Acquisition - Schedule of Business Acquisition, Pro Forma (Details) - MB Aerospace Holdings, Inc. - USD ($) $ in Thousands	3 Months Ended	9 Months Ended
Sep. 30, 2023	Sep. 30, 2023
Business Acquisition, Pro Forma Information, Nonrecurring Adjustment [Line Items]
Net sales	$ 416,088	$ 1,249,129
Net (loss) income	$ (5,013)	$ 6,701

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

No definition available.

+ Details

Name:	us-gaap_BusinessAcquisitionProFormaInformationNonrecurringAdjustmentLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

The pro forma net Income or Loss for the period as if the business combination or combinations had been completed at the beginning of a period.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 805
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (h)(2)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147479328/805-10-50-2

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 805
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 2
-Subparagraph (h)(3)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147479328/805-10-50-2

+ Details

Name:	us-gaap_BusinessAcquisitionsProFormaNetIncomeLoss
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

The pro forma revenue for a period as if the business combination or combinations had been completed at the beginning of the period.

+ References

+ Details

Name:	us-gaap_BusinessAcquisitionsProFormaRevenue
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Details

Name:	us-gaap_BusinessAcquisitionAxis=b_MBAerospaceHoldingsIncMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: