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I have trouble understanding clauses start with ING forms. Blow is some examples and my analysis of the sentences. Please correct me.

1. The legends and symbols meant nothing to him, being far too complex for his own engineering background.
"being far too complex" is an adjective clause start with an gerund 'being', which describes "The legends and symbols."

2. Not moving, he lay thrust forward on his elbows, looking into the hall and knowing as surely as he knew that he had heard something.
I guess that "looking into the hall" and "knowing as surely as he knew" are adverb clauses.

3. Just before dust on September 26, 1986, a short man with a bushy beard slipped into position near the Jalalabad airport in Soviet-occupied Afghanistan, nervously balancing a bazooka-like "Stinger" weapon on his shoulder.
I think this one is the same as the second one, and I'm not sure why use the verb gerund form.
Comments  
Akdom: You need to understand the difference between a clause and a phrase. Your questions relate to (present) participle phrases. Present participles act as adjectives, modifying a noun or pronoun.
1- OK
2- the last part of this sentence (beginning with "knowing") does not make sense - is part of it missing? Who is "looking into the hall"?
3- who was balancing the bazooka on his shoulder?
[EDIT]:
There is a difference between a gerund and a participle. Here is an example of clauses with gerunds:
Because loving him had been so joyous for so long, losing him swept her into a deep depression.
AlpheccaStars2- the last part of this sentence (beginning with "knowing") does not make sense - is part of it missing? Who is "looking into the hall"?

Thank you, AlpheccaStars! Sentence 2 is an excerpt I quoted from a novel. Here is a word-for-word quote:

"Not moving, he lay thrust forward on his elbows, looking into the hall and knowing as surely as he knew that he had extinguished the light two hours earlier that someone was sitting in the living room."

I just have some trouble understanding the composition of this kind of sentence. I wish I could get more familar with these 'gerund' or 'participle' concepts, and one day be able to use them correctly in my writings.
Students: We have free audio pronunciation exercises.
AlpheccaStars 3- who was balancing the bazooka on his shoulder?
Sentence 3 is also a word-for-word quote from a readers digest article.

Just before dusk on September 26, 1986, a short man with a bushy beard slipped into position near the Jalalabad airport in Soviet-occupied Afghanistan, nervously balancing a bazooka-like "Stinger" weapon on his shoulder.

So, if I understand you correctly, you mean "balancing a bazooka-like Stinger weapon on his shoulder" is a participle phrase? But why a participle can be used like this, without was/had? I mean I'm only used to sentence like: A man slipped into position, and he balanced a bazooka on his shoulder.
Akdom:
The present participle is one of the verb forms. Others are past participle and (bare) infinitive. In English we use these forms for different functions - in verb phrases, as adjectives and even nouns.
Have you ever heard a singing yellow canary? "Singing" is an adjective, modifying "canary". It is also the present participle of the verb "sing". Sometimes this type of modifier can follow the noun: Have you ever heard a yellow canary singing?

he balanced a bazooka on his shoulder. - this is simple past tense. The next examples are adjective phrases with the present participle:
Balancing a bazooka on his shoulder, the man climbed over the wall.
Running pell-mell down the stairs, Gail tripped and fell and broke her arm.

Misuse of the participle phrase leads to a frequent grammatical error: the dangling (misplaced) modifier. This mistake is easy to make. Modifiers (adjectival phrases or clauses) can dangle in very funny ways and sometimes make a great joke. Here are some examples: Emotion: smile
Hanging on the wall, my mom really liked the picture.
Covered in chili, mustard, ketchup and relish, I enjoyed the hot dog.
We saved the scraps of meat for the dog that had been left on our plates.
Flying over the African landscape, the elephant herd looked magnificent.
For sale: An antique desk suitable for a lady with thick legs and large drawers. .
i have no idea,so i cant underastand
Students: Are you brave enough to let our tutors analyse your pronunciation?
Hi,

i have no idea,so i cant underastand

Maybe you are not ready to learn about this topic yet.

Do you know what a gerund is?

Best wishges, Clive
akdomI'm not sure why use the verb gerund form of the verb is used. ... I wish I could get more familar with these 'gerund' or 'participle' concepts, and one day be able to use them correctly in my writings.
Actually, it's a participle. The problem in learning to use these is that the participial construction has no paraphrase that works consistently from one case to another. The only general remarks one can make is that this construction presents a secondary observation related to the main clause, and that it is used much more in writing than in conversation.
akdomThe legends and symbols meant nothing to him, being far too complex for his own engineering background.
Sometimes the participial construction gives the meaning "because".

The legends and symbols meant nothing to him, because they were far too complex for his own engineering background.

Compare:

The package was left exactly where it had been set down, being too heavy for John to lift without help. (because it was too heavy)
akdomNot moving, he lay thrust forward on his elbows, looking into the hall and knowing as surely as he knew that he had heard something.
Sometimes the participial construction gives the meaning "while" or "as", that is, "at the same time as".

Not moving, he lay thrust forward on his elbows [while / as] he looked into the hall, knowing ... that he had heard something.

Sometimes the participial construction gives a meaning that is paraphrasable with an adjective or noun form.

Motionless, he lay thrust forward on this elbows as he looked into the hall with the certain knowledge that he had heard something.
akdomJust before dust on September 26, 1986, a short man with a bushy beard slipped into position near the Jalalabad airport in Soviet-occupied Afghanistan, nervously balancing a bazooka-like "Stinger" weapon on his shoulder.
Again, "while" or "as", meaning "at the same time as". And often, not even the conjunction is necessary, because the reader can figure this out for himself.

... a short man ... slipped into position .... [While / As] he did this, he was nervously balancing a ... weapon on his shoulder.

A short man slipped into position. He was balancing a weapon on his shoulder.

Compare:

A well dressed man entered the room carrying an oddly shaped lamp.
Lucy sat up late, listening for the sound of the car in the driveway. Her guests were due to arrive at any moment.

CJ

P.S. Because some grammarians use the term participial phrase while others use participial clause or non-finite clause, I have used the neutral term participial construction.