The smash is the hardest stroke in badminton. Learn to hit in front of you, use trunk rotation and control the angle before you go for speed.
The honest answer: The smash is a steep overhead stroke with which you hit the shuttle hard downwards into your opponent's court. It is the fastest stroke in badminton and, in studies, accounts for roughly one fifth of all attacking strokes in a game [5]. But if you try it with arm strength alone, you often hit into the net or play it weak and flat.
How fast is fast? The elite players in a study of 19 Malaysians reached on average 97 m/s with their fastest jump smashes, the peak was 105 m/s [1]. That is roughly 350 km/h, faster than most high-speed trains. The authors cite a record of over 111 m/s [1]. Don't worry: for you it is not about records. It is about getting the shuttle into the court steeply and under control.
And the good news: the speed comes from technique. In a study with 24 players, skilled players used clearly more trunk rotation than beginners, and exactly this rotation gave the shuttle more speed [2].
Run sideways under the shuttle and take the racket behind your head, exactly as for the clear. The elbow is high, the free arm points at the shuttle. If you prepare differently from the clear, you make the smash readable.
The contact point is in front of your body and high above the racket shoulder. In a study with 29 players, body position had a direct influence on launch angle and clearance over the net [5]. For beginners it worked well to stand comfortably under the shuttle and then step one foot back [5].
Rotate hips and shoulders back and then explosively forward. The larger trunk rotation stretches the chest muscle more beforehand, which enables a stronger contraction, and carries the movement like a whip from the trunk into the racket [2]. Advanced players (over 7 years of experience) additionally co-activated the deep abdominal muscles together with the forearm muscles at contact, beginners (under 3 years) did not [4]. That is the main difference between beginner and skilled player.
Milestone: You feel the power coming from the trunk, not from the arm.
Hit the shuttle steeply downward, the racket face pointing slightly down at contact. The authors of the 19-player study recommend: do not fully extend the elbow at contact and do not lift the upper arm excessively; faster smashes had a more internally rotated shoulder, a less elevated shoulder and a less extended elbow [1]. Wrist and forearm muscles control accuracy: in skilled players activity of triceps and wrist flexor dropped right after contact [3]. Forearm and hand snap through, then you let go.
Start with about 70 % of the speed. More important than speed is the contact zone: even for elite players, hits just one standard deviation away from the average contact point lowered speed by up to 5.3 % and deflected the shuttle by up to 2.9° [6]. Count ten smashes at a towel.
Milestone: Seven of ten half smashes land in court without ending in the net. Only then do you add speed.
Film your smash from the side in slow motion and freeze the frame at contact. Look at three things: is the contact point in front of the body? Is the elbow slightly bent? Is the upper arm not yanked up? These three points come directly from the study findings [1] [5]. A two-minute check often brings more than ten smashes on a hunch.
Run sideways under the shuttle, take the racket behind the head and hit the shuttle in front of you, high above the racket shoulder. Rotate the trunk up and through, hit steeply downward and relax after contact. Practise first at 70 % speed and count hits.
Elite players reached on average 97 m/s in one study, peak 105 m/s [1]. That is roughly 350 km/h. According to the authors the record is over 111 m/s. For you, the angle counts first, not the speed.
Usually the stroke comes from the arm only. Use trunk rotation; skilled players rotate the upper body clearly more than beginners and hit faster [2]. A contact point too late behind the head also takes speed.
I found no direct study on pure shoulder strength. What is documented: the fastest smashes in one study were associated with arm position at contact (shoulder, elbow), not with ground force of the jump [1]. Trunk rotation and contact point are the documented levers. Strength does not replace technique (coaching practice).
Badminton Jump Smash: Build-Up, Technique and a Safe Landing