Revenue - Contract Balances (Details) - USD ($) $ in Thousands	3 Months Ended
Mar. 31, 2023	Mar. 31, 2022	Dec. 31, 2022
Revenue from Contract with Customer [Abstract]
Unbilled receivables (contract assets)	$ 46,900		$ 42,423
Unbilled receivables (contract assets), $ Change	$ 4,477
Unbilled receivables (contract assets), % Change	11.00%
Contract liabilities	$ (31,127)		(27,857)
Contract liabilities, $ Change	$ (3,270)
Contract liabilities, % Change	12.00%
Net contract assets	$ 15,773		14,566
Net contract assets, $ Change	$ 1,207
Net contract assets, % change	8.00%
Customer advances	$ 12,116		$ 9,593
Revenue recognized	40.00%	60.00%

Contract with Customer, Asset (Liability), Percent Change

No definition available.

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- Definition

Contract with Customer, Asset, Percent Change

+ References

No definition available.

+ Details

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- Definition

Contract with Customer, Customer Advance Liability

+ References

No definition available.

+ Details

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

- Definition

Contract with Customer, Liability, Percent Change

+ References

No definition available.

+ Details

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Period Type:	duration

- Definition

Contract with Customer, Liability, Revenue Recognized, Percent

+ References

No definition available.

+ Details

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- Definition

Contract with Customer, Net Asset (Liability)

+ References

No definition available.

+ Details

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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

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Namespace Prefix:	b_
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Balance Type:	debit
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- 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
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 10
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479806/606-10-50-10

+ Details

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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
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 3
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479837/606-10-45-3

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479837/606-10-45-1

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 8
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479806/606-10-50-8

+ Details

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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
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 1
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479837/606-10-45-1

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 8
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479806/606-10-50-8

Reference 3: http://www.xbrl.org/2003/role/disclosureRef
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 2
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479837/606-10-45-2

+ Details

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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
-Topic 606
-SubTopic 10
-Name Accounting Standards Codification
-Section 50
-Paragraph 10
-Subparagraph (a)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147479806/606-10-50-10

+ Details

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- References

No definition available.

+ Details

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