrust reverser on the left engine and stepped a little hard on the brakes to slow us down. We exited the runway. After crossing XXR I noticed the right brake temps were registering 8 and 9 and a bunch of sevens. We told fire department of the hot brakes and they told us that a little smoke was emanating from that side. I taxied the aircraft slowly to the gate while fire [department] followed closely monitoring the smoke. I only used the left brakes for most of the taxi to the gate. We parked the aircraft and did not set the parking brake even after we were chalked and told the company this. We wrote up the engine shutdown and overweight landing in the maintenance log in ACARS.We read the shutdown checklists. Fire [department] came to the cockpit and told us there was no more smoke. Maintenance came to the cockpit and told us to go ahead and set the parking brake. They also told us a fuse plug on one of the hot brakes melted and the tire was flat. I kind of suspected that when taxiing in because it felt a little bumpy on that side. Maintenance also told us they found metal shavings in the exhaust area of the right engine.I left the cockpit and talked to the passengers with the PA explaining what happened and told them that customer service would help them from here.


Second narrative: [Report narrative contained no additional information.]


Synopsis: B757 flight crew reported an engine vibration anomaly; resulting in an engine shutdown and diversion.