Islam allows every one to express his feelings of happiness or sadness in his own way as far as it does include a sinful act like drinking alcohol or dancing etc. Human beings all over the world have different cultures and they are allowed to express their feelings according to what they feel suitable within the frame of Shariah.
Mourning the tragedies of Imam Husain (AS) and all Ahlul Bayt (AS) is a sign of love of the Prophet (SAWA) and his Progeny Ahlul Bayt (SAWA). Those who do not mourn such tragedies reflect that they do not feel a degree of love. The Prophetic Hadeeth says: No one from you O Muslims will be a believer till he loves me and my Progeny more than his love to himself and his family.
We ask those Muslims who do not mourn nor care for the tragedies of Imam Husain (AS): If any of your beloved ones faces a tragedy, what will be your feeling for him? Then why do you ignore the tragedies of Imam Husain?
The Prophet himself mourned and wept on this tragedy before it took place ( Al-Mo'jam Al-Kabeer by Al-Tabarani ( 2811) Sunan al-Tirmithi 3774, and Musnad Ahmad Ibn Hanbal , V.1, P.85 and Al-Mustadrak 'Alaa Saheehain by al-Haakim al-Nisabori , V.3, P. 176, and Al-Bidayah WAL Nihayah by Ibn al-Atheer , 8:200, and many other Sunni books).
The basic fact is this : Every lover of the Prophet (SAWA) must feel sad for the tragedies of the Prophet and his Progeny, otherwise his claim to be a lover of the Prophet (SAWA) will be questionable.
The way of expressing the sad feelings depends on the person himself or herself and his culture and society. We have no right to dictate on people from different cultures what we think suitable according to our own culture and society.
Beating the chest to express sadness and grief is used by people in many cultures. Even Ay'sha daughter of Abu Bakr used to beat her chest on the death of beloved person as you can find it in many Sunni books e.g. Musnad Ahmad ibn Hanbal ; Hadeeth number 25144, and Musnad Abi Ya'la ;V.8, P. 63, and Al-Bayhaqi Dala'il Al-Nubowwah , Hadeeth 3151 and Ibn Sa'ad in al-Tabaqaat al-Kubra,V2, P.261 and Ibn al-Atheer in al-Kaamil fil Rareekh, V.2, P. 322 and many other Sunni books..
Zanjeer involves hitting yourself with chains or sometimes blades as part of a mourning ceremony for Imam Husain. It is customary in some areas.
There is a difference of opinion on this these days. Some Shi'is consider it to be a good expression of dedication or loyalty to Imam Husain or as a practice they associate with their identity. (Personally I think some people associate it with being manly, as women do not usually do it, but admittedly no one around me has ever put this into words.)
Other Shi'is consider it wrong because it does not have a solid basis in Qur'an/hadith, because there is evidence that it is a later addition to Shi'i practice, because they feel it gives Islam a bad image, because it could distract from doing acts that the Qur'an teaches about, or because it might lead to self-harm.
Anyway there are plenty of enthusiastic debates about this online which you can read.
God knows best!