I'll try to (late) answer your question, sorry if it doesn't satisfy you. Most of the time it was economic and evangelistic
The first contact between the USA and the Ottomans occurred indirectly in North Africa. America, signed an agreement with Hasan Pasha, the bey of Algeria on September 5, 1795 accepted to pay ransom to and to recognize two States mutually as the "most-favored nations". American ships had also raise the Ottoman flag in order to navigate in Mediterranean. America also made an agreement with Libya in 1796, and with Tunisia in 1797 both with Algerian mediation. During this time, America also traded with the Ottomans during this time, but not establishing diplomatic relations just yet. Count Vergenes objected sending mission to Istanbul, arguing that the cost of signing the treaty would outweigh the benefits. In 1799, John Adams and Rufus King finally established a mission intended to negotiate a treaty. However, the mission never received official instructions and didn't even leave for Istanbul. Later on, as the Napoleonic Wars broke out, the mission was abandoned because The Ottoman Empire was an ally of Britain and Russia to drive Napoleon from Egypt. So negotiations with the Ottomans were avoided because it was feared that it would hinder Franco-American relations
In the 1820s, the Ottomans delayed negotiations for a treaty to appease Britain and because they were suspicious of the Americans, since many American groups provided support via ships to Greece without state support during the Greek revolt. Moreover, rapid Russian expansion kept the Ottomans busy during this period. However, this would change with the destruction of Ottoman navy in Navarino on October 27, 1827. As the Ottomans lost many ships in Navarino, they seek out foreign aid in order to rebuild the navy and finally signed a treaty with the Americans in May 7, 1830. The treaty included trade agreements, capitulations, and privileges just like European countries. Additionally, the treaty included a secret clause where if the Ottomans requested, America would start to build the ships in the required size in the American dockyard using the American timbers purchased at the cost price and the ships would be delivered to Ottoman after the contract was signed. However, the US senate was against this clause. After Mahmud II stated his disapproval, the US government guaranteed that David Porter, the US ambassador would provide all aids and support on the matters of buying and building of battleships. From 1840 onwards, the Ottomans bought ships and sent officers from imperial school of naval engineering to the US in order to learn shipbuilding. The Ottomans also purchased rifles for the Ottoman army and rifle surplus from the American Civil War. As time went on, America also planned to invest in the Ottoman capital, when in 1908 the Ottoman-American Development Company was created. They initiated the so-called "Chester Project", building railway networks in Eastern Anatolia. The project was halted when World War I broke out, and even though the company was re-established, the company could not finance the project and Turkey cancelled the project at the end of 1923.
Another American influence and activities on the Ottoman Empire are missionary activities. In 1810, Congregationalist ministers from Massachusetts and Connecticut in the formed the Board of Commissioners for Foreign Missions in Boston to spread their brand of evangelical Christianity around the globe. At first, the mission was to convert the Jews of Jerusalem, however the Jews resisted the conversion, and facing hostilities in Jerusalem, the Board moved the mission to Beirut in 1823 and began to proselytize local Christians there. Obtaining some success, the board established mission stations throughout the Ottoman Empire. In addition to promoting their own version of Christianity, these stations offered schools and medical clinics and played a role in promoting Ottoman immigration to the US. By the middle of the 19th century, the directors of the Board realized that actual conversion was proceeding at a slow pace and made a decision to increase their efforts in education in the hope that the American missionaries would convince the people of the Ottoman Empire to accept their version of Christianity. To further that plan, the Board approved the opening of Robert College, today Boğaziçi University, in Istanbul in 1863, and the Syrian Protestant College, later the American University, in Beirut in 1866. Missionaries continued to try to win converts and had some limited success among the Armenians of central and southern Anatolia. While their evangelical work can be considered a failure, the secular schools they build still left a mark on the Ottoman Empire
Sources
Ottoman-American Relations, Francis Hopkins Smith and Armenian Issue by Berrin Akalin
Turkish-American Relations by Şuhnaz Yılmaz
Encyclopedia of Ottoman Empire by Bruce Masters et al