Revenue - Contract Balances (Details) - USD ($) $ in Thousands	12 Months Ended
Dec. 31, 2021	Dec. 31, 2020
Revenue from Contract with Customer [Abstract]
Unbilled receivables (contract assets)	$ 33,522	$ 33,009
Unbilled receivables (contract assets), $ Change	$ 513
Unbilled receivables (contract assets), % Change	2.00%
Contract liabilities	$ (25,374)	(39,865)
Contract liabilities, $ Change	$ 14,491
Contract liabilities, % Change	(36.00%)
Net contract liabilities	$ 8,148	(6,856)
Net contract liabilities, $ Change	$ 15,004
Net contract liabilities, % Change	0.00%
Customer advances	$ 9,364	$ 12,750
Revenue recognized	100.00%	100.00%

Contract with Customer, Customer Advance Liability

No definition available.

Name:	b_ContractWithCustomerCustomerAdvanceLiability
Namespace Prefix:	b_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Contract with Customer, Asset (Liability), Percent Change

+ References

No definition available.

+ Details

Name:	b_ContractwithCustomerAssetLiabilityPercentChange
Namespace Prefix:	b_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Asset, Percent Change

+ References

No definition available.

+ Details

Name:	b_ContractwithCustomerAssetPercentChange
Namespace Prefix:	b_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Liability, Percent Change

+ References

No definition available.

+ Details

Name:	b_ContractwithCustomerLiabilityPercentChange
Namespace Prefix:	b_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Liability, Revenue Recognized, Percent

+ References

No definition available.

+ Details

Name:	b_ContractwithCustomerLiabilityRevenueRecognizedPercent
Namespace Prefix:	b_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Net Asset (Liability)

+ References

No definition available.

+ Details

Name:	b_ContractwithCustomerNetAssetLiability
Namespace Prefix:	b_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Contract with Customer, Net Asset (Liability), Cumulative Catch-up Adjustment to Revenue, Change in Measure of Progress

+ References

No definition available.

+ Details

Name:	b_ContractwithCustomerNetAssetLiabilityCumulativeCatchupAdjustmenttoRevenueChangeinMeasureofProgress
Namespace Prefix:	b_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Amount of increase (decrease) in revenue recognized for cumulative catch-up adjustment from change in measure of progress which increases (decreases) right to consideration in exchange for good or service transferred to customer when right is conditioned on something other than passage of time.

+ References

Reference 1: http://www.xbrl.org/2003/role/exampleRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 50
-Paragraph 10
-Subparagraph (b)
-URI http://asc.fasb.org/extlink&oid=123351226&loc=SL49130551-203045

+ Details

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

- Definition

Amount, after allowance for credit loss, of right to consideration in exchange for good or service transferred to customer when right is conditioned on something other than passage of time, classified as current.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 45
-Paragraph 1
-URI http://asc.fasb.org/extlink&oid=123360276&loc=SL49130531-203044

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 45
-Paragraph 3
-URI http://asc.fasb.org/extlink&oid=123360276&loc=SL49130533-203044

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 50
-Paragraph 8
-Subparagraph (a)
-URI http://asc.fasb.org/extlink&oid=123351226&loc=SL49130549-203045

+ Details

Name:	us-gaap_ContractWithCustomerAssetNetCurrent
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Amount of obligation to transfer good or service to customer for which consideration has been received or is receivable.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 45
-Paragraph 1
-URI http://asc.fasb.org/extlink&oid=123360276&loc=SL49130531-203044

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 50
-Paragraph 8
-Subparagraph (a)
-URI http://asc.fasb.org/extlink&oid=123351226&loc=SL49130549-203045

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 45
-Paragraph 2
-URI http://asc.fasb.org/extlink&oid=123360276&loc=SL49130532-203044

+ Details

Name:	us-gaap_ContractWithCustomerLiability
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Amount of increase (decrease) in obligation to transfer good or service to customer for which consideration from customer has been received or is due, from business combination.

+ References

Reference 1: http://www.xbrl.org/2003/role/exampleRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 50
-Paragraph 10
-Subparagraph (a)
-URI http://asc.fasb.org/extlink&oid=123351226&loc=SL49130551-203045

+ Details

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

- References

No definition available.

+ Details

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