you can use subplot_tight or subtightplot from the FEX. to remove all x-tick and labels use:
set(gca, 'XTickLabel', [],'XTick',[])
in the appropriate subplot...
Edit:
Since you do want to include the labels etc, you can so it using the position
handle in axes
:
t = 0:0.01:10;
y1 = sin(t);
y2 = cos(t);
left= 0.15;
bottom1=0.5;
bottom2=0.05;
width=0.8;
height=0.45; % which is also bottom1-bottom2
axes('Position',[left bottom1 width height]);
plot(t, y1);
ystr = {'sin(x)','(dimensionless)'}
hy1 = ylabel(ystr);
set(gca, 'fontsize', 14);
set(hy1, 'fontsize', 14);
set(gca, 'XTickLabel', [],'XTick',[])
axes('Position',[left bottom2 width height])
plot(t, y2, 'r-o');
hx2=xlabel('frequency');
hy2=ylabel('amplitude');
set(gca, 'fontsize', 14);
set(hx2, 'fontsize', 14);
set(hy2, 'fontsize', 14);