Revenue - Contract Balances (Details) - USD ($) $ in Thousands	12 Months Ended
Dec. 31, 2022	Dec. 31, 2021
Revenue from Contract with Customer [Abstract]
Unbilled receivables (contract assets)	$ 42,423	$ 33,522
Unbilled receivables (contract assets), $ Change	$ 8,901
Unbilled receivables (contract assets), % Change	27.00%
Contract liabilities	$ (27,857)	(25,374)
Contract liabilities, $ Change	$ (2,483)
Contract liabilities, % Change	10.00%
Net contract liabilities	$ 14,566	8,148
Net contract liabilities, $ Change	$ 6,418
Net contract assets, $Change	79.00%
Customer advances	$ 9,593	$ 9,364
Revenue recognized	90.00%	100.00%

Contract with Customer, Asset (Liability), Percent Change

No definition available.

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Balance Type:	na
Period Type:	duration

- 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, Asset, Percent Change

+ References

No definition available.

+ Details

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Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Liability, Percent Change

+ References

No definition available.

+ Details

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Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Liability, Revenue Recognized, Percent

+ References

No definition available.

+ Details

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Balance Type:	na
Period Type:	duration

- Definition

Contract with Customer, Net Asset (Liability)

+ References

No definition available.

+ Details

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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

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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 https://asc.fasb.org/extlink&oid=126920106&loc=SL49130551-203045

+ Details

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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 https://asc.fasb.org/extlink&oid=126919976&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 https://asc.fasb.org/extlink&oid=126919976&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 https://asc.fasb.org/extlink&oid=126920106&loc=SL49130549-203045

+ 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
-Publisher FASB
-Name Accounting Standards Codification
-Topic 606
-SubTopic 10
-Section 45
-Paragraph 1
-URI https://asc.fasb.org/extlink&oid=126919976&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 https://asc.fasb.org/extlink&oid=126920106&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 https://asc.fasb.org/extlink&oid=126919976&loc=SL49130532-203044

+ Details

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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 https://asc.fasb.org/extlink&oid=126920106&loc=SL49130551-203045

+ Details

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

- References

No definition available.

+ Details

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