why speed is not directly proportional to torque in dc machine?
All of this applies to permanent magnet DC brushed motors -- different kinds of motors behave differently.  Be warned.


Speed is not proportional to torque.  Rotational acceleration is proportional to torque over moment of inertia, though: \$\alpha = \frac{T}{J}\$, where \$\alpha\$ is acceleration in \$\frac{\mathrm{radians}}{\mathrm{sec}^2}\$, \$T\$ is torque in N-m/s, and \$J\$ is moment of inertia in \$\frac{\mathrm{kg}}{\mathrm{m^2}}\$.
Armature current is a function of the motor's back EMF, the applied voltage, and the armature resistance: \$i_a = \frac{V_a - E_{back}}{R_a}k_t\$, where \$i_a\$ is the armature current, \$R_a\$ is the armature resistance, \$V_a\$ is the applied armature voltage, and \$E_{back}\$ is the back EMF
Back EMF is a function of the motor's speed: \$E_{back} = \omega_a k_t\$, where \$\omega_a\$ is the armature speed and \$k_t\$ is the motor's torque constant.
Torque is a function of the motor's armature current: \$T_a = i_a k_t\$


If you put that all together, you'll find that a DC motor with a constant voltage on it will "try" to spin up to a speed proportional to the applied voltage, and that its torque (and current) will be proportional to how much slower it is than that ideal unloaded speed.

A DC motor that's driven with a constant current will behave differently than one driven with a constant voltage -- and it's fairly common in servo applications to drive a motor with a current command, rather than a voltage command.  But it'll still follow the equations above.