You can enter a linux system without entring a password using below steps. We will be creating a ssh jey which will be share between the servers, which will be used to authenticate. This is also a secure way of connecting to server, as the private key is inside the user home directory and can only be accessed by the user.

In the exmaple below we are make connecting to SERVER02 and SERVER03 passwordless from SERVER01.

SERVER01 ----(no password) ----- > SERVER02
                   |
                    ------------ > SERVER03

Creating passwordless entry from (SERVER01) to other servers.

Create a rsa key on SERVER01

ssh-keygen -t rsa

Create .ssh directory on other 2 servers.

ssh ahmed@SERVER02 mkdir -p .ssh
ssh ahmed@SERVER03 mkdir -p .ssh

Add the id_rsa.pub to authorized_keys

cat ~/.ssh/id_rsa.pub | ssh ahmed@SERVER02 'cat >> .ssh/authorized_keys'
cat ~/.ssh/id_rsa.pub | ssh ahmed@SERVER03 'cat >> .ssh/authorized_keys'

Make sure we have the right permissions.

ssh ahmed@SERVER02 chmod 744 -R .ssh 
ssh ahmed@SERVER03 chmod 744 -R .ssh 

Testing.

ssh ahmed@SERVER02
ssh ahmed@SERVER03